Effect of stress triaxiality corrected plastic flow on ductile damage evolution in the framework of continuum damage mechanics

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Abstract

Most of the damage models derived from the theory of Lemaı̂tre have been developed using the hypothesis based on “proportional loading” for the determination of some material parameters. This hypothesis is adopted in such a way that a material coefficient (a scale factor for damage values) can be evaluated from tensile tests on unnotched specimens, considering the triaxiality of the stress state to be that corresponding to perfect uniaxiality, over the entire load history, and so ignoring necking induced variations in the triaxiality.

The present paper analyses the response of some continuum damage mechanics models which incorporate a correlation between the triaxiality factor and logarithmic plastic strain. These correlation laws, affecting also the true stress–true strain material curve, were derived from experimental data processed using the Bridgman method. Due to the lack of experimental data specifically related to damage evolution, the aim of the study was to evaluate the sensitivity of these models to the variability of the triaxiality parameter during the deformation history.

Introduction

In the field of the continuum damage mechanics (CDM), many models have been derived from the concepts proposed by Chaboche and Lemaı̂tre which allow the prediction of damage evolution under different stress state conditions.

The models considered here were proposed by Lemaı̂tre and Chaboche [3], [4], [5], [6], [7], [8], Wang [9], [10], Chandrakanth et al. [14] and Bonora [11], [12], [13]. The differences between them are mainly in the form of the dissipative potential from which the kinetic law of damage is derived, and in assumptions made regarding some material parameters.

The aspect investigated in all these models is the hypothesis that the triaxiality factor characterising the stress state during a tensile test is assumed to be constant and its value that due to the initial, non-deformed geometry.

This consideration allows the ready evaluation of some calibration or factor-scale like constants which act on the entire damage law finally derived, but it is acceptable only in the first phase of the deformation history, when the geometry of the specimen undergoes only small or negligible transversal deformations.

Many papers have reported that the triaxiality of the stress state during a tensile test is constant only until the necking phenomenon begins [1], [2], [15], [16], [18], [22], [23]. This event usually starts when the material is still far from failure, in fact for the steels tested here, the logarithmic plastic strain at which necking starts ranges from less than 1% up to 30% of each failure strain.

After necking begins during a tensile test, the material in the necking region (the only zone where plastic and damage characterisation is meaningful) experiences an increasing triaxiality of the stress state. The increase of the triaxiality ratio near the failure of the specimen can lead to values considerably greater than those predicted ignoring the necking effect.

These considerations justify the present study, which consists of an evaluation of the sensitivity of some CDM models with respect to necking-induced variations in the triaxiality ratio during a tensile test. The triaxiality evolution laws adopted here were evaluated using the Bridgman model for the stress and strain distributions in the neck, and a set of experimental data from tensile tests performed on some commercial steels.

Given the comparative nature of this study, experimental data specifically dealing with damage evolution was not evaluated, so that a set of values found in the literature, typical of steels similar to those tested by the authors, were used to obtain some damage parameters required by the models in order to predict the damage kinetic.

Section snippets

Overview of the models

As mentioned above, all of the models examined have a common physical root, introduced by Chaboche in the late seventies, which relates the damage growth rate Ḋ to the damage-induced elastic energy release rate Y through a damage potential FD. The sum of the potential FD and the yielding function constitutes the complete dissipation potential which, in the framework of a thermodynamic approach, connects the associated plasticity to the kinetic-state coupling theory. Other damage models, such

Experimental

Different materials and specimen geometries were tested in order to evaluate the constitutive law and triaxiality evolution during tensile stress-deformation histories, according to the data reported in Table 1 and Fig. 1.

All the specimens were subjected to nominal deformation rates varying between 0.001 and 0.007 s−1. Clearly, when necking begins, the real deformation rate near the minimum section increases considerably with respect to its nominal value. However, given that in terms of true

Analysis of results

The curves in Fig. 8, Fig. 9 concern the case where p0=10%pcr, the initial and critical damage values were considered to be zero and one, respectively. The results regarding the influence of the necking effect are quite similar to those obtained for different values of the threshold strains and of the damage range.

Conclusions

Four CDM models, all developed under the hypothesis of triaxiality constancy during uniaxial loading, were tested in terms of their sensitivity to the increasing triaxiality found in real geometries when subjected to tensile loading until failure (and thus well above the necking strain).

In order to evaluate the triaxiality evolution in notched and unnotched tensile specimens, the Bridgman approach was applied to a set of experimental data. The stress–strain law, resulting from the fitting of

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