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 引用本文: 罗阳军. 基于D-P准则的压力相关材料结构拓扑优化[J]. 力学学报, 2011, 43(5): 878-885.
Yangjun Luo. Topology optimization of pressure-dependent material structures based on D-P criterion[J]. Chinese Journal of Theoretical and Applied Mechanics, 2011, 43(5): 878-885.
 Citation: Yangjun Luo. Topology optimization of pressure-dependent material structures based on D-P criterion[J]. Chinese Journal of Theoretical and Applied Mechanics, 2011, 43(5): 878-885.

## Topology optimization of pressure-dependent material structures based on D-P criterion

• 摘要: 基于描述材料力学行为的Drucker-Prager(D-P)屈服准则, 研究了压力相关材料连续体结构拓扑优化设计问题的数学模型和数值算法. 以单元材料人工密度为设计变量, 结合SIMP惩罚模型和多孔微结构局部应力插值模型, 建立了以材料体积最小化为目标、考虑材料D-P屈服条件约束的优化问题数学模型. 利用\varepsilon-松弛方法消除奇异解现象, 采用伴随法有效推导约束函数灵敏度计算公式, 运用基于梯度的连续变量优化算法迭代求解优化问题. 数值算例验证了优化模型的正确性及数值算法的有效性, 并通过与von Mises应力约束优化结果的比较, 说明了材料的压力相关特性会对结构最优拓扑产生重要影响. 该方法设计出的最优拓扑由于充分利用了压力相关材料的抗压能力, 因而更为合理和实际.

Abstract: Many widely used materials, such as concrete, rocks,ceramics and polymers, have the feature of increasing shear strength as aresult of hydrostatic pressure increases. Structures made of thesepressure-dependent materials would typically hold the characteristic ofbetter stress limit in tension than in compression. In this case, the vonMises criterion is incompetent while the D-P criterion describedin terms of stress invariants is available as one of the simplest plasticityyield models. To take into account the asymmetrical compression and tensionbehaviors in the conceptual design of continuum structures, a practicabletopology optimization strategy for pressure-dependent materials based onD-P yield criterion is presented in this paper. By using theelement artificial relative densities as design variables, the optimizationproblem is formulated as to minimize the total material volume underD-P yield constraints on each element. In this optimizationmodel, the SIMP interpolation for element stiffness and the power-lawinterpolation for the local stress of porous microstructures are adopted. Inorder to circumvent the stress singularity phenomenon, the \varepsilon-relaxation strategy is applied for relaxing the local yield constraintsinvolved in the low-density elements. In this context, the sensitivity ofthe element constraints with respect to the design variables is efficientlyderived by the adjoint variable method. Then, the optimal design is obtainedby employing the gradient-based optimization algorithm. Finally, threenumerical examples with different strength limits in compression and tensionhave been solved to illustrate the validity of the proposed optimizationmodel as well as the efficiency of the numerical techniques. It is observedthat the optimal material distribution designed by the present method mayhave a significant difference compared with one designed by the conventionalvon Mises stress constraint approach. The obtained optimization solutionsare reasonable since they can make the best use of their strength inwithstanding the compression. The meaning of the proposed method forpressure-dependent material structures is thus demonstrated.

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