Modeling stress state dependent damage evolution in a cast Al-Si-Mg aluminum alloy
Internal state variable rate equations are cast in a continuum framework to model void nucleation, growth, and coalescence in a cast Al-Si-Mg aluminum alloy. The kinematics and constitutive relations for damage resulting from void nucleation, growth, and coalescence are discussed. Because damage evolution is intimately coupled with the stress state, internal state variable hardening rate equations are developed to distinguish between compression, tension, and torsion straining conditions. The scalar isotropic hardening equation and second rank tensorial kinematic hardening equation from the Bammann-Chiesa-Johnson (BCJ) Plasticity model are modified to account for hardening rate differences under tension, compression, and torsion. A method for determining the material constants for the plasticity and damage equations is presented. Parameter determination for the proposed phenomenological nucleation rate equation, motivated from fracture mechanics and microscale physical observations, involves counting nucleation sites as a function of strain from optical micrographs. Although different void growth models can be included, the McClintock void growth model is used in this study. A coalescence model is also introduced. The damage framework is then evaluated with respect to experimental tensile data of notched Al-Si-Mg cast aluminum alloy specimens. Finite element results employing the damage framework are shown to illustrate its usefulness. ®2000 Elsevier Science Ltd. All rights reserved.
Author(s): Mark F. Horstemeyer, J. Lathrop, A.M. Gokhale, M. Dighe
Corresponding Author: Mark F. Horstmeyer
Kinematics in the continuum damage mechanics framework
The kinematics follows that in [4,5]. The material point motion is described by elastic straining, inelastic flow, and formation and growth of damage and is illustrated by the multiplicative decomposition of the deformation gradient shown in Fig. 1. The deformation gradient, , is decomposed into the isochoric inelastic, or plastic, (), dilational inelastic (), and elastic parts () given by
Eq. (1) assumes that the motion of the body is described by a smooth displacement function. This precludes the initiation of discrete failure surfaces but still allows a continuum description of damage. The elastic deformation gradient,, represents lattice displacements from equilibrium. The inelastic deformation gradient,, represents a continuous distribution of dislocations whose volume preserving motion produces permanent shape changes. The volumetric inelastic deformation gradient, , represents a continuous distribution of voids causing the volume change of the material from that arises from inelastic deformation. It is assumed to have the form , where is a function to be determined from kinematics (or conservation of mass).
and must be positive. The change in volume from the reference configuration (State 0) to the intermediate configuration (State 2) is assuming that the volume in State 0 equals that in State 1 because of inelastic incompressibility. The volume and density in the reference configuration are given by and , respectively. In transforming the configuration from State 0 to State 2, an added volume from the voids, ), is introduced to the total volume, but the volume of the solid matter remains unchanged at its reference value, because the material is unstressed in this configuration. The intermediate configuration in State 2 then designates when elastic unloading has occurred. Damage, ), can be defined as the ratio of the change in volume of an element in the elastically unloaded state (State 2) from its volume in the initial reference state to its volume in the elastically unloaded state
From this definition, it follows that
where now the Jacobian is determined by the damage parameter, , as
Consequently, the restriction that damage is assumed to produce isotropic dilatation gives the volumetric part of the deformation gradient as
where . The velocity gradient associated with the deformation gradient, from Eq. (1) is given by
where and with analogous formulas holding for the elastic, volumetric plastic, and deviatoric plastic parts of the velocity gradients expressed as , , and . The volumetric part of the velocity gradient is then given by
which defines the plastic volumetric rate of deformation as
Also note here that vanishes. The trace of the volumetric part, Eq. (9), is given as
so the damage parameter, , directly relates to the volumetric rate of deformation. The elastic rate of deformation relates to the volumetric rate of deformation by the additive decomposition of the deformation rates similar to Eq. (7),
Similarly, the elastic velocity gradient can be decomposed into components like Eqs. (7) and (11), where the elastic spin equals the total spin when no plastic spin is prescribed. Recall that no volumetric component exists for the spin tensor, that is ,
By combining this definition and inserting them into Eq. (3), the damage parameter, , can be written as
This framework for damage was employed in . If the number of voids per unit volume is defined in the intermediate configuration, then
Recalling that the unstressed intermediate configuration has the volume and employing Eq. (3), there prevails
The density of voids is counted after the specimen is loaded to a certain strain level and then unloaded. From this point, the specimen is machined and the number counting of voids nucleated is performed representing the elastically unloaded intermediate configuration; hence, is experimentally determined.
Void nucleation, growth, and coalescence parameters
where is the void nucleation density, the strain at time t, and is a material constant. The material parameters a, b, and c relate to the volume fraction of nucleation events arising from local microstresses in the material. These constants are determined experimentally from tension, compression, and torsion tests in which the number density of void sites is measured at different strain levels. The stress state dependence on damage evolution is captured in Eq. (21) by using the stress invariants denoted by , , and , respectively. is the first invariant of stress . is the second invariant of deviatoric stress , where . is the third invariant of deviatoric stress . The rationale and motivation for using these three invariants of stress is discussed in . The volume fraction of the second phase material is f, the average silicon particle size is d, and the bulk fracture toughness is .
For the cast A356 aluminum alloy in our study, , , and . The volume fraction and average size were determined from optical images of the sectioned test specimens. Fracture toughness tests were performed to determine . The stress state parameters were determined to be , , , and . In tension, compression, and torsion, specimens were strained to various levels and then stopped. The number of damaged sites were then counted. The details of determining the stress state parameters are given in Appendix A. Fig. 9 shows the nucleation model compared to the experimental results for compression, tension and torsion, respectively. Note that torsion incurred the highest number of voids nucleated followed by tension and then by compression. It is emphasized here that compression does indeed induce fracture sites in which damage accumulates.
In Eq. (22) the void volume grows as the strain and/or stress triaxiality increases. The material constant n is related to the strain hardening exponent and is determined from the tension tests. is taken to be the initial radius of the voids. As with most void growth models, the McClintock model allows voids to grow in tension, but not in compression or torsion. This complies with physical observations from measurements of this cast Al-Si-Mg aluminum alloy.
In the limiting case when the function in Eq. (23) equals zero, simple coalescence occurs and Eq. (18) results. When some function is used for , microvoid linking is said to have occurred and the rate of damage is increased. The parametric trend of this effect is shown in Fig. 11. For implementation into the constitutive relations, we assume . It has been observed [1,21] that the microvoid sheet mechanism is related to particles initiating small voids in between two larger voids as the larger voids impose their influence on the surrounding region. As such, coalescence is a function of both nucleation and void growth. Obviously, other forms can be used depending on the material being analyzed.
and are generally written as objective rates with indifference to the continuum frame of reference assuming a Jaumann rate in which the continuum spin equals the elastic spin . The elastic Lame` constants are denoted by and . The elastic rate of deformation results when the total deformation , which is defined by the boundary conditions, is subtracted from the deviatoric and volumetric components of the flow rule. The deviatoric plastic flow rule, , is a function of the temperature, the kinematic hardening internal state variable , the isotropic hardening internal state variable (R), the volume fraction of damaged material , and the functions , , and , which are related to yielding with Arrhenius-type temperature dependence. The function is the rate-independent yield stress. The function determines when the rate dependence affects initial yielding. The function determines the magnitude of rate dependence on yielding. These functions are determined from simple isothermal compression tests with different strain rates and temperatures
The kinematic hardening internal state variable, , reflects the effect of anisotropic dislocation density, and the isotropic hardening internal state variable R, reflects the effect of the global dislocation density. As such, the hardening equations (24) and (25) are cast in a hardening-recovery format that includes dynamic and static recovery. The functions and are scalar in nature and describe the diffusion-controlled static or thermal recovery, while and are scalar functions describing dynamic recovery. The anisotropic hardening modulus is , and the isotropic hardening modulus is .
Because attention is focused on stress state dependent damage in this study, we will reduce the BCJ equations to quasi-static strain rates for examining the phenomena at ambient temperature. As such, many of the BCJ constants will be zero. The constant values are given in Appendix A for the Al-Si-Mg cast material discussed in this paper.
Here, the backstress, , is the thermodynamic force related to kinematic hardening; the isotropic stress, , is the thermodynamic force related to isotropic hardening; and the energy release rate, , is the thermodynamic force related to the kinematic damage variable, . The damage ISV encompasses dissipation according to Eq. (38) from the void nucleation, growth, and coalescence terms.
Model implementation into finite element code
A few comments are warranted in regard to the implementation of the plasticity-damage model. When damage approaches unity, failure is assumed to occur. The goal is to implement the damage framework into a finite element code for solving complex boundary value problems, so failure occurs as within an element. Damage accumulation less than unity would be designated as failed material by engineers. In fact, it was stated  that a damage level of 50% is the limitation on the degraded elastic moduli. Practically, the total damage for final failure should perhaps be even less than 50%, but in applications  using the void growth rule [29,30], the damage goes rapidly to unity just after a few percent void volume fraction. More complicated functions for the damaged elastic moduli can be used such as that in , but these are computationally expensive.
At the beginning of each time step, determine the values for and from the previous step to modify , , h, and H. Employed then is a radial return method to determine the plastic part of the strain by assuming the strain to be all elastic (i.e. ). This gives the following trial values for the deviatoric stress and internal hardening variables:
Note that a differential equation of the form has a solution that tends to zero, but if we take too large a time step, our approximate integration could oscillate about zero. To avoid this behavior, we limit the terms multiplying (al) and in Eqs. (42) and (43) to be greater than or equal to zero.
By taking the norm of both sides, Eq. (44) can be inverted to give
This leads to corrections to the trial values of:
Substituting these corrected values back into the inverted flow rule, it is easy to show that is satisfied by choosing as:
Eq. (49) is then used to correct the trial values. We then calculate the total effective strain as
Finally, we add the pressure term to update the total stress as
Fig. 12 shows a comparison of the model to stress-strain data from compression, tension, and torsion tests at ambient temperature and quasistatic loading conditions . These curves reflect the inclusion of the void nucleation, growth, and coalescence terms. The hardening rate differences arising from these different global stress states is driven more by void nucleation than the growth and coalescence. It can be seen from Fig. 9 that the void nucleation rate is increasing as you go from compression to tension to torsion. The void nucleation relaxes the local dislocation density around the particles to relieve the local microstresses. As such, the global hardening rate is increasing as you go from torsion to tension to compression, the reverse order of the void nucleation rate.
The support and interest of USCAR/USAMP Lightweight Metals Group under the leadership of Richard Osborne is appreciated. Gerry Shulke provided the cast plates and Westmoreland Mechanical Testing and Research performed the tests. The work by M.F. Horstemeyer and J. Lathrop was performed at the US Department of Energy, Sandia National Laboratories under contract DE-AC04-94AL85000. The authors thank Dan Mosher, Doug Bammann, Ganesh Ramaswamy, Dave McDowell, and Jinghong Fan for their comments regarding this paper.
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