Dynamic finite deformation viscoelasticity and energy‐consistent time integration
Dynamic finite deformation viscoelasticity and energy‐consistent time integration
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动态有限变形粘弹性和能量一致时间积分
DOI：
10.1002/pamm.200910156
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发表时间：
2009
期刊：
影响因子：
--
通讯作者：
Betsch P
中科院分区：
文献类型：
--
作者：
Müller M;Groß M;Betsch P
本文研究了有限变形粘弹性体的守恒积分问题。各向同性粘弹性材料可以通过使用对称粘性内变量\calC_i来测量非弹性应变，并使用右柯西绿色张量\calC来测量总应变来描述（参见Reese & Govindjee 1）。然后，通过使用非对称乘积张量\calC\calC_i^-1，纯弹性应变进入各向同性自由能函数。或者，将对称右拉伸张量\calU_i作为内部变量，可以定义一个进入自由能的对称弹性应变张量\calC_e（参见Miehe 2）。这两种不同的方法导致不同的发展方程的粘性内变量。在本讲座中，两个演化方程都是通过空间中标准非线性位移基有限元的每个高斯点处的普通中点规则进行离散的。对于离散的半离散的汉密尔顿的运动方程的时间，我们使用数值时间积分保持基本守恒定律的基本系统。特别是，我们利用修改后的中点规则，根据离散梯度法，在冈萨雷斯3。数值例子表明这两种配方之间的差异。（© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA，魏因海姆）
In the present work we deal with the conserving integration of viscoelastic bodies undergoing finite deformations. Isotropic viscoelastic materials can be described by using a symmetric viscous internal variable \calC_i for measuring the inelastic strains, and the right Cauchy‐Green tensor \calC as measure of the total strain (see Reese & Govindjee 1). Then, by using the unsymmetric product tensor \calC\calC_i^-1, the purely elastic strains enter the isotropic free energy function. Alternatively, the i application of the symmetric right stretch tensor \calU_i as internal variable allows to define a symmetric elastic strain tensor \calC_e which enters the free energy (see also Miehe 2). These two different approaches lead to different evolution equations for the viscous internal variable. In this lecture, both evolution equations are discretised by an ordinary midpoint rule at each Gauss point of a standard nonlinear displacement‐based finite element in space. For discretising the semidiscrete Hamilton's equations of motion in time, we use numerical time integrators which preserve the fundamental conservation laws of the underlying system. In particular, we make use of a modified midpoint rule according to the discrete gradient method, proposed in Gonzalez 3. Numerical examples demonstrate the difference between both formulations.(© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)