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J. Kaul, H. E. Pettermann:
"A constitutive model for fiber reinforced polymer plies - transition from hardening to softening behavior during anisotropic damage evolution";
Vortrag: 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Vienna; 18.02.2019 - 22.02.2019.

In the present work an extension to the damage formulation for the constitutive model for fiber reinforced polymers, proposed in [1], is presented. The existing model predicts plastic strain accumulation and uses continuum damage mechanics to model the stiffness degradation accompanied by strain hardening as well as strain softening behavior. In contrast to the majority of constitutive models for FRP plies, the damage evolution for the hardening behavior is formulated in terms of stresses. On one hand, this enables the use of well established fracture criteria, but on the other hand it poses difficulties combining the formulation for hardening with one for softening in a unified manner. Firstly, this is due the fact that the damage variables cannot be described as injective functions in terms of the stresses for both hardening and softening. And secondly, this is due to the fact that the softening behavior has to be mesh-adjusted, while the hardening has not. To overcome these difficulties, a piecewise formulation is used in [1], with a damage evolution law based on the Puck fracture criterion for the hardening and a damage evolution law based on equivalent strains for the softening behavior.
The aim of the present work is to unify the formulation by employing the framework presented in [2], where the damage evolution is controlled by a set of loading threshold functions. To identify critical states for the transition from hardening to softening, for multiple evolving internal damage variables, a transition condition is introduced. Furthermore the softening branches of the loading threshold functions are adjusted to provide meshsize-independent results. The extension to the constitutive model as well as the consistent Jacobian are implemented in a material subroutine for ABAQUS/Standard (SIMULIA, Providence, RI, USA).
The capabilities of the proposed damage formulation are demonstrated on various angle-ply laminates. The transition from hardening to softening is studied by using laminates with different ply angles to intentionally induce different levels of strain hardening before the onset of softening.

References:
[1] Th. Flatscher, H. E. Pettermann, A constitutive model for fiber-reinforced polymer plies accounting for plasticity and brittle damage including softening-implementation for implicit FEM, Composite Structures 93.9 (2011): 2241-2249.
[2] J. Kaul, H. E. Pettermann, A constitutive model for fiber reinforced polymer plies - coupling of plastic and elasto-damage mechanisms, Proceedings of the 6th European Conference on Computational Mechanics (ECCM 6), R. de Borst, J. Reese, C. Pearce, (2018)