Abstract
The paper deals with the effect of stress state on the dynamic damage behavior of ductile materials. The rate and temperaturedependent continuum damage model has been enhanced to take into account the influence of the stress triaxiality and the Lode parameter on damage condition and on rate equations of damage strains. Different branches of these criteria depending on the current stress state are considered based on different damage and failure processes on the microlevel. To get more insight in the dynamic damage and fracture processes micromechanical behavior of voidcontaining representative volume elements have been analyzed numerically. These threedimensional numerical simulations based on different dynamic loading conditions take into account a wide range of stress states in tension, shear and compression domains. Based on the numerical results general trends of dynamic damage and failure behavior can be shown and stressstatedependent equations for damage criteria and for the formation of damage strains can be proposed.
Keywords
Ductile damage; stress state dependence; micromechanical numerical simulations; dynamic loading
1 Introduction
During the last decades the use of high quality ductile metals like high strength steels, advanced high strength steels and various improved aluminum alloys in complex engineering structures has been remarkably increased especially caused by demands on economic, environmental and material strength requirements. Therefore, accurate and realistic modeling of ratedependent and rateindependent inelastic deformation and failure behavior of these materials is essential for the numerical simulation of quasistatic and dynamic loading processes of structures. In addition, high strain rate deformation processes such as impact problems, dynamic shear banding or high speed machining have to be analyzed in many industrial applications. This leads to an increasing request on accurate, robust and efficient material models to be able to realistically simulate the mechanical response of engineering structures under dynamic loading conditions. Thus, constitutive approaches taking into account large strains, high strain rates and thermal softening as well as stressstatedependent damage and fracture criteria have to be developed to realistically predict the failure mechanisms observed in experiments and in various engineering applications.
Different models for analyzing the occurrence of damage and failure in metals under general loading conditions have been proposed to predict heterogeneous material behavior and failure. Motivated by experimental observations phenomenological approaches have been developed using internal damage variables whose growth is described by evolution laws. General frameworks of internal variable theories of high strain rate deformation processes in metals based on ratedependent continuum mechanics have been published, for example, by Bruhns and Diehl (1989Bruhns, O.T., Diehl, H., 1989. An internal variable theory of inelastic behaviour at high rates of strain. Archive of Mechanics 41, 427460.); Voyiadjis et al. (2003Voyiadjis, G.Z., Abu AlRub, R., Palazotto, R.K., 2003. Nonlocal coupling of viscoplasticity and anisotropic viscodamage for impact problems using the gradient theory. Archives of Mechanics 55, 3989.); Brünig (2006Brünig, M., 2006. Continuum framework for ratedependent behavior of anisotropic damaged ductile metals. Acta Mechanica 186, 3753.); Brünig and Gerke (2011)Brünig, M., Gerke, S., 2011. Simulation of damage evolution in ductile metals undergoing dynamic loading conditions. International Journal of Plasticity 27, 15981617.. These models are able to predict the evolution of damage in high strain rate processes but the calibration of material properties and their stress state dependence seems to be difficult. However, accurate and efficient constitutive approaches for stressstatedependent formation of damage are needed as a basis for theories of ductile fracture.
The main mechanisms leading to damage and fracture in metals are nucleation, growth and coalescence of microdefects as well as formation of macrocracks. In this context, analysis of behavior of individual microdefects can be seen as a straightforward approach for modeling and simulation of ductile failure processes. Thus, features on the scale of the microdefects have to be included in studies in order to simulate damage and fracture behavior in an accurate manner. For example, numerical results of unitcell analyses can be used to develop damage evolution equations in continuum models as well as to identify corresponding micromechanically motivated material parameters (Brünig et al. (2013Brünig, M., Gerke, S., Hagenbrock, V., 2013. Micromechanical studies on the effect of the stress triaxiality and the Lode parameter on ductile damage. International Journal of Plasticity 50, 4965., ^{2014}Brünig, M., Gerke, S., Hagenbrock, V., 2014. Stressstatedependence of damage strain rate tensors caused by growth and coalescence of microdefects. International Journal of Plasticity 63, 4963., ^{2018b}Brünig, M., Hagenbrock, V., Gerke, S., 2018b. Macroscopic damage laws based on analysis of microscopic unit cells. ZAMM  Zeitschrift für Angewandte Mathematik und Mechanik 98, 181194.)). Therefore, combination of analyses on both the micro and the macrolevel has widely enriched the understanding of damage and failure processes in ductile metals.
Detailed information on stress state dependence of the microscopic mechanisms can be obtained by systematic finite element analyses with microdefect containing unitcells loaded by different macroscopic stress states (Barsoum and Faleskog (2011Barsoum, I., Faleskog, J., 2011. Micromechanical analysis on the influence of the Lode parameter on void growth and coalescence. International Journal of Solids and Structures 48, 925938.); Brünig et al. (2013Brünig, M., Gerke, S., Hagenbrock, V., 2013. Micromechanical studies on the effect of the stress triaxiality and the Lode parameter on ductile damage. International Journal of Plasticity 50, 4965., ^{2014}Brünig, M., Gerke, S., Hagenbrock, V., 2014. Stressstatedependence of damage strain rate tensors caused by growth and coalescence of microdefects. International Journal of Plasticity 63, 4963., ^{2018b}Brünig, M., Hagenbrock, V., Gerke, S., 2018b. Macroscopic damage laws based on analysis of microscopic unit cells. ZAMM  Zeitschrift für Angewandte Mathematik und Mechanik 98, 181194.); Gao and Kim (2006Gao, X., Kim, J., 2006. Modeling of ductile fracture: Significance of void coalescence. International Journal of Solids and Structures 43, 62776293.); Gao et al. (2005Gao, X., Wang, T., Kim, J., 2005. On ductile fracture initiation toughness: Effects of void volume fraction, void shape and void distribution. International Journal of Solids and Structures 42, 50975117., ^{2010}Gao, X., Zhang, G., Roe, C., 2010. A study on the effect of the stress state on ductile fracture. International Journal of Damage Mechanics 19, 7594.); Kim et al. (2003)Kim, J., Gao, X., Srivatsan, T., 2003. Modeling of crack growth in ductile solids: a threedimensional analysis. International Journal of Solids and Structures 40, 73577374.; Nielsen et al. (2012Nielsen, K., Dahl, J., Tvergaard, V., 2012. Collapse and coalescence of spherical voids subject to intense shearing: studies in full 3D. International Journal of Fracture 177, 97108.); Scheyvaerts et al. (2011Scheyvaerts, F., Onck, P., Tekoğlu, C., Pardoen, T., 2011. The growth and coalescence of ellipsoidal voids in plane strain under combined shear and tension. Journal of the Mechanics and Physics of Solids 59, 373397.); Zhang et al. (2001Zhang, K., Bai, J., Francois, D., 2001. Numerical analysis of the influence of the Lode parameter on void growth. International Journal of Solids and Structures 38, 58475856.)). Their quasistatic numerical studies have shown that the stress state remarkably affects the macroscopic deformation of the representative volume elements, the growth of the microdefects and the amount of the fracture strain. Even if these numerical calculations are restricted to materials with periodic microstructure they can be seen as an important tool in gaining better understanding of the processes on the microlevel of damage and failure in ductile metals.
Although many papers dealing with quasistatic loading of void containing volume elements exist only minor attention has been drawn on growth of microdefects under dynamic loading conditions. For example, Carroll and Holt (1972Carroll, M., Holt, A., 1972. Static and dynamic porecollapse relations for ductile porous materials. Journal of Applied Physics 43, 16261636.) studied the dynamic void collapse in an ideally plastic material using a hollow sphere model and Johnson (1981Johnson, J., 1981. Dynamic fracture and spallation in ductile solids. Journal of Applied Physics 52, 28122825.) extended this approach considering ratedependent materials under dynamic hydrostatic tension. Furthermore, Ortiz and Molinari (1992Ortiz, M., Molinari, A., 1992. Effect of strain hardening and rate sensitivity on the dynamic growth of a void in a plastic material. Journal of Applied Mechanics 59, 4853.) examined the dynamic growth of a single void in an unbounded solid under rapidly varying remote hydrostatic tension loading. They studied the effects of inertia, strain hardening and rate sensitivity on the short and long time behavior of the porous material under dynamic loading conditions as well as the influence of ramp loading. At the early stage of the deformation process the growth of the single pore was affected by strain hardening and strain rate sensitivity whereas inertia dominated the long term part of void growth. Benson (1993Benson, D., 1993. An analysis of void distribution effects on the dynamic growth and coalescence of voids in ductile metals. Journal of the Mechanics and Physics of Solids 41, 12851308.) investigated the effect of void distribution on dynamic failure of copper and 4340 steel. Numerical studies with a representative volume element with discrete set of randomly distributed voids loaded by a tension wave showed that the peak transmitted stress was insensitive to the magnitude of the tension wave but dependent on void distribution. The dynamic behavior of porous materials under hydrostatic tension and compression as well as axisymmetric loading conditions has been analyzed by Molinari and Mercier (2001)Molinari, A., Mercier, S., 2001. Micromechanic modelling of porous materials under dynamic loading. Journal of the Mechanics and Physics of Solids 49, 14971516. to take into account the effect of the stress triaxiality. They proposed a ratedependent material model for voidcontaining solids based on averaging methods and homogeneous kinematic boundary conditions. In addition, Wu et al. (2003aWu, X., Ramesh, K., Wright, T., 2003a. The dynamic growth of a single void in a viscoplastic material under transient hydrostatic loading. Journal of the Mechanics and Physics of Solids 51, 126.,bWu, X., Ramesh, K., Wright, T., 2003b. The effects of thermal softening and heat conduction on the dynamic growth of voids. International Journal of Solids and Structures 40, 44614478.) investigated the dynamic growth of a single void in an elasticviscoplastic material under transient hydrostatic loading conditions with focus on inertia, thermal and ratedependent effects. They found out that the influence of inertia was small in the early stages of void growth but strongly depended on the initial size of the pore and on the loading rate. The process of coalescence of voids in ductile materials subjected to dynamic loading has been examined numerically by Jaques et al. (2012Jaques, N., Mercier, S., Molinari, A., 2012. Void coalescence in a porous solid under dynamic loading conditions. International Journal of Fracture 173, 203213.). Their finite element simulations showed that inertia led to stabilizing effects and slowed down the coalescence process. This stabilizing influence of inertia is enhanced by higher stress triaxialities, larger initial sizes of the voids and higher loading rates.
The numerical analyses discussed above have clearly shown that the mechanisms of damage and failure processes on the microlevel under dynamic loading conditions are different to those observed under quasistatic loading. These results may give important aspects for ratedependent constitutive damage approaches. Therefore, the stressstatedependent continuum damage model proposed by Brünig et al. (2013Brünig, M., Gerke, S., Hagenbrock, V., 2013. Micromechanical studies on the effect of the stress triaxiality and the Lode parameter on ductile damage. International Journal of Plasticity 50, 4965., ^{2018b}Brünig, M., Hagenbrock, V., Gerke, S., 2018b. Macroscopic damage laws based on analysis of microscopic unit cells. ZAMM  Zeitschrift für Angewandte Mathematik und Mechanik 98, 181194.) will be reconsidered and modified in the present paper and assessed to ensure its suitability for dynamic problems.
2 Fundamental governing equations
The anisotropic continuum damage model proposed by Brünig (2003Brünig, M., 2003. An anisotropic ductile damage model based on irreversible thermodynamics. International Journal of Plasticity 19, 16791713.,^{2004}Brünig, M., 2004. An anisotropic continuum damage model: Theory and numerical analyses. Latin American Journal of Solids and Structures 1, 185219., ^{2006}Brünig, M., 2006. Continuum framework for ratedependent behavior of anisotropic damaged ductile metals. Acta Mechanica 186, 3753.); Brünig and Gerke (2011)Brünig, M., Gerke, S., 2011. Simulation of damage evolution in ductile metals undergoing dynamic loading conditions. International Journal of Plasticity 27, 15981617. is used to predict the ratedependent inelastic deformation behavior as well as the evolution of ductile damage and failure in aluminum alloys caused by dynamic loading. Phenomenological modeling of macroscopic damage and failure behavior corresponding to different mechanisms acting on the microlevel caused by various stress states is based on combination of microscopic and macroscopic investigations. They take into account results of numerical simulations on the microlevel analyzing deformation and failure behavior of microvoidcontaining unitcells (Brünig et al. (2013Brünig, M., Gerke, S., Hagenbrock, V., 2013. Micromechanical studies on the effect of the stress triaxiality and the Lode parameter on ductile damage. International Journal of Plasticity 50, 4965., ^{2014}Brünig, M., Gerke, S., Hagenbrock, V., 2014. Stressstatedependence of damage strain rate tensors caused by growth and coalescence of microdefects. International Journal of Plasticity 63, 4963., ^{2018b}Brünig, M., Hagenbrock, V., Gerke, S., 2018b. Macroscopic damage laws based on analysis of microscopic unit cells. ZAMM  Zeitschrift für Angewandte Mathematik und Mechanik 98, 181194.)) as well as results of various experiments with carefully designed specimens under different loading conditions (Brünig et al. (2008)Brünig, M., Chyra, O., Albrecht, D., Driemeier, L., Alves, M., 2008. A ductile damage criterion at various stress triaxialities. International Journal of Plasticity 24, 17311755.; Driemeier et al. (2010Driemeier, L., Brünig, M., Micheli, G., Alves, M., 2010. Experiments on stresstriaxiality dependence of material behavior of aluminum alloys. Mechanics of Materials 42, 207217.); Brünig et al. (2015Brünig, M., Brenner, D., Gerke, S., 2015. Stress state dependence of ductile damage and fracture behavior: Experiments and numerical simulations. Engineering Fracture Mechanics 141, 152169., ^{2016}Brünig, M., Gerke, S., Schmidt, M., 2016. Biaxial experiments and phenomenological modeling of stressstatedependent ductile damage and fracture. International Journal of Fracture 200, 6376.)). This phenomenological approach is briefly summarized and discussed in the present paper.
The continuum model is based on the introduction of damaged and corresponding fictitious undamaged configurations. Considering undamaged configurations the temperature dependent elastic law for the effective stress tensor of an undamaged material sample is given. In addition, ratedependent plastic behavior is described using a yield criterion and a flow rule. Furthermore, considering the damaged configurations, the dynamic elastic law for the stress tensor of a damaged material sample also depending on current state of damage is proposed. In addition, ratedependent damage behavior is governed by a damage criterion and a damage rule both depending on stress triaxiality and the Lode parameter.
Considering the undamaged configurations, the temperaturedependent elastic behavior of the undamaged ductile matrix material is modeled by an isotropic hyperelastic law leading to the effective Kirchhoff stress tensor
where
with the first and second deviatoric stress invariants
depends on the current equivalent plastic strain measure,
Furthermore, the isochoric effective plastic strain rate
is taken to model the evolution of plastic deformations in an accurate manner. In Eq. (4)
Considering the damaged configurations of the continuum approach, constitutive equations for the anisotropically damaged material sample undergoing dynamic loading conditions are formulated. Since formation of damage leads to deterioration of elastic material properties the elastic law takes into account both the elastic and the damage strain tensors,
where the additional material parameters
where
denotes the material toughness to microdefect propagation depending on the equivalent damage strain measure
defined as the ratio of the mean stress
written in terms of the principal Kirchhoff stress components
Furthermore, irreversible deformations caused by damage can be determined by the damage strain rate tensor
where the normalized stress related deviatoric tensors
have been used and
3 Numerical simulations on the microscale
Dynamic numerical calculations on the microlevel considering a representative volume element (RVE) of ductile metals containing a microvoid have been performed with the commercial finite element software LSDyna. The investigated material is an aluminum alloy of series 2017 and its behavior is modeled by the LSDyna material model 12. Thus, the constitutive parameters mass density
The numerical studies performed by Benson (1993Benson, D., 1993. An analysis of void distribution effects on the dynamic growth and coalescence of voids in ductile metals. Journal of the Mechanics and Physics of Solids 41, 12851308.) have shown that the range of the peak transmitted stress was not affected by the number of voids in the representative volume element for both copper and 4340 steel. The mean values were comparable to the values predicted for unitcells with a single pore with the same void volume fraction. Therefore, in the present dynamic analysis only the effect of a single void on the macroscopic deformation and damage behavior is considered. Figure 1 shows the finite element mesh of one half of the RVE containing 21,384 standard underintegrated LSDyna brick elements. The mesh size is rather homogeneous without mesh refinement near the voids since in explicit finite element analysis the smallest element edge determines the time step size.
In contrast to quasistatic analysis on the microscale (Brünig et al. (2013Brünig, M., Gerke, S., Hagenbrock, V., 2013. Micromechanical studies on the effect of the stress triaxiality and the Lode parameter on ductile damage. International Journal of Plasticity 50, 4965.,^{2014}Brünig, M., Gerke, S., Hagenbrock, V., 2014. Stressstatedependence of damage strain rate tensors caused by growth and coalescence of microdefects. International Journal of Plasticity 63, 4963., ^{2018b}Brünig, M., Hagenbrock, V., Gerke, S., 2018b. Macroscopic damage laws based on analysis of microscopic unit cells. ZAMM  Zeitschrift für Angewandte Mathematik und Mechanik 98, 181194.)) for microscopic numerical simulations under dynamic loading conditions the physical size of the investigated RVE is of significant importance because material inertia, wave propagation and reflection of waves affect the numerical results. Therefore, the micropore structure of the material studied by Becker et al. (1988Becker, R., Needleman, A., Richmond, O., Tvergaard, V., 1988. Void growth and failure in notched bars. Journal of the Mechanics and Physics of Solids 36, 317351.) has been chosen as reference initially damaged material. Thus, the RVE with initial porosity of
For numerical analysis of the inelastic deformation behavior of RVEs under various multiaxial dynamic loadings the choice of appropriate boundary conditions is a challenging task since a straight forward approach from static simulations is not possible. Benson (1993Benson, D., 1993. An analysis of void distribution effects on the dynamic growth and coalescence of voids in ductile metals. Journal of the Mechanics and Physics of Solids 41, 12851308.) proposed for twodimensional studies under onedimensional tension loading the enlargement of the RVE with homogeneous material in loading direction to facilitate an interferencefree launching of the applied tension pulse because the applied load is reflected at free surfaces, especially at voids. In his studies, zero displacements are taken into account perpendicular to the unloaded surfaces. Preliminary numerical calculations by Gerke et al. (2014Gerke, S., Kuhnt, K., Brünig, M., 2014. Micromechanical numerical analysis of ductile damage under dynamic loading conditions. In: E. Oñate, J. Oliver, A. Huerta (Eds.), Proc. 11th World Congress on Computational Mechanics (WCCM XI), CIMNE Barcelona, 11871198.) have shown that this approach could be extended to threedimensional studies under tension loading. However, generalization of this proceeding to further threedimensional loading conditions was not possible. Embedding the RVE in all three directions into homogeneous material leads to restriction of inelastic deformations in zones close to the void and softening of the surrounding material results into interferences at material boundaries. Therefore, this approach was not followed up and a new access has to be developed.
Extensive numerical studies of various possible boundary conditions of RVEs with one single void performed by Gerke et al. (2014Gerke, S., Kuhnt, K., Brünig, M., 2014. Micromechanical numerical analysis of ductile damage under dynamic loading conditions. In: E. Oñate, J. Oliver, A. Huerta (Eds.), Proc. 11th World Congress on Computational Mechanics (WCCM XI), CIMNE Barcelona, 11871198.) have given the following indications: Zero displacements perpendicular to the loading surface may reflect the situation inside of a bigger material sample most appropriatly, but do not allow the application of loading in this direction. Coupled displacements perpendicular to the loading surface guarantee that no gaps between neighboring unitcells arise, but restrict the deformation during wave propagation and result into not required stress states. Flowout boundary conditions avoid reflections at the boundaries of the RVE, but consequently the stress intensity within the RVE drops remarkably during the numerical simulation and only smaller deformations are reached which do not reflect the situation in a bigger material sample. Furthermore, for analysis of stressstatedependent dynamic damage and failure behavior it has to be discussed if the stress pulse is applied on one side of the RVE or simultaneously on both opposite sides. Whereas the application on one side seems to be more appropriate reflecting the real situation, the application on both sides has the advantage that the RVE deforms symmetrically and that the macrostressstate and, consequently, the stress triaxiality
Based on these numerical studies the boundary and loading conditions shown in Fig. 2 are taken to be most appropriate for the objective of the present analysis and have been applied in this paper. In particular, the displacements of the nodes located on each side are not coupled allowing noncontinuous displacements of neighboring unitcells whereas no flowout boundary conditions are applied. The pulse indicated in Fig. 3 is applied as tension or compression loading simultaneously on opposite sides of the RVE whereas the shape of the pulse is similar to the one occurring in SplitHopkinsonPressureBar (SHPB) experiments (Brünig and Gerke (2011Brünig, M., Gerke, S., 2011. Simulation of damage evolution in ductile metals undergoing dynamic loading conditions. International Journal of Plasticity 27, 15981617.)). The sum of the absolute values of the applied maximum loads is
The scope of all numerical calculations discussed below is to keep the stress triaxiality coefficient and the Lode parameter nearly constant during the entire loading history of the unitcell to be able to accurately analyze their effect on damage and failure behavior of ductile metals under dynamic loading conditions. Therefore, for given parameters
with
and
It is worthy to note that only the boundary conditions shown in Fig. 2 lead to almost constant stress triaxialities and Lode parameters during the dynamic loading process allowing analysis of the stressstatedependent damage behavior of the RVE.
Various loading conditions with principal macroscopic stresses acting on the outer bounds of the representative volume elements are taken into account to be able to cover the wide range of the stress triaxiality coefficients (Eq. (8))
The effect of stress state on damage during dynamic loading can be demonstrated by the deformation of the microdefects. Fig. 4 shows the ratios of the principal stress components
In order to quantify the state of damage in the dynamically loaded RVE macroscopic damage strain components are analyzed. In particular, the current volume of the unitcell is approximated by a cuboid with side lengths
where the respective lengths
The shape of the void is approximated by an ellipsoid with principal semiaxes
Here, the respective length of the semiaxes is calculated from the initial radius
is calculated. The evolution of the void volume fraction
Progress of principal damage strain components
Furthermore, with the principal stretches
of the initially spherical void the principal components of the damage strain tensor
are directly extracted.
The evolution of respective components of the damage strain tensor
4 Damage equations and identification of associated parameters
The numerical results based on dynamic unitcellmodel analyses shown in Figs. 46 represent the development of anisotropic damage behavior in ductile materials undergoing different proportional dynamic loading conditions. These numerical results are used to propose stressstatedependent dynamic damage equations and to identify corresponding constitutive parameters. Stress state dependence is here characterized by the stress triaxiality
In particular, for the dynamic damage criterion (6) the damage mode parameter
and the parameter
with
and
In addition, the parameters
which is shown in Fig. 7. The parameter
with the Lodeparameterdependent part
which is illustrated in Fig. 8. The parameter
Threedimensional representation of strain rate parameter
Threedimensional representation of strain rate parameter
Threedimensional representation of strain rate parameter
Furthermore, the parameter
which is shown in Fig. 9. This parameter
Moreover, the damage strain rate parameters
Damage strain rate parameters
Based on the stressstatedependent parameters discussed above numerical simulation of deformation and damage behavior of dynamically loaded homogeneous representative volume elements using the proposed continuum damage model shows good agreement with the numerical results obtained from voidcontaining unitcell calculations, see Fig. 11. For all investigated stress states the continuum damage model accurately predicts the anisotropic damage behavior for dynamically loaded ductile metal samples.
5 Conclusions
The main objective of this paper was to investigate the effect of dynamic loading conditions on damage behavior of ductile metals. For this purpose, series of numerical calculations on the microlevel considering voidcontaining representative volume elements under different threedimensional dynamic loading scenarios have been performed. They enabled insight in the complex stressstate and ratedependent damage and failure processes on the microscale and their effect on the macroscopic behavior of ductile material samples. The numerical results have been used to discuss general mechanisms of dynamic damage for different stress states and to propose equations for phenomenological damage criteria corresponding to different processes on the microlevel. The results have also been used to develop ratedependent evolution equations for macroscopic damage strains corresponding to anisotropic formation of microdefects and to identify stressstatedependent parameters of the dynamic continuum model. The accuracy of the proposed model has been demonstrated by comparison of numerical results based on the macroscopic continuum model with data obtained by unitcell calculations for different stressstatedependent dynamic loading conditions. It has been shown that for dynamic loading scenarios the parameters in the damage rule characterizing the dependence on the stress state are similar to those modeling static behavior.
Acknowledgement
Financial support from the Deutsche Forschungsgemeinschaft DFG (German Research Foundation, BR1793/122) is greatfully acknowledged.
References
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 Wu, X., Ramesh, K., Wright, T., 2003b. The effects of thermal softening and heat conduction on the dynamic growth of voids. International Journal of Solids and Structures 40, 44614478.
 Zhang, K., Bai, J., Francois, D., 2001. Numerical analysis of the influence of the Lode parameter on void growth. International Journal of Solids and Structures 38, 58475856.

Available Online: May 25, 2018
Publication Dates

Publication in this collection
2018
History

Received
01 Mar 2018 
Reviewed
11 Mar 2018 
Accepted
31 Mar 2018