黏弹-双曲线Drucker-Prager塑性模型应力更新隐式算法

A return mapping implicit algorithm for coupled viscoelastic and hyperbolic Drucker-Prager plastic modeling

黏弹-塑性模型是由黏弹性模型与塑性元件串联而成,可视为黏弹-黏塑性模型中黏塑性黏度参数趋于0的一种极限情况,为黏弹性材料的结构破坏分析提供了一个途径。把黏弹-塑性模型的应变增量分解为黏弹和塑性增量两部分,考虑黏弹性应变历史,把黏弹性积分型本构关系在一个时步内线性化,定义与时间增量相关的剪切模量和体积模量,导出应力递推公式,把黏弹-塑性本构积分转化为与弹塑性相似的形式。针对由黏弹性和双曲线Drucker-Prager塑性、各向同性硬化的黏弹-塑性模型,通过黏弹性预估和塑性校正"二步"算法实现对应力的更新,给出完全隐式算法和最终的计算公式。算例比较分析表明,由于迭代过程中仅需要简单的函数计算,该算法具有很好的收敛性。一般经过2次迭代运算后,屈服函数值已达到10-10的量级,应力点便返回到屈服面上。

A coupled viscoelastic-plastic model is composed of viscoelastic model and a series of plastic elements, which can be regarded as a limit condition of the viscoelastic-viscoplastic model when the viscosity-related parameter of viscoplasticity is close to O. This model provides an alternative scheme for analysing the structural collapse of viscoelastic materials by numerical solution at the certain circumstance. First, the strain increment was decomposed into viscoelastic part and plastic part in this viscoelastic-plastic model. Then the integral type viscoelastic constitutive equations were linearized over the time interval, by taking the history of viscoelastic strains into consideration. Meanwhile, the shear and bulk modules were clearly defined, which are fimctions of the time increment. The recurrence formulas for stresses on viscoelastic strains were deduced as well. The numerical integration of viscoelastic-plastic constitutive equation was transformed into the similar format with the general elastic-plastic circumstance. The plastic part of the viscoelastic-plastic model was assumed as the hyperbolic Drucker-Prager plasticity with isotropic hardening. Finally, the fully implicit stress update algorithm and the associated consistent tangent operator, as well as the final formulas, were derived by the combination of the viscoelastic predictor and the plastic return mapping. The comparison and analysis of numerical examples indicate that the algorithm had a good convergence, since only the simple function calculation was performed in each iteration process. After two iterations, the value of yield function reached to 10-10 degree, and the stress point returned to the yield surface.