【UMAT6】上篇-Chaboche非线性随动硬化本构理论

大家好，我是九千CAE。

我们将分上（本构理论）、中（雅可比矩阵）、下（UMAT实现）三篇讲解 Chaboche 非线性随动硬化本构 UMAT 实现。 我们注意到 Chaboche 非线性随动硬化本构与线性随动硬化本构在屈服准则、关联流动法则、等效塑性应变等的计算上基本相同，感兴趣的同学可回看 【UMAT3】上篇 线性随动硬化弹塑性UMAT中的内容。 

Elastic Behavior

Linear elastic constitutive model, namely generalized Hooke's law, is adopted as

where      is shear modulus,      is bulk modulus,      is the lame constant,      is deviatoric strain tensor,      is the deviatoric stress tensor, the superscript 'e' means elastic.

Yield Criterion

The yield criterion of combined (kinematic-isotropic) hardening is

where      is the equivalent stress (not the Mises stress which is still      ),      is the deviatoric back stress tensor.

Associative Flow Rule

By choosing the yield function as the plastic potential, one has

where      is the rate of plastic strain tensor,      is the plastic multiplier and is identity to equivalent plastic strain rate, the superscript 'p' means plastic.

Isotropic Hardening Equation

The following isotropic hardening equation is used

where      is the initial yield stress,      and      are constants.

Note that other isotropic hardening equations can also be used, it is not a hard job to replace it when implementing this in UMAT.

Non-linear Kinematic Hardening Equation

The non-linear kinematic hardening equation proposed by Chaboche is adopted

where      is 'k' th component of deviatoric back stress tensor,      and      are constants.

Predictor - Return Mapping Algorithm

Strain Decomposition

As the time discretized in finite element solving, consider a time increment start at time      and end at time     , we have the strain decomposition

where      is the strain increment, and note that we will omit the subscription      in the following for brevity.

Stress Predictor

If we assume that the strain increment is totally elastic, then we have the strain predictor (or trial strain)

Accordingly, the stress predictor is

and the yield stress predictor remains the same

Plastic  Correction

Based on the stress predictor, if the corresponding yield function     , then this increment is purely elastic, we can just update the stress with the predictor one and return elastic Jacobian matrix when implementing in UMAT.

On the other hand, if     , the return mapping method should be applied to obtain the increment plastic strain. The better is to solve for the equivalent one (    ) first, to do so, we have to represent all the terms in yield criterion by     .

Consider     

Consider     , since

we have

where we define     .

With       and      at hand

This show that      and      are co-linear, thus     , so

Then

Consider     

Substitute these 2 equation into the yield function

Newton Iteration

We use Newton iteration method to solve the previous equation

where      mean m'th iteration, the derivative

Since

The 1st term is

Then the 2nd term is

The 3rd term is defined as

Combine all we have

Once      is solved, we can update all the strain and stress as