摘要
正交各向异性体弹塑性分析是一个具有理论意义和工程实用价值的问题,现代工业的发展促使正交各向异性弹塑性问题逐渐为人们所关注,而计算机技术的发展则使得有关数值计算方法成为求解这类问题的有效手段。
本文提出了正交各向异性平面问题弹塑性分析的一般边界元方法。根据功的互等定理(Betti定理),首先建立了正交各向异性平面弹塑性问题的边界积分方程,然后根据几何关系和本构方程分别给出了内点位移以及内点应力公式。针对这些公式中由于塑性变形影响而产生的域内积分项,通过Mikhlin奇异积分的随体微分原理分析了域内奇异积分的性质,并给出了内点应力公式中由于域内强奇异而形成的自由项的解析表达式。利用正交各向异性平面弹性问题中的基本解以及建立上述积分方程中形成的关系式,推导了相应弹塑性问题中边界元法使用的基本解,并以矩阵形式给出了这些基本解之间的统一关系。
根据上述方程和相应基本解,通过对边界和域内分别离散,建立了初应力形式的正交各向异性平面弹塑性问题的离散方程和迭代方程。通过对数积分、坐标变换、刚体位移、常塑性应变场等方法,分别给出了积分方程中各类奇异积分的具体数值计算方法。弹塑性计算中采用了Hill-Tsai屈服准则以及增量形式的正交各向异性理想弹塑性本构关系;通过初应力增量迭代法求解非线性本构方程时,根据切向预测径向返回法确定了本构方程中的实际应力状态。最后通过数值算例分析了具体的正交各向异性平面弹塑性问题,计算结果表明了本文提出的边界元方法在分析这类问题上的有效性和可靠性。
由于正交各向异性平面弹塑性问题的边界元方法需要建立在相应弹性问题的基础之上,因此在进行以上的弹塑性分析之前,本文还对已有的正交各向异性平面弹性问题的边界元法进行了探讨。通过对已有的弹性问题中的位移基本解的改进,给出了弹性问题中的应变、应力以及面力基本解,改进后的基本解可以用于各向同性问题以及正交各向异性问题。数值算例表明了这些改进的基本解以及相关数值方法的有效性和可靠性,从而为弹塑性分析打下基础。
Elasto-plastic analysis of orthotropic bodies is a theoretical and practical problem, which has attracted many interests on study due to the development of modern sciences and technologies. On the other hand, the development of computer technologies permits computational methods an effective approach to solve this kind of complicated problems.
A general boundary element method (BEM) is proposed for elasto-plastic analysis of plane orthotropic bodies. Based on the Betti’s reciprocal work theorem, the boundary integral equation of plane orthotropic elasto-plastic problem is established firstly, and then the displacement and stress formulae in internal points are derived according to the geometric relations and constitutive equations. For the integrals in domain resulted from the plastic deformation, the convected differentiation concept of singular integral by Mikhlin is adopted to analyze the kind of singularity, and the free term induced by the strong singularity in domain is introduced with analytical expressions. According to the fundamental solutions in the plane orthotropic elastic problem and expressions from the above integral equations, the stress, strain and traction fundamental solutions in the plane orthotropic elasto-plastic problems are derived, respectively, and the matrix denotation is introduced to formulate the unified relations among all the fundamental solutions.
On the basis of above integral equations and fundamental solutions, the boundary and domain are discretized, respectively, and the discretized equations and iterative equations are established with the initial plastic stress to analyze the plane orthotropic elasto-plastic problems. All the kinds of singular integrals in the boundary and domain integral equations are numerically dealt with the schemes such as logarithmic quadratures, coordinate transformations, rigid-body displacement solutions, constant plastic strain solutions, etc, respectively. Hill-Tsai yielding criterion and the orthotropic elastic-perfectly plastic constitutive equations with incremental formulations are adopted in the numerical elasto-plastic analysis. The tangent
predictor-radial return algorithm is used to determine the stress state in solving the nonlinear constitutive equations with the initial stress and incremental iteration method. Finally, numerical examples show that the BEM is effective and reliable in analyzing elasto-plastic problems of orthotropic bodies.
As the BEM for plane orthotropic elasto-plastic problems is established on the basis of the corresponding elastic analysis, the elastic analysis of plane orthotropic problems with the BEM is also discussed before implementation of the elasto-plastic analysis. The existing displacement fundamental solutions are improved, and then the strain, stress and traction fundamental solutions in the plane orthotropic elastic problems are derived, respectively. All of the improved fundamental solutions can be exploited in the isotropic and orthotropic problems. Numerical examples show the improved fundamental solutions and BEM schemes are effective and reliable in the elastic analysis, which provides the foundations for the elasto-plastic analysis of orthotropic bodies.
引文
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