International Journal of Impact Engineering

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1 International Journal of Impact Engineering 5 (21) 29e4 Contents lists available at SciVerse ScienceDirect International Journal of Impact Engineering journal homepage: Dynamic crushing of cellular materials: A unified framework of plastic shock wave models Zhijun Zheng *, Jilin Yu, Changfeng Wang, Shenfei Liao, Yaodong Liu CAS Key Laboratory of Mechanical Behavior Design of Materials, University of Science Technology of China, Hefei, Anhui 226, PR China article info abstract Article history: Received 14 November 211 Received in revised form 18 March 212 Accepted 1 June 212 Available online 26 September 212 Keywords: Cellular material Deformation localisation Shock front Strength enhancement Voronoi structure Strength enhancement deformation localisation are typical features of the dynamic response of cellular materials. Several one-dimensional shock models have been developed to explain these features. A unified framework of one-dimensional plastic shock wave models was established in this paper. Based on an arbitrary plastic hardening constitutive model for cellular materials, general solutions, although implicit, have been derived for two impact scenarios. For a rigidepower-law hardening () idealisation involved in three material parameters, namely the yield stress, the strength index the strainhardening index, closed-form/semi-closed-form solutions of the physical quantities across the shock front have been derived. The linearly hardening locking idealisations are found to correspond to the two opposite limit cases with the strain-hardening index of one infinity, respectively. The shock models based on three different idealisations are verified with cell-based finite element models including an irregular honeycomb a closed-cell foam. It is found that the force responses predicted by the shock models are not very sensitive to the choice of the idealisations they are in good agreement with the cell-based finite element results. Deformation features predicted by the shock models are compared well with the cell-based results when the impact velocity is not very low. The comparisons show that using more realistic constitutive models such as the idealisation may present more accurate predictions. Ó 212 Elsevier Ltd. All rights reserved. 1. Introduction Cellular materials such as metallic foams honeycombs have been extensively used as impact energy absorbers blast protectors. Strength enhancement deformation localisation are typical features of their dynamic responses. These features are mainly caused by inertia well explained by several crushing models, especially the continuum-based shock models as reviewed below. A simple one-dimensional shock model was first proposed by Reid Peng [1] to explain the strength enhancement of wood specimens subjected to impact against a rigid target. For simplicity, they modelled the wood using a rate-independent, rigideperfectly plasticelocking (R-P-P-L) idealisation including two key parameters, i.e. the plateau stress s pl the locking strain L. The later parameter is usually assumed to equal the densification strain D. By analysing the stress discontinuity across the shock front, an * Corresponding author. Tel.: þ ; fax: þ address: zjzheng@ustc.edu.cn (Z. Zheng). elegant solution for the stress behind the shock front has been presented as [1e4] s B ¼ s pl þ r v 2 = L; (1) where r is the initial density of the cellular material v the particle velocity behind the shock front. A numerical scheme, e.g. a fourth-order Runge-Kutta numerical scheme [4], should be used to determine the velocity v when considering its temporal change. Based on this simple model, a first order understing has been produced for the dynamic crushing strength enhancement of cellular materials, such as wood [1], aluminium honeycombs [2] aluminium foams [,4]. The R-P-P-L idealisation was also used to model the crushing of a foam with blast loading [5,6], with mass loading [4] or impacting a free mass [7]. The deformation patterns of cellular materials under compression are very complex, but they can be catalogued into three deformation modes according to the deformation mechanisms under different impact velocities, namely the Homogeneous Mode, the Transitional Mode the Shock Mode [8,9]. A rigidelinear hardening plasticelocking (R-LHP-L) idealisation was employed by 74-74X/$ e see front matter Ó 212 Elsevier Ltd. All rights reserved.

2 Z. Zheng et al. / International Journal of Impact Engineering 5 (21) 29e4 Zheng et al. [1] to develop a Shock-Mode model for high-velocity impact a Transitional-Mode model for moderate-velocity impact. This idealisation has three parameters including the yield stress s, the plastic hardening modulus E 1 the locking strain L. Two impact scenarios considered in the literature [1,4] have been revisited: the rod-target impact (a cellular material rod with a backing mass striking axially against a stationary target) the striker-rod impact (a rigid striker striking axially against a stationary cellular material rod with a fixed wall at the distal end). Some closedform solutions such as those for the striker-rod impact scenario (see Eqs. (5) (54) in the present paper) have been derived. The elastic property may strongly affect the response of cellular materials under low-velocity impact. An elastic-perfectly plastice rigid (E-P-P-R) idealisation an elasticeplasticerigid (E-P-R) idealisation have been employed by Lopatnikov et al. [11e1] to consider the elastic effect. They used the E-P-P-R idealisation, which has three key parameters (the elastic modulus E, the plateau stress s pl the maximum possible strain max), to investigate both Taylor impact plate impact of aluminium foams [11,12]. Four significantly different possibilities of impact velocity were suggested by Lopatnikov et al. [12] improved by Li Reid [14]. The E-P-R idealisation was used to derive the energy absorbed under quasi-static compression of cellular materials when the wave propagation effects were neglected [1]. All the above idealisations mentioned have a locking stage, but different locking strains have been suggested. In fact, predictions are very sensitive to the locking strain [15,16]. The dynamic densification strain increases with the increase of impact velocity as found in regular honeycombs [17] its limit is much higher than the quasi-static one, which was used as the locking strain in most papers mentioned above. Several idealisations with a nonlinear plastic hardening stage have been proposed to improve the prediction precision of shock models [16,18,19]. A rigide softeningehardening (R-S-H) idealisation an elastice softeningehardening (E-S-H) idealisation have been used by Harrigan et al. [18] to model the rod-target impact the striker-rod impact of wood along the grain. Harrigan et al. [19] also used an elastic-perfectly plastic hardening (E-P-P-H) idealisation to model the striker-rod impact of cellular materials, but unfortunately, it is not possible to obtain closed-form solutions. A power-law hardening stage has been suggested by Pattofatto et al. [16] to replace the plastic stages of the R-P-P-L idealisation. They derived solutions for the shock strain the shock speed of a constant-velocity impact scenario. This paper aims to present general solutions for a unified framework of plastic shock wave models, which may help to obtain high-precision predictions of shock-induced stress enhancement. Shock models are developed by using a constitutive relation with arbitrary plastic hardening for cellular materials general solutions for two impact scenarios are presented in Section 2. As desired for applications, closed-form/semi-closedform solutions for a specified idealisation are derived in Section. Comparison between the shock models cell-based finite element models is presented in Section 4, followed by conclusions in Section Continuum-based plastic shock wave models 2.1. Basic equations based on one-dimensional stress wave theory The constitutive behaviour of cellular materials is considered to be rate-independent, rigideplastic hardening (RePH). The stresse strain relation is given by s ¼ sð Þ (2) with its first derivation s ( ), which is required as a monotonically increasing function. In this paper, we consider only a plastic shock wave front propagating through a cellular material rod aim to obtain analytical solutions, especially in closed form, for the plastic shock wave models. Involving the elastic property makes closed-form solutions impossible does not give much impressive understing when the impact velocity is not very low, thereby we do not take the elastic property into consideration. One can refer to the work of Lopatnikov et al. [11e1] Harrigan et al. [18,19] for the influence of the elastic property. Using continuumbased finite element models to consider the influence of the elastic property is also recommended. The striker-rod impact scenario the rod-target impact scenario are revisited. The striker mass is M b the initial velocity is V. The rod with length l cross-sectional area A is made of a cellular material with density r. A plastic shock front propagates along the rod, as represented schematically in Fig. 1. At time t, the shock front locates at Lagrangian position F(t) travels with Lagrangian speed FðtÞ, _ where the dot symbol (.) denotes the derivative with respect to the time. One-dimensional loading rigid unloading assumptions are adopted here with the continuumbased stress wave theory. The cross-sectional area of the rod is a b Fig. 1. Schematic diagrams of shock front for (a) the striker-rod impact scenario (b) the rod-target impact scenario. The location F(t) of the shock front its speed _ FðtÞ are measured in the Lagrangian coordinate system.

3 Z. Zheng et al. / International Journal of Impact Engineering 5 (21) 29e4 1 considered as a constant during the crushing process, so the density r at Lagrangian location X after deformation is related to its local strain by r ¼ r =ð1 Þ: () We use {v,, s} to list the physical quantities including the particle velocity, strain stress [1]. Stress strain are referred to engineering ones taken as positive in compression. Across the shock front, the physical quantities such as the particle velocity, strain stress are essentially discontinuous. The jump the mean of physical quantity Q across the shock front are defined as ½QŠ ¼Q B Q A (4) hqi ¼ðQ B þ Q A Þ=2 (5) respectively, where subscripts A B refer to values closely ahead of behind the shock front. Conservation conditions of mass, momentum energy across the shock front lead to the RankineeHugoniot relations [18,2] 8 >< ½vŠ ¼ r F _ r 1 ¼ F½ _ Š; ½sŠ ¼r >: _ F½vŠ; ½UŠ ¼ hsi (6) r 1 ¼ r 1 hsi½ Š; where U is the internal energy per unit mass. It is no need to use the third relation in Eq. (6) as a basic equation in constructing the shock models. A discussion on the energy conservation has been carried out by Harrigan et al. [19]. The first second relations in Eq. (6) are known as the kinematic kinetic compatibility conditions across the shock front. Associating these two relations leads to the velocity jump sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ½vŠ ¼ ½sŠ½ Š r the shock speed sffiffiffiffiffiffiffiffiffiffiffiffi 1 _F ¼ $ ½sŠ ; (8) r ½ Š where the plus sign corresponds to a rightward travelling front the minus sign to a leftward travelling front. A discussion on Eqs. (7) (8) has been presented by Li Reid [14]. We use Eqs. (7) (8) as the basic equations to carry out the present work Solution for the striker-rod impact scenario For the striker-rod impact scenario, the part of the cellular material behind the shock front travels together with the mass according to the rigid unloading assumption. Their common velocity is denoted as v b (t). The physical quantities closely ahead of the front (i.e., at Lagrangian location X A ¼ F(t)þ) are {v,, s} A ¼ {,, s }, while those closely behind the front (i.e., at Lagrangian location X B ¼ F(t) ) are {v,, s} B ¼ {v b (t), B(t), s B (t)}, in which s B is related to B by the stressestrain relation of Eq. (2). Hence, the jumps of physical quantities {v,, s} across the shock front are (7) easily have the expressions for the velocity of the mass the shock speed related to the unknown strain B(t), which is determined as follows. Substituting the initial velocity of the mass v b () ¼ V into Eq. (7) with terms in Eq. (9) leads to the following relation ðsð Þ s Þ ¼ r V 2 ; (1) where h B() is the initial shock strain. This relation has been obtained by Pattofatto et al. [16] for the case of constant-velocity impact, see Eq. (5a) in Ref. [16]. Here, Eq. (1) is a condition to determine the initial shock strain the corresponding initial shock enhancement is s( ) s. As in the previous work [1], we introduce a dimensionless parameter to characterize the initial stress enhancement, namely the stress enhancement parameter, ah sð Þ s ¼ r V 2 : (11) s s On the one h, the acceleration of the mass can be determined by carrying out the differentiation of its velocity with respect to time t, i.e. a b ðtþ ¼ dv bðtþ dt ¼ kð BðtÞÞ BðtÞ r _ ; (12) FðtÞ where k( ) is a combined slope by measuring the average of the slopes of the tangent line (corresponding to the first derivative, s ( )) the Rayleigh line (i.e. the chord line that joins the two stress states (, s ) (, s)) at the point on the curve s( ), kð Þh 1 s ð Þþ sð Þ s : (1) 2 On the other h, the part of the rod behind the front travels together with the mass, so according to the inertia law the mass has an acceleration of s a b ðtþ ¼ B ðtþa M b þ r A FðtÞ ¼ sð BðtÞÞ m þ r FðtÞ ; (14) where m is the specific mass mhm b =A : (15) Eq. (14) also can be derived by applying the momentum conservation, as referred to Refs. [18] [1]. Combining Eqs. (12) (14) will lead to a differential equation, see Eq. (B.1). Solving this differential equation with the initial conditions of F() ¼ B() ¼, we obtain the Lagrangian location of the shock front 2 1 FðtÞ ¼ m Z BðtÞ 6 B kð Þ 4exp@ r sð Þ d C 7 A 15: (16) From Eq. (16) Eq. (8), wefinally have an implicit solution for B(t) t ¼ T 1 Z BðtÞ pffiffiffiffiffiffiffi f 1 ð Þ s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ; (17) a ðsð Þ s Þ f½vš; ½ Š; ½sŠg¼fv b ðtþ; BðtÞ; sð BðtÞÞ s g: (9) where T 1 is a characteristic time parameter, defined as Substituting Eq. (9) into Eqs. (7) (8) with remaining the plus sign due to the convention of the rightward travelling front, one can T 1 hmv =s ; (18)

4 2 Z. Zheng et al. / International Journal of Impact Engineering 5 (21) 29e4 f 1 is a dimensionless function, defined as 2 f 1 ð Þ¼ kð Z Þ sð Þ exp 4 kðxþ sðxþ dx 5: (19) For more details on the derivations, one is referred to Appendix B. The solution in Eq. (17) can be converted to its explicit form of B(t) for some special constitutive relations, see Section Solution for the rod-target impact scenario For the rod-target impact scenario, the part of the cellular material ahead of the shock front travels together with the mass according to the rigid assumption of the elastic behaviour. The physical quantities closely ahead of the front (i.e., at X A ¼ F(t) ) are {v,, s} A ¼ {v b (t),, s } those closely behind the front (i.e., at X B ¼ F(t)þ) are {v,, s} B ¼ {, B(t), s( B(t))}, thereby the jumps of physical quantities {v,, s} across the shock front are f½vš; ½ Š; ½sŠg ¼ f v b ðtþ; BðtÞ; sð BðtÞÞ s g: (2) From Eqs. (7) (8) with remaining the negative sign, one can obtain the expressions for the velocity of the mass the shock speed related to the unknown strain B(t). The initial shock strain B() ¼ is identical with that from Eq. (1). Similarly, we can determine the Lagrangian location of the shock front as (see Appendix B) 2 FðtÞ ¼ m Z 6 B 4ð1 þ mþexp@ r BðtÞ kð Þ C 7 d A 15; (21) s where m is the ratio of the rod s mass to the striker mass mhr l A =M b ¼ r l =m: (22) From Eq. (21) Eq. (8), as derived in Appendix B, wefinally have an implicit solution for B(t) t ¼ T 2 Z BðtÞ 1 pffiffiffiffiffiffiffi f 2 ð Þ s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ; (2) a ðsð Þ s Þ where T 2 f 2 are defined by the definitions of f i ( ) T i, i ¼ 1, 2. The functions f 1 ( ) f 2 ( )are defined in Eqs. (19) (25), respectively. They are dependent on the constitutive relation of cellular materials. The time parameters T 1 T 2 are defined in Eqs. (18) (24), respectively. The time parameters characterize the ratios of the initial momentum of the system per unit area to the yield stress of the rod. General solutions for the plate impact the Taylor impact can then be determined, since they are equivalent to the striker-rod impact with l / N the rod-target impact with M b ¼, respectively. The initial solution of the shock strain is identical for all of the impact scenarios mentioned above, which depends on the initial impact velocity but not the striker mass. At the initial stage of impact, we present the first-order asymptotic solutions as 8 >< BðtÞ ¼ sð Þ at 2 kð Þ $at þ O T 1 T 1 >: s B ðtþ ¼sð Þ s ð Þsð Þ $ at at 2 (26) þ O kð Þ T 1 T 1 for the striker-rod impact scenario 8 >< BðtÞ ¼ s at 2 kð Þ $at þ O T 2 T 2 >: s B ðtþ ¼sð Þ s ð Þs at 2 (27) kð Þ $at þ O T 2 T 2 for the rod-target impact scenario.. Application to a special constitutive relation The shock models may have simple solutions for some special constitutive relations, such as the R-P-P-L idealisation the R- LHP-L idealisation [1]. Here, a particular type of constitutive relation was employed to obtain some closed-form/semi-closedform solutions. The shock models for the two impact scenarios are related to the combined slope defined in Eq. (1). For convenience, we consider the slope of the tangent line at the point on the curve s( ) proportional to the slope of the Rayleigh line, denoted by s ð Þ¼nðsðÞ s Þ= ; (28) where n is the proportionality coefficient (n 1). Thus, the combined slope defined in Eq. (1) can be simplified as T 2 hðm þ r l ÞV =s ¼ T 1 ð1 þ mþ (24) kð Þ¼qs ð Þ; (29) 1 f 2 ð Þ¼ kð Z Þ exp@ kðxþ dxa; (25) s s respectively. Few constitutive relations can make Eq. (2) convert to its explicit form of B(t), so we will focus only on its semi-closedform solutions, see Section A discussion on the general solutions General solutions of B(t), although implicit, have been derived for the two impact scenarios with an arbitrary plastic hardening constitutive relation of cellular material. For the striker-rod impact scenario, the strain behind the shock front B(t) is related to t by Eq. (17), while for the rod-target impact scenario, it is related to t by Eq. (2). These two relations are identical in the form but different in where q is related to n by qhðn þ 1Þ=2n: () As n 1, we have 1/2 q 1. From Eqs. (16) (21) with Eq. (29), the locations of the front in the two impact scenarios can be integrated as FðtÞ ¼ m sð q BðtÞÞ 1 (1) r sð Þ FðtÞ ¼ m q ð1 þ mþexp ðsð r BðtÞÞ sð ÞÞ 1 ; (2) s respectively. Solving the simple differential equation Eq. (28) with s() ¼ s, we are led to a constitutive relation, namely the rigide power-law hardening () idealisation

5 Z. Zheng et al. / International Journal of Impact Engineering 5 (21) 29e4 sð Þ¼s þ K n ; () where s is the yield stress, K the strength index n the strainhardening index. Pattofatto et al. [16] have used this constitutive model instead of the R-P-P-L idealisation to improve the predictions of the shock strain the shock speed for the case of constant-velocity impact. In this paper, we focus on the case of initial-velocity impact. The shock strain predicted by Pattofatto et al. [16] is equivalent to the initial shock strain of the present work, which is written as BðtÞ ¼ h ð1 þ aþð1 t=t 1 Þ 1=q a i 1=n : (41) h 1; s B ðtþ ¼s ð1 þ aþ 1 þ a að1 t=t 1 Þqi 1 (42) respectively. From Eqs. (7), (9) (12), the velocity acceleration of the mass can be determined as ¼ r V 2 =K 1=ðnþ1Þ: (4) v b ðtþ ¼V hð1 þ aþð1 t=t 1 Þ 1=q a i q : (4) This strain can be easily determined by substituting Eq. () into Eq. (1). The initial shock stress is determined as n=ðnþ1þ: sð Þ ¼s þ K r V 2 =K (5) Thus, the stresseenhancement parameter can be given by n a ¼ K =s ¼ K r V 2 n 1=ðnþ1Þ. s : (6) Solutions for the two impact scenarios are derived as follows..1. The striker-rod impact scenario.1.1. Closed-form solutions Substituting Eqs. (29) () into Eq. (19), we obtain f 1 ðxþ ¼ nq $ að1 þ aþq ðx= x n qþ1 ; (7) 1 þ a Þ n 1 where q is defined in Eq. (), which is an equivalent strainhardening index. Hence, Eq. (17) can be given in the form t ¼ T 1 nq ð1 þ aþ q Z B 1 þ a ðn 1Þ=2 n qþ1 d : (8) By taking h ¼ ( / ) n, Eq. (8) can be simplified as t ¼ T 1 qð1 þ aþ q ð B= Z 1 Þ n ðh þ aþ q 1 dh: (9) Carrying out the definite integral in Eq. (9), we finally have t ¼ T 1 "1! q # 1 þ a ð B= Þ n : (4) þa This equation is an implicit form of B with respect to time t, but it can be easily converted to the explicit form. Thus, we have the strain stress closely behind the front a b ðtþ ¼ s m ð1 þ aþ h 1 þ a að1 t=t 1 Þ 1=qi q 1 ; (44) respectively. From Eqs. (B.) (1), the speed location of the front can be given by _FðtÞ ¼ V h ð1 þ aþð1 t=t 1 Þ 1=q a i q 1 : (45) FðtÞ ¼ m r h 1 þ a að1 t=t 1 Þ 1=q q 1 i; (46) respectively Some transition parameters The deformation patterns of a cellular material under different impact velocities are very different, which has been catalogued into three deformation modes: a Homogeneous Mode, a Transitional Mode a Shock Mode [8,9,21]. There exist two critical impact velocities: the first critical velocity, V c1, which corresponds to the mode transition between the Homogeneous the Transitional modes, the second critical velocity, V c2, which corresponds to the mode transition between the Transitional the Shock modes. A stress uniformity index (SUI), defined as the ratio of the plateau stresses at the support the impact ends, has been introduced to determine the first critical velocity when SUI ¼ 9% for constantvelocity impact [21,22]. Here, at the initial stage of crushing, the stress at the support end is s the stress at the impact end is s( ) as given in Eq. (5). Hence, from s /s( ) ¼ 9% with V ¼ V c1, we obtain the first critical velocity s V c1 ¼ ðn þ 1Þ=2npffiffiffiffiffiffiffiffiffiffiffi K=r 9K : (47) The second critical velocity can be determined by the fact when the shock strain reaches the densification strain [1]. From Eq. (4) with ¼ D V ¼ V c2,wehave V c2 ¼ ðnþ1þ=2 D pffiffiffiffiffiffiffiffiffiffiffi K=r : (48) When the initial impact velocity is very high the rod is long enough, both mode transitions occur during the crushing process. The deformation mode changes from the Shock Mode to the Transitional Mode at time t S T, which is determined from Eq. (4) when v b (t S T ) ¼ V c2, i.e.

6 4 Z. Zheng et al. / International Journal of Impact Engineering 5 (21) 29e4 1 þ 1=a q t S T ¼ T 1 1 ; (49) 1 þ 1=b where b ¼ K D n /s. The deformation mode changes from the Transitional Mode to the Homogeneous Mode at time t T-H, which is determined from Eq. (4) when v b (t T H ) ¼ V c1, i.e. t T H ¼ T 1 1 ð:1ð1 þ 1=aÞÞ q : (5) According to this shock model, when the rod is long enough, the striker will stop at time t end ¼ T 1 since v b (T 1 ) ¼. Therefore, the time parameter T 1 characterises the time when the kinetic energy of the mass converts totally to the internal energy of the rod. It is interesting that t end does not depend on the hardening parameters except of the yield stress. At this time, the location of the front is Fðt end Þ¼ m r ð1 þ aþ q 1 ; (51).2. The rod-target impact scenario.2.1. Semi-closed-form solutions Similarly, substituting Eqs. () (29) into Eq. (25), we have f 2 ðxþ ¼ aqn x n x n 1 exp aq 1 ; (55) where q is defined in Eq. (). Thus, Eq. (2) can be expressed by t ¼ T 2 nq Z BðtÞ exp aq n 1 ðn 1Þ 2 d : (56) by taking u ¼ aq( / ) n, the above equation can be rewritten as T 2 t ¼ aðaqþ q 1 e aq aqð ZB= Þ n aq u q 1 e u du; (57) so l F(t end ) as required. However, if the rod is not long enough, i.e. l < F(t end ), the front will be reflected from the distal end at time t 1 ¼ T 1 h1 1 þ a 1 a 1 ð1 þ mþ 1 q i q ; (52) or in the form t ¼ T 2 aðaqþ q 1 Fðq; aqþ F q; aq e aq where F(q, z) is a function defined by B n ; (58) which is derived from F(t 1 ) ¼ l. The case of wave reflection has not been considered in this paper, but it can be extended, as presented in Ref. [1]..1.. Limit cases of the strain-hardening index For the case of n ¼ 1, we have q ¼ 1 K ¼ E 1 with E 1 being the plastic hardening modulus. Eqs. (41)e(4) lead to 8 >< BðtÞ ¼ ð1 t=t 1 Þð1 þ at=t 1 Þ 1 s B ðtþ ¼s ð1 þ aþð1 þ at=t 1 Þ >: ; (5) v b ðtþ ¼V ð1 t=t 1 Þð1 þ at=t 1 Þ 1 where ¼ V /C 1 a ¼ r C 1 V /s with C 1 ¼ (E 1 /r ) 1/2 being the plastic wave speed. These results have been predicted in Ref. [1] for the R-LHP-L idealisation subjected to the mass impact with a moderate initial velocity. For the case of n / N, we obtain q ¼ 1/2. The strain closely behind the front reaches the densification strain, i.e. B(t) ¼ ¼ D. Eqs. (42) (4) lead to ( sb ðtþ ¼s ð1 þ aþh1 þ a að1 t=t 1 Þ 2i 1 v b ðtþ ¼V ð1 t=t 1 Þ h1 þ a að1 t=t 1 Þ 2i 1=2 ; (54) where a ¼ r V 2 /(s D). These predictions are valid for both the R-P- P-L the R-LHP-L idealisations when the impact velocity is very high, as discussed in Ref. [1]. It should be noticed that the corresponding predictions in the original R-P-P-L shock model [1,4] are expressed in implicit form. Therefore, the predictions involved in the R-P-P-L the R- LHP-L idealisations can be easily deduced from the present limit cases of the strain-hardening index. The related critical velocities of modes transition have been discussed in Ref. [1]. Z z Fðq; zþh u q 1 e u du (59) with its first derivative F ðq; zþ ¼ d dz Fðq; zþ ¼zq 1 e z : (6) the function F(q, z) may have some closed-form solutions, for example F(1, z) ¼ e z 1. Let shfðq; aqþ aðaqþ q 1 e aq t=t 2 ; (61) Eq. (58) can be rewritten as F q; aq B n ¼ s; (62) so we have the strain closely behind the front BðtÞ ¼ F 1!1 n ðq; sþ ; (6) aq where F 1 (q, s) is the inverse of the function F(q, z), for example F 1 (1, s) ¼ ln(1 þ s). The function F 1 (q, s) can not be expressed in a closed form for almost all values of q, but it can be easily determined by solving F(q, z) ¼ s for z through Newton s method with the iterative formula z k ¼ z k 1 þ s Fðq; z k 1Þ F ; k ¼ 1; 2;.; (64) ðq; z k 1 Þ where z is a first guess value. By substituting Eq. (6) into Eq. (), the stress closely behind the front is given by

7 s B ðtþ ¼s 1 þ F 1 ðq; sþ=q : (65) Z. Zheng et al. / International Journal of Impact Engineering 5 (21) 29e4 5 which is derived from F(t 1 ) ¼. Thus, the shock speed the velocity of the mass can be expressed by _FðtÞ ¼ V v b ðtþ ¼V! 1 q F 1 ðq; sþ (66) aq! q F 1 ðq; sþ ; (67) aq respectively. The location of the shock front Eq. (2) is determined as FðtÞ ¼ m i hð1 þ mþexp F r 1 ðq; sþ aq 1 (68) then the acceleration of the mass Eq. (B.7) is given by a b ðtþ ¼ s mð1 þ mþ exp F 1 ðq; sþþaq : (69).2.2. Some transition parameters The two critical impact velocities presented in Eqs. (47) (48) are also adequate for the rod-target impact scenario. Mode transitions may occur during the crushing process when the initial impact velocity is very high the rod is long enough. The time of mode transition from the Shock Mode to the Transitional Mode is determined from Eq. (67) when v b (t S T ) ¼ V c2, i.e. t S T ¼ T 2 Fðq; aqþ Fðq; bqþ aðaqþ q 1 e aq ; (7) where b ¼ K D n /s. The time of mode transition from the Transitional Mode to the Homogeneous Mode is determined from Eq. (67) when v b (t T H ) ¼ V c1, i.e. t T H ¼ T 2 Fðq; aqþ Fðq; q=9þ aðaqþ q 1 e aq : (71) If the rod is long enough, from Eq. (67) with v b (t end ) ¼, we can determine the time t end when the crushing stops, i.e. Fðq; aqþ t end ¼ T 2 aðaqþ q 1 : (72) eaq Given n or q, the dimensionless time t end /T 2 reduces with increasing the stresseenhancement parameter a. Different from that of the striker-rod impact scenario, the time t end in the present impact scenario is related to all the hardening parameters including the yield stress, the strength index the strain-hardening index. At time t end, the location of the front is Fðt end Þ¼ m i hð1 þ mþe r aq 1 : (7) when F(t end ), i.e. l (e aq 1)m/r, the rod is long enough. However, if the rod is not long enough, the front will be reflected from the distal end at time t 1, t 1 ¼ Fðq; aqþ Fðq; aq lnð1 þ mþþ aðaqþ q 1 e aq T 2 ; (74).2.. Limit cases of the strain-hardening index For the case of n ¼ 1, we have q ¼ 1 K ¼ E 1. Since F(1, z) ¼ e z 1, Eq. (61) gives s ¼ e a 1 ae a t=t 2 ; (75) so we obtain F 1 ð1; sþ ¼lnð1 þ sþ ¼a þ lnð1 at=t 2 Þ; (76) where a ¼ r C 1 V /s. Eqs. (6), (65) (67) lead to closed-form expressions 8 < BðtÞ ¼ 1 þ a 1 lnð1 at=t 2 Þ s : B ðtþ ¼s ½1 þ a þ lnð1 at=t 2 ÞŠ v b ðtþ ¼V 1 þ a 1 lnð1 at=t 2 Þ ; (77) where ¼ V /C 1. These results correspond to Eqs. (2), (5) (6) in Ref. [1]. For the case of n / N, we obtain q ¼ 1/2. Since 1 F 2 ; z p ¼ p ffiffiffi pffiffi p $i$erf i z ¼ p ffiffiffi pffiffi $erfi z ; (78) where i is the imaginary unit, erf the error function erfi the imaginary error function with erfðxþhp 2 ffiffiffi p Z x e u2 du (79) erfiðhþ ¼ i$erfðihþ; (8) p Eq. (61) gives s ¼ s ffiffiffi p with pffiffiffiffiffiffiffiffi p s ¼ erfi a=2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ae a =p t=t 2 ; (81) where ap ¼ffiffi r V 2 /(s D). Solving F(1/2, z) ¼ s, which can be simplified as erfið z Þ¼s, for z, wehave 1 F 1 2 ; s ¼ z ¼ erfi 1 ðsþ 2: (82) The inverse of the imaginary error function erfi 1 (s) can be determined by solving erfi(x) ¼ s for x through Newton s method with the iterative formula x k ¼ x k 1 þ s erfiðx k 1Þ p 2= p ffiffiffi ; k ¼ 1; 2;.; (8) e x2 k 1 where x is a first guess value. The strain closely behind the front reaches the densification strain, i.e. B(t) ¼ ¼ D. Eqs. (65) (67) lead to 8 h >< s B ðtþ ¼s 1 þ 2 erfi 1 ðsþ 2 i rffiffiffi >: 2 ; (84) v b ðtþ ¼V $erfi 1 ðsþ a which have also been predicted in Ref. [1].

8 6 Z. Zheng et al. / International Journal of Impact Engineering 5 (21) 29e4 4. Comparisons with cell-based finite element simulations Two types of virtual cellular structures are employed to demonstrate the validity of the continuum-based shock models. One type is an irregular honeycomb the other is a closed-cell foam. They are constructed by the 2D D rom Voronoi techniques, respectively. The 2D Voronoi method was introduced in Ref. [8] the D Voronoi method was introduced in Ref. [2]. The methods are briefly introduced here with four steps. Step 1, N nuclei are romly generated in a given region (an area or a box) by constraining the minimum allowable distance between any two nuclei. Step 2, the nuclei are copied to the surrounding neighbouring regions by transitions. Step, the Delaunay triangulation the Voronoi diagram are constructed. Step 4, the part of the Voronoi diagram in the given region is reserved to form the desired specimen. The quasi-static compression the dynamic impact of specimens were implemented by the finite element (FE) simulations using ABAQUS/Explicit code. The specimens are swiched between two rigid platens. One platen is fixed the other moves along the x-direction. Cell walls of specimens are modelled with shell elements as used in Refs. [8] [2]. Contact with a slight friction is specified between any possible surfaces. The irregular honeycomb the closed-cell foam used in this paper have the relative density of.1 the cell irregularity of.5. The specimen of honeycomb for quasi-static compression is constructed in an area of 1 mm 1 mm with 4 nuclei (the length of specimen in the out-of-plane direction is 1 mm), while that of foam is constructed in a box of 25 mm 25 mm 25 mm with 4 nuclei. The geometric dimensions of specimens for dynamic impact are given later, which have the same number densities of nucleation. The average cell size of the honeycomb is about 5.4 mm that of the foam is about 4.1 mm. The cell-wall thickness of the honeycomb is about.254 mm that of the foam is about.1 mm. The average element length of the honeycomb is about.5 mm that of the foam is about.6 mm. The cell-wall material of all specimens is taken to be elasticperfectly plastic with Young s modulus, yield stress, Possion s ratio mass density being 69 GPa, 17 MPa,. 27 kg/m, respectively. The friction coefficient is set to be.2. The quasi-static stressestrain curves for the two types of cellular materials the corresponding idealisations are shown in Fig. 2. InFig. 2a, the material parameters of the irregular honeycomb are presented here: s pl ¼.862 MPa L ¼.579 for the R- P-P-L idealisation; s ¼.668 MPa, E 1 ¼.981 MPa L ¼.579 for the R-LHP-L idealisation; s ¼.668 MPa, K ¼ MPa n ¼ 5.69 for the idealisation. In Fig. 2b, the material parameters for the closed-cell foam are presented as follows: s pl ¼ 7.9 MPa L ¼.62 for the R-P-P-L idealisation; s ¼ 6.19 MPa, E 1 ¼ 5.2 MPa L ¼.62 for the R-LHP-L idealisation; s ¼ 6.19 MPa, K ¼ 5.2 MPa n ¼ 5.68 for the idealisation. The locking strain L is assumed to be equal to the densification strain D, which corresponds to the maximum of the energy absorption efficiency [24,25]. Some parameters here are estimated because of the oscillation of the numerical FE results. The yield strain is taken as.1 the yield stress s is estimated near this strain. The plateau stress s pl is calculated from the average stress over the strain range of.1 to D. The strength index K the strain-hardening index n are determined by the method of least squares. The hardening modulus E 1 is determined as (s D s )/ D, in which s D ¼ s( D) is estimated from the idealisation. These parameters are used in the following comparisons. Two dynamic impact cases are considered in this section. Case 1, an irregular honeycomb is stationary with a fixed rigid wall at its Nominal Stress (MPa) b Nominal Stress (MPa) a 2 Quasi-static compression Honeycomb R-P-P-L R-LHP-L E 1 σ 1 pl σ Quasi-static compression Foam R-P-P-L R-LHP-L 1 E σ 1 pl σ 5 Nominal Strain Nominal Strain Fig. 2. The quasi-static stressestrain relations with three different idealisations for two types of cellular materials: (a) The irregular honeycomb (b) the closed-cell foam. right end, while another rigid wall with mass M b ¼ 5 g initial velocity V ¼ 1 m/s comes from the left end of the honeycomb. The size of the honeycomb is 4 mm 1 mm 1 mm. In this simulation case, the number of nodes is the number of elements is 28,84. Case 2, a closed-cell foam with a rigid wall (M b ¼ 1.25 g) at its left end travels at an initial velocity of V ¼ 1 m/s strikes axially against a fixed rigid wall at its right end. The size of the foam is 1 mm 25 mm 25 mm. In this case, the number of nodes is 18,15 the number of elements is 172,525. The dynamic impact is along the longest dimension of each specimen. Force responses deformation features are presented below Case 1: A striker-rod impact of an irregular honeycomb Force responses The forces detected at the two ends of the honeycomb specimen are shown in Fig.. The numerical FE results show that at the initial stage of crushing, the force at the left end of the specimen is much larger than that at the right end, which was well known as the strength enhancement. During the crushing process, the amplitude of the force at the left end reduces rapidly, but the force at the right end almost remains at a stable level. From the shock models, the force response at the left end of the specimen is given by ε D ε D

9 Z. Zheng et al. / International Journal of Impact Engineering 5 (21) 29e4 7 Fig.. Forces at the two ends of the irregular honeycomb under dynamic impact versus the impact time. F left ¼ M b a b ¼ ma s B ðtþ m þ r FðtÞ the one at the right end of the specimen is written as (85) F right ¼ A s A ðtþ; (86) which are also shown in Fig.. The force at the left end of the specimen given in Eq. (85) is determined by considering the motion governed by the inertia law. This is reasonable because we have taken into account the influence of the unloading wave behind the shock front given up the assumption used by Tan et al. [4], who assumed only two stress states in the specimen. The comparison shows that the theoretical predictions are in good agreement with the numerical FE results. The force responses are not very sensitive to the choice of the idealisations, but the deformation features may be very different when using different idealisations. The model predicts that the mass stops at time t end ¼ 7.49 ms, which is close to the numerical FE result, i.e ms Deformation features Some deformation patterns of the irregular honeycomb under crushing are shown in Fig. 4. The deformation localisation is observed in the specimen. Cells collapse layer-by-layer a deformation front propagates from the left end to the right end of the specimen, see the pattern at t ¼ 1 ms. As the impact energy is absorbed by the collapsed cells, the impact velocity of the mass is reduced. When this velocity is not very high, cells are not fully collapsed (see the pattern at t ¼ 2.2 ms, which corresponds to v b ¼ 8.2 m/s) then deformation bs are rom (see the pattern at t ¼ 5.4 ms, which corresponds to v b ¼ 1.8 m/s). To characterise the deformation features, we employed the discrete deformation gradient [26,27] to determine the local strain in the X-direction, see Appendix C. The local engineering strain fields corresponding to the above deformation patterns are also shown in Fig. 4. It should be noticed that this local strain is resulted from statistical average with measuring over the range of 1.5 times of cell size. The strain field at t ¼ 1 ms shows that the shock front is macroscopic even there is almost no deformation ahead of the shock front. The strain field at t ¼ 2.2 ms shows that the shock front turns not very even. This is due to the influence of the irregularity of cells when the impact velocity is not high. At t ¼ 5.4 ms, the large strain locals at rom deformation bs. Based on the one-dimensional assumption, the local strain at location X is defined as the average strain along the Y-direction. An example of the strain distribution along the X-direction is shown in Fig. 5a, which corresponds to the time when the mass has zero velocity (t end ¼ ms). The magnitude of the strain reduces along the X- direction. The strain distributions predicted by the shock models are plotted also in Fig. 5a. The R-P-P-L model predicts t end ¼ 5.8 ms the model predicts t end ¼ 7.49 ms that the shock front has not reached the right end of the specimen. However, the R-LHP-L model predicts the front reaches the right end of the specimen at time t 1 ¼ 5.9 ms. Compared to the numerical FE result, the strain distribution has been well predicted by the shock models. The comparison also shows that using more realistic constitutive model such as the idealisation has more accurate predictions. The strain distributions at different times predicted by the cell-based FE model the model Fig. 4. Deformation patterns (left column) their corresponding local strain fields in the Lagragian coordinate system (right column) for the irregular honeycomb under dynamic impact.

10 8 Z. Zheng et al. / International Journal of Impact Engineering 5 (21) 29e4 a Local Engineering Strain b Local Engineering Strain Shock Mode Striker-rod impact with V = 1 m/s Honeycomb R-P-P-L R-LHP-L (t 1 ) Transitional Mode Homogeneous Mode Φ S-T Φ T-H Lagrangian Location (mm) Lagrangian Location (mm) Honeycomb: t (ms) Fig. 7. Forces at the two ends of the closed-cell foam under dynamic impact versus the impact time. scale, the physical quantities below this scale are macroscopically meaningless not reasonable. On the other h, there is no intrinsic length scale in the continuum-based shock models so the jump across the shock front is described as a mathematical discontinuity. The magnitude of the strain behind the front reduces as the impact time increases. The strain ahead of the front becomes apparent after ms, which is because of the influence of Fig. 5. Local strain distributions along the X-direction of the irregular honeycomb: (a) when the striker stops (b) at different times. are plotted in Fig. 5b. The shock fronts obtained from the cellbased model seem not as steep as those from the shock models, but concerning about this difference has no substantive significance. The difference is due to the mathematical treatments. On one h, the cell size in the cell-based model is an intrinsic length Time (ms) Striker-rod impact with V = 1 m/s Honeycomb R-P-P-L R-LHP-L Location (mm) Fig. 6. The impact time versus the Lagrangian location of shock front in the striker-rod impact scenario of the irregular honeycomb. Fig. 8. The rod-target impact of a closed-cell foam: (a) the undeformed configuration (b) a deformed configuration at t ¼.4 ms.

11 Z. Zheng et al. / International Journal of Impact Engineering 5 (21) 29e4 9 the reflection wave from the fixed end. In real experiments, the deformation process of cellular structures under impact is difficult to observe due to the limit of experimental conditions. Here, the difficulty is solved by using the cell-based simulations. This observation of the deformation process provides important deformation mechanisms of cellular structures, which strongly supports the basis of the shock models. A comparison of the Lagrangian location of shock front between the cell-based FE model the shock models is shown in Fig. 6. The Lagrangian location of shock front based on the cell-based FE model is defined as the position near the shock front where the strain declines most rapid. Thus, it can be determined by finding the Lagrangian location along the X axis corresponding to the maximum of the absolute strain gradient within a pre-estimated Fig. 9. The undeformed deformed configurations of the closed-cell foam in the sectional plane of z ¼ 12.5 mm, which is a middle plane of the specimen.

12 4 Z. Zheng et al. / International Journal of Impact Engineering 5 (21) 29e4 interval, in which the strain is very different from that of previous time. For the cell-based model, we only present the results before 4. ms because after that time the shock front is not easy to determine. These results compare well with those by the shock models (see Eq. (46)). Hence, the shock models provide well understing of the high-velocity deformation mechanism of cellular structures. The low-velocity deformation mechanism of irregular honeycomb is through the emergence of rom deformation bs. This deformation mechanism of cellular structures is different from that of the shock models, but the shock models still give well prediction of force responses, as shown in Section It means that the force responses are not sensitive to the location of the deformation b when the impact velocity is low Case 2: A rod-target impact of a closed-cell foam The choice of the types of the cellular structures the impact scenario does not affect our main conclusions. As a change, here we focus on the rod-target impact scenario of a much real cellular structure, namely a closed-cell foam Force responses For the rod-target impact scenario, the force responses predicted by the shock models are given by F left ¼ M b a b ¼ ma s A ðtþ m þ r FðtÞ (87) F right ¼ A s B ðtþ: (88) The inertia law is also used to determine the force at the left end of the specimen. These predictions based on three idealisations together with the simulation results of the cell-based model are compared in Fig. 7. The strength enhancement is found at the right end of the specimen, which reduces as the impact time increases. The theoretical predictions are also in good agreement with the numerical FE results Deformation features The deformed configuration of the closed-cell foam at t ¼.4 ms is shown in Fig. 8b. As a comparison, the undeformed configuration is presented in Fig. 8a. This comparison shows that the deformation localisation emerges at the right end of the specimen. To get a much impressive understing, we only consider the configuration in the plane of z ¼ 12.5 mm, which is a middle plane of the specimen. The undeformed deformed configurations in this plane are shown in Fig. 9. Cells collapse layer-by-layer from the right end of the specimen. The cell walls closely parallel to the impact direction deform almost in a buckling manner, while those approximately perpendicular to the impact direction deform likely in a bending manner. A shock front can be detected the crushing cells locate at this front. The cells far from the front close to the left end of the specimen almost have no deformation. The local strain fields have also been calculated using the technique introduced in Appendix C. With the average processing in the YeZ plane, the strain distribution along the X-direction is determined, where the coordinate system is Lagrangian. The strain distribution when the mass stops (t end ¼.89 ms) is shown in Fig. 1a. For comparison, the strain distributions predicted by the shock models are plotted also in Fig. 1a. In this case, the R-P-P-L model predicts t end ¼.86 ms the model predicts t end ¼.98 ms. It should be noticed that the R-LHP-L model predicts the shock front reaching the left end of the specimen at time a Local Engineering Strain b Local Engineering Strain Homogeneous Mode Φ T-H Transitional Mode.4 Rod-target impact with V = 1 m/s Foam.2 R-P-P-L R-LHP-L (t 1 ) Lagrangian Location (mm) t 1 ¼.71 ms. The result of the cell-based model shows that the strain near the left end is very small. Additionally, a comparison of the strain distributions at different times between the cell-based model the model is shown in Fig. 1b. Compared to the cell-based result, the shock models predict slight small strains near the right end of the specimen. This difference may be due to Φ S-T Foam: t (ms) Shock Mode Lagrangian Location (mm) Fig. 1. Local strain distributions along the X-direction of the closed-cell foam: (a) when the striker stops (b) at different times. Time (ms) Location (mm).1 Rod-target impact with V = 1 m/s Foam R-P-P-L R-LHP-L Fig. 11. The impact time versus the Lagrangian location of shock front in the rod-target impact scenario of the closed-cell foam.

13 Z. Zheng et al. / International Journal of Impact Engineering 5 (21) 29e4 41 some deformation mechanisms such as the micro-inertia effect the strain-rate effect, which have not been introduced in the present shock models. It may need much evidence further researches to answer this interesting point. Here, the shock models have well predicted the strength enhancement induced by inertia presented a first-order understing of the dynamic crushing behaviour of the closed-cell foam. A comparison of the Lagrangian location of shock front between the cell-based model the shock models is shown in Fig. 11. The theoretical predictions (see Eq. (68)) are in good agreement with the numerical FE results when the impact time is short (e.g. t <. ms), the impact velocity is high enough. With the increase of the impact time, the impact velocity reduces the shock front moves slowly. Compared to the cell-based results, the shock models overestimate the shock speed when the impact velocity is not high. 5. Conclusions Different idealisations have been used by many authors in the literature to construct one-dimensional shock models, which can well explain the strength enhancement the deformation localisation involved in the dynamic responses of cellular materials. A unified framework of one-dimensional plastic shock wave models was presented in this paper. Based on an arbitrary plastic hardening constitutive model for cellular materials, general solutions, although implicit, have been derived for two impact scenarios, namely the striker-rod impact the rod-target impact, which also can be deduced to two other common impact scenarios, i.e. the plate impact the Taylor impact. Conservation conditions of mass momentum across the shock front based on onedimensional stress wave theory have been used as the basic equations. A combined slope by measuring the average of the slopes of the tangent line the Rayleigh line a stresse enhancement parameter characterising the initial stress enhancement were introduced to make the solutions concise. Based on a power-law hardening constitutive relation of cellular material, closed-form solutions of the physical quantities across the shock front have been derived for the striker-rod impact scenario semi-closed-form solutions for the rod-target impact scenario. These solutions have taken into account the temporal change of impact velocity. Some transition parameters including the critical impact velocities of modes transitions several time parameters have been determined. For each impact scenario, the model was deduced to two limit cases, i.e. the R-P-P-L model [4] the R- LHP-L model [1]. The shock models are verified by comparing with cell-based FE simulations. Based on 2D D Voronoi techniques, the strikerrod impact of an irregular honeycomb the rod-target impact of a closed-cell foam have been considered. Results show that the force responses predicted by the shock models are not very sensitive to the choice of the idealisations they are in good agreement with the numerical FE results. To characterise the deformation features from the cell-based simulations, the discrete deformation gradient [26,27] was introduced to determine the local strain field. Deformation features predicted by the shock models are compared well with the cell-based results when the impact velocity is high. For low-velocity impact, shock models predict different deformation features, which are also different from the cell-based results. The comparisons show that using more realistic constitutive model such as the idealisation may present more accurate predictions. For the closed-cell foam, some other possible deformation mechanisms such as the micro-inertia effect the strain-rate effect may need to be taken into consideration in further researches. Acknowledgements This work is supported by the National Natural Science Foundation of China (Projects Nos ), the China Postdoctoral Science Foundation (Project No ) the Chinese Academy of Sciences (Grant No. KJCX2-EW-L). Appendix A. Nomenclature A cross-sectional area of the cellular material rod a b acceleration of the rigid mass C 1 plastic wave speed, C 1 ¼ (E 1 /r ) 1/2 E elastic modulus E 1 plastic hardening modulus F a function F(q, z) defined in Eq. (59) F 1 the inverse of the function F(q, z) F left force measured at the left end of specimen F right force measured at the right end of specimen f 1 a function defined in Eq. (19) f 2 a function defined in Eq. (25) K strength index l length of the cellular material rod M b mass of the striker m specific mass defined in Eq. (15) n strain-hardening index q equivalent strain-hardening index, see Eq. () s dimensionless variable defined in Eq. (61) T 1 characteristic time parameter, see Eq. (18) T 2 characteristic time parameter, see Eq. (24) t impact time t 1 time when the shock front reaches the distal end t end time when the striker stops t S T time of mode transition from the shock mode to the transitional mode t T H time of mode transition from the transitional mode to the homogeneous mode U internal energy per unit mass V initial velocity of the striker V c1 the first critical velocity V c2 the second critical velocity v velocity v b velocity of the striker x, y, z Eulerian coordinates X, Y, Z Lagrangian coordinates a the stresseenhancement parameter, see Eq. (11) strain initial strain B strain behind the shock front D densification strain L locking strain max full densification strain k combined slope, see Eq. (1) m ratio of the rod s mass to the striker mass, see Eq. (22) r local density in the cellular material rod after deformation r initial density of the cellular material rod s stress s yield stress of the rod s B stress behind the shock front s pl plateau stress s dimensionless variable defined in Eq. (81) F location of the shock front Q A physical parameter ahead of the shock front Q B physical parameter behind the shock front _QðtÞ a time rate of change of physical parameter Q, i.e. dq/dt