Charalampos Tsakmakis

Continuum Damage Models based on Energy Equivalence: Part I — Isotropic Material Response

An energy equivalence method for modeling damage effects in material response is proposed. In the present article, the main issues of the method are discussed for the less complicated case of isotropic constitutive functions. Otherwise, the material response addressed is supposed to be (rate-independent) elasto-plastic exhibiting isotropic and kinematic hardening. In order to make clear the difference to other continuum damage models, it suffices to deal here with isotropic damage expressed in terms of a scalar state variable. Our approach is based on the concept of effective stress and effective strain combined with a principle of energy equivalence as explained in the article. As a result, both the yield function and the evolution equations governing the hardening response of the damaged material are obtained from a given undamaged model material. Characteristic properties of the damage theory proposed are illustrated by comparing predicted responses with those according to damage models based on the principle of strain equivalence.

Predictions of microtorsional experiments by micropolar plasticity

Kinematic hardening rules are employed in classical plasticity to capture the so–called Bauschinger effect. They are important when describing the material response during reloading. In the framework of thermodynamically consistent gradient plasticity theories, kinematic hardening effects were first incorporated into a micropolar plasticity model by Grammenoudis and Tsakmakis. The aim of the present paper is to investigate this model by predicting size effects in torsional loading of circular cylinders. It is shown that kinematic hardening rules compared with isotropic hardening rules, as adopted in the paper, provide more possibilities for modelling size effects in the material response, even if only monotonous loading conditions are considered.

The Principle of Generalized Energy Equivalence in Continuum Damage Mechanics

Using the methods of continuum damage mechanics, constitutive models for elasto-plastic response and isotropic damage are derived. The approach is based on the concept of effective stress combined with a principle of generalized energy equivalence as explained in the paper. A family of yield functions parameterized by n, is obtained. If n = 1, then the yield function is identical to that commonly used in the damage model based on the strain equivalence hypothesis. Results of the analysis are so interpreted, that n = 1 should be favored, when modeling the damage response of metallic materials. This case is further investigated, by comparing predicted responses with those due to the model established in the framework of the strain equivalence principle. To this end the numerical method of finite elements is employed.

Use of the Elastic Predictor-Plastic Corrector Method for Integrating Finite Deformation Plasticity Laws

Classical plasticity theories are formulated by means of ordinary differential equations coupled with algebraic equations, so that the whole system of equations governing the material response is highly nonlinear. To integrate these equations, particular algorithms have been developed, the method of elastic predictor and plastic corrector being often used. This method has turned out to be a very efficient tool, when small deformation plasticity is considered. Especially, the usual constraint condition of plastic incompressibility is preserved exactly. However, when nonlinear geometry is involved, there are several possibilities to employ the method of elastic predictor and plastic corrector. Moreover, some effort has to be made, in order to ensure plastic incompressibility. The exponential map approach is one possibility to overcome the difficulties, but this approach is not suitable when deformation induced anisotropy is considered. The present work addresses the numerical integration of finite deformation plasticity models exhibiting both, isotropic and kinematic hardening. The integration of the evolution equations is based on the elastic predictor and plastic corrector procedure, appropriately adjusted to the structure of the adopted constitutive theory. Plastic incompressibility is preserved by introducing a further unknown into the system of equations to be solved numerically.

A Simple Model for Describing Yield Surface Evolution During Plastic Flow

It has been recognized for a long time that inelastic deformations may induce anisotropy in the material response, even if this is initially isotropic. For metallic materials, deformation induced anisotropy is reflected above all by translation, rotation and distortion of the yield surface. This has been confirmed by several experimental investigations independent of the way the yield point is defined. In the present paper a simple, thermodynamically consistent model is proposed., describing the evolving anisotropy of the yield surface. The model is first theoretically established, based on a sufficient condition for the dissipation inequality to be satisfied. Then, it is applied to predict the subsequent yield surfaces, after various prestressings, which have been observed experimentally by Ishikawa.