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Analytical stiffness matrices for tetrahedral elements

机译：四面体单元的解析刚度矩阵

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摘要

Stiffness matrices based on the non-linear Green-Lagrange strain definition may seem too complicated for analytical treatment. However, for the case of a linear displacement tetrahedron element no numerical integrations are needed. Closed form explicit analytical results are presented, making it possible to see the influence of each individual parameter and the results are directly suited for coding in a finite element program. The analytical secant and tangent element stiffness matrices are obtained by separating the dependence of the material constitutive parameters and of the stress/strain state from the dependence of the initial geometry and of the displacement assumption. The nodal positions of an element and the displacement assumption give six basic matrices of fourth order. These matrices do not depend on the material and the stress/strain state, and are thus unchanged during the necessary iterations for obtaining a solution based on the Green-Lagrange strain measure. This basic matrix approach on the directional level diminish the order of the involved matrices from 12 to 4. The presented resulting stiffness matrices are especially useful for design optimization, because analytical sensitivity analysis can then be performed. Another aspect of the paper is that linear strains imply erroneous displacement fields when rotations are involved, and we quantify these errors in relation to results based on Green-Lagrange strains.
机译：基于非线性格林-拉格朗日应变定义的刚度矩阵对于分析处理而言似乎太复杂了。但是，对于线性位移的四面体单元，不需要数值积分。给出了封闭形式的明确分析结果，从而可以查看每个单独参数的影响，并且该结果直接适合于在有限元程序中进行编码。通过将材料本构参数和应力/应变状态的依赖性与初始几何形状和位移假设的依赖性分开，可以得到分析割线和切线刚度矩阵。元素的节点位置和位移假设给出六个四阶基本矩阵。这些矩阵不依赖于材料和应力/应变状态，因此在进行必要的迭代以获取基于Green-Lagrange应变量的解时不变。这种在方向级别上的基本矩阵方法将所涉及的矩阵的阶数从12减少到4。呈现的结果刚度矩阵对于设计优化特别有用，因为可以执行分析灵敏度分析。本文的另一个方面是，当涉及旋转时，线性应变暗示了错误的位移场，并且我们根据格林-拉格朗日应变对这些误差进行了量化。

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