Dear Community,(adsbygoogle = window.adsbygoogle || []).push({});

I am currently implementing the Lemaitre Damage Model as part of my Bachelor's thesis. The general procedure is a fully implicit backward Euler integration scheme. The presence of back-stress (a consequence of the Bauschinger effect) prohibits the simplification of the state variable update procedure to one equation. For those of you not familiar with continuum damage mechanics, I will try to keep my question in the realm of mathematics.

I need to solve a system of four coupled equations (A(σ,D,Δλ,β) = 0) of four variables (see the attached PDF (eq. 5.13) (variable explanation follows in the next paragraph). In order to find the solution, the Newton-Raphson scheme is employed. I therefore need to find the derivatives of these functions with respect to the variables σ,D,Δλ,β. I have tried to the best of my knowledge. Since literature on derivatives of tensor valued functions of tensors is fairly scarce (if you know of any, please tell me!), I am now on my own. Although everything appears fine to me, and the convergence of the NR scheme is fairly good, I know I must have done something wrong. Why do I know this? Well, for those of you not familiar with the finite element method, this would be lengthy to explain. For those of you who do have knowledge: global convergence is severely affected as damage evolves at the Gauss points. This is evidence of an erroneous consistent elasto-plastic tangent, which is a function of the inverse of the Jacobian used for the incremental solution of the internal Newon-Raphson scheme (elastic predictor, corrector).

The variables:

σ is the Cauchy stress tensor (2nd order, symmetric), D (damage) and Δλ (plastic multiplicator) are scalar, β is the back stress tensor (2nd order, symmetric, deviatoric). Note that η = dev(σ)-β (relative stress), where dev(σ) is the deviatoric part of σ. Therefore η is also of second order, symmetric, and deviatoric. C is the standard fourth-order isotropic elasticity tensor (minor and major symmetric). Anything you will encounter with index 0 is constant. Bold-face is second order tensor, stylized is fourth-order tensor. κ(R+Δλ) is a function you need not worry about. σ_vM is the von Mises equivalent stress, given by √-3J(η), where J(η) is the second invariant. σ_h is simply the hydrostatic component of σ. Everything else should be defined. If you have any questions, I will be happy to answer them!

I would be so grateful if someone could help me find what I did wrong! I am not asking for solutions (please do not post as I do value my own academic integrity), just for a push in the right direction!

Thanks in advance,

Robert

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Lemaitre Damage Model: Derivatives of Tensor Valued Functions with Respect to Tensors

Loading...

Similar Threads for Lemaitre Damage Model |
---|

I A question regarding Logistic population model |

**Physics Forums | Science Articles, Homework Help, Discussion**