基于渐进均匀化的力-电-热耦合光滑有限元法研究

基于渐进均匀化的力-电-热耦合光滑有限元法研究 Research on thermo-electro-mechanical coupling smoothed finite element method based on asymptotic homogenization 1 first-author 1 2 corresp 周立明，博士，教授，博士生导师，从事压电复合材料性能研究；E-mail：lmzhou@jlu.edu.cn 周立明，博士，教授，博士生导师，从事压电复合材料性能研究；E-mail：lmzhou@jlu.edu.cn lmzhou@jlu.edu.cn 西安建筑科技大学 机电工程学院，陕西 西安，710055 西安建筑科技大学 机电工程学院，陕西 西安，710055 School of Mechanical and Electrical Engineering, Xi'an University of Architecture and Technology, Xi'an 710055, China School of Mechanical and Electrical Engineering, Xi'an University of Architecture and Technology, Xi'an 710055, China 吉林大学 机械与航空航天工程学院，吉林 长春，130025 吉林大学 机械与航空航天工程学院，吉林 长春，130025 School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130025, China School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130025, China 伍锦花 针对基于微观结构的力-电-热多物理场耦合压电复合材料的力学特性分析问题，基于压电复合材料的基本方程和力-电-热多物理场耦合效应，结合渐进均匀化方法预测的压电复合材料有效性能参数，提出基于渐进均匀化的力-电-热耦合光滑有限元法；推导力-电-热耦合光滑有限元的动力学控制方程，并运用Wilson- θ 法对压电复合材料结构动力学问题进行求解，研究温度变化对结构固有频率和动力学响应的影响，并将其结果与有限元法的求解结果进行对比，验证该方法的正确性和有效性。研究结果表明：本文方法所得结果与有限元法所得结果较一致，且具有较好的收敛性和较高的精度；该方法对分析压电复合材料元器件的多物理场耦合力学特性具有广阔的应用前景。 Aiming at the mechanical property analysis problem of piezoelectric composite materials with microstructure-based thermo-electro-mechanical multi-physical-field coupling, the thermo-electro-mechanical coupling smoothed finite element method based on asymptotic homogenization was proposed. The theoretical basis of this method includes the basic equation, the thermo-electro-mechanical coupling effect and effective performance parameters of piezoelectric composite materials predicted by asymptotic homogenization. The dynamic control equation of this method was deduced, and the structural dynamic problems of piezoelectric composite materials were solved by applying the Wilson- θ method. The effects of temperature variation on the structure natural frequency and dynamic response were studied. The results of the proposed mechod were compared with those of the finite method to verify the correctness and validity of the proposed method. The results show that the proposed method produces consistent results with the finite element method, and has better convergence and higher precision, and thus this method has a broad application prospect for analyzing the multi-field coupling mechanical properties of piezoelectric composite components. 固体力学 有限元法 力-电-热耦合 渐进理论 solid mechanics finite element method thermo-electro-mechanical coupling asymptotic method http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680427&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680429&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680428&type= 14.95016479 14.95016479 随着复合材料、压电材料和智能材料在各技术领域的应用日益广泛，其服役环境越来越复杂，这对材料的力学性能提出了新要求，因此，智能压电复合材料应运而生。压电复合材料具有高灵敏度、高频响、高力电耦合系数、高稳定性等优良特性，在传感器、存储器、制动器等智能元件中得到广泛应用。压电复合材料在服役环境中表现出多物理场耦合特性，涉及力学、电学、热学等领域。压电复合材料多物理场耦合问题伴有多尺度关联、迟滞特性和力-电-热耦合等科学难点，属于复杂多场力学问题，故压电复合材料的多物理场耦合机理研究对提高相关元器件的力学性能具有非常重要的意义。此外，温度是工程结构设计的必要因素，通常温度变化会导致元器件结构产生热应力；随着时间推移，温度引起的热应力会对压电复合材料的力学特性、可靠性和稳定性产生较大的影响，但目前对于压电复合材料的力-电-热多物理耦合作用的研究还很少，有待于进一步分析。 目前，已发展出多种针对压电复合材料有效性能参数的预测模型和方法，主要有直接法、近似法、微分方法、基于变分原理的界限法和基于双尺度展开的均匀化法等。CHAN等 [ 1 1 ] 提出了1-3型压电复合材料参数预测的理论模型，预测了速度、声阻抗、机电耦合系数和电压系数，并通过实验验证了方法的正确性；HAGOOD等 [ 2 2 ] 提出了两种预测压电纤维复合材料性能的微机电模型，研究了不同相组成的压电复合材料性能随纤维体积分数的变化趋势；SAFARI等 [ 3 3 ] 分析了1-3型压电纤维复合材料的性能参数。BISEGNA等 [ 4 4 ] 基于非线性、非局部压电本构关系，研究了周期性非均匀压电复合材料整体性能的变分边界，预测了压电复合材料的整体有效性能。随后，各国学者开始对压电纤维复合材料的力学性能进行探究 [ 5 5 - 7 7 5-7 ] 。 为改善有限元计算模型的求解精度，LIU等 [ 8 8 ] 将梯度光滑技术引入有限元法中，发展了多种光滑有限元法，建立并完善了光滑有限元法的理论体系。光滑有限元法克服了有限元法中单元质量要求高的缺陷，具有计算简单、精度高、效率高等优势 [ 9 9 - 11 11 9-11 ] 。力-电-热多物理场耦合条件下压电智能元器件的力学特性研究对增大压电复合材料的应用范围具有重要意义。MINDLIN等 [ 12 12 ] 探索了耦合物理场中应变、电场和温度之间的耦合关系，并建立了压电复合材料的基本方程，对压电复合材料的力-电-热耦合力学特性进行了分析。RAO等 [ 13 13 ] 利用准静态力-电-热方程分析了压电传感和控制系统中的热效应，建立了力-电-热有限元方程，对分布式压电材料智能结构的热效应进行了分析，发现热效应引起的位移对分布式压电材料系统的传感和控制精确具有重要意义。SHEN等 [ 14 14 ] 基于三阶剪切变形理论提出了一种力-电-热耦合条件下层合板结构的主动控制方程，但该研究仅考虑了物理场间的单向耦合作用。CHATTOPADHYAY等 [ 15 15 ] 建立了力-电-热耦合计算模型，分析了热环境作用下压电材料结构的双向耦合行为。GU等 [ 16 16 ] 提出了一种适用于力-电-热耦合问题的高阶温度场理论。由此可见，基于微观结构的力-电-热多物理场耦合复合压电材料的力学行为研究还未见报道。 本文作者基于压电复合材料的压电方程、广义几何方程、广义平衡方程、广义边界条件及力-电-热耦合效应，结合渐进均匀化方法预测的压电复合材料有效性能参数，提出基于渐进均匀化的力-电-热耦合光滑有限元法；推导出力-电-热耦合光滑有限元的动力学控制方程，并将其改写成统一形式；运用Wilson- θ 法对压电复合材料结构动力学问题进行求解，研究温度变化对结构固有频率和动力学响应的影响，并将本文方法所得结果与有限元法的求解结果进行对比，验证方法的正确性和有效性。 1 均匀化渐进展开技术 1 对于由压电陶瓷相(纤维)和聚合物相(基体)组成的具有周期性的压电复合材料，若基体的弹性刚度张量为 <math id="M1"><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>i</mi><mi>j</mi><mi>k</mi><mi>l</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679454&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679453&type= 5.50333309 4.65666676 ，压电常数张量为 <math id="M2"><msubsup><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>k</mi><mi>i</mi><mi>j</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679456&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679455&type= 4.23333359 4.65666676 ，介电常数张量为 <math id="M3"><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">1</mn></mrow></mfenced></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679469&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679457&type= 4.48733330 4.23333359 ；纤维的弹性刚度张量为 <math id="M4"><msubsup><mrow><mi mathvariant="bold-italic">C</mi></mrow><mrow><mi>i</mi><mi>j</mi><mi>k</mi><mi>l</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn></mrow></mfenced></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679473&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679470&type= 5.84200001 4.65666676 ，压电常数张量为 <math id="M5"><msubsup><mrow><mi mathvariant="bold-italic">e</mi></mrow><mrow><mi>k</mi><mi>i</mi><mi>j</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn></mrow></mfenced></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679472&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679471&type= 4.65666676 4.65666676 ，介电常数张量为 <math id="M6"><msubsup><mrow><mi mathvariant="bold-italic">λ</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow><mrow><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn></mrow></mfenced></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679475&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679474&type= 4.82600021 4.23333359 ；纤维沿 x 3 方向排列且呈 l 个周期对称分布，则可取一个代表体积元进行分析。考虑到1-3型压电复合材料中单胞的周期性和对称性，取其中一个代表性体积元进行分析。典型1-3型压电复合材料的结构、代表性体积元(RVE)及其横截面如 图1 图1 所示。 图1 图1 中， o 1 x 1 x 2 x 3 为压电复合结构的坐标系，极化方向沿 x 3 方向； o 2 y 1 y 2 y 3 为单胞的坐标系； R 为压电纤维的半径； S 1 为压电纤维横截面的面积； S 2 为基体横截面的面积； l 为 y 1 方向的单胞长度； m 为 y 2 方向的单胞长度； ω 1 为 y 1 方向的周期函数； ω 2 为 y 2 方向的周期函数； Γ 为压电纤维和基体之间的界面。 (a) 压电复合结构；(b) 代表性体积元(RVE)； (c) RVE的横截面 压电复合材料中细观单胞通常很小，其包含了材料的基本特性，故可借助一个细观单胞(代表性体积元)来获得整个宏观复合材料的力学特性。 若存在一个非常小的变量 δ 为宏观尺度坐标 x 与微观尺度坐标 y 之间的比值，则有 <math display="block" id="M7"><mi mathvariant="bold-italic">y</mi><mo>=</mo><mfrac><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>δ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679492&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679491&type= 8.63599968 7.28133297 由于 δ ( <math id="M8"><mn mathvariant="normal">0</mn><mo><</mo><mi>δ</mi><mo>≪</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679493&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679476&type= 13.20800018 2.62466669 )为一个微量，因此，在整个宏观范畴内，对于任一场函数 <math id="M9"><msup><mrow><mi>ψ</mi></mrow><mrow><mi>δ</mi></mrow></msup><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679477&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679494&type= 9.05933285 3.89466691 ，有 <math display="block" id="M10"><msup><mrow><mi>ψ</mi></mrow><mrow><mi>δ</mi></mrow></msup><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi></mrow></mfenced><mo>=</mo><mi>ψ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi mathvariant="bold-italic">y</mi></mrow></mfenced><mo>=</mo><mi>ψ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mfrac><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>δ</mi></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679478&type= 40.30133438 8.29733276 基于渐进均匀化方法( δ →0)，将压电材料本构关系等效成2类局部问题即 <math id="M11"><mi>L</mi><mmultiscripts><mrow/><mprescripts/><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">q</mi></mrow><mrow/></mmultiscripts></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679485&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679484&type= 5.24933338 3.72533321 和 <math id="M12"><mi>L</mi><mmultiscripts><mrow/><mprescripts/><mrow><mi mathvariant="normal">p</mi></mrow><mrow/></mmultiscripts></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679487&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679486&type= 4.14866638 3.72533321 局部问题。考虑两相压电复合材料，压电陶瓷纤维沿 x 3 方向均匀排列，其呈现出横观各向同性，代表性体积元的纵向与 x 3 方向保持平行，长度取为单位长度，代表体积元中压电陶瓷纤维与聚合物基体的交界面为 Γ ， <math id="M13"><mi>L</mi><mmultiscripts><mrow/><mprescripts/><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">q</mi></mrow><mrow/></mmultiscripts></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679485&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679484&type= 5.24933338 3.72533321 和 <math id="M14"><mi>L</mi><mmultiscripts><mrow/><mprescripts/><mrow><mi mathvariant="normal">p</mi></mrow><mrow/></mmultiscripts></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679487&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679486&type= 4.14866638 3.72533321 局部问题应在代表性体积元 Y 中进行求解 [ 17 17 - 19 19 17-19 ] 。 2 压电复合材料的基本方程 1 2.1 　压电复合材料的广义平衡方程 2 压电复合材料中任意一点的应力张量 <math id="M15"><msub><mrow><mi mathvariant="bold-italic">σ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679504&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679503&type= 11.26066685 4.40266657 、电位移矢量 <math id="M16"><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679506&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679505&type= 11.43000031 3.97933316 和温度差 θ 分别满足动力学平衡方程、电场平衡方程和热场平衡方程(无热源、稳态导热)。 <math display="block" id="M17"><mfrac><mrow><mo>∂</mo><msub><mrow><mi mathvariant="bold-italic">σ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mrow><mo>∂</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">f</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mover accent="true"><mi>ρ</mi><mo>¯</mo></mover><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mrow><mo>∂</mo><msup><mrow><mi>t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679508&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679507&type= 48.76800156 9.14400005 <math display="block" id="M18"><mfrac><mrow><mo>∂</mo><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mrow><mo>∂</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mo>-</mo><mi>Q</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679510&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679509&type= 34.62866592 8.72066593 <math display="block" id="M19"><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mfenced separators="|"><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub><mfrac><mrow><mo>∂</mo><mi>θ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mrow><mo>∂</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679512&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679511&type= 30.31066895 9.48266602 式中： <math id="M20"><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679514&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679513&type= 10.75266743 4.40266657 为位移张量的分量； <math id="M21"><mi>Q</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679516&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679515&type= 10.92199993 3.97933316 为体电荷密度； <math id="M22"><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679518&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679517&type= 2.45533323 3.72533321 为热传导系数； <math id="M23"><mi>θ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679520&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679519&type= 10.07533360 3.97933316 为温度的变化量，满足 <math id="M24"><mi>θ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679522&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679521&type= 10.07533360 3.97933316 = T 0 ‒ T t ，其中 T 0 为初始温度， T t 为 t 时刻下的温度。 2.2 　压电复合材料的广义几何方程 2 在线弹性小变形条件下，压电复合材料中任意一点的应变张量 <math id="M25"><msub><mrow><mi mathvariant="bold-italic">ε</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679524&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679523&type= 10.75266743 4.40266657 、电场强度矢量 <math id="M26"><msub><mrow><mi mathvariant="bold-italic">E</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679496&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679495&type= 11.26066685 3.97933316 和热流密度 <math id="M27"><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679498&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679497&type= 10.49866676 3.97933316 分别满足力学几何方程、电学几何方程和热场几何方程，有 <math display="block" id="M28"><msub><mrow><mi mathvariant="bold-italic">ε</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced separators="|"><mrow><mfrac><mrow><mo>∂</mo><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mrow><mo>∂</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>∂</mo><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mrow><mo>∂</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679526&type= 53.17066193 10.41399956 <math display="block" id="M29"><msub><mrow><mi mathvariant="bold-italic">E</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mfrac><mrow><mo>∂</mo><mi mathvariant="bold-italic">Φ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mrow><mo>∂</mo><msub><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679531&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679529&type= 31.66533470 8.72066593 <math display="block" id="M30"><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mo>-</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>⋅</mo><mi mathvariant="normal">g</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">d</mi><mfenced separators="|"><mrow><mi>θ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679500&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679533&type= 41.48666382 4.74133301 式中： <math id="M31"><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679535&type= 10.75266743 3.97933316 为位移张量的分量； <math id="M32"><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679551&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679549&type= 10.49866676 3.97933316 为导热热流密度； <math id="M33"><mi mathvariant="bold-italic">Φ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679538&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679553&type= 11.09133339 3.97933316 为电势； <math id="M34"><mi mathvariant="normal">g</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679540&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679555&type= 6.60400009 3.38666677 为向量微分算子。 2.3 　压电复合材料的力-电-热耦合场本构方程 2 压电复合材料的力-电-热耦合场本构方程为 <math display="block" id="M35"><msub><mrow><mi mathvariant="bold-italic">σ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>j</mi><mi>k</mi><mi>l</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">ε</mi></mrow><mrow><mi>k</mi><mi>l</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>-</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">e</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>k</mi><mi>l</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">E</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>-</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">η</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mi>θ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679543&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679542&type= 69.68067169 4.48733330 <math display="block" id="M36"><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">e</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>k</mi><mi>l</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">ε</mi></mrow><mrow><mi>k</mi><mi>l</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>+</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">λ</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">E</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>+</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">p</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mi>θ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679545&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679557&type= 67.30999756 3.97933316 <math display="block" id="M37"><mi>K</mi><mo>=</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">η</mi><mo>¯</mo></mover></mrow><mrow><mi>k</mi><mi>l</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">ε</mi></mrow><mrow><mi>k</mi><mi>l</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>+</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">p</mi><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">E</mi></mrow><mrow><mi>k</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>+</mo><mfrac><mrow><mover accent="true"><mi>ρ</mi><mo>¯</mo></mover><mi>c</mi></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><mi>θ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679561&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679559&type= 59.68999863 8.12800026 式中： <math id="M38"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">λ</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679565&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679563&type= 3.30200005 4.23333359 为介电常数，是等温与自由形变下的二阶张量； <math id="M39"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>j</mi><mi>k</mi><mi>l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679578&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679546&type= 5.50333309 4.23333359 为弹性刚度常数，是等温和自由电场下四阶张量； <math id="M40"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">e</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>k</mi><mi>l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679580&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679579&type= 4.06400013 3.72533321 为压电常数，与压电效应对应，是等温和自由形变下的三阶张量； <math id="M41"><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679570&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679581&type= 2.79399991 2.62466669 为熵； <math id="M42"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">p</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679571&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679584&type= 3.13266683 4.48733330 和 <math id="M43"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">p</mi><mo>¯</mo></mover></mrow><mrow><mi>k</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679575&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679573&type= 3.47133350 4.48733330 为热电常数，与热电效应对应，是自由形变下的一阶张量； <math id="M44"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">η</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679585&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679577&type= 3.47133350 4.48733330 为热力系数，与热弹效应对应，是自由电场下的二阶张量； <math id="M45"><mover accent="true"><mi>ρ</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679595&type= 2.03200006 3.38666677 为密度； <math id="M46"><mi>c</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679599&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679587&type= 1.69333339 2.62466669 为比热容。 2.4 　压电复合材料的力-电-热耦合场边界条件 2 在考虑稳态热传导时，力-电-热耦合压电复合材料结构力学问题的边界条件主要包含力学边界条件、电学边界条件和热学边界条件。 力学边界条件为 <math display="block" id="M47"><msub><mrow><mi mathvariant="bold-italic">σ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><msub><mrow><mi>n</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi>T</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679591&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679600&type= 28.53266525 4.48733330 ，在 <math display="block" id="M48"><msub><mrow><mi>Γ</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679603&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679601&type= 3.04800010 3.72533321 上 <math display="block" id="M49"><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">u</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679606&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679592&type= 25.39999962 3.97933316 ，在 <math display="block" id="M50"><msub><mrow><mi>Γ</mi></mrow><mrow><mi>u</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679609&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679608&type= 3.55599999 3.72533321 上 电学边界条件为 <math display="block" id="M51"><msub><mrow><mi mathvariant="bold-italic">D</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mo>-</mo><mover accent="true"><mi>Q</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679611&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679624&type= 30.98800087 3.97933316 ，在 <math display="block" id="M52"><msub><mrow><mi>Γ</mi></mrow><mrow><mi>D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679630&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679627&type= 4.06400013 3.72533321 上 <math display="block" id="M53"><mi mathvariant="bold-italic">Φ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mover accent="true"><mi mathvariant="bold-italic">Φ</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679613&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679632&type= 26.16200066 3.97933316 ，在 <math display="block" id="M54"><msub><mrow><mi>Γ</mi></mrow><mrow><mi>Φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679617&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679615&type= 4.14866638 3.72533321 上 热学边界条件为 <math display="block" id="M55"><mi>J</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>θ</mi></mrow><mrow><mo>,</mo><mi>i</mi></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><mover accent="true"><mi>J</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679619&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679633&type= 36.83000183 4.31799984 ，在 <math display="block" id="M56"><msub><mrow><mi>Γ</mi></mrow><mrow><mi>a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679620&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679636&type= 3.55599999 3.72533321 上 <math display="block" id="M57"><mi>θ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mover accent="true"><mi>θ</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679622&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679638&type= 24.21466637 3.97933316 ，在 <math display="block" id="M58"><msub><mrow><mi>Γ</mi></mrow><mrow><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679655&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679653&type= 3.55599999 3.72533321 上 式中： <math id="M59"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">T</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679659&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679657&type= 10.92199993 3.97933316 ， <math id="M60"><msub><mrow><mover accent="true"><mi mathvariant="bold-italic">u</mi><mo>¯</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679663&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679661&type= 10.75266743 3.97933316 ， <math id="M61"><mover accent="true"><mi>Q</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679666&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679641&type= 10.92199993 3.97933316 ， <math id="M62"><mover accent="true"><mi mathvariant="bold-italic">Φ</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679667&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679643&type= 11.09133339 3.97933316 ， <math id="M63"><mi>J</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679646&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679668&type= 10.32933331 3.97933316 和 <math id="M64"><mover accent="true"><mi>θ</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679669&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679648&type= 10.07533360 3.97933316 分别为相应边界条件上的给定表面力、给定位移、给定面电荷密度、给定电势、给定热流通量和给定温度差。 <math id="M65"><msub><mrow><mi>n</mi></mrow><mrow><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679671&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679650&type= 2.70933342 3.72533321 为边界 <math id="M66"><mi>Γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679684&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679652&type= 2.62466669 2.62466669 的方向余弦， <math id="M67"><msub><mrow><mi>Γ</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679672&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679686&type= 3.04800010 3.72533321 ， <math id="M68"><msub><mrow><mi>Γ</mi></mrow><mrow><mi>u</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679674&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679688&type= 3.55599999 3.72533321 ， <math id="M69"><msub><mrow><mi>Γ</mi></mrow><mrow><mi>D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679677&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679690&type= 4.06400013 3.72533321 ， <math id="M70"><msub><mrow><mi>Γ</mi></mrow><mrow><mi>Φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679694&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679692&type= 4.14866638 3.72533321 ， <math id="M71"><msub><mrow><mi>Γ</mi></mrow><mrow><mi>a</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679697&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679696&type= 3.55599999 3.72533321 和 <math id="M72"><msub><mrow><mi>Γ</mi></mrow><mrow><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679682&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679681&type= 3.55599999 3.72533321 分别为力边界、位移边界、电位移边界、电势边界、热流通量边界和温度边界，满足 <math id="M73"><msub><mrow><mi>Γ</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>=</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>+</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mi>Φ</mi></mrow></msub><mo>=</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>+</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>=</mo><mi>Γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679701&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679714&type= 43.26466751 3.72533321 。 3 力-电-热耦合光滑有限元法 1 基于变分原理，力-电-热耦合光滑有限元的积分弱形式为 <math display="block" id="M74"><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><mfenced close="" separators="|"><mrow><mi>δ</mi><mi mathvariant="bold-italic">ε</mi><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">σ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>-</mo><mi>δ</mi><mi mathvariant="bold-italic">E</mi><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">D</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>+</mo></mrow></mfenced></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679718&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679704&type= 62.31467056 7.28133297 <math display="block" id="M75"><mfenced open="" separators="|"><mrow><mi>δ</mi><msup><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi>'</mi></mrow></msup><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">q</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow></mfenced><mi mathvariant="normal">d</mi><mi>Ω</mi><mo>=</mo><mo>-</mo><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><mi>δ</mi><mi mathvariant="bold-italic">u</mi><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mover accent="true"><mi>ρ</mi><mo>¯</mo></mover><mover accent="true"><mrow><mi mathvariant="bold-italic">u</mi></mrow><mi mathvariant="bold-italic">¨</mi></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mi mathvariant="normal">d</mi><mi>Ω</mi><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679723&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679708&type= 75.09933472 7.28133297 <math display="block" id="M76"><msub><mo>∫</mo><mrow><mi mathvariant="normal">Ω</mi></mrow></msub><mrow><mi>δ</mi><mi mathvariant="bold-italic">u</mi><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mi mathvariant="bold-italic">f</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mi mathvariant="normal">d</mi><mi>Ω</mi><mo>+</mo><msub><mo>∫</mo><mrow><msub><mrow><mi>Γ</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub></mrow></msub><mrow><mi>δ</mi><mi mathvariant="bold-italic">u</mi><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mover accent="true"><mi mathvariant="bold-italic">T</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mi mathvariant="normal">d</mi><mi>Γ</mi><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679744&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679711&type= 74.16799927 7.95866632 <math display="block" id="M77"><msub><mo>∫</mo><mrow><msub><mrow><mi>Γ</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></mrow></msub><mrow><mi>δ</mi><mi mathvariant="bold-italic">Φ</mi><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mover accent="true"><mi mathvariant="bold-italic">Q</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mi mathvariant="normal">d</mi><mi>Γ</mi><mo>+</mo><msub><mo>∫</mo><mrow><msub><mrow><mi>Γ</mi></mrow><mrow><mi mathvariant="normal">a</mi></mrow></msub></mrow></msub><mrow><mi>δ</mi><mi mathvariant="bold-italic">θ</mi><msup><mrow><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mover accent="true"><mi mathvariant="bold-italic">J</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mi mathvariant="normal">d</mi><mi>Γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679733&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679747&type= 73.23667145 7.95866632 式中： <math id="M78"><mi>Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679741&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679748&type= 2.70933342 2.62466669 为求解域； <math id="M79"><msup><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679761&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679758&type= 2.62466669 2.96333337 为求解域内温度差； <math id="M80"><mover accent="true"><mrow><mi mathvariant="bold-italic">u</mi></mrow><mi mathvariant="bold-italic">¨</mi></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679764&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679754&type= 2.11666679 2.70933342 为加速度张量。 光滑单元中任意点的位移 <math id="M81"><mi mathvariant="bold-italic">u</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679776&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679767&type= 10.24466610 3.97933316 、电势 <math id="M82"><mi mathvariant="bold-italic">Φ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679781&type= 11.09133339 3.97933316 和温度变化 <math id="M83"><mi mathvariant="bold-italic">θ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679787&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679785&type= 10.15999985 3.97933316 可表示为： <math display="block" id="M84"><mi mathvariant="bold-italic">u</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>N</mi></mrow><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msubsup></mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>u</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mi>u</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679806&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679803&type= 53.25533295 9.65200043 <math display="block" id="M85"><mi mathvariant="bold-italic">Φ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>N</mi></mrow><mrow><mi>Φ</mi></mrow><mrow><mi>i</mi></mrow></msubsup></mrow><msub><mrow><mi>Φ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>Φ</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>u</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679800&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679808&type= 48.34466553 9.65200043 <math display="block" id="M86"><mi mathvariant="bold-italic">θ</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></mrow></munderover></mstyle><mrow><msubsup><mrow><mi>N</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msubsup></mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>θ</mi></mrow></msub><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi>θ</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679820&type= 44.45000076 9.65200043 式中： <math id="M87"><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679833&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679826&type= 3.21733332 3.72533321 为单元总数； <math id="M88"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>u</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679832&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679829&type= 4.14866638 3.72533321 ， <math id="M89"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>Φ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679841&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679839&type= 4.65666676 3.72533321 和 <math id="M90"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>θ</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679845&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679854&type= 4.06400013 3.72533321 分别为光滑有限元的位移形函数矩阵、电势形函数矩阵和温度场形函数矩阵； <math id="M91"><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mi>u</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679847&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679846&type= 11.17599964 3.97933316 ， <math id="M92"><msub><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mrow><mi>u</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679870&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679866&type= 12.10733318 3.97933316 和 <math id="M93"><msub><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mrow><mi>θ</mi></mrow></msub><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679875&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679872&type= 11.09133339 3.97933316 分别为节点位移向量、节点电势向量和节点温度变化向量。 图2 图2 所示为光滑单元划分及相关节点信息。 图2 图2 中， <math id="M94"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3,4</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679894&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679877&type= 22.26733208 3.97933316 为第 i 个光滑子元的面积，并标记了场节点、边中心光滑节点和高斯点的位置及子域边界的外法线向量分布。 在光滑单元 <math id="M95"><msup><mrow><mi mathvariant="bold-italic">Ω</mi></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679911&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679905&type= 3.97933316 2.96333337 中的任意点 <math id="M96"><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679916&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679913&type= 3.21733332 2.96333337 处，光滑应变 <math id="M97"><mover accent="true"><mi mathvariant="bold-italic">ε</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679922&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679926&type= 11.34533310 4.74133301 、光滑电场强度 <math id="M98"><mover accent="true"><mi mathvariant="bold-italic">E</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679933&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679930&type= 12.36133385 4.74133301 和光滑温度 <math id="M99"><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679944&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679937&type= 11.76866722 4.74133301 变化为 <math display="block" id="M100"><mover accent="true"><mi mathvariant="bold-italic">ε</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><msub><mo>∫</mo><mrow><msup><mrow><mi>Ω</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></msub><mrow><mi mathvariant="bold-italic">ε</mi><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mi>r</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfenced><mi mathvariant="normal">d</mi><mi>Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679948&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679946&type= 51.64666748 7.70466709 <math display="block" id="M101"><mover accent="true"><mi mathvariant="bold-italic">E</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><msub><mo>∫</mo><mrow><msup><mrow><mi>Ω</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></msub><mrow><mi mathvariant="bold-italic">E</mi><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mi>r</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfenced></mrow><mi mathvariant="normal">d</mi><mi>Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679968&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679951&type= 53.67866135 7.70466709 <math display="block" id="M102"><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><msub><mo>∫</mo><mrow><msup><mrow><mi>Ω</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></msub><mrow><mi mathvariant="bold-italic">θ</mi><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced></mrow><mi>r</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfenced><mi mathvariant="normal">d</mi><mi>Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679972&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679969&type= 52.49333191 7.70466709 式中： <math id="M103"><mi>r</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679978&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679976&type= 13.71599960 4.74133301 为光滑函数，其表达式为 <math display="block" id="M104"><mi>r</mi><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>-</mo><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfenced><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mn mathvariant="normal">1</mn><mo>/</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>,</mo><mi mathvariant="bold-italic">x</mi><mo>∈</mo><msup><mrow><mi>Ω</mi></mrow><mrow><mi>k</mi></mrow></msup></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="bold-italic">x</mi><mo>∉</mo><msup><mrow><mi>Ω</mi></mrow><mrow><mi>k</mi></mrow></msup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679982&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43679991&type= 37.33800125 10.24466610 <math display="block" id="M105"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>=</mo><msub><mo>∫</mo><mrow><msup><mrow><mi>Ω</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></msub><mrow><mi>d</mi><mi>Ω</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680001&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680000&type= 17.01799965 7.70466709 将 式(25) 式(25) 代入到式(22)~ 式(24) 式(24) 中，可得： <math display="block" id="M106"><mover accent="true"><mi mathvariant="bold-italic">ε</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></mfrac><msub><mo>∫</mo><mrow><msubsup><mrow><mi>Γ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></msub><mrow><msubsup><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow><mi mathvariant="bold-italic">u</mi><mi mathvariant="normal">d</mi><mi>Γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680006&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680016&type= 36.06800079 8.72066593 <math display="block" id="M107"><mover accent="true"><mi mathvariant="bold-italic">E</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></mfrac><msub><mo>∫</mo><mrow><msubsup><mrow><mi>Γ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></msub><mrow><msubsup><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi>Φ</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow><mi mathvariant="bold-italic">Φ</mi><mi mathvariant="normal">d</mi><mi>Γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680011&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680022&type= 38.52333450 8.72066593 <math display="block" id="M108"><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></mfrac><msub><mo>∫</mo><mrow><msubsup><mrow><mi>Γ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></msub><mrow><msubsup><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow><mi mathvariant="bold-italic">θ</mi><mi mathvariant="normal">d</mi><mi>Γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680045&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680028&type= 36.40666580 8.72066593 式中： <math id="M109"><msubsup><mrow><mi>Γ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680034&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680032&type= 3.97933316 4.06400013 为光滑区域 <math id="M110"><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680037&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680053&type= 3.89466691 4.06400013 的边界； <math id="M111"><msubsup><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680061&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680059&type= 3.38666677 4.06400013 ， <math id="M112"><msubsup><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi>Φ</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680076&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680074&type= 3.97933316 4.06400013 和 <math id="M113"><msubsup><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680072&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680066&type= 3.38666677 4.06400013 分别为力场、电场和热场对应的光滑区域边界的外法向矢量。 <math display="block" id="M114"><msubsup><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msubsup><mrow><mi>n</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msubsup><mrow><mi>n</mi></mrow><mrow><mi>z</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mi>n</mi></mrow><mrow><mi>z</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mtd><mtd><msubsup><mrow><mi>n</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680086&type= 20.15066719 13.03866673 <math display="block" id="M115"><msubsup><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi>Φ</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msubsup><mrow><mi>n</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mi>n</mi></mrow><mrow><mi>z</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680113&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680112&type= 14.05466747 8.21266651 <math display="block" id="M116"><msubsup><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msubsup><mrow><mi>n</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msubsup><mrow><mi>n</mi></mrow><mrow><mi>z</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680102&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680098&type= 13.46199989 8.21266651 式中： <math id="M117"><msubsup><mrow><mi>n</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680122&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680133&type= 3.04800010 4.06400013 和 <math id="M118"><msubsup><mrow><mi>n</mi></mrow><mrow><mi>z</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680125&type= 3.04800010 4.06400013 均为单元外法线向量的分量。 式( 27 27 )~( 29 29 )可改写为如下形式： <math display="block" id="M119"><mover accent="true"><mi mathvariant="bold-italic">ε</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></mrow></munderover></mstyle><mrow><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msubsup></mrow><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><msub><mrow><mi mathvariant="bold-italic">u</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680149&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680131&type= 37.08399963 9.39799976 <math display="block" id="M120"><mover accent="true"><mi mathvariant="bold-italic">E</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></mrow></munderover></mstyle><mrow><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>Φ</mi></mrow><mrow><mi>i</mi></mrow></msubsup></mrow><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><msub><mrow><mi>Φ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680152&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680135&type= 39.36999893 9.39799976 <math display="block" id="M121"><mover accent="true"><mi mathvariant="bold-italic">θ</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></mrow></munderover></mstyle><mrow><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msubsup></mrow><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680154&type= 37.25333405 9.39799976 式中： <math id="M122"><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680158&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680142&type= 3.04800010 3.72533321 为光滑子元的个数； <math id="M123"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680145&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680160&type= 13.29266548 4.74133301 为光滑应变-位移矩阵； <math id="M124"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>Φ</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680164&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680162&type= 13.80066681 4.74133301 为光滑电场强度矩阵； <math id="M125"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680180&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680179&type= 13.20800018 4.74133301 为光滑温度变化量矩阵。 <math display="block" id="M126"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>b</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow></munderover></mstyle><mrow><msubsup><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msubsup></mrow></mfenced><msubsup><mrow><mi>l</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680173&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680182&type= 46.14333344 9.73666668 <math display="block" id="M127"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>Φ</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>b</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow></munderover></mstyle><mrow><msubsup><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi>Φ</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>Φ</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msubsup></mrow></mfenced><msubsup><mrow><mi>l</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680175&type= 47.92133331 9.73666668 <math display="block" id="M128"><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></mfrac><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>b</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>b</mi></mrow></msub></mrow></munderover></mstyle><mrow><msubsup><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="normal">θ</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mfenced separators="|"><mrow><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msubsup></mrow></mfenced><msubsup><mrow><mi>l</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680193&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680177&type= 46.14333344 9.73666668 式中： <math id="M129"><msub><mrow><mi>n</mi></mrow><mrow><mi>b</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680194&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680190&type= 3.21733332 3.72533321 为光滑单元的边界总数； <math id="M130"><msubsup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi mathvariant="normal">G</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680198&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680196&type= 4.06400013 4.06400013 和 <math id="M131"><msubsup><mrow><mi>l</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680201&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680199&type= 2.53999996 4.06400013 为高斯点和光滑边界的长度。 忽略体力的影响，力-电-热耦合光滑有限元的控制方程为 <math display="block" id="M132"><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>u</mi><mi>u</mi></mrow><mrow/></msubsup><mover accent="true"><mrow><mi mathvariant="bold-italic">u</mi></mrow><mi mathvariant="bold-italic">¨</mi></mover><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>u</mi><mi>u</mi></mrow><mrow/></msubsup><mi mathvariant="bold-italic">u</mi><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>u</mi><mi>Φ</mi></mrow><mrow/></msubsup><mi mathvariant="bold-italic">Φ</mi><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>u</mi><mi>θ</mi></mrow><mrow/></msubsup><mi mathvariant="bold-italic">θ</mi><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>u</mi></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680209&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680202&type= 47.66733551 4.06400013 <math display="block" id="M133"><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>Φ</mi><mi>u</mi></mrow><mrow/></msubsup><mi mathvariant="bold-italic">u</mi><mo>-</mo><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>Φ</mi><mi>Φ</mi></mrow><mrow/></msubsup><mi mathvariant="bold-italic">Φ</mi><mo>+</mo><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>Φ</mi><mi>θ</mi></mrow><mrow/></msubsup><mi mathvariant="bold-italic">θ</mi><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>Φ</mi></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680211&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680205&type= 38.69266510 4.06400013 <math display="block" id="M134"><msubsup><mrow><mi>K</mi></mrow><mrow><mi>θ</mi><mi>θ</mi></mrow><mrow/></msubsup><mi mathvariant="bold-italic">θ</mi><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>θ</mi></mrow><mrow/></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680207&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680214&type= 14.22399998 4.06400013 式( 39 39 )~( 41 41 )的矩阵形式为： <math display="block" id="M135"><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>u</mi><mi>u</mi></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mover accent="true"><mrow><mi mathvariant="bold-italic">u</mi></mrow><mi mathvariant="bold-italic">¨</mi></mover></mtd></mtr><mtr><mtd><mover accent="true"><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mi mathvariant="bold-italic">¨</mi></mover></mtd></mtr><mtr><mtd><mover accent="true"><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mi mathvariant="bold-italic">¨</mi></mover></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>u</mi><mi>u</mi></mrow></msub></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>u</mi><mi>Φ</mi></mrow></msub></mtd><mtd><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>u</mi><mi>θ</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>Φ</mi><mi>u</mi></mrow></msub></mtd><mtd><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>Φ</mi><mi>Φ</mi></mrow></msub></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>Φ</mi><mi>θ</mi></mrow></msub></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>θ</mi><mi>θ</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">u</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">Φ</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">θ</mi></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>u</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>Φ</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>θ</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680218&type= 81.11067200 14.64733315 式中： <math id="M136"><mover accent="true"><mrow><mi mathvariant="bold-italic">u</mi></mrow><mi mathvariant="bold-italic">¨</mi></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680221&type= 2.11666679 2.70933342 为位移张量的二阶导数； <math id="M137"><mover accent="true"><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mi mathvariant="bold-italic">¨</mi></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680241&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680240&type= 2.96333337 3.38666677 为电势张量的二阶导数； <math id="M138"><mover accent="true"><mrow><mi mathvariant="bold-italic">θ</mi></mrow><mi mathvariant="bold-italic">¨</mi></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680226&type= 2.03200006 3.38666677 为温度差的二阶导数。 <math display="block" id="M139"><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>u</mi><mi>u</mi></mrow><mrow/></msubsup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></mrow></munderover></mstyle><mrow><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mover accent="true"><mi mathvariant="bold-italic">C</mi><mo>¯</mo></mover><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>u</mi></mrow><mrow/></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680231&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680229&type= 46.90533447 9.39799976 <math display="block" id="M140"><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>u</mi><mi>Φ</mi></mrow><mrow/></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>Φ</mi><mi>u</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></mrow></munderover></mstyle><mrow><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mover accent="true"><mi mathvariant="bold-italic">e</mi><mo>¯</mo></mover><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>Φ</mi></mrow><mrow/></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680234&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680233&type= 56.13399887 9.39799976 <math display="block" id="M141"><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>Φ</mi><mi>Φ</mi></mrow><mrow/></msubsup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></mrow></munderover></mstyle><mrow><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mover accent="true"><mi mathvariant="bold-italic">e</mi><mo>¯</mo></mover><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>Φ</mi></mrow><mrow/></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680236&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680235&type= 48.34466553 9.39799976 <math display="block" id="M142"><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>θ</mi><mi>θ</mi></mrow><mrow/></msubsup><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></mrow></munderover></mstyle><mrow><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>θ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mi mathvariant="bold-italic">k</mi><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>θ</mi></mrow><mrow/></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680247&type= 46.48199844 9.39799976 <math display="block" id="M143"><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>u</mi><mi>θ</mi></mrow><mrow/></msubsup><mo>=</mo><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>u</mi></mrow><mrow><mi>T</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mover accent="true"><mi mathvariant="bold-italic">η</mi><mo>¯</mo></mover><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>θ</mi></mrow><mrow/></msubsup><mi>d</mi><mi>Ω</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680237&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680251&type= 38.01533508 7.28133297 <math display="block" id="M144"><msubsup><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>Φ</mi><mi>θ</mi></mrow><mrow/></msubsup><mo>=</mo><msub><mo>∫</mo><mrow><mi>Ω</mi></mrow></msub><mrow><msubsup><mrow><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>¯</mo></mover></mrow><mrow><mi>Φ</mi></mrow><mrow><mi>T</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mover accent="true"><mi mathvariant="bold-italic">p</mi><mo>¯</mo></mover><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>θ</mi></mrow><mrow/></msubsup><mi>d</mi><mi>Ω</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680269&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680268&type= 39.28533173 7.28133297 <math display="block" id="M145"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>u</mi></mrow><mrow/></msubsup><mo>=</mo><msub><mo>∫</mo><mrow><msub><mrow><mi>Γ</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></msub><mrow><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mover accent="true"><mi mathvariant="bold-italic">T</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mi mathvariant="normal">d</mi><mi>Γ</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680257&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680256&type= 32.25799942 7.95866632 <math display="block" id="M146"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>Φ</mi></mrow><mrow/></msubsup><mo>=</mo><mo>-</mo><msub><mo>∫</mo><mrow><msub><mrow><mi>Γ</mi></mrow><mrow><mi>D</mi></mrow></msub></mrow></msub><mrow><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>Φ</mi></mrow><mrow/></msubsup><mover accent="true"><mi mathvariant="bold-italic">Q</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mi mathvariant="normal">d</mi><mi>Γ</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680259&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680258&type= 35.56000137 7.95866632 <math display="block" id="M147"><msubsup><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>θ</mi></mrow><mrow/></msubsup><mo>=</mo><mo>-</mo><msub><mo>∫</mo><mrow><msub><mrow><mi>Γ</mi></mrow><mrow><mi>a</mi></mrow></msub></mrow></msub><mrow><msubsup><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi>θ</mi></mrow><mrow/></msubsup><mover accent="true"><mi mathvariant="bold-italic">J</mi><mo>¯</mo></mover><mfenced separators="|"><mrow><mi mathvariant="bold-italic">x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mi mathvariant="normal">d</mi><mi>Γ</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680262&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680260&type= 34.03599930 7.95866632 <math display="block" id="M148"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>u</mi><mi>u</mi></mrow></msub><mo>=</mo><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>e</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>e</mi></mrow></msubsup><mo>=</mo><mi mathvariant="normal">d</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi><mfenced open="{" close="}" separators="|"><mrow><msubsup><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>m</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>k</mi></mrow></msubsup><mo>,</mo><mo>⋯</mo><mo>,</mo><msubsup><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msubsup></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680275&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680264&type= 71.03533173 4.74133301 式中： <math id="M149"><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">p</mi></mrow></msub><mo>×</mo><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680277&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680276&type= 15.66333294 3.72533321 ； <math id="M150"><msubsup><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mo>=</mo><mi>ρ</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup><mfenced separators="|"><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680278&type= 39.53932953 3.97933316 ； i 为第 k 个光滑单元中第 i 个光滑子元； <math id="M151"><msub><mrow><mi>T</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680281&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680280&type= 2.87866688 3.72533321 为第 i 个光滑子元的厚度。 当仅考虑稳态热传导问题时， 式(42) 式(42) 可改写为 <math display="block" id="M152"><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>u</mi><mi>u</mi></mrow></msub></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mover accent="true"><mrow><mi mathvariant="bold-italic">u</mi></mrow><mi mathvariant="bold-italic">¨</mi></mover></mtd></mtr><mtr><mtd><mover accent="true"><mrow><mi mathvariant="bold-italic">Φ</mi></mrow><mi mathvariant="bold-italic">¨</mi></mover></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>u</mi><mi>u</mi></mrow></msub></mtd><mtd><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>u</mi><mi>Φ</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>Φ</mi><mi>u</mi></mrow></msub></mtd><mtd><mo>-</mo><msub><mrow><mi mathvariant="bold-italic">K</mi></mrow><mrow><mi>Φ</mi><mi>Φ</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mi mathvariant="bold-italic">u</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">Φ</mi></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>u</mi><mi>θ</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>Φ</mi></mrow></msub><mo>+</mo><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>Φ</mi><mi>θ</mi></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680283&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680282&type= 73.66000366 9.05933285 式中： <math display="block" id="M153"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>u</mi><mi>θ</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></mrow></munderover></mstyle><mrow><msubsup><mrow><mover accent="true"><mi>B</mi><mo>¯</mo></mover></mrow><mrow><mi>u</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mover accent="true"><mi>η</mi><mo>¯</mo></mover></mrow><mi mathvariant="bold-italic">θ</mi><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680265&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680284&type= 34.96733475 9.39799976 <math display="block" id="M154"><msub><mrow><mi mathvariant="bold-italic">F</mi></mrow><mrow><mi>Φ</mi><mi>θ</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></mrow></munderover></mstyle><mrow><msubsup><mrow><mover accent="true"><mi>B</mi><mo>¯</mo></mover></mrow><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msubsup><mfenced separators="|"><mrow><msup><mrow><mi mathvariant="bold-italic">x</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>,</mo><mi>t</mi></mrow></mfenced><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mi>θ</mi></mrow><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680285&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680267&type= 36.57600021 9.39799976 4 数值模拟算例 1 4.1 　压电复合材料悬臂梁 2 温度变化条件下压电复合材料悬臂梁的尺寸示意图如 图3 图3 所示。压电复合材料由BaTiO 3 基体和PZT-7纤维复合而成，基体和纤维的材料参数 [ 20 20 ] 如 表1 表1 所示，其中，纤维体积分数为66.67%的压电复合材料应用广泛 [ 21 21 ] ，故选取该类压电复合材料进行研究。 表2 表2 所示为基于渐近均匀化方法预测的压电复合材料悬臂梁的等效性能参数。边界条件如下：左端固定端处满足 u x =0， u z =0， Φ =0。受温度的影响，压电复合材料弹性常数 <math id="M155"><msubsup><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">q</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mover accent="true"><mi>C</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">q</mi></mrow><mrow/></msubsup><mo>×</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>α</mi></mrow></msub><mi mathvariant="normal">Δ</mi><mi>T</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680286&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680300&type= 31.15733528 5.24933338 ，温度变化量Δ T =5 K，系数 <math id="M156"><msub><mrow><mi>S</mi></mrow><mrow><mi>α</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680287&type= 3.21733332 3.72533321 =-0.01。 图4 图4 所示为压电复合材料悬臂梁的四节点畸变单元的离散模型。 注： <math id="M160"><msubsup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>θ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680316&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680315&type= 3.13266683 3.64066648 为热膨胀系数。 2 材料 5 C ij /GPa 3 e ij /(C∙m -2 ) 2 λ ij /(10 -9 C 2 ∙N -1 ∙m -2 ) 2 p i /(10 -4 C∙m -2 ∙K -1 ) 2 <math id="M157"><msubsup><mrow><mi mathvariant="bold-italic">α</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>θ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680292&type= 3.13266683 3.64066648 /(10 -6 K -1 ) 2 ρ / (kg∙m -3 ) C 11 C 12 C 13 C 33 C 44 e 31 e 33 e 15 λ 11 λ 33 p 1 p 2 <math id="M158"><msubsup><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680295&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680294&type= 3.04800010 3.64066648 <math id="M159"><msubsup><mrow><mi>α</mi></mrow><mrow><mn>3</mn></mrow><mrow><mi>θ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680297&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680296&type= 3.04800010 3.64066648 BaTiO 3 (基体) 150.00 65.63 65.94 145.50 43.86 -5.600 17.36 11.404 12.84 15.05 — — — — 7 600 PZT-7 (纤维) 133.40 83.20 78.80 101.20 16.70 -4.924 13.96 13.310 13.04 9.76 -2.5 -2.5 2.0 2.0 7 600 等效性能参数 数值 等效性能参数 数值 等效性能参数 数值 <math id="M161"><msub><mrow><mover accent="true"><mi>C</mi><mo>¯</mo></mover></mrow><mrow><mn>11</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680318&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680317&type= 4.57200003 3.89466691 /GPa 139.0 <math id="M162"><msub><mrow><mover accent="true"><mi>e</mi><mo>¯</mo></mover></mrow><mrow><mn>15</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680409&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680408&type= 3.80999994 3.38666677 /(C∙m -2 ) 12.72 <math id="M163"><msub><mrow><mover accent="true"><mi>λ</mi><mo>¯</mo></mover></mrow><mrow><mn>11</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680322&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680321&type= 3.89466691 3.80999994 /(10 -9 C 2 ∙N -1 ∙m -1 ) 13.05 <math id="M164"><msub><mrow><mover accent="true"><mi>C</mi><mo>¯</mo></mover></mrow><mrow><mn>12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680389&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680376&type= 4.57200003 3.89466691 /GPa 77.5 <math id="M165"><msub><mrow><mover accent="true"><mi>e</mi><mo>¯</mo></mover></mrow><mrow><mn>33</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680374&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680373&type= 3.80999994 3.38666677 /(C∙m -2 ) 15.06 <math id="M166"><msub><mrow><mover accent="true"><mi>λ</mi><mo>¯</mo></mover></mrow><mrow><mn>33</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680388&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680375&type= 3.89466691 3.80999994 /(10 -9 C 2 ∙N -1 ∙m -1 ) 11.51 <math id="M167"><msub><mrow><mover accent="true"><mi>C</mi><mo>¯</mo></mover></mrow><mrow><mn>13</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680329&type= 4.57200003 3.89466691 /GPa 74.1 <math id="M168"><msub><mrow><mover accent="true"><mi>e</mi><mo>¯</mo></mover></mrow><mrow><mn>13</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680378&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680377&type= 3.80999994 3.38666677 /(C∙m -2 ) ‒5.20 <math id="M169"><msub><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680390&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680379&type= 3.30200005 4.06400013 /(10 -4 C∙m -2 ∙K -1 ) ‒2.50 <math id="M170"><msub><mrow><mover accent="true"><mi>C</mi><mo>¯</mo></mover></mrow><mrow><mn>33</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680397&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680396&type= 4.57200003 3.89466691 /GPa 115.2 <math id="M171"><msub><mrow><mover accent="true"><mi>β</mi><mo>¯</mo></mover></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680406&type= 3.13266683 4.48733330 = <math id="M172"><msub><mrow><mover accent="true"><mi>β</mi><mo>¯</mo></mover></mrow><mrow><mn>3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680306&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680305&type= 3.13266683 4.48733330 /(m∙C∙kg -1 ) 0 <math id="M173"><msub><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover></mrow><mrow><mn>3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680307&type= 3.30200005 4.06400013 /(10 -4 C∙m -2 ∙K -1 ) ‒2.50 <math id="M174"><msub><mrow><mover accent="true"><mi>C</mi><mo>¯</mo></mover></mrow><mrow><mn>44</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680310&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680309&type= 4.57200003 3.89466691 /GPa 25.6 <math id="M175"><msubsup><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mn>1</mn></mrow><mrow><mi mathvariant="normal">M</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680312&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680311&type= 3.89466691 3.64066648 /(m 3 ∙kg -1 ) 0 <math id="M176"><msubsup><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mn>1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680343&type= 3.04800010 3.64066648 /(10 -6 K -1 ) 2.00 <math id="M177"><mover accent="true"><mi>ρ</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680405&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680404&type= 1.86266661 3.04800010 /(kg∙m -3 ) 7 600 <math id="M178"><msubsup><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mn>3</mn></mrow><mrow><mi mathvariant="normal">M</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680382&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680381&type= 3.89466691 3.64066648 /(10 -4 m 3 ∙kg -1 ) 1.10 <math id="M179"><msubsup><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mn>3</mn></mrow><mrow><mi>θ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680350&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680349&type= 3.04800010 3.64066648 /(10 -6 K -1 ) 2.00 基于渐进均匀化的力-电-热耦合光滑有限元法(CS-FEM)对温度变化条件下压电复合材料悬臂梁的固有频率进行求解，单元网格采用30×2，60×4和90×6这3种网格形式，并将其求解结果与有限元法(FEM)计算结果以及参考解进行对比。 图5 图5 所示为温度变化时压电复合材料悬臂梁的前10阶固有频率。由 图5 图5 可见，3种单元离散条件下CS-FEM的求解结果与FEM的求解结果基本一致，验证了CS-FEM的正确性和有效性；不同单元条件下CS-FEM的计算结果基本相同，验证了CS-FEM具有较好的收敛性。 图6 图6 所示为采用畸变单元的CS-FEM和采用规则四边形单元的FEM求解得到的压电复合材料悬臂梁的前10阶固有频率。由 图6 图6 可见：在60个单元(含畸变单元)下的CS-FEM求解结果与540个规则单元下的FEM求解基本相同，进一步验证了CS-FEM对离散单元的质量要求较低；而FEM采用畸变单元往往导致结果精度低、错误或无法计算，进一步验证了在网格划分方面CS-FEM较FEM具有优势。 在CPU为Intel Xeon E3-1220 v3 @ 3.10 GHz，32 G RAM条件下，对不同单元数下CS-FEM和FEM的计算效率进行对比，结果如 表3 表3 所示。由 表3 表3 可见，不同单元数下2种方法的计算时间基本相同，这是由于基于光滑单元的CS-FEM刚度矩阵的带宽与FEM的相同；随着单元数量增加，CS-FEM的计算时间逐渐比FEM的少，这是因为CS-FEM不需要对形函数求导，并可将面积积分转化为线积分。 单元数/个 FEM计算时间/s CS-FEM计算时间/s 60 0.127 0.159 240 2.375 2.462 480 3.862 3.906 2 160 203.173 201.968 4.2 　受瞬态集中力作用的压电复合材料俘能器 2 受瞬态集中力作用的压电复合材料俘能器在变温环境中服役，结构尺寸示意图如 图7 图7 所示。压电复合材料由压电陶瓷纤维PZT-5和聚合物基体复合而成，纤维体积分数66.67%，基体和纤维的材料参数 [ 21 21 ] 如 表4 表4 所示。边界条件如下：左端固定端处满足 u x =0， u z =0，底部电势 Φ =0，初始时刻位移和速度均为0，时间步长Δ t 取0.2 s。受温度的影响，压电复合材料弹性常数 <math id="M180"><msubsup><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">q</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mover accent="true"><mi>C</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">q</mi></mrow><mrow/></msubsup><mo>×</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>α</mi></mrow></msub><mi mathvariant="normal">Δ</mi><mi>T</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680364&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680363&type= 31.15733528 5.24933338 ，温度变化量Δ T =0.01 K，系数 <math id="M181"><msub><mrow><mi>S</mi></mrow><mrow><mi>α</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680366&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680365&type= 3.21733332 3.72533321 =-0.005。俘能器的长度为 l ，在 A 点处受集中力载荷 F =1 000 sin(0.4π t ) N，计算节点 B 处的瞬态广义位移。采用Wilson- θ 法求解广义位移 [ 22 22 - 28 28 22-28 ] 。 2 材料 5 C ij /GPa 3 e ij /(C∙m -2 ) 2 λ ij /(10 -9 C 2 ∙N -1 ∙m -2 ) 2 α i θ / (10 -6 K -1 ) 2 p i /(10 -4 C∙m -2 ∙K -1 ) 2 ρ /(kg∙m -3 ) C 11 C 12 C 13 C 33 C 44 e 31 e 33 e 15 λ 11 λ 33 <math id="M182"><msubsup><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi mathvariant="normal">θ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680340&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680368&type= 3.04800010 3.64066648 <math id="M183"><msubsup><mrow><mi>α</mi></mrow><mrow><mn>3</mn></mrow><mrow><mi mathvariant="normal">θ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680342&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680341&type= 3.04800010 3.64066648 p 1 p 2 聚合物 (基体) 3.86 2.57 2.57 3.86 0.64 — — — — — — — — — 7 600 PZT-5 (纤维) 121.00 75.40 75.20 111.00 21.10 -5.40 15.80 12.30 8.11 7.35 8.53 1.99 -2.50 -2.50 7 600 在不考虑聚合物的热胀系数、湿胀系数、热电系数和湿电系数情况下，基于渐近均匀化方法预测的压电复合材料俘能器的等效性能参数见 表5 表5 。 等效性能参数 数值 等效性能参数 数值 等效性能参数 数值 <math id="M184"><msub><mrow><mover accent="true"><mi>C</mi><mo>¯</mo></mover></mrow><mrow><mn>11</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680372&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680371&type= 4.57200003 3.89466691 /GPa 13.278 01 <math id="M185"><msub><mrow><mover accent="true"><mi>e</mi><mo>¯</mo></mover></mrow><mrow><mn>33</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680374&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680373&type= 3.80999994 3.38666677 /(C∙m -2 ) 12.984 637 <math id="M186"><msub><mrow><mover accent="true"><mi>λ</mi><mo>¯</mo></mover></mrow><mrow><mn>33</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680388&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680375&type= 3.89466691 3.80999994 /(nF∙m -1 ) 5.105 80 <math id="M187"><msub><mrow><mover accent="true"><mi>C</mi><mo>¯</mo></mover></mrow><mrow><mn>12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680389&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680376&type= 4.57200003 3.89466691 /GPa 7.019 85 <math id="M188"><msub><mrow><mover accent="true"><mi>e</mi><mo>¯</mo></mover></mrow><mrow><mn>13</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680378&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680377&type= 3.80999994 3.38666677 /(C∙m -2 ) ‒0.394 201 <math id="M189"><msub><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680390&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680379&type= 3.30200005 4.06400013 /(10 -5 C∙m -2 ∙K -1 ) ‒2.50 <math id="M190"><msub><mrow><mover accent="true"><mi>C</mi><mo>¯</mo></mover></mrow><mrow><mn>13</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680392&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680391&type= 4.57200003 3.89466691 /GPa 7.872 01 <math id="M191"><msubsup><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mn>1</mn></mrow><mrow><mi mathvariant="normal">M</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680394&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680393&type= 3.89466691 3.64066648 /(10 -4 m 3 ∙kg -1 ) 0 <math id="M192"><msub><mrow><mover accent="true"><mi>p</mi><mo>¯</mo></mover></mrow><mrow><mn>3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680380&type= 3.30200005 4.06400013 /(10 -5 C∙m -2 ∙K -1 ) ‒2.50 <math id="M193"><msub><mrow><mover accent="true"><mi>C</mi><mo>¯</mo></mover></mrow><mrow><mn>33</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680397&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680396&type= 4.57200003 3.89466691 /GPa 42.316 62 <math id="M194"><msubsup><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mn>3</mn></mrow><mrow><mi mathvariant="normal">M</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680382&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680381&type= 3.89466691 3.64066648 /(10 -4 m 3 ∙kg -1 ) 0.44 <math id="M195"><msubsup><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mn>1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680383&type= 3.04800010 3.64066648 /(10 -6 K -1 ) 8.53 <math id="M196"><msub><mrow><mover accent="true"><mi>C</mi><mo>¯</mo></mover></mrow><mrow><mn>44</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680399&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680398&type= 4.57200003 3.89466691 /GPa 3.188 67 <math id="M197"><msub><mrow><mover accent="true"><mi>β</mi><mo>¯</mo></mover></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680406&type= 3.13266683 4.48733330 /(m∙C∙kg -1 ) 0 <math id="M198"><msubsup><mrow><mover accent="true"><mi>α</mi><mo>¯</mo></mover></mrow><mrow><mn>3</mn></mrow><mrow><mi>θ</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680402&type= 3.04800010 3.64066648 /(10 -6 K -1 ) 1.99 <math id="M199"><mover accent="true"><mi>ρ</mi><mo>¯</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680405&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680404&type= 1.86266661 3.04800010 /(kg∙m -3 ) 7 600 <math id="M200"><msub><mrow><mover accent="true"><mi>β</mi><mo>¯</mo></mover></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680406&type= 3.13266683 4.48733330 /(m∙C∙kg -1 ) 0 — — <math id="M201"><msub><mrow><mover accent="true"><mi>e</mi><mo>¯</mo></mover></mrow><mrow><mn>15</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680409&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680408&type= 3.80999994 3.38666677 /(C∙m -2 ) 0.052 676 <math id="M202"><msub><mrow><mover accent="true"><mi>λ</mi><mo>¯</mo></mover></mrow><mrow><mn>11</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680411&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680410&type= 3.89466691 3.80999994 /(nF∙m -1 ) 0.420 31 — — CS-FEM采用5 mm×5 mm的单元网格(112个单元)进行离散，FEM采用2 mm×2 mm的单元网格(700个单元)进行离散。 图8 图8 所示为基于CS-FEM和FEM离散模型，在温度变化及瞬态集中力作用下压电复合材料俘能器在 B 点处的位移 u x 、位移 u z 和电势 Φ 的求解结果。由 图8 图8 可见，采用112个单元的CS-FEM计算结果与采用700个单元的FEM计算结果基本相同，进一步验证了CS-FEM较FEM具有更高的精度。 (a) 位移 u x ；(b) 位移 u z ；(c) 电势 Φ 4.3 　受温度连续变化作用的压电复合材料俘能器 2 受温度连续变化作用的压电复合材料俘能器的尺寸示意图如 图9 图9 所示。压电复合材料由压电陶瓷纤维PZT-5和聚合物基体复合而成，材料参数及其等效性能参数分别见 表4 表4 和 表5 表5 。边界条件如下：左端固定端处满足 u x =0 nm， u z =0 nm，底部电势 Φ =0 V，初始时刻位移和速度均为0，时间步长Δ t 取0.2 s。受温度的影响，压电复合材料弹性常数 <math id="M203"><msubsup><mrow><mi>C</mi></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">q</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mover accent="true"><mi>C</mi><mo>¯</mo></mover></mrow><mrow><mi mathvariant="normal">p</mi><mi mathvariant="normal">q</mi></mrow><mrow/></msubsup><mo>×</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>α</mi></mrow></msub><mi mathvariant="normal">Δ</mi><mi>T</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680386&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680385&type= 30.14133072 5.24933338 ，系数 <math id="M204"><msub><mrow><mi>S</mi></mrow><mrow><mi>α</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680418&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680387&type= 3.21733332 3.72533321 =-0.01。在 B 点处受集中力载荷 F =1 000 N，温度改变量 <math id="M205"><mi mathvariant="normal">Δ</mi><mi>T</mi><mo>=</mo><mn mathvariant="normal">5</mn><mo>×</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><mn mathvariant="normal">0.4</mn><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfenced><mi mathvariant="normal">K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680420&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43680419&type= 31.07266808 3.72533321 。 图10 图10 所示为受温度连续变化作用的压电复合材料俘能器在 B 点处的位移 u x 、位移 u z 和电势 Φ 的求解结果。CS-FEM采用4 mm×4 mm的网格进行离散，共60个单元；FEM采用2 mm×2 mm的网格进行离散，共240个单元。由 图10 图10 可见，采用60个单元的CS-FEM求解结果与采用240个单元的FEM求解结果基本相同，进一步证明了CS-FEM具有较高的精度。 (a) 位移 u x ；(b) 位移 u z ；( c ) 电势 Φ 5 结论 1 1) 由压电复合材料的本征特性出发，基于材料的基本方程以及渐进均匀化方法预测的压电复合材料有效性能参数，提出了基于渐进均匀化的力-电-热耦合光滑有限元法。 2) 推导出了力-电-热耦合光滑有限元的动力学控制方程，并将其改写成统一形式，运用Wilson- θ 法对压电复合材料结构瞬态响应问题进行了求解，研究了温度变化对结构固有频率和动力学响应的影响。 3) 基于渐进均匀化的力-电-热耦合光滑有限元法的求解结果与有限元法的求解结果较一致，且具有较好的收敛性和较高的精度，验证了方法的正确性和有效性。 参考文献 CHAN H W , UNSWORTH J . 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Finite element analysis and energy recovery calculation of carbon fiber composite based on ANSYS software 10.11817/j.issn.1672-7207.2023.04.011 TB115 A 1672-7207（2023）04-1325-11 郑建校, 段志善, 周立明. 基于渐进均匀化的力-电-热耦合光滑有限元法研究[J]. 中南大学学报(自然科学版), 2023, 54(4): 1325-1335. 郑建校, 段志善, 周立明. 基于渐进均匀化的力-电-热耦合光滑有限元法研究[J]. 中南大学学报(自然科学版), 2023, 54(4): 1325-1335. ZHENG Jianxiao, DUAN Zhishan, ZHOU Liming. Research on thermo-electro-mechanical coupling smoothed finite element method based on asymptotic homogenization[J]. Journal of Central South University(Science and Technology), 2023, 54(4): 1325-1335. ZHENG Jianxiao, DUAN Zhishan, ZHOU Liming. Research on thermo-electro-mechanical coupling smoothed finite element method based on asymptotic homogenization[J]. Journal of Central South University(Science and Technology), 2023, 54(4): 1325-1335. 陕西省自然科学基础研究计划-一般项目(面上)(2023-JC-YB-313)；国家自然科学基金资助项目(52002309)；陕西省自然科学基础研究计划-青年项目(2020JQ-669) Project(2023-JC-YB-313) supported by the Natural Science Basic Research Plan-General Project of Shaanxi Province; Project(52002309) supported by the National Natural Science Foundation of China; Project(2020JQ-669) supported by the Natural Science Basic Research Plan-Youth Project of Shaanxi Province 2022-04-24 2022-06-18 2023-04-26 2023-05-22 37691432 中南大学学报(自然科学版) 4 54