Application of a Dung’s Model to Predict Ductile Fracture of Aluminum Alloy Sheets Subjected to Deep Drawing

th , 2015) ABSTRACT: In this paper, prediction of failed evolution of anisotropic voided ductile materials will be developed based on Dung’s microscopic damage model. An isotropic and anisotropic formulation of the Dung’s damage model that using von Mises yield criterion and Hill’s quadratic anisotropic yield criterion (1948) integrated with isotropic hardening rules of matrix material used to simulate the deep drawing process of aluminum alloy sheets. The model is implemented as a vectorized user-defined material subroutine (VUMAT) in the ABAQUS/Explicit commercial finite element code. The predictions of ductile crack behavior in the specimens based on void nucleation, growth and coelescence are compared with Gurson – Tvergaard – Needleman (GTN) model and experiment


ABSTRACT:
In this paper, prediction of failed evolution of anisotropic voided ductile materials will be developed based on Dung

INTRODUCTION
Recently, the aluminum alloy materials widely applied in automotive and aerospace industry since their light weight and excellent strength characteristics. The sheet metals made from aluminum alloys by rolling process is usually induced anisotropy. Therefore, investigation of plastic fractured behavior of these materials play an important role in industrial applications. The plastic micro-crack mechanism in the metal materials are based on assuming that matrix material contain inclusions and second phase particles. During matrix material under deformation then micro-crack will appear because of void nucleation, growth and coalescences. Gurson [1] proposed a yield function that isotropic matrix material contains spherical voids that including a special damage parameter of void volume fraction (f). Tvergaard [2,3] [8] supplied the criterions for void nucleation into Gurson model. There is limit to anisotropic material of original porous plastic material model. Therefore, in recent years, some reseachers extended original Gurson model to anisotropic materials. Liao et al [9] integrated Gurson model with the Hill quadratic and nonquadratic anisotropic yield criteria to describe the matrix normal anisotropy and planar isotropy. Wang et el [10] formed a closed-form anisotropic Gurson yield criterion based on an average anisotropy parameter. Tanguy et al [11] developed a constitutive model based on Gurson model that integrating anisotropic behaviour and ductile damage for a X100 pipeline steel. Grange et al [12] predicted ductile fracture of Zircaloy-4 sheets based on the Gurson-Tvergaard-Needleman model which is extended to take into account plastic anisotropy and viscoplasticity. Chen and Dong [13] developed an implicit stress integration procedure to adapt the explicit dynamic solver for GTN model with equivalent stress is Hill's quadratic anisotropic yield criterion (1948). Morgeneyer et al [14] investigated fracture mechanisms of AA2139 Alalloy sheet by experiments and GTN model to describe and predict deformation behaviour, crack propagation and toughness anisotropy.
Kami et al [15] predicted plastic fractue of AA6016-T4 metallic sheet of deep-drawing by using GTN model and Hill'48 quadratic anisotropic yield function.
In this paper, Dung's model based on Hill'48 expression of the equivalent stress is implemented by a VUMAT subroutine in the finite element software (ABAQUS) to investigate ductile fracture process of deep drawing in aluminum alloy materials. The predictions of ductile crack behavior in the specimens based on void nucleation, growth and coelescence are compared with GTN model and experiment results from referenced documents.

DUNG'S DAMAGE MODEL FOR ANISOTROPIC METAILLIC MATERIAL
Since original Dung's model constituted based on assume that matrix material is isotropy. Therefore, to apply the Dung's model on anisotropic material then von Mises equivalent stress in yield funtion will be replaced by Hill'48 quadratic anisotropic yield criterion.
For von Mises yield criterion The Lankford's coefficients r0, r45 and r90 are determined by unaxial tensile tests at 0 o , 45 o and 90 o in rolling direction.
σf is the yield stress of matrix material.
The equivalent plastic strain rate of matrix material p   is dominated by equivalent plastic work:  is plastic strain rate tensor.
The void volume fraction growth is computed as follow: Here, the void volume fraction growth of the presence voids in matrix material: The nucleated volume void fraction growth during matrix material under deformation: The number of nucleated voids A is a function of equivalent plastic strain of matrix Where, fn, sN, εN are the parameters relative to the void nucleation during matrix material under deformation.

NUMERICAL IM PLEMENTATIO N
A numerical algorithm based on the Euler backward method has been developed for a class of pressure-dependent plasticity models by Aravas [17] used to solve of the constitutive equations via a VUMAT subroutine in ABAQUS/Explicit software.
The steps of implementation procedures as follow: Step1: Initialize the variables at initial time Step 2: Calculate trial state of stresses Calculate stress tensor The fourth order tensor D is the elastic stiffness matrix. Isotropic elasticity is assumed so that Where, K is the elastic bulk modulus, G is the shear modulus and δij is the Kronecker delta Calculate hydrostatic stress Here, I is second order unit tensor

Step 4: Calculate plastic correction
The plastic strain increment is divided into spherical and deviatoric parts:  is the plastic multiplier and is the flow direction.
Eliminating λ from equations (15) and (16) leads to: Using Newton-Raphson iterative method to solve the nonlinear system of equations (18) and (19), the consistency condition equation (20) must be met at the same time. 1 1 1 Here, is current strain hardening of the matrix material.
The algorithm stops iterations when the values of |E1| and |E2| are less than a specified tolerance  = 1E-08 Step

NUMERICAL ANALYSIS 4.1. Tensile tests on single element
The subroutine is verified using a single 8node brick element (C3D8R) for hydrostatic tensile test and plane strain element (CPE4R) for unaxial tensile test. The boundary conditions and loading as shown in figure 1. The initial size of each element edge is 1 mm. The loading velocity for tension is set to 15 mm/s.

Deep drawing
In this section, cylindrical cup and square cup deep drawing process was be investigated. The forming behavior of Dung's damage model was compared with GTN model in Abaqus and experiment results from refercences.

Cylindrical cup deep drawing
The material of sheet is AA6111-T4 aluminum alloy. The properties of porous material is refered to Chen et al [13] as table 1. The isotropic hardening rule of matrix material: Here, σf is the equivalent stress of matrix material, p  is the equivalent plastic strain, σ0 is the initial yield stress, a and b are the material constants. The material properties of AA6111-T4 alloy in the unxiaxial tensile test as table 2. Figure 6 shows tooling setup for cylindrical cup drawing. The sheet thickness is 1 mm. The punch stroke is 50 mm. The blank holding force is 50 kN. Element type of blank is eight-node linear brick, reduced integration with hourglass control continuum element (C3D8R). The rigid element (R3D4) of tools is chosen. The friction coefficient has been set to a value of 0.0096 on all the contact surfaces.

Square cup deep drawing
The AA6016-T4 aluminum alloy sheet was used to predict plastic fracture in deep drawing process. The parameters of damage model given in table 3.  The yield stress versus plastic strain curve that fitting from experiment data in tensile test of AA6016-T4 aluminum alloy by means of Swift's hardening rule: The material properties as table 4 The deep drawing tools was installed as figure 9. The diameter of circular blank is 85 mm. The holding force is 10 kN. Punch stroke is 25 mm. The rigid shell elements (R3D4) was used to mesh punch, die and blank holder, while 8-node hexahedral (C3D8R) solid elements have been meshed blank. The size of initial length of element is 0.5 mm. The friction coefficient has been set to a value of 0.05 on all the contact surfaces.

CONCLUSIONS
In this paper, the Dung's model is implemented using the commercial code Abaqus/Explicit with the user-difined material subroutine (VUMAT).