Physically-motivated elasto-visco-plastic model for the large strain-rate behavior of steels

A physically based elasto-visco-plastic constitutive model is presented and compared to experimental results for a DD14 mild steel. The model requires significantly fewer material parameters compared to other visco-plasticity models from the literature while exhibiting very good accuracy. Accordingly, the parameter identification is simple and intuitive, requiring a relatively small set of experiments. The strain-rate sensitivity modeling is not restricted to a particular hardening law and thus provides a general framework in which advanced hardening equations can be adopted and compared. The model has been implemented in the commercial finite element code Abaqus/Explicit. First predictions compared to experiments are analyzed and underline the effect of hardening law and strain-rate sensitivity on 3D finite element simulations. The model has been also applied as the basis for a homogenization approach at the phase scale; preliminary investigations showed the benefits of coupling such an approach with scale-transition technique where microstructure-relevant data can explicitly enter the model and may be used for material design simulations. Introduction Main industrial processes such as rolling, stamping or metal forming involve large deformations under non quasi-static loading conditions and frequent strain-path changes. In such circumstances, accurate description of the material behavior relies on the ability of the model to capture physical phenomena such as strain-rate sensitivity. This paper presents a physically-motivated elasto-viscoplastic model suitable to take into account main features of elasto-visco-plasticity such as strain-rate and temperature sensitivity. In its original form [1], the physically-based model is derived from the coupling between the Edgar Rauch’s formalism [2] and a phenomenological physically-based constitutive elasto-visco-plastic law. In this paper, a tensorial approach, not restricted to any hardening law, is proposed for 3D finite element simulations of metal forming processes. A first basic 3D simulation of a tensile test is analyzed and compared to experiments. Then, within the framework of a homogenization approach, the complete formulation is used at the phase scale. Preliminary investigations on a V-bending simulation show the possible benefits of downgrading the scale of description. Formulation of the elasto-visco-plastic constitutive model Elasto-visco-plastic formulation for large deformations. Frame invariance of the material is ensured by reformulating constitutive equations in terms of rotation-compensated variables. Tensor quantities are written in a convenient rotating frame in which simple material time derivatives can be used in the constitutive equations. Therefore, the strain rate tensor ε can be simply decomposed into an elastic part e ε and a visco-plastic part vp ε . The elastic behavior is thus expressed as: ( ) : vp = − σ C ε ε . (1) where σ is the Cauchy stress rate tensor and C the fourth-order elasticity tensor. The flow rule is defined by: ( ) 3 , 2 eq vp vp eq eq σ ε σ ′ ∂ = = = ∂ σ X ε V V σ (2) where 32 ( ) : ( ) eq σ ′ ′ = σ X σ X is for the sake of simplicity the von Mises equivalent stress, ′ σ denotes the deviatoric part of the Cauchy stress tensor and X the backstress variable, for kinematic hardening description. Obviously, any other yield function can be used without restriction. Flow stress and flow rule adopted for BCC materials. The flow stress is the stress required to move the mobile dislocations according to the elastic fields generated by the different obstacles in the microstructure. Generally considered as the sum of three independent contributions for FCC materials, the flow stress for BCC materials is described in Rauch formalism as [2]: 2 * * 2 0 2 2 R σ σ σ σ   = + + +     . (3) where * σ is the effective stress required for the mobile dislocations to overcome the local obstacles with the help of thermal fluctuations, 0 σ the initial athermal flow stress depending of the alloying elements and R the long range internal stress due to microstructure obstacles. The Rauch formalism, based on energetic considerations, implies a coupling between the effective thermal activated stress * σ and R. 2 * * 2 0 0 2 2 eq R σ σ σ σ     − + − + ≤         . (4) The relationship between the scalar overstress * σ and the equivalent visco-plastic strain-rate vp eq ε is expressed by the following relationship [1]: * * 1 * sinh vp eq K ε σ ε −   =       . (5) where K* and * ε are material parameters. The strain-rate sensitivity affects both the initial yield stress and the hardening term because of the coupling used in the Rauch formalism (Eq. 3). It is worth noting that any isotropic hardening model can be adopted. In this paper, a six-parameter law is used, corresponding to a linear combination between Voce and Swift hardening laws: ( ) 1 V S R R R α α = + − . (6) where ( ) V vp R sat eq R C R R ε = − and ( ) 1 1 0 n S n vp n n eq R nK K R ε ε − = + are respectively the Voce and Swift hardening formulations. R C , sat R , K, n, 0 ε , and α are material parameters. For the sake of simplicity, kinematic hardening represented by the backstress tensor X is described by the nonlinear model of Armstrong–Frederick: ( ) , vp X sat eq C X ε = − = V X N X N V . (7) where X C et sat X are material parameters. Experimental validations and application Parameter identification and comparison with experiments. The proposed constitutive model has been previously applied to three different mild steels and compared to tensile tests performed at different strain rates, quasi-static reverse shear tests and two-step sequential loadings [3]. Results showed the ability of the model to capture the main features of elasto-visco-plasticity: the increase of the tensile flow stress with strain rate is well predicted. Description of the temperature sensitivity of the mean flow stress is also claimed by adding two fitting parameters. Figure 1 illustrates the capability of the model to capture strain-rate sensitivity: tensile tests at four different strain rates are compared with model predictions for a DC05 steel sheet. Identification of material parameters, simple and intuitive, is given on Table 1. Despite the small number of parameters that require identification, the model showed similar ability to reproduce the main features of elasto-viscoplasticity of mild steels when compared to other physically-motivated models from the literature [4,5]. Figure 1. Comparison between experiments (dashed lines) and model predictions (solid lines) for a DC05 steel sheet under monotonic tensile tests (strain-rate sensitivity effect). Table 1. Material parameters for the DC05 mild steel. 0 [MPa] σ R C [MPa] sat R [MPa] K 0 ε n α * [MPa] K * 1 [s ] ε − 155 13 200 1 (Voce) 36 0.3 0 100 200 300 400 500 600 700 0,00 0,05 0,10 0,15 983s -1