含压电纤维复合材料的电—弹场及其有效性能预测
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摘要
压电材料因其固有的机—电耦合效应在工程中得到了广泛应用。然而，压电材料的脆性使其难以制成大尺寸的结构。解决这个问题的办法之一是将压电材料（如压电纤维）与弹性基体合成为复合材料。针对这种多相材料，由于物理上的不连续将在界面附近形成局部的应力集中，这往往引起纤维与基体脱粘，甚至导致材料失效。提高界面强度的方法之一是在基体和压电纤维之间加入合适的界面相，以减少界面附近的应力集中。另一方面，压电纤维的合理排布同样可以有效地降低界面附近的应力集中。
基于复变函数理论和含有界面相的广义自洽方法，本文研究了含压电纤维复合材料内的电-弹性场及其有效性能，重点分析了纤维排布方式以及界面相对电-弹性场和有效性能的影响。全文由七章组成，具体各章主要内容概括如下：
第一章介绍了压电复合材料的研究现状以及有待进一步探讨的问题。
第二章推导了本文所涉及的基本方程。基于Stroh理论首先引入了以x_3轴为极化方向的压电材料反平面问题的电弹场基本公式。其次，通过复变函数理论以及线弹性压电方程得到了极化方向为x_3方向的压电复合材料平面问题的电弹场基本公式。
第三章主要研究了含有功能梯度界面相压电纤维复合材料反平面问题与平面问题的电弹场。基于复变函数理论，将功能梯度界面相分层均匀化，然后将基体、分层均匀化后的界面相以及压电纤维的复势函数设定为含有待定系数的级数求和形式，通过边界条件建立其相应的方程，最后求得在各种载荷下的电弹场。
第四章分析了多个含界面相压电纤维反平面问题的电弹场。基于复变函数理论以及Stroh理论，研究了复变函数中多连通域的弹性平衡问题，得到了压电复合材料各组分内电弹场的理论解。最后用数值方法分析了界面相的材料常数、厚度以及纤维间的距离对压电复合材料电弹场的影响。
第五章考虑了在远处面内机械载荷和沿着压电纤维方向电载荷共同作用下任意排列的含界面相压电纤维复合材料平面问题的电弹场，利用数值结果讨论了不同的界面相对压电复合材料电弹场的影响。
第六章基于不同的模型分析了含有界面相压电纤维复合材料的有效性能。针对纤维统计均匀分布的情况，根据含有界面相广义自洽理论以及第三章至第五章求解得到的复合材料各组分内的电弹场获得了多种含有界面相压电复合材料模型的有效性能。利用数值结果讨论了不同的界面相对压电复合材料有效性能的影响。
最后，在第七章，对本文所做的工作做了总结并对未来的工作做了展望。
Due to the mechanical-electric coupling behaviors, piezoelectric materials have been widely applied inengineering applications. However, the brittleness of piezoelectric materials has limited their sizes.One way to solve this problem is to embed the piezoelectric materials into the elastic matrix and makeup piezoelectric composites. The physical discontinuity in the composites will lead to local stressconcentration along the interface, causing the separation of the fibers from the matrix and finallyfailures. One way to improve the interface strength is to add an appropriate interphase between thepiezoelectric fibers and the matrix. In addition, the reasonable array type of piezoelectric fibers canalso effectively reduce the stress concentration near the interphase.
Based on the complex variable theory and the generalized self-consistent method with interphase, inthe paper we study the local mechanical-electric fields and the effective properties of the piezoelectriccomposites. The effect of array type of fibers and interphase on the mechanical-electric fields andeffective properties are analyzed, respectively. This paper consists of seven chapters and the maincontents are as follows:
In the first Chapter, a brief introduction to piezoelectric composites is given and the problems neededto be solved are outlined.
In the second Chapter, the basic equations in this paper are obtained. Based on the Stroh theory, thebasic equations of the anti-plane problem for the piezoelectric materials are firstly derived when thepolar direction is assumed to be along the x_3axis. Then according to the complex variable theory andlinear elastic piezoelectric equations, the mechanic-electric equations of the plane problems for thepiezoelectric composites are also derived when the polar direction of piezoelectric fiber is assumed tobe along the x_3axis.
In the third Chapter, the mechanical-electric field under anti-plane and plane deformation for thepiezoelectric fiber with a functionally graded interphase embedded into the matrix is studied,respectively. The functionally graded interphase is divided into be uniform, and the complex potentialsof matrix, the after-homogenization of interphase and piezoelectric fiber are set to power series with undetermined coefficients. Then according to the corresponding boundary conditions, themechanical-electric fields of the composites are obtained under different kinds of known remote loads.
In the fourth Capter, studied is an anti-plane problem for mechanical-electric field of the multiplepiezoelectric fibers with an interphase in the matrix. Based on the complex variable theory and theStroh formalism, the elastic equilibrium of the multiply-connected problem in the complex potential issolved and the analytical solutions for all the constituents of the piezoelectric composites are derived.Then numerical results are presented to discuss the influences of interphase properties, interphasethickness and distances on the mechanical-electric fields of the piezoelectric composites.
In the fifth Chapter, addressed is a plane problem for mechanical-electric field of the multiplepiezoelectric fibers with interphase randomly-distributed in the elastic matrix under remote in-planemechanical load and electric load along the piezoelectric fiber. Numerical solutions are made toexplore the influence of different interphase on the mechanical-local field of the piezoelectriccomposites.
In the sixth Chapter, investigated is the effective properties problem for a variety of an interphaseembedded into the piezoelectric composites’ models. According to the generalized self-consistentmethod with interphase and the solutions obtained from the last three Chapters, the effectiveproperties of piezoelectric composites under statistics homogeneity are derived by different kinds ofmodels. Numerical results are presented to discuss the influence of interphase on the effectiveproperties of piezoelectric composites.
Finally, in the last Chapter, the present work is summarized and some future works are proposed onthe topic.
基于复变函数理论和含有界面相的广义自洽方法，本文研究了含压电纤维复合材料内的电-弹性场及其有效性能，重点分析了纤维排布方式以及界面相对电-弹性场和有效性能的影响。全文由七章组成，具体各章主要内容概括如下：
第一章介绍了压电复合材料的研究现状以及有待进一步探讨的问题。
第二章推导了本文所涉及的基本方程。基于Stroh理论首先引入了以x_3轴为极化方向的压电材料反平面问题的电弹场基本公式。其次，通过复变函数理论以及线弹性压电方程得到了极化方向为x_3方向的压电复合材料平面问题的电弹场基本公式。
第三章主要研究了含有功能梯度界面相压电纤维复合材料反平面问题与平面问题的电弹场。基于复变函数理论，将功能梯度界面相分层均匀化，然后将基体、分层均匀化后的界面相以及压电纤维的复势函数设定为含有待定系数的级数求和形式，通过边界条件建立其相应的方程，最后求得在各种载荷下的电弹场。
第四章分析了多个含界面相压电纤维反平面问题的电弹场。基于复变函数理论以及Stroh理论，研究了复变函数中多连通域的弹性平衡问题，得到了压电复合材料各组分内电弹场的理论解。最后用数值方法分析了界面相的材料常数、厚度以及纤维间的距离对压电复合材料电弹场的影响。
第五章考虑了在远处面内机械载荷和沿着压电纤维方向电载荷共同作用下任意排列的含界面相压电纤维复合材料平面问题的电弹场，利用数值结果讨论了不同的界面相对压电复合材料电弹场的影响。
第六章基于不同的模型分析了含有界面相压电纤维复合材料的有效性能。针对纤维统计均匀分布的情况，根据含有界面相广义自洽理论以及第三章至第五章求解得到的复合材料各组分内的电弹场获得了多种含有界面相压电复合材料模型的有效性能。利用数值结果讨论了不同的界面相对压电复合材料有效性能的影响。
最后，在第七章，对本文所做的工作做了总结并对未来的工作做了展望。
Due to the mechanical-electric coupling behaviors, piezoelectric materials have been widely applied inengineering applications. However, the brittleness of piezoelectric materials has limited their sizes.One way to solve this problem is to embed the piezoelectric materials into the elastic matrix and makeup piezoelectric composites. The physical discontinuity in the composites will lead to local stressconcentration along the interface, causing the separation of the fibers from the matrix and finallyfailures. One way to improve the interface strength is to add an appropriate interphase between thepiezoelectric fibers and the matrix. In addition, the reasonable array type of piezoelectric fibers canalso effectively reduce the stress concentration near the interphase.
Based on the complex variable theory and the generalized self-consistent method with interphase, inthe paper we study the local mechanical-electric fields and the effective properties of the piezoelectriccomposites. The effect of array type of fibers and interphase on the mechanical-electric fields andeffective properties are analyzed, respectively. This paper consists of seven chapters and the maincontents are as follows:
In the first Chapter, a brief introduction to piezoelectric composites is given and the problems neededto be solved are outlined.
In the second Chapter, the basic equations in this paper are obtained. Based on the Stroh theory, thebasic equations of the anti-plane problem for the piezoelectric materials are firstly derived when thepolar direction is assumed to be along the x_3axis. Then according to the complex variable theory andlinear elastic piezoelectric equations, the mechanic-electric equations of the plane problems for thepiezoelectric composites are also derived when the polar direction of piezoelectric fiber is assumed tobe along the x_3axis.
In the third Chapter, the mechanical-electric field under anti-plane and plane deformation for thepiezoelectric fiber with a functionally graded interphase embedded into the matrix is studied,respectively. The functionally graded interphase is divided into be uniform, and the complex potentialsof matrix, the after-homogenization of interphase and piezoelectric fiber are set to power series with undetermined coefficients. Then according to the corresponding boundary conditions, themechanical-electric fields of the composites are obtained under different kinds of known remote loads.
In the fourth Capter, studied is an anti-plane problem for mechanical-electric field of the multiplepiezoelectric fibers with an interphase in the matrix. Based on the complex variable theory and theStroh formalism, the elastic equilibrium of the multiply-connected problem in the complex potential issolved and the analytical solutions for all the constituents of the piezoelectric composites are derived.Then numerical results are presented to discuss the influences of interphase properties, interphasethickness and distances on the mechanical-electric fields of the piezoelectric composites.
In the fifth Chapter, addressed is a plane problem for mechanical-electric field of the multiplepiezoelectric fibers with interphase randomly-distributed in the elastic matrix under remote in-planemechanical load and electric load along the piezoelectric fiber. Numerical solutions are made toexplore the influence of different interphase on the mechanical-local field of the piezoelectriccomposites.
In the sixth Chapter, investigated is the effective properties problem for a variety of an interphaseembedded into the piezoelectric composites’ models. According to the generalized self-consistentmethod with interphase and the solutions obtained from the last three Chapters, the effectiveproperties of piezoelectric composites under statistics homogeneity are derived by different kinds ofmodels. Numerical results are presented to discuss the influence of interphase on the effectiveproperties of piezoelectric composites.
Finally, in the last Chapter, the present work is summarized and some future works are proposed onthe topic.
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