Constitutive Model Of Metal Foam Under Complex Stress/dynamic Compression

Cellular materials usually refer to the combination of solid edges or solid walls and holes,and they have a certain cellular structure.Metal foams are a kind of cellular materials.Due to the excellent capability of energy absorption,metal foams have a wide range of applications in the field of impact energy absorption.Metal foams usually suffer in complex and dynamic load in engineering application.Therefore,it is necessary to understand its constitutive behavior under complex stress state and dynamic impact.There are macroscopically visible holes in metal foams.It is doubtful whether such inhomogeneous and non-continuum materials can be treated as macro homogeneous or as equivalent continuum materials.Metal foams can yield under hydrostatic pressure due to their compressibility,which is very different from that of dense metals.So,the traditional theory of metal plasticity fails to extend to metal foams.The mean stress should be considered to understand the constitutive behaviour of metal foams.Under dynamic loading,the deformation mode of metal foams will change from quasi-static shear collapse band to a layer-wise collapse band.Therefore,there are many difficulties in studying the constitutive behavior of metal foams under complex stress state and dynamic impact.Many yield criteria and subsequent yield surfaces were developed to describe the stress state of cellular material under complex loads.Some suggested forms of the relation between the von Mises effective stress ?e and the mean stress ?m have been proposed to describe the yield surface of cellular materials.A self-similar isotropic hardening model was developed for metal foams by Deshpande and Fleck.The Deshpande-Fleck foam model attracts a lot of attention because it can describe the response of metal foams under multi-axial loading approximately and it only has two parameters,i.e.the ellipticity ? and the uniaxial yield stress Y.Cell-based finite element models based on 3D Voronoi technique are used to verify this model in this study.The loading scenarios include uniaxial,biaxial and triaxial compressions.The ellipticity is obtained by fitting the results of numerical simulations with the ellipse standard equation.It is found that the ellipticity varies with the equivalent plastic strain.The data of uniaxial and hydrostatic compression tests(two-point fitting)were used to determine approximately the ellipticity ?.A fitting relation between the ellipticity and the equivalent plastic strain is suggested.The ellipticity a is found to be independent of the relative density of cellular material.A modified Deshpande-Fleck foam model with a variable value of ellipticity is suggested.The modified Deshpande-Fleck foam model provides a satisfactory prediction based on the cell-based FE model.The ellipticity a determined experimentally by uniaxial compression tests and hydrostatic compression tests is not constant.Good agreement is also observed between the experimentally measured stress-strain responses and the predictions of the modified Deshpande-Fleck foam model using a non-associated flow rule.The prediction becomes better when considering the effect of the plastic Poisson's ratio.It is important to measure the plastic Poisson's ratio to determine the plastic flow potential accurately.Further study is required to measure the plastic Poisson's ratio effectively,or calculate the plastic Poisson's ratio by some characteristic experiments.Under dynamic loading,the deformation mode of metal foams will change from quasi-static shear collapse band to a layer-wise collapse band.The stress-strain states under dynamic loading are different from the quasi-static curve.This phenomenon is often referred to as the loading-rate effect and used to be distinguished by critical velocity.However,constitutive models can not contain velocity.In this paper,a critical strain-rate is proposed to distinguish quasi-static and dynamic densification behavior.A complete strain-rate dependent constitutive model is proposed,based on the strain-rate sensitivity of initial crushing stress.The model can not only describe quasi-static behavior,but also initial crushing stress and hardening behavior under densification different strain-rates.A test method for determining the constitutive parameters of foams is proposed.The material parameters without strain-rate effect are obtained by quasi-static uniaxial compression based on the cell-based FE model.Some dynamic material parameters are obtained directly based on the constant-velocity impact and direct impact by using the cell-based FE model.While the parameters that are difficult to determine directly are obtained by using the cell-based FE model and the continuum-based FE model,thus all parameters can be determined.Finally,the rate-dependent model is validated under constant-velocity impact and mass block impact by using the continuum-based FE model the cell-based FE model.