Abstract
In this package, the continuum damage mechanics framework for ductile materials developed by Lemaitre is implemented and developed in ABAQUS by VUMAT Subroutine. A wide range of element types, material models and other facilities, such as efficient equation solvers, are available for the user in the ABAQUS software. A fully coupled constitutive elastic-plastic-damage model is used.
Constitutive modeling is treated within the framework of continuum damage mechanics (CDM) and the effect of micro-crack closure, which may decrease the rate of damage growth under compression, is incorporated and implemented.
The present package has been organized as follows. In the Introduction section, the basis of the CDM in ductile materials is explained, and the applications of the CDM are stated. In the Theory section, the CDM model formulation is briefly reviewed and with micro-crack closure, the effect is described. In the Implementation section, an algorithm for the numerical integration of the damage constitutive equations is presented. In the VUMAT Subroutine section, the flowchart of the subroutine, and the subroutine structure, step by step, are explained in detail. How to run the VUMAT Subroutine in ABAQUS will be presented in this section. In the Verification section, the validation and verification of the numerical implementation will be evaluated, and the stability, convergence and accuracy of the results will be investigated. In the Application section, the applications of using the ductile damage model in mechanical processes are presented, and the prediction of damage growth and failure in mechanical processes are investigated.
Introduction
The damage of materials is the progressive physical process by which they break. Damage mechanics is the study, through mechanical variables, of the mechanisms involved in this deterioration when the materials are subjected to loading. In the section, the phenomenological aspects of damage with a description of the different kinds of damage are introduced. Particularly damage models in ductile metals are proposed by Chaboche [1], and Lemaitre [2] are discussed. Under compressive loading, voids and micro-cracks that would grow under tension will partially close, reducing the damage growth rate. This phenomenon can be important in the simulation of forming processes. In this section, all the necessary conditions and parameters affecting ductile damage modeling are introduced.
Theory
The theory assumes that the state of damage at any material point is isotropic and is represented by a single scalar internal variable, D, associated with the fraction of load carrying area across any surface at that point. The damage variable assumes values between 0 (for the undamaged material) and 1 (for the completely damaged material).
The evolution of the damage internal variable is assumed to be governed by the relation:
The effect of damage on plastic behaviour can be presented by von Mises yield function:
where 
 is the accumulated plastic strain, r, s are material constants, The quantity
and Y, is the damage energy release rate, with E and n denoting, respectively, Young’s modulus and Poisson’s ratio of the undamaged material.
For the crack closure effect; the compressive/tensile additive split of the stress tensor:
where and  are, respectively, the tensile and compressive components of . The numerical prediction of material degradation, based on the damage model without crack closure effects, is not in agreement with experimental evidence, which shows that only a relatively small damage accumulation results from the process.
The theory required for the implementation of the ductile damage model according to the modified Lemaitre model will be presented
Implementation
This section yields to describe an algorithm for the implementation of the elastic-plastic-damage constitutive equations including the effect of crack closure. Algorithms based on the operator split concept, resulting in the standard elastic predictor/plastic corrector format, are widely used in computational plasticity. The flow chart of the elastic predictor/return mapping algorithm for the elastic-plastic-damage model is implemented as shown in figure 1 based on article titled” Numerical analysis of damage evolution in ductile solids“:
Figure 1: Flow chart of elastic predictor/return mapping algorithm for the elastic-plastic-damage model.
VUMAT Subroutine
To introduce the ductile damage model, a user material subroutine VUMAT is developed in ABAQUS software package. In the VUMAT subroutine, the damage variable will be calculated in each the integration points (or Gauss points) locally. The element deletion option is used with VUMAT to induce ductile crack growth. The developed VUMAT subroutine has the ability to analyze ductile damage in three-dimensional and plane strain cases. 3D and 2D plane strain problems. According that the VUMAT subroutine is run in the ABAQUS/Explicit, it can be used in various problems that require complex contact algorithms. How to run the VUMAT subprogram in ABAQUS will be explained in this section for loading in simple models, such as one element.
Verification VUMAT Subroutine
Validation and verification of the numerical implementation will be evaluated in various examples. The ductile damage model has been verified by solving two numerical examples using ABAQUS/Explicit in two- and three-dimensional solid elements. The first example is the simulation of a two- and three-dimensional tensile test on a specimen subjected to monotonic axial loading. In the second example, the simulation of a three-point bending test is investigated (figure 2). The stability, convergence, and accuracy of the results in these two examples are also evaluated.
Figure 2: Comparison of experimental crack initiation location in three-point bending test
Application of the VUMAT Subroutine
The modified ductile damage model can be used to predict prediction of damage growth and failure in various mechanical processes. For example; the modified ductile damage model is applied to predict initiation of the micro-void as a central burst along the bar axis during the forward extrusion (Figure 3), and prediction of fracture initiation area in the upsetting test of tapered axisymmetric specimen (Figure 4).
Figure 3: Prediction of central burst along the bar axis during the forward extrusion
Figure 4: Evolution of damage in tapered axisymmetric specimen
What is the Lemaitre damage model?
This model is a model whose hardening is isotropic and rate-dependent. Also, the yield is based on the von Mises criterion, which is combined with the Lemaitre damage model.
Is the Lemaitre damage model very different from the ductile damage models? And what is the difference between them?
Yes, this model is different from the normal damage model that exists in ABAQUS and other software. In the Lemaitre damage model, yielding and plasticization occur simultaneously with damage, but in the normal models that exist, the damaged area and the plasticity area are two separate areas.
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