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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

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

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