International Journal of Plasticity

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1 International Journal of Plasticity 7 (011) Contents lists available at ScienceDirect International Journal of Plasticity journal homepage: Simulation of damage evolution in ductile metals undergoing dynamic loading conditions Michael Brünig, Steffen Gerke Institut für Mechanik und Statik, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, D Neubiberg, Germany article info abstract Article history: Received 14 June 010 Received in final revised form 4 February 011 Available online 16 February 011 Keywords: Damage and fracture Stress triaxiality dependence Dynamic loading Numerical algorithm Numerical simulations A set of constitutive equations for large rate-dependent elastic plastic-damage materials at elevated temperatures is presented to be able to analyze adiabatic high strain rate deformation processes for a wide range of stress triaxialities. The model is based on the concepts of continuum damage mechanics. Since the material macroscopic thermo-mechanical response under large strain and high strain rate deformation loading is governed by different physical mechanisms, a multi-dissipative approach is proposed. It incorporates thermo-mechanical coupling effects as well as internal dissipative mechanisms through rate-dependent constitutive relations with a set of internal variables. In addition, the effect of stress triaxiality on the onset and evolution of plastic flow, damage and failure is discussed. Furthermore, the algorithm for numerical integration of the coupled constitutive rate equations is presented. It relies on operator split methodology resulting in an inelastic predictor elastic corrector technique. The explicit finite element program LS-DYNA augmented by an user-defined material subroutine is used to approximate boundary-value problems under dynamic loading conditions. Numerical simulations of dynamic experiments with different specimens are performed and good correlation of numerical results and published experimental data is achieved. Based on numerical studies modified specimens geometries are proposed to be able to detect complex damage and failure mechanisms in Hopkinson-Bar experiments. Ó 011 Elsevier Ltd. All rights reserved. 1. Introduction Due to rapid simultaneous progress in hardware technologies as well as in computational mechanics and material modeling during the last decades, engineers are nowadays able to analyze complex structures undergoing static and dynamic loading conditions. Consequently, the accurate and realistic modeling of rate-independent and of rate-dependent inelastic behavior of ductile metals as well as the development of associated efficient and stable numerical solution techniques are essential for the solution of numerous boundary-value problems occurring in many engineering disciplines. In addition, high strain rate deformations such as dynamic shear banding, impact problems or high-speed machining have become increasingly important for industrial applications. Another problem of practical interest is the temperature rise at the tips of dynamically propagating cracks. The prediction of such complex thermo-mechanical processes can only be satisfactorily performed using numerical solution procedures such as the finite element method for solving strongly coupled mechanical and thermal boundary-value problems. This leads to an increasing demand on accurate, robust and efficient numerical models to be able to realistically analyze the mechanical response of engineering structures under dynamic loading conditions. The analysis Corresponding author. Tel.: ; fax: address: michael.bruenig@unibw.de (M. Brünig) /$ - see front matter Ó 011 Elsevier Ltd. All rights reserved. doi: /j.ijplas

2 M. Brünig, S. Gerke / International Journal of Plasticity 7 (011) requires constitutive models accounting for large strains, high strain rates and thermal softening as well as generalized damage and fracture criteria to be able to realistically predict the failure mechanisms. Currently, there are many computer codes for numerical analyses of these problems. Although these codes can nowadays be used to perform computations of structures even under dynamic loading conditions, it is generally agreed that there is a need for improved strength models applicable for a wide range of stress triaxialities as well as corresponding procedures to identify the parameters for these models. Within the context of dynamic modeling this requires generalized constitutive equations describing the rate-dependent inelastic behavior of metals as functions of strains, strain rates and temperature. For example, due to simplicity many numerical simulations are based on the Johnson Cook material model (Johnson and Cook, 1983) accounting for isotropic strain hardening, strain rate sensitivity and thermal softening in an uncoupled form in one single equation. Rohr et al. (005), however, have shown that the Johnson Cook model might lead to questionable results in the high-dynamic region and, thus, there is a necessity of more reliable and adequate material models. In the literature, thermo-elastic plastic frameworks involving large strains have been developed particularly with respect to the phenomenological description of the deformation behavior of poly-crystalline metals. The standard applications are thermo-mechanical modeling of material testing problems, metal forming processes and structural impact. Theoretical approaches are well established in the literature in, for example, Taylor and Quinney (1937), Anand (1985), Brown et al. (1989), Khan and Huang (199), Miehe (1995), Rule and Jones (1998), Khan and Liang (1999), Khan et al. (004), Austin and McDowell (011), Sung et al. (010). Numerical implementations of large-strain thermo-elasto-plasticity have been presented by Argyris and Doltsinis (1981), Lemonds and Needleman (1986), Simo and Miehe (199), Wriggers et al. (199), Canadija and Brnic (004), Brünig and Driemeier (007). In addition, continuum thermo-elasto-plasticity approaches for applications with isotropic damage have been discussed by Simo and Ju (1987a), Lestriez et al. (004), among others. On the other hand, different models for predicting the occurrence of damage and failure in materials and structures under general loading conditions have been proposed to simulate heterogeneous material behavior and failure. Motivated by experimental observations, phenomenological approaches introduce internal scalar or tensorial damage variables whose growth is governed by evolution laws. Thus, within the general framework of continuum thermodynamics of irreversible processes several continuum damage models have been published which are either phenomenologically based or micromechanically motivated (see e.g. Chaboche, 1988a,b; Tvergaard, 1990; Lemaitre, 1996; Brünig, 003a, 006; Bammann and Solanki, 010). Rajendran et al. (1988) presented an approach for dynamic plasticity and failure in ductile metals. To be able to simulate dynamic void growth, further modifications have been discussed by Addessio et al. (1993) and Worswick and Pick (1995). In addition, Wang (1997) and Wang and Jiang (1997) studied the effect of inertia on the yield condition for porous solids for the simulation of high strain rate processes. Furthermore, Zavaliangos and Anand (1993) presented a complete rate and temperature dependent constitutive model for moderately porous metallic materials also characterizing the porosity by the void volume fraction but using evolution equations different from Gurson (1977) and Taylor and Quinney (1937). Predictions from their numerical calculations agree well with experimental results. In addition, Børvik et al. (003) carried out non-linear finite element analyses of high rate tensile tests of steel specimens using Johnson Cook constitutive model and fracture criterion (Johnson and Cook, 1985). Their numerical results show good agreement with experiments performed by Hopperstad et al. (003). General frameworks of internal variable theory of high strain rate deformation processes in metals are given, for example, by Bruhns and Diehl (1989) and Voyiadjis et al. (003). These models are able to describe the evolution of damage in high strain rate processes but the calibration for the different material properties seems to be difficult leading to problems in numerical applications. However, accurate and efficient constitutive models of damaged materials are needed as the basis for an accurate theory of ductile fracture. Critical values of proposed continuum damage variables may be viewed as major parameters characterizing the onset of failure. Moreover, besides the stress intensity, the stress triaxiality has been shown to be the most important factor that controls initiation of ductile damage and fracture. Attempts to take into account the effect of stress triaxiality in their continuum approaches have been presented by Hancock and Mackenzie (1976), Børvik et al. (003), Bonora et al. (005), Oh et al. (007), Bao and Wierzbicki (008), Brünig et al. (008, 011a), Gao et al. (009), Mirone and Corallo (010) and Li et al. (011) based on static tension tests with notched specimens. Bao and Wierzbicki (004) proposed fracture strain criteria based on three different branches corresponding to different failure mechanisms on the micro-scale. The objective of the present paper is to discuss a generalized and extended version of Brünig s damage model (Brünig, 003a,b, 006) for high strain rate problems as well as its numerical implementation. A set of constitutive equations for large rate-dependent elastic plastic-damage materials at elevated temperatures is proposed which may be useful for application to the analysis of adiabatic deformation processes for a wide range of stress triaxialities. The rate- and temperature-dependent deformation behavior of anisotropically damaged elastic plastic materials subjected to impact loading conditions is based on the concepts of continuum damage mechanics. Since the material macroscopic thermo-mechanical response under large strain and high strain rate deformation loading is governed by different physical mechanisms, a multi-dissipative model is introduced with manifold structure accounting for dislocations along crystal slip planes (isotropic plasticity) as well as void and micro-crack interactions (anisotropic damage). It incorporates thermo-mechanical coupling effects as well as internal dissipative mechanisms through rate-dependent constitutive relations with a set of internal variables. The proposed high strain rate model takes into account the effect of stress triaxiality on the onset and evolution of plastic flow, damage and failure. It is based on a stress-triaxiality dependent yield condition as well as on damage and failure criteria with different

3 1600 M. Brünig, S. Gerke / International Journal of Plasticity 7 (011) branches corresponding to various high strain rate damage and failure modes depending on stress triaxiality. In addition, the algorithm for numerical integration of the coupled constitutive rate equations is presented. It relies on operator split methodology resulting in an inelastic predictor elastic corrector technique for rate dependent material behavior. The explicit finite element program LS-DYNA augmented by an user-defined material subroutine is used to approximate boundary-value problems under dynamic loading conditions. Numerical simulations of dynamic experiments with different specimens are performed and good correlation of published experimental data and numerical results is achieved. Based on numerical studies modified specimens geometries are proposed to be able to detect complex damage and failure mechanisms in Hopkinson- Bar experiments.. Constitutive equations A macroscopic continuum model is used to predict the irreversible material behavior caused by dynamic loading conditions while ignoring details of the mechanisms on the microscopic scale of individual grains, voids and micro-cracks as well as their interaction. Brünig (003a) proposed a phenomenological framework to describe the inelastic deformations including anisotropic damage by micro-defects. The continuum model has been extended by Brünig (006) to be able to take into account rate and temperature effects. The basic ideas of the approach employing the introduction of damaged as well as corresponding fictitious undamaged configurations are also used here and will be briefly summarized and extended for general dynamic loading conditions in the following subsections..1. Undamaged configurations The effective undamaged configurations are considered to formulate the constitutive equations describing the effective elastic plastic deformation behavior of the undamaged matrix material. In particular, the effective specific Helmholtz free energy / is taken to be additively decomposed into a thermo-elastic and a thermo-plastic part q o /ða el ; c; hþ ¼q o / el ða el ; hþþq o / pl ðc; hþ; ð1þ where q o represents the initial density, A el is the elastic part of the logarithmic strain tensor, c denotes the scalar internal mechanical state variable characterizing the current plastic strain state, and h represents the absolute temperature. The thermo-elastic part / el is the effective specific free energy which can be instantaneously recovered upon unloading whereas the thermo-plastic part / pl is the effective specific free energy corresponding to residual stresses and dislocations in the matrix material. In particular, the finite thermo-elastic part of the undamaged material behavior is assumed to be governed by the effective free energy function q o/ el ða el ; hþ ¼GA el A el þ 1 K 3 G ðtra el Þ 3Ka T ðh h o ÞtrA el þ q ohðhþ; ðþ where G and K represent the shear and bulk modulus of the matrix material, a T is the coefficient of thermal expansion, h o means the reference temperature, and h denotes an explicit function of temperature. Taking into account isotropic hyperelastic constitutive behavior the effective stress tensor is expressed in the T ¼ q / el ¼ el GAel þ K 3 G tra el 1 3Ka T ðh h o Þ1: ð3þ Furthermore, the rate-dependent plastic behavior of the considered ductile metals is assumed to be governed by the dynamic representation of the Drucker Prager-type yield condition (Brünig, 1999b, 003a) qffiffiffiffi f pl ¼ ai 1 þ J cðc; _c; hþ ¼0: ð4þ In Eq. (4) a represents the hydrostatic stress coefficient while I 1 ¼ trt is the first invariant of the effective stress tensor (3) and J ¼ 1 devt devt denotes the second invariant of the effective stress deviator. In addition, cðc; _c; hþ means the rate- and temperature-dependent equivalent stress measure. The coefficient a is stress-dependent whereas experimental data have shown that the ratio a/c is constant (Spitzig et al., 1975, 1976). Therefore, the yield condition (4) can alternatively be written in the form qffiffiffiffi f pl ði 1 ; J ; cþ ¼ J 1 a c I 1 c ¼ 0: ð5þ In addition, the consistency condition 0 1 _f pl B 1 C qffiffiffiffi devt þ a1a _ T 1 a c I 1 _c ¼ 0 J ð6þ ensures that the subsequent yield condition remains satisfied during any incremental deformation.

4 M. Brünig, S. Gerke / International Journal of Plasticity 7 (011) In the present analysis, the equivalent stress is expressed as a specific function of plastic strain, plastic strain rate and temperature using the multiplicative decomposition cðc; _c; hþ ¼~cðcÞf 1 ð_cþf ðhþ which is assumed to be the basis of empirical arguments (Johnson and Cook, 1983; Johnson et al., 1983; Klopp et al., 1985; Zerilli and Armstrong, 1987). In Eq. (7) the reference equivalent effective stress can be numerically simulated by the power law H o c n ~cðcþ ¼~c o þ 1 ; ð8þ n~c o where ~c o represents the initial yield stress, H o denotes the initial hardening modulus and n means the hardening exponent. These material parameters are determined performing quasi-static tension tests with smooth specimens leading to homogeneous one-dimensional stress-states at constant reference temperature h o. The experimentally observed increasing dependence of the equivalent stress on the inelastic strain rate _c is described by the strain rate hardening function c f 1 ð_cþ ¼1 þ d _ _c m o ð9þ _c o with the constitutive parameters d and m which have to be determined by a series of dynamic experiments at different strain rates and constant temperature. The reference inelastic strain rate _c o is given by the quasi-static reference tests. It should be noted that in cases with _c < _c o the cutting off value f 1 ð _cþ ¼1 is used in the numerical calculations. The proposed rate-dependent power-law description (9) based on two material parameters can be seen as a generalization of the power laws presented by Asaro (1983), Pan and Rice (1983) and Johnson and Cook (1983) who only take into account one constitutive parameter. In addition, the decrease in equivalent stress with increasing temperature as observed in experiments is taken into account by the thermal softening function f ðhþ ¼1 b h h o h m q ð10þ with the constitutive parameters b and q which have to be determined by a series of isothermal tests at different constant temperatures and constant strain rates, and h m denotes the melting temperature of the matrix material. Moreover, to be able to compute plastic strain rates, the temperature-dependent plastic potential function qffiffiffiffi g pl ðt; c; hþ ¼ c o ðc; hþ with c o ðc; hþ ¼~cðcÞf ðhþ ð11þ J is introduced taking into account the second invariant of the effective stress deviator J and the temperature-dependent scalar effective stress measure c o. This leads to the isochoric plastic strain rate _H pl ¼ ¼ k _ 1 1 qffiffiffiffi devt ¼ _c q ffiffiffiffiffiffiffi devt ¼ _cn; ð1þ J J where _ k and _c are non-negative scalar-valued factors and N represents the normalized deviatoric stress direction. It should be noted that the parameter _c is taken to be the equivalent plastic strain rate. In the present paper, only adiabatic processes are considered and, as a consequence, no constitutive equation for the effective heat flux vector is required (Brünig, 006). The evolution equation for the temperature is approximated by c F _ h ¼ nt _ H pl only taking into account the inelastic contribution to heating validated by the high-deformation experiments performed by Mićunović et al. (003). In Eq. (13) c F denotes the specific heat and n defines the fraction of plastic work rate converted to heating. This latter parameter has been controversially discussed in the literature (see, for example, Brünig and Driemeier (007) for further details), and in the present paper it is taken to be constant n = 0.8. Using the equivalence of plastic work rates expressed in terms of general effective stress states and the corresponding equivalent-measures formulation _w pl ¼ T H _ qffiffiffiffiffiffiffi pl ¼ J _c ð14þ the temperature increase (13) can be rewritten in the form _h ¼ n 1 qffiffiffiffiffiffiffi J _c: c F ð7þ ð13þ ð15þ.. Anisotropically damaged configurations Furthermore, anisotropically damaged configurations are considered and used to formulate damaged elastic and damage constitutive equations characterizing the inelastic deformation behavior of the damaged aggregate. In particular, in order to

5 160 M. Brünig, S. Gerke / International Journal of Plasticity 7 (011) take into account thermo-plasticity and thermo-damage phenomena in an adequate manner two sets of internal variables are chosen to be able to characterize the formation of dislocations (plastic internal variable: the equivalent plastic strain c) as well as the nucleation and propagation of micro-defects (damage internal variable: the equivalent damage strain l). The specific Helmholtz free energy of the damaged material sample is assumed to be additively decomposed into three parts: q o /ða el ; A da ; c; l; hþ ¼q o / el ða el ; A da ; hþþq o / pl ðc; hþþq o / da ðl; hþ: ð16þ It is well known that the existence of micro-defects causes a decrease of the stress level in the aggregate and a decrease of the elastic material properties when compared to the response of the virgin undamaged material. To be able to model these phenomena, the thermo-elastic part of the free energy of the damaged material / el is expressed in terms of the elastic and damage strain tensors, A el and A da, whereas the thermo-plastic part, / pl, and the thermo-damage part, / da, take into account the respective internal equivalent plastic and damage state variables, c and l. In addition, all parts depend on the current temperature h. In particular, the finite thermo-elastic part of the material behavior is governed by the free energy density function q o / el ða el ; A da ; hþ ¼GA el A el þ 1 K 3 G tra el 3KaT ðh h o ÞtrA el þ q o hðhþþg 1 tra da tra el þ g tra da A el A el þ g 3 tra el A da A el þ g 4 A el A el A da ; ð17þ where g 1 g 4 represent material parameters describing the deterioration of the elastic properties by the occurrence of damage and h(h) denotes an explicit function of temperature. Taking into account hyper-elastic constitutive behavior, the stress tensor is expressed in the form T ¼ q el ¼ G þ g tra da A el 3Ka T ðh h o Þ1 þ K 3 G þ g 1 trada tra el þ g 3 A da A el 1 þ g 3 tra el A da þ g 4 A el A da þ A da A el : ð18þ In addition, constitutive equations for damage evolution are required and the determination of onset and continuation of damage is based on the concept of damage surface at the macroscopic level formulated in stress space (Chow and Wang, 1988). Taking into account different damage mechanisms in ductile metals leading to final fracture (Bao and Wierzbicki, 004; Brünig et al., 008, 011a) considered a wide range of stress triaxialities g (ratio of mean stress and equivalent von Mises stress) with different branches (Fig. 1): damage is characterized by shear modes for negative stress triaxialities (g c 6 g60), by void-growth-dominated modes for large positive triaxialities (gpg t ) and by mixed modes for lower positive stress triaxialities (0 < g < g t ). In the hydrostatic pressure regime (g < g c ), Bao and Wierzbicki (005) proposed a cut-off value of stress triaxiality below which damage and fracture do not occur in ductile metals. This stress-triaxiality-dependent concept schematically illustrated in Fig. 1 is also used in the present rate- and temperature-dependent continuum damage and failure approach. In particular, onset and continuation of damage is characterized by the dynamic damage condition f da ðt; rþ ¼~aI 1 þ b ~ p ffiffiffiffi J rðl; _l; hþ ¼0 ð19þ expressed in terms of the first stress invariant I 1 =trt and the second invariant of the stress deviator J ¼ 1 devt devt, the equivalent stress of the damaged configuration, r denotes the damage threshold depending on the equivalent damage strain, equivalent damage strain rate and temperature, and the parameters ~a and b ~ describe the influence of stress triaxiality on the damage condition: Fig. 1. Different damage mechanisms depending on stress triaxiality g.

6 ~a ¼ 0 for g c 6 g60; 1=3 for g > 0; M. Brünig, S. Gerke / International Journal of Plasticity 7 (011) for g c 6 g60; >< ~b ¼ 1 dg ~ ~m for 0 < g < gt ; ð1þ >: 0 for gpg t : In Eq. (1), ~m and d ~ represent material parameters which are identified using numerical simulations of experiments with differently notched specimens (Brünig et al., 008). In addition, using the definition of the stress triaxiality in the damaged configurations ð0þ g ¼ I 1 p 3 ffiffiffiffiffiffiffi 3J ðþ the damage condition Eq. (19) can be rewritten in the form p f da ¼ 3 ffiffiffi 3ag ~ þ b ~ pffiffiffiffi r ¼ 0 ð3þ J to exhibit the assumed stress-triaxiality dependence of damage. In the case of dynamic applications, the equivalent damage stress, r, is expressed as a specific function of damage accumulation, its rate and temperature using the multiplicative decomposition rðl; _l; hþ ¼~rðlÞf 3 ð _lþf ðhþ: In Eq. (4) the reference stress is numerically simulated by the quadratic relation ~r l e þ ~r o l e l þ ~r o shown in Fig. where ~r o represents the initial equivalent damage stress, l e the fictitious value where the damage softening relation reaches zero the slope of the static plastic hardening function indicated in Eq. (8) at the onset of damage. The equivalent stress (5) is determined using quasi-static tests at constant temperature h o. The increasing dependence of the equivalent damage stress on the equivalent damage strain rate _l is described by the damage rate hardening function r f 3 ð _lþ ¼1 þ h ð6þ _l _l o _l o with the constant parameters h and r obtained from an inverse identification procedure based on numerical simulations of a series of dynamic experiments at different strain rates and constant temperature, and the reference damage rate _l o is given by the quasi-static reference test. It should be noted that only experiments at different strain rates can be performed and, thus, it is difficult to determine the damage parameters h, r and _l o. To bypass these difficulties, the parameters are taken to be those of the plastic strain rate dependence h = d, r = m and _l o ¼ _c o, see Eq. (9), and the cutting off value f 3 ð _l o Þ¼1 for _l < _l o is used in the numerical calculations presented here. Furthermore, to be able to compute damage strain rates, the damage potential function p g da ðt; l; hþ ¼aI 1 þ b ffiffiffiffi J go ðl; hþ ð7þ is also formulated in the terms of invariants of the stress tensor T discussed above, and ð4þ ð5þ g o ¼ ~rðlþf ðhþ ð8þ Fig.. Damage softening law.

7 1604 M. Brünig, S. Gerke / International Journal of Plasticity 7 (011) represents a temperature-dependent scalar stress measure. In Eq. (7), a and b denote kinematically motivated damage parameters (Brünig, 003a). This leads to the non-associated damage rule _H da 1 ¼ _la1 þ _lb p ffiffiffiffi devt: ð9þ J The first term on the right-hand side in Eq. (9) represents the rate of inelastic volumetric deformations caused by the isotropic growth of voids whereas the second term corresponds to the anisotropic evolution of damage strain rates caused by micro-shear-cracks. In the present approach, the kinematic variables are taken to be a ¼ 1 b and b ¼ 0:9~ b. 3ð1 f Þ Moreover, the internal damage variable l can be used to define a simple triaxiality-dependent fracture criterion. This fracture condition can be written in the form f cr ¼ l l cr ¼ 0; where l cr is the triaxiality-dependent critical equivalent damage strain: 8 l f for g > g f ; >< l l cr ¼ f l o g f g þ l o for 06g6g f ; >: l o for g < 0 and the material parameters l f and l o represent the critical tension and compression values of the equivalent damage strain, respectively. The propagation of the macro-crack is numerically realized through an element erosion technique. 3. Numerical integration Numerical solution of the discretized equilibrium equations is based on an incremental finite element procedure. In the present paper, the commercial explicit finite element code LS-DYNA augmented by an user-defined material subroutine has been used. In particular, the results of each iteration step are employed as estimates of the incremental displacements from which the current stress state as well as the other field variables are computed at the integration points of each finite element. In the current formulation, these variables are considered to be known at time t n = t together with an estimate of the current displacement field at time t n+1 = t + Dt. The constitutive rate equations discussed above are numerically integrated by an algorithm based on operator split methodology. In finite strain inelastic theories the numerical integration of constitutive rate equations in an accurate and efficient manner is often performed using radial return techniques, see e.g. Simo and Ju (1987b), Ju (1989), de Souza Neto et al. (1994), Doghri (1995), de Souza Neto and Peric (1996). These elastic predictor plastic corrector algorithms work quite well for smooth yield conditions and associated flow rules as long as time increments remain small. However, since the trade-off between accuracy and computational efficiency is an issue of current interest, Nemat-Nasser and Li (199) proposed a modification of the radial return method to be able to get a very accurate and efficient technique even for large scale codes. Alternatively, Nemat-Nasser (1991) and Nemat-Nasser and Li (199) have presented an algorithm for large strain elastic plastic problems. In this algorithm the total incremental deformation is first assumed to be entirely plastic, and only the subsequent correction accounts for the generally small elastic part. In addition Wang and Atluri (1994) proposed a modification of Nemat Nasser s algorithm which shows better convergence in cases where the direction of stress cannot well follow the direction of deformation rate. Furthermore, Brünig (1999a, 003b) presented finite element analyses using a generalized version of Nemat Nasser s method for the integration of the inelastic rate equations permitting remarkably large load increments with almost no loss in accuracy. This algorithm is generalized in the present paper to be able to solve complex boundary value problems undergoing dynamic loading conditions. The numerical technique takes into account rate- and temperature-dependent constitutive relations valid for a wide range of stress triaxialities. Based on the plastic consistency condition (6) and making use of the rate-dependent constitutive relations discussed above one arrives at the following scalar-valued rate constitutive equation related to the undamaged configurations pffiffiffi G1 ð_e 1 k 1 _c k _lþ ¼_c; ð3þ see (Brünig, 003b) for further details. In Eq. (3) the weighted shear modulus of the matrix material G 1 ¼ G ; ð33þ 1 a I c 1 the equivalent strain rate measures _e 1 ¼ 1 pffiffiffi pffiffiffi GN þ 3aK1 H _ G with the total strain rate H _ (see (Brünig, 003a) for further details) as well as the abbreviations qffiffiffiffi 9aKa Tn J k 1 ¼ RNR 1 N þ Gc F ð30þ ð31þ ð34þ ð35þ

8 M. Brünig, S. Gerke / International Journal of Plasticity 7 (011) and k ¼ 1 pffiffiffi 1 pffiffiffi Q el GN þ 3aK1 Q el b a1 þ p ffiffiffi N G have been introduced. In Eqs. (35) and (36), R and Q el are metric transformation tensors representing the damage and elastic kinematics, respectively, see (Brünig, 003a) for further details. Furthermore, based on the damage consistency condition f _ da ¼ 0 the second scalar-valued rate constitutive equation related to the damaged configurations can be written in the form ffiffiffi ð37þ p G ð_e k 3 _c k 4 _lþ ¼ _r where the shear modulus of the damaged material ð36þ G ¼ G þ g tra da ; the second equivalent strain measure _e ¼ pffiffiffi 1 C 1 H; _ G the factors 0 qffiffiffiffiffiffiffi1 k 3 ¼ pffiffiffi 1 RC 1 R 1 N þ 9Ka T n A G c F ð38þ ð39þ ð40þ and k 4 ¼ ffiffiffi 1 h i p Q el C 1 ðq el Þ 1 C a1 þ b p ffiffiffi N ~ ; ð41þ G as well as the abbreviations p C 1 ¼ ffiffiffi ~bg N þ b 3K þ p ~ ffiffiffi g 3 N A da b!1 þ p ~ ffiffiffi g 4 ðna da þ A da NÞþð3g 3 þ g 4 ÞA da ð4þ and b C ¼ p ~ ffiffiffi g 3 tra el N þ " b ~ # p ffiffiffi g N A el þ ð6g 1 þ g þ g 3 ÞtrA el b 1 þ p ~ ffiffiffi g 4 ðna el þ A el NÞþð3g 3 þ g 4 ÞA el ð43þ with the shear modulus of the damaged material tra da K ¼ K þ g 1 þ 3 g þ 1 3 g 3 ð44þ and the normalized deviatoric tensor N ¼ p 1 ffiffiffiffiffiffiffi devt ð45þ J have been used. In the present context, the integration algorithm starts from the scalar rate Eqs. (3) and (37) describing the entire rateand temperature-dependent constitutive behavior of the ductile, anisotropically damaged material for arbitrary loading paths. The complete stress and deformation history is obtained from a simultaneous straightforward numerical integration of these equations. In particular, to approximate the evolution equations for plasticity and damage, a one step midpoint integration scheme is employed in a standard manner. This leads to the increment of the equivalent effective stress p Dc ¼ c nþ1 c n ¼ ffiffiffi G1 ðde 1 k 1 Dc k DlÞ ð46þ and to the increment of the equivalent aggregate stress p ffiffiffi G ðde k 3 Dc k 4 DlÞ; Dr ¼ r nþ1 r n ¼ ð47þ respectively. In the inelastic predictor step elastic strain increments are neglected. This leads to " # k 1 k Dcpr k 3 k 4 Dl pr ¼ De 1 De ð48þ

9 1606 M. Brünig, S. Gerke / International Journal of Plasticity 7 (011) which can be rewritten in an abbreviated manner kdc pr ¼ De ð49þ where k denotes the matrix of the coefficients k i, Dc pr represents the vector of the equivalent inelastic predictor strain increments and De is the vector of the incremental strain increments representing the incremental deformation of the matrix material and the damaged material, respectively. This leads to the predictor strain increments Dc pr ¼ k 1 De: ð50þ Computing the corresponding equivalent predictor strains c pr ¼ c n þ Dc pr and l pr ¼ l n þ Dl pr ; ð51þ predictor strain rates _c pr ¼ Dc pr Dt and _l pr ¼ Dl pr Dt and the predictor temperature qffiffiffiffi J Dh pr ¼ n Dc c pr and h pr ¼ h n þ Dh pr ð53þ F leads to the respective equivalent predictor stresses at the end of the time interval c pr ¼ cðc pr ; _c pr ; h pr Þ and r pr ¼ rðl pr ; _l pr ; h pr Þ which are determined from the respective equivalent stress equivalent inelastic strain relationships. Since these assumptions lead to over estimations of the equivalent inelastic strain increments and the corresponding current equivalent stress measures, the respective errors are estimated by the elastic corrector steps. Considering the undamaged configuration, this leads to the error in the effective stress measure D er c ¼ c pr c nþ1 ¼ c pr c n Dc: ð5þ ð54þ Taking into account the constitutive relation for the error in stress increment 0 qffiffiffiffiffiffiffi1 D er þ n AD er c c cd er c; this leads to the effective stress increment p Dc ¼ ffiffiffi G1 ðk 1 D er c þ k D er lþ¼c pr c c cd er c: c F ð55þ ð56þ Similarly, the error in the damage stress measure is given by D er r ¼ r pr r nþ1 ¼ r prh r n Dc: With the corresponding constitutive relation qffiffiffiffiffiffiffi J D er þ _l D er n one arrives at the stress increment ffiffiffi ð57þ c F D er c l rd er l c rd er c ð58þ p Dr ¼ G ðk 3 D er c þ k 4 D er lþ ¼r pr r l rd er c rd er c ð59þ where the c l r c r given by the slopes of the equivalent stress equivalent inelastic ¼ f H o c ðn 1Þ 1ð_cÞf ðhþh o þ 1 ð60þ n~c l e þ ~r o l e 3 l 3 ð _lþf ðhþ; ð61þ by the slopes of the equivalent stress equivalent inelastic strain _c ¼ ~cðcþf ðhþ md _c _c ðm 1Þ o ð6þ _c o _c o

10 M. Brünig, S. Gerke / International Journal of Plasticity 7 (011) _l ¼ ~r ðlþf ðhþ hr l _l ðr 1Þ o ð63þ l o _l o as well as by the slopes of the equivalent stress ¼ ~cðcþf 1 ðc _ Þ bq ðq 1Þ h h o ð64þ h m h ¼ ~r l _l ð Þf 3 ð Þbq h m ðq 1Þ h h o ð65þ h m are predicted at the end of the inelastic predictor increment. Using, now, Eqs. (56) and (59) the estimates of the errors in the scalar inelastic strain measures D er c and D er l are computed from " pffiffiffi pffiffiffi G1 k 1 c c G1 k # pffiffiffi pffiffiffi Der c ¼ c pr c n ð66þ G k 3 c r G k 4 l r D er l r pr r n which can also be rewritten in abbreviated manner ad er c ¼ Dc pr : ð67þ This leads to the errors D er c ¼ a 1 Dc pr : Then, the respective incremental equivalent strains are given by Dc ¼ Dc pr D er c: ð68þ ð69þ leading to the current values c nþ1 ¼ c n þ Dc and l nþ1 ¼ l n þ Dl; ð70þ respectively, as well as to the temperature qffiffiffiffiffiffiffi J Dh ¼ Dh pr n D er c and h nþ1 ¼ h n þ Dh: ð71þ c F The corresponding current equivalent stress measures are estimated from the constitutive relations c nþ1 ðc nþ1 ; _c nþ1 ; h nþ1 Þ and r nþ1 ðl nþ1 ; _l nþ1 ; h nþ1 Þ where the current equivalent strain rates are given by _c nþ1 ¼ Dc Dt and _l nþ1 ¼ Dl Dt : Corresponding estimates of the respective tensorial quantities are derived using the fundamental relationships discussed above. In particular, based on Eq. (1) the incremental effective plastic strain tensor is computed from ð7þ DH pl ¼ DcN m ; where the effective deviatoric orientation tensor N m ¼ 1 N n þ N nþ1 ð73þ ð74þ is evaluated at the midpoint of the time increment. In addition, using Eq. (9) the increment of the damage strain tensor is given by p N m ð75þ DH da ¼ Dl a1 þ b ffiffiffi with the mean value of the orientation tensor N m ¼ 1 ðn n þ N nþ1 Þ: ð76þ 4. Numerical examples The split Hopkinson-Bar method is a well-known testing technique to achieve very high strain rates. Several modifications of the method have been presented since its introduction by Kolsky (1949), special modifications to preform tension

11 1608 M. Brünig, S. Gerke / International Journal of Plasticity 7 (011) test have been discussed, for instance, by Lindholm and Yeakley (1968) for hat-shaped specimens and by Nicholas (1981) for axisymmetric specimens. The experimental setup basically consists of two relatively long bars, called input and output bar in-between which a relatively small specimen is located. At the input bar a wave is generated which causes major deformation of the specimen and is partly transmitted to the output bar. Børvik et al. (003) presented numerical simulations of dynamic tension tests with unnotched and notched axisymmetric specimens used in split Hopkinson-Bar experiments. For their numerical simulations the model was roughly reduced and only the specimens were considered which were modeled by four-node axisymmetric finite elements. For the numerical studies discussed in the present paper the explicit finite element code LS-DYNA, enhanced by user-defined material subroutine, has been used. Discretization of the specimen is preformed by eight-node brick elements with one integration point. Following the ideas of Børvik et al. (003), only the specimens are considered and the boundary conditions are described by a dynamic surface impulse (see Fig. 3), similar to those occurring in split Hopkinson-Bar experiments performed by Mohr and Gary (007), while the other end is fixed. With this approach an excess of numerical efforts is avoided and reasonable results can be expected. However, in the future it is desirable to realize simulations of the complete experimental setup to study further effects in more detail. The material characteristics are taken from the aluminum alloy described by Brünig et al. (008, 011a,b) where the identification of the corresponding material parameters has been discussed in detail. A summary of the parameters used in the present study can be found in Table Flat unnotched specimens Flat unnotched specimens taken from sheets of metal or other materials are frequently used for split Hopkinson-Bar experiments. These specimens are generally glued in a thin slit of the bars whereas the thin part of the specimen and the transmission zone stays free. The specimen geometry is taken from Zhou and Xia (000) and is shown in Fig. 4; similar specimens were used, for example, by Wang and Xia (000), Eskandari and Nemes (000) and Verleysen et al. (008). For better legibility Fig. 4 shows a coarser mesh than used for the simulations where 801,791 elements are used; the thin part of the specimen has 7 elements over the width and 4 elements in thickness direction. In these numerical calculations, clamped ends on the left side are considered and the pulse acts on the right side. First simulations neglecting the rate and temperature dependence of the material, i.e. f 1 =1,f = 1 and f 3 = 1, are carried out. Fig. 5 shows the corresponding damage evolution. Here first damage is numerically predicted after major necking, i.e. after remarkable plastic deformations, at the center-line of the specimen but closer to the right boundary caused by the wave Fig. 3. Applied pressure. Table 1 Summary of parameters, corresponding in mm, s, N and K. q o G K h o h m ,800 6, a T c F n ~c o a/c H o n d m _c o b q h r _l o g 1 g g 3 g 4 g c 30,000 0,000 5,000 0,000 1/3 g t d ~ ~m ~r o l e p 1= ffiffiffi 3 pffiffiffi g f l f l o p 1= ffiffiffi

12 M. Brünig, S. Gerke / International Journal of Plasticity 7 (011) Fig. 4. Specimen geometry and mesh, thickness 1.4 mm, width of thinner part 4.0 mm. Fig. 5. Damage evolution for rate- and temperature-independent material behavior, shown in red the elements with damage occurrence, first image at t =68ls, time in-between images 1.4 ls. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 6. Damage evolution for rate- and temperature-dependent material behavior, shown in red the elements with damage occurrence, first image at t =5ls, time in-between images 4.0 ls. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) of the dynamic pulse. The evolution of damage occurs then very localized and rather fast; in the right image the maximum value of the internal damage variable already reaches l = Furthermore simulations taking into account the rate- and temperature-dependence of the material have been carried out. First damage is predicted without necking of the specimen at its outside, see Fig. 6. In continuation damage evolves over the complete thinner part during a quite long period. In the last image the maximum value of the equivalent damage strain is l = In addition, Fig. 7 shows the corresponding temperature distribution. Figs. 5 and 6 clearly show that the numerical simulation taking into account the proposed rate- and temperature-dependent model predicts more extended damage zones than those based on the rate- and temperature-independent approach. In conclusion, realistic constitutive models validated by experiments have to be used to accurately simulate the rate- and temperature-dependent deformation process. 4.. Axisymmetric specimen Nicholas (1981) proposed an experimental technique for axisymmetric tension specimens in split Hopkinson-Bar apparatus. Hopperstad et al. (003) used similar specimens and, furthermore, extended the testing program with notched specimens with notch radii r n of 0.4, 0.8 and.0 mm whereas the inclination angle for the specimens with smaller notch radii was chosen to be a =15, see Fig. 8. The corresponding numerical simulations were presented by Børvik et al. (003) who used an axisymmetric finite element model. Again to avoid numerical costs in the present study the reduced numerical model only considering a part of the specimen as shown in Fig. 8 is used. It is assumed that this approach does not remarkably affect neither the inelastic

13 1610 M. Brünig, S. Gerke / International Journal of Plasticity 7 (011) Fig. 7. Temperature distribution. Fig. 8. Axisymmetric specimens taken from Hopperstad et al. (003), all dimensions in mm. Fig. 9. Mesh of 1/8th of notched axisymmetric specimens. deformation behavior nor the stress state of the specimen since the material remains elastic in these thicker parts. The numerical calculations take into account clamped ends on the left boundary and dynamic loading is given by a pulse acting on the right side. The simulations are carried without applying symmetry conditions but Fig. 9 shows for demonstration purposes the mesh of 1/8th of the specimen. For the unnotched and the notched specimen with r n =.0 mm a total of 17 elements have been used over the radius leading to a total of 54,079 (unnotched) and accordingly 8,354 (r n =.0 mm) elements. For the smaller notch radii of r n = 0.8 and 0.4 mm 0 elements have been taken into account over the radius which leads to a total of 37,789 (r n = 0.8 mm) and 50,44 (r n = 0.4 mm) elements, respectively. Herewith the mesh densities are comparable to those used by Børvik et al. (003). Simulations with the unnotched specimen considering the rate and temperature dependence of the material, show an almost homogeneous evolution of damage over the thinner part, see Fig. 10. This behavior is comparable to the evolution of damage shown in Fig. 6 of the flat specimen although the last image of Fig. 10 shows a very slight necking. Comparing this necking phenomenon with the one described by Børvik et al. (003) it is remarkably less due to the fact that the considered aluminum alloy is less ductile than the Weldox steel tested by Hopperstad et al. (003). As described in Section 4.1 a notable necking is numerically predicted if the rate and temperature dependence of the material is neglected in the calculations. Fig. 11 displays the triaxiality g, the plastified region at the indicated damage state for the different notch radii 0.4, 0.8 and.0 mm. As expected the triaxiality g decreases with increasing notch radius whereas the damaged zone has lower values of triaxiality. In other words, damage causes the material to respond softer. For the biggest notch radius of r n =.0 mm the complete notched zone is plastified, which is not the case for the smaller notch radii where the center is not fully plastified at the onset of damage, but with increasing damage zone the center fully plastifies. The onset of damage is for the smallest notch radius of r n = 0.4 mm located at the notch and moves with increasing notch radius towards the center.

14 M. Brünig, S. Gerke / International Journal of Plasticity 7 (011) Fig. 10. Evolution of damage variable l for unnotched specimen, first image at t =90 ls, time in-between images.0 ls. Fig. 11. Column: triaxiality g with corresponding legend, plastic zone at the shown damage state in yellow and damage state in red; row: notch radii 0.4, 0.8, and.0 mm. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Comparing the locus of the onset of damage predicted here with the locus of the onset of fracture described by Børvik et al. (003) one can notice that they coincide for 0.4 as well as for.0 mm but that Børvik et al. (003) indicates the onset of fracture for r n = 0.8 mm also at the center, i.e. here a fully plastified center was assumed. Corresponding simulations with rate- and temperature-independent material behavior have been carried out. The onset of damage is shown in Fig. 1 for the different notch radii. Comparing the locus of the onset of damage with that one obtained for rate- and temperature-dependent material behavior shown in Fig. 11 it is located more to the center of the specimen. In a series of experiments generally the radius at minimum cross section r i is kept constant while the notch radius r n is varied to achieve different ratios r i /r n, see Mackenzie et al. (1977) and Hopperstad et al. (003). But motivated by the insights found in the numerical simulations it also seems to be reasonable to vary the radius at minimum cross section r i and keep the notch radius r n constant. Due to the applied boundary conditions in form of surface tension, the ratio in-between the maximum and minimum cross-section, i.e. r i =r a has to be kept constant thus comparable results can be obtained, but in experimental realization a different solution can be found. In Fig. 13 results of numerical simulations with a constant notch radius r n = 0.4 mm and different radii at minimum crosssection r i varying from 0.5 to 1.0 mm are shown. Here it can be clearly seen, that with decreasing radius r i the specimen

15 161 M. Brünig, S. Gerke / International Journal of Plasticity 7 (011) Fig. 1. Location of damage occurrence for rate and temperature independent behavior. Fig. 13. Plastified zone (top row in yellow) and location of onset of damage (bottom row in red) for r n = 0.4 mm and different r i (from left to right) 0.5 mm (t = 64ls), 0.5 mm (t = 58ls) and 1.0 mm (t = 50ls). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Fig. 14. Fracture surface of axisymmetric specimen with r n = 0.4 mm. fully plastifies before first damage occurs whereas with larger radius r i, at onset of damage the plastic zones are concentrated in the region of the notch tips. Furthermore, first damage is numerically predicted to occur later with decreasing radius r i, i.e. at higher load. With these results in mind a new series of numerical simulations where the ratios r i /r n and r i =r n, respectively, are kept constant seems to be an interesting task for future studies but would exceed the scope of this paper. Applying the fracture criteria described in Eq. (30) to the specimen with r n = 0.4 mm the final fracture surface is cup-coneshaped, similar to the one experimentally obtained shown in Hopperstad et al. (003) (see Fig. 14) M-shape specimen Mohr and Gary (007) proposed a new M-shaped specimen which can be used in split Hopkinson-Bar apparatus to perform tensile experiments. This specimen is specially designed to transform compressive loading conditions into tensile loading in the critical part. The complete specimen is shown in Fig. 15a. Again to avoid extensive numerical costs, the specimen is

16 M. Brünig, S. Gerke / International Journal of Plasticity 7 (011) Fig. 15. M-shape specimen taken from Mohr and Gary (007). Fig. 16. Mesh of M-shape specimen taken from Mohr and Gary (007). Fig. 17. Evolution of internal damage variable l and fracture for rate- and temperature-dependent material behavior, thickness 7.0 mm, first image at t = 58 ls, time in-between images 6.5 ls. simplified to only the tension part shown in red. The exact geometry can be seen in Fig. 15b d. Mohr and Gary (007) determined their static material parameters using a 3.5 mm thick specimen and dynamic experiments were performed with thicknesses being a multiple of 3.5 mm. This systematic is maintained in the present calculations and first studies with unnotched specimens have been performed with 7.0 mm thickness. For better legibility Fig. 16 shows a coarser mesh than the one used for the simulations with a total of 46,58 elements; the thin part of the specimen has 1 elements over the width and 7 elements in thickness direction. The boundary conditions are clamped ends on the lower side and dynamic loading is given by a pulse acting on the upper boundary. With the fracture criterion defined in Eq. (30) the complete deformation process, i.e. elastic, elastic plastic, elastic plastic-damaged and finally fracture behavior, can be described. Fig. 17 shows the evolution of the equivalent damage strain l for rate- and temperature-dependent material behavior. As illustrated in the previous examples (Figs. 6 and 10) for as well unnotched specimens damage evolves over the complete thinner part of the specimen over a quite long period of time forming maxima at the center as well as on the edge close to the thicker part of the specimen, see Fig. 17b. Here first fracture occurs almost simultaneously and evolves from the center horizontally towards the outside and from the edge horizontally towards the center. This causes major shear behavior with low positive triaxialities in-between the advancing cracks, see Fig. 18 where the internal damage variable already reached a remarkable value of approximately 4%. In combination with the triaxiality dependence of the fracture criterion this leads to the shear fracture shown in Fig. 19 giving detailed impression of the fracture surface.

17 1614 M. Brünig, S. Gerke / International Journal of Plasticity 7 (011) Fig. 18. Damage evolution with propagating crack and corresponding triaxiality g distribution. Fig. 19. Fracture surface of unnotched specimen with thickness 7.0 mm, rate- and temperature-dependent material behavior. Fig. 0. Evolution of internal damage variable l and fracture for rate and temperature independent material behavior, thickness 7.0 mm, first image at t = 59ls, time in-between first and second image 1.0 ls and second and third 3.0 ls. Corresponding simulations with rate- and temperature-independent material behavior have been performed and numerical results are shown in Fig. 0. It can be clearly seen that completely different damage and fracture behavior is predicted. Thus, since rate and temperature dependence remarkably affect the damage and fracture processes (Figs. 17 and 0) these models have to be taken into account performing numerical simulations of Hopkinson-Bar experiments. Motivated by the insights found through the tension experiments with notched specimens performed by Hopperstad et al. (003) and the corresponding numerical simulations published by Børvik et al. (003) as well as the advantages of the new dynamic test with the M-shaped specimens, see (Mohr and Gary, 007), the authors propose a new series of experiments with notched M-shaped specimens (see Fig. 15e). Therefore in the present paper, first results of numerical studies of a series of specimens with notch radii r n (0.5, 0.5, 0.75 and 1.0 mm) are discussed. Fig. 1 shows for demonstration purposes the mesh of 1/8th of the specimen whereas all simulations were performed without applying symmetry conditions. In Fig., it is clearly seen that remarkable concentration of damage is observed near the notch tips of small radius whereas large damage zones are characteristic for larger notch radii. The final fracture line is straight and, compared with that one of the unnotched specimen (Fig. 19), completely different. Thus different failure modes are numerically predicted to occur allowing analysis for stress triaxiality dependence. Hence, a series of experiments with variously notched M-shape specimens compared with corresponding numerical simulations will give new insights in damage and failure mechanisms under dynamic loading conditions.

18 M. Brünig, S. Gerke / International Journal of Plasticity 7 (011) Fig. 1. Mesh of 1/8th of proposed new notched specimen. Fig.. Evolution of internal damage variable l and fracture, time in-between images 3.5 ls. 5. Conclusions A rate- and temperature-dependent continuum damage model has been discussed. The multi-dissipative approach takes into account stress triaxiality dependent plastic flow, damage and failure. The corresponding evolution equations for respective equivalent stress measures are based on multiplicative decomposition of functions simulating equivalent strain, strain rate and temperature effects. The algorithm for numerical integration of coupled constitutive rate equations relies on operator split technology leading to inelastic predictor elastic corrector technique. It has been implemented into the commercial finite element program LS-DYNA as an user-defined material subroutine. Numerical simulation of Split-Hopkinson-Bar experiments with different specimens have been performed. Evolution of plastic flow, damage, failure and temperature has been discussed in detail for these adiabatic deformation processes. Numerical results have shown good agreement with available experimental data published in the open literature. To be able to study the effect of stress triaxiality on rate- and temperature-dependent material behavior in more detail, modified specimen geometries have been proposed. This may lead to new series of standard experiments for identification of material parameters in dynamic models for a wide range of stress triaxialities.