Constitutive model for casting magnesium alloy involving spherical void evolution
CHEN Bin(陈 斌), PENG Xiang-he(彭向和), FAN Jing-hong(范镜泓), SUN Shi-tao(孙士涛)
Department of Engineering Mechanics, College of Resource and Environment Science,
Chongqing University, Chongqing 400030, China
Received 12 June 2008; accepted 5 September 2008
Abstract: Casting magnesium alloys are highly heterogeneous materials inevitably containing numerous voids. These voids will evolve during material deformation and markedly affect material behaviors, so it is important to investigate the equation of the void evolution and the constitutive relation involving the void evolution. By assuming the voids in casting magnesium alloys were spherical, the growth equation of the voids was obtained from the incompressibility and continuity conditions of material matrix. Through combining the obtained void-growth equation with the void-nucleation equation relative to the increment of intrinsic-time measure, the evolution equation of the voids was presented. By introducing the presented void-evolution equation to a nonclassical elastoplastic constitutive equation, a constitutive model involving the void evolution was put forward. The corresponding numerical algorithm and finite element procedure of the model were developed and applied to the analysis of the elastoplastic response and the porosity change of casting magnesium alloy ZL305. Computed results show satisfactory agreement with those of the corresponding experiments.
Key words: casting magnesium alloy; spherical void; void evolution; constitutive model
1 Introduction
Casting magnesium alloys have received increasing attention in transportation vehicle applications due to their desirable characteristics in reducing vehicle mass and amenability to recycling[1]. For example, casting magnesium alloy ZL305 is currently considered for automotive applications such as engine blocks and cylinder heads. However, because casting magnesium alloys contract during solidification, hydrogen dissolves easily in the melted metals. So the materials inevitably contain a certain amount of voids. These voids may grow and coalesce during a thermomechanical loading process, which may degrade the ductility of the materials and cause ductile fracture of the materials. The investigations on the evolution rule of these voids and the effect of the void evolution on the ductile fracture of the materials are significant to make more efficient use of the materials.
Great effort has been made in the analysis of the void evolution and the ductile fracture in various porous materials, which can be used in the analysis of the void evolution and elastoplastic behaviors of cast magnesium alloys. It is known that voids usually distribute randomly in porous materials and void growth plays an important role in material properties[2]. A feasible method used frequently for the ductile fracture of porous materials is based on the analysis of a simple representative void-cell model of the materials. In this aspect, GURSON[3] made a pioneering contribution. He assumed that the void-matrix aggregate of a porous ductile media could be represented with a rigid-plastic cell containing a single void, and the void volume fraction (porosity) f of the cell is equal to that of the aggregate. REUSCH et al[4] extended GURSON model to the case of isotropic ductile damage and crack growth. LICHT and SUQUQUET[5] investigated the growth of a cylindrical void in a finite shell of a nonlinear viscous material and obtained a closed solution. SIRUGUET and LEBLOND[6] modeled the effect of inclusions on void growth in porous ductile metal. LI et al[7] investigated the growth of voids embedded in elastic-plastic matrix materials with a finite element approach. MARIANI and CORIGLIANO[8] built an orthotropic constitutive model for porous ductile media based on the micromechanical analysis of a cylindrical representative volume element. TVERGAARD and NIORDSON[9] analyzed the effects of nonlocal plasticity on the interactions of voids of different sizes based on an axisymmetric unit cell model with special boundary conditions. BAASER and GROSS [10] analyzed the growth of microvoids in a ductile material in a crack tip loaded by a remote KI field. HORSTEMEYER et al[11] investigated the internal state variable rate equations in continuum framework to model void nucleation, growth and coalescence in a casting Al-Si-Mg aluminum alloy. The work presented here is to extend the foregoing research results in porous ductile media to the research of the void evolution and the elastoplastic behavior of casting magnesium alloy. The void-evolution equation and the nonclassical elastoplastic constitutive model involving void evolution are presented. The obtained evolution equation and the constitutive model are applied to the investigation of the elastoplastic response and the porosity change of casting magnesium alloy ZL305.
2 Spherical void-cell model and void evolution rule
A representative material element of casting magnesium alloys that include numerous voids with different shapes is shown in Fig.1. Because spherical voids are often found in the materials and have special significance, it is supposed that the voids in the materials are of spherical shape and a spherical void-cell model is presented. The spherical void-cell model is a spherical cell with a spherical void at its center (Fig.2). The radii of the void and the cell are a and b, respectively. The radius of an arbitrary point in the matrix of void-cell model is r. The volumes of the void and the matrix are Vv and Vm, respectively. The volume of the cell is then V=Vv+Vm. The matrix is assumed to be a homogeneous, elastoplastic and incompressible media. Given a velocity
Fig.1 Material element with numerous voids
Fig.2 Spherical void-cell model
field vr, vψ, and vθ in the matrix of the void-cell model in a spherical coordinates, the corresponding strain rate field 
in the matrix can be expressed as
(1a)
(1b)
(1c)
(1d)
Since the matrix is incompressible and continuous, from the incompressibility and continuity conditions of the matrix[3], the rate equation of void growth during the deformation of the material can be given by
(2)
where 
is the current volume fraction of the voids. The increment of intrinsic-time measure of the spherical void-cell model can be expressed as[12]
(3)
Similar to the work of NEEDLEMAN and TVERGAARD [13], the following formula controlled nucleation of the new voids during the deformation of the materials by the increment of the intrinsic-time measure is put forward:
(4)
where 
is the current volume fraction of void nucleating part, zN is the mean intrinsic time measure of the nucleation, and sN is the corresponding standard deviation[13].
The increase of the void volume fraction is due to both the growth of the existing voids and the nucleation of the new ones during the material deformation. The evolution-rate equation of the spherical voids can be expressed as
(5)
3 Constitutive model
The evolution of the voids may reduce the resistance at the boundary between the grains and the local residual stress in the crystals of the material, and in turn, reduce the resistance to the motions of dislocations that are closely related to the elastoplastic and creep deformations of the material. This softening effect can be taken into account by a softening function w(f), which may depend on the void volume fraction. An incremental constitutive equation, which considers the effect of the void evolution, can be given as follows:
(6)
where
(7a)
(7b)
(7c)
(7d)
where Cr, αr and k are material parameters that can be determined form σ—εp curve using a nonlinear curve fitting approach[14-15]. w(f) takes following simple form:
(8)
It can be explicitly seen from Eqn.(6) that the marked effect of the void evolution is made on the stress response of the material. Taking tensile loading as an example, because w(f) is a monotonically decreasing function, and all other quantities in Eqn.(6) are positive, the increase in the void volume fraction f results in a reduction in the deviatoric stress 
.
4 Comparison between theoretical and experimental results
The presented constitutive model and the corresponding finite element approach were used to analyze two kinds of examples. One is the stress—strain and the porosity—deformation relationships of cylindrical specimens, and the other is the porosity on the cross section of a notched cylindrical specimen. The computed results were compared with test data.
The material used is casting magnesium alloy ZL305. The geometry of the cylindrical specimens is 250 mm in length, and 10 mm in the diameter of the working section. The geometry of the notched cylindrical specimen is shown in Fig.3. The upper and right quarter of the two kinds of the cylinder specimens was taken for the analysis due to the symmetry of the problems. The eight-node isoparametric element with 2×2 Gaussian points was adopted. The axial displacements were prescribed at the end of the specimens with the incremental step of 0.02 mm, and no radial constraint was applied to the surface of the specimens. The experimental data were obtained from tensile tests conducted on an Instron 1342 material testing system.
Fig.3 Cylindrical tensile specimen with notch
Fig.4 shows the stress—strain curves obtained by both computation and experiment, and the comparison shows satisfactory agreement. In order to experimentally determine the relationship between the porosity and deformation, a set of cylindrical specimens were loaded to different stress levels and then unloaded. After experiment, these specimens were cut along the minimum cross section of the tested specimens and polished, and a photo-interpreter was used to quantify the porosity on the minimum cross sections of these specimens. Since the void distribution is stochastic, a quantitative metallographic method was adopted. Fig.5
Fig.4 Relationship between stress and strain
Fig.5 Relationship between void volume fraction and plastic strain
shows the relationships of the porosity and plastic deformation obtained respectively by experiment and computation. The computed results agree reasonably with the experimental results. The porosity increases with the increase of the plastic deformation, approximately following an exponential rule. Fig.6 shows the metallograph at the fracture section of the specimen obtained with a scanning electron microscope, from which many dimples can be observed. These dimples indicate ductile fracture of the material. A notched cylindrical tensile specimen was loaded to 30 kN followed by unloading; then the tested specimen was cut along the radius at its minimum cross section and polished for porosity measurement using a photo- interpreter. The distribution of the porosity along the radius of the specimen was recorded. Fig.7 shows the computed and the experimental distributions of the porosity, in which the coordinate “0” denotes the location of the smallest cross section of the specimen, 5 mm corresponds to the center of the cross section of the specimen, and a denotes the distance from the surface. It
Fig.6 SEM image of fracture section showing dimples
Fig.7 Distribution of void volume fraction along notch line of specimen
can be seen from Fig.7 that the computed and the experimental results also show reasonable agreement. Moreover, the porosity reaches a maximum value at the root of the notch and decreases rapidly toward the center of the specimen.
5 Conclusions
1) Spherical void-cell model isolated from casting magnesium alloys is investigated for obtaining the void-evolution equation.
2) A formula controlled nucleation of new voids during the deformation of the materials by the increment of the intrinsic-time measure is put forward.
3) An elastoplastic constitutive equation taking into account the effects of voids is derived. The corresponding numerical algorithm and finite element approach are developed.
4) The comparison between the computed and the experimental results shows reasonable agreement. The porosity increases with the increase of the plastic deformation, approximately following an exponential rule. The porosity reaches a maximum value at the notch root of the specimen and decreases rapidly toward the center of the specimen.
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(Edited by CHEN Wei-ping)
Foundation item: Project(10872221) supported by the National Natural Science Foundation of China
Corresponding author: CHEN Bin; Tel: +86-23-65103597; E-mail: bchen@cqu.edu.cn
