MATHEMATICAL PROBLEMS IN THE INTEGRAL-TRANSFORMATION METHOD OF DYNAMIC CRACK ~

In the investigation on fracture mechanics, the potential function was introduced,and the moving differential equation was constructed. By making Laplace and Fourier transformation as well as sine and cosine transformation to moving differential equations and various responses, the dual equation which is constructed from boundary conditions lastly was solved. This method of investigating dynamic crack has become a more systematic one that is used widely. Some problems are encountered when the dynamic crack is studied.After the large investigation on the problems, it is discovered that during the process of mathematic derivation, the method is short of precision, and the derived results in this method are accidental and have no credibility. A model for example is taken to explain the problems existing in initial deriving process of the integral-transformation method of dynamic crack.     

基于能量原理确定系泊碰撞载荷的有限元法

准确地确定出船舶系泊时的撞击载荷, 对船舶及海洋平台的强度研究都是极其重要的. 提出了通过能量原理和有限元分析确定系泊撞击载荷的方法. 碰撞时, 动能和变形能满足能量守恒, 动能是包括附连水质量在内的船体总动能, 此动能在撞击时转化为变形能, 根据变形能与载荷的关系, 求出碰撞载荷. 通过具有碰撞载荷理论解的梁的验证, 表明该方法是完全可行的. 为实际工程提供了确定护舷撞击力的理论依据和计算方法.     

空间静不定桁架的变形协调条件及求解

根据桁架受载后各杆件的伸长量、各杆件夹角的变化之间的几何关系，以各杆的伸长量和夹角的微小变化为未知量，建立了空间静不定桁架的变形协调条件，并结合实例给出了求解空间静不定桁架的一种计算方法．     

动态裂纹积分变换法中的数学问题

引入势函数，形成运动微分方程，对运动微分方程和各种响应进行Laplace变换及Fourier正弦、余弦变换，最后求解由边界条件形成的对偶方程——这种研究动态裂纹的方法已经被广泛使用并成为比较系统的方法，以一种模型为例，对其推演过程进行了研究，最后发现：此方法在数学推演时，存在着不严密的问题，推演结果带有偶然性，不具可信性。     

平面应力应变问题与转轴公式的群表示

从坐标变换出发,讨论了材料力学中形如任意斜截面应力的公式.结果表明待求量可以表示为已知量的变换,变换关系为 SO(2)群的直积,并且此变换也适用于应变问题与转轴公式.     

压电材料Ⅰ型裂纹动态问题的对偶方程组及其求解

首先引入势函数,用势函数表示压电材料的基本微分方程,并采用Laplace变换、半无限对称Fourier正弦变换和Fourier余弦变换,对微分方程进行变换和初步求解;然后通过Fourier反演和引入边界条件,建立了二维压电材料动态裂纹问题的对偶方程组;再根据Bessel函数性质,利用Abel型积分方程及其反演,将对偶方程组化为第二类Fredholm积分方程组.结果表明,方法是可行的,可以成为研究此类问题的一种有效方法.     

哈密顿体系下机电耦合问题的基本方程

利用多变量变分原理，针对具有机电耦合特性的压电材料，在哈密顿体系下推导出了机电耦合问题的对偶方程．求出了问题的零本征解和非零本征解的一般表达式。并就“二维压电平板对边受均布载荷”这一具体问题进行了完整的分析计算，得出了应力和电位移的具体解答，结果与一般弹性力学所得的结论相吻合，而且精度极高。表明哈密顿体系适用于压电材料的力学分析。     

超静定桁架变形协调方程 的新方法

在三角形桁架变形协调的基础上，提出了建立超静定桁架的变形协调方程的新方法，此法简便易行，更利于数值计算。     

Dual equations and solutions of I-type crack of dynamic problems in piezoelectric materials

This paper firstly works out basic differential equations of piezoelectric materials expressed in terms of potential functions, which are introduced in the very beginning. These equations are primarily solved through Laplace transformation, semi-infinite Fourier sine transformation and cosine transformation. Secondly, dual equations of dynamic cracks problem in 2D piezoelectric materials are established with the help of Fourier reverse transformation and the introduction of boundary conditions. Finally, according to the character of the Bessel function and by making full use of the Abel integral equation and its reverse transform, the dual equations are changed into the second type of Fredholm integral equations. The investigation indicates that the study approach taken is feasible and has potential to be an effective method to do research on issues of this kind.