## Abstract

Propagation of longitudinal deformation in hardening elastoplastic solids is investigated by the way of analogy with the shock tube pattern in fluid mechanics. Conditions for the appearance of a shock discontinuity are formulated and steady-state continuous and discontinuous solutions are derived. Field characteristics are then investigated for a representative family of hardening elastoplastic Mises solids accounting for finite strains. A critical limit value of the imposed velocity for the emergence of a shock wave is found and sensitivity to material parameters is assessed. Evaluation of dissipation effects is conducted and field response is compared with other uniaxial stress fields. In agreement with available experimental results, it is established that the field may consist of both an elastic precursor and a plastic shock separated by a continuous elastoplastic range. Or, alternatively, when the imposed velocity is higher, the plastic shock overtakes those regions allowing for a variety of resulting fields.

## 1. Introduction

At the height of the Second World War, during a train ride to Pasadena, Theodore von Karman formulated, on a piece of stationery, what is known today as ‘The theory of Plastic Waves’. That brainstorm is recorded in his autobiography [1] with further reference to parallel yet independent work by G. I. Taylor and by ‘a Russian scholar’ (apparently Rakhmatulin). Though Karman's interest in the propagation of plastic deformation was motivated by the war time need of safety regulations for hangars and other buildings, a simple analysis of a one-dimensional dynamically stretched bar [2] preserves the physical essence of plastic wave propagation. In that spirit, the present study aims at capturing the fundamental nature of shock wave behaviour in elastoplastic solids at finite strain. A benchmark problem, analogous to the piston shock model in fluid mechanics, is therefore chosen to represent the physical response while facilitating a simplified analysis of shock conditions and field characteristics, as well as enabling a complete solution of the steady-state field. Experimental investigation of shock wave behaviour in solids had been conducted during the war within the Manhattan project and later in Los Alamos Scientific Laboratories where more than 5000 high-pressure shock compression tests were performed to evaluate dynamic material properties [3]. Though shock testing requires specialized knowledge and facilities, the number of such experiments has steadily increased since then, as reviewed by Davison & Graham [4]. While the amount of experimental data on shock compression in solids that has accumulated over the years is sufficient to delineate the physical nature of shock processes, the theoretical background has yet to mature. An extensive survey can be found in the recent paper by Howell *et al.* [5] and earlier in Craggs [6] and Morland & Cox [7]. The complexity of the problem, which involves thermomechanical coupling and strain-rate effects at hypervelocities, and stresses well above yield, has held back theoretical investigation. In this study, we attempt a simplified analytical solution which captures the basic physical picture using rate-independent stress–strain constitutive relations and neglecting thermomechanical coupling. Differences from the true physical response are then evaluated via energy balance considerations similar to those applied by Knowles [8] for a rubber-like solid, and sensitivity to material parameters is assessed.

In compressible fluid dynamics, the elementary problem involving propagation of shock waves is the piston shock problem. However, in solid dynamics the analogous pattern of the appearance of longitudinal shock waves in a dynamically compressed semi-infinite body is less commonly addressed. The solution by von Karman & Duwez [2], which deals with a tensioned semi-infinite elastoplastic bar, has recently been generalized by Knowles [8] for large strains in a tensioned rubber-like solid. The small-strain linearly elastic solution of a one-dimensional field is given in Timoshenko & Goodier [9]. Cristescu [10] presented a similar solution for a bilinear elastoplastic bar.

In this analysis, we consider the steady response of a semi-infinite solid allowing displacement only in the longitudinal direction. This geometry is chosen to maintain the analogue with the piston shock problem and allows analysis of shock behaviour in tension and compression with no jump in cross-sectional area or possible buckling. We begin with the analysis of field characteristics and shock conditions applicable to longitudinal fields, suggesting purely continuous and discontinuous steady solutions. Balance of power input and rate of energy consumption by the deformation process is applied and the shock dissipation efficiency is considered as a measure for the significance of dissipation effects. The finite strain response is then specified for the hardening/softening elastoplastic Mises solid. It is found that material response varies with imposed surface velocity. The deformation field may consist of both an elastic shock and a plastic shock separated by a continuous elastoplastic region or, as imposed velocity increases, the plastic shock accelerates and overtakes the precursors, displaying considerable sensitivity to material hardening characteristics and compressibility. This behaviour has been observed in experimental studies [11,12].

Recently, Howell *et al.* [5] presented an analysis of the time-dependent response of an elastic-viscoplastic solid, with the viscous branch activated beyond the plastic yield, in a similar longitudinal field. Their numerical simulation shows a different yet rich response with elastic precursor and continuous plastic region separated by a shock. However, steady response is not detected and an expansion wave forms behind the plastic shock as time progresses. Molinari & Ravichandran [13] examined the structure of steady plastic shock waves in viscoplastic solids.

## 2. Problem formulation

The surface of a semi-infinite body is put to motion at constant velocity *V* in the normal direction, as if being pushed by a piston with rigid walls (figure 1). The induced deformation field is assumed to be in uniaxial motion with no transverse gradients, implying that strains develop only in the longitudinal direction parallel to the velocity vector and that both longitudinal and transverse stresses are active. That piston model can be viewed as representing an elemental cell of the half-space.

Dynamic fields of a one-dimensional bar under uniaxial tension have been studied in the pioneering work by von Karman & Duwez [2] for small-strain plasticity, and by Knowles [8] for finite elasticity. It would be instructive therefore to formulate, at the outset, the governing equations appropriate for the uniaxial strain field considered here, by following the uniaxial tension analysis of von Karman & Duwez [2], Knowles [8].

Thus, with (*x*,*u*,*t*) denoting the reference Lagrangian material coordinate, the displacement and the time, respectively, we have the stretch
2.1and particle velocity
2.2where a comma followed by a subscript denotes partial differentiation. Relations (2.1)–(2.2) imply the compatibility equation
2.3

The only active equation of motion can be written as
2.4where is the wave speed in a long linearly elastic rod, (*E*,*ρ*_{o}) are the referential elastic modulus and density, respectively, and *σ* denotes the longitudinal Cauchy stress, non-dimensionalized with respect to *E*. Note, in the context of (2.4), that the deformed state density *ρ* is related to *ρ*_{o} by *ρ*=*ρ*_{o}/*a*. Subsequently, we shall use *C*_{E} as a scaling reference value.

Now, we focus our attention on solids with constitutive response that admits a relation of type *σ*=*σ*(*a*), implying that *σ* depends only on the stretch. Thus (2.4) can be recast, with the aid of (2.1), in the form
2.5with
2.6where *C* denotes the characteristic velocity and the superposed prime represents differentiation with respect to *a*. We limit our analysis to materials with real characteristic velocity, thus maintaining the hyperbolic nature of the equation, namely
2.7Solids for which condition (2.7) does not hold are known as phase transforming materials [14]. In these solids, the compressive response may include a discontinuity owing to a jump to an energetically favourable state. Such jumps are not shock waves and they are referred to in the literature as kinks [15].

Considering the steady response, equation (2.5) is supplemented by the boundary condition
2.8where *V* is the constant imposed velocity as illustrated in figure 1.

The existence of the relation *σ*=*σ*(*a*), even if not explicit, is a key ingredient of this study. Thus, we limit the analysis to rate independent solids and neglect thermal effects. However, wide families of solids are in the scope of this investigation, including elastic polymers and elastoplastic materials at large strains. It is also worth noting that the stress–stretch response used here is different from that employed in von Karman & Duwez [2] and Knowles [8]. The latter allows for transverse strain while here we impose the constraint of vanishing transverse displacements and gradients.

In the next subsections, we will show that solutions of the nonlinear wave equation (2.5) converge to a discontinuous shock wave solution, a similarity solution or a combination of both. Conditions for the appearance of each response are examined over a range of material parameters.

### (a) Shock condition

The gradual evolution of a shock is extensively discussed in Courant & Friedrichs [16], with different treatments for compression and tension. As the material is ‘pushed’ (*V* >0), a compression wave, originating at the surface, is formed and propagates into the undisturbed material. If the characteristic curves become steeper as the material is compressed (*C*′<0) then they will eventually intersect and after some transient period the solution is expected to become discontinuous. Intuitively, the disturbance in the more compressed zone propagates faster than the disturbance in the less compressed zone. This causes a gradual increase in gradient until eventually the two zones will overlap and create a discontinuity. If the tendency is inverted (*C*′>0) the wave will gradually disperse and a self-similar solution exists. In tension however (*V* <0) a rarefaction shock will appear if the characteristic curves become steeper as the density decreases.

From (2.6) we find the shock condition
2.9which implies the convexity/concavity of the stress–stretch curve in compression/tension, respectively. Likewise, the small-strain shock condition discussed by Cristescu [10], with the earlier reference to White & Griffis [17], implies the convexity/concavity of the stress–strain curve in compression/tension, respectively, see also Craggs [6], Drumheller [18]. The slope of the stress–stretch relation (*σ*′=d*σ*/d*a*) describes the instantaneous effective hardening modulus and for common solids remains positive in compression with *σ*<0 and 0<*a*<1.

### (b) The discontinuous field

If condition (2.9) applies and the field is purely discontinuous, then the steady-state response dictates constant stretch and velocity in each distinct region. The displacement then varies linearly with both *x* and *t*
2.10as deduced from (2.1)–(2.2). That displacement satisfies wave equation (2.5) identically. Now we consider a field with one shock wave propagating with (Lagrangian) velocity *s*, namely a field consisting of two distinct regions; a dynamically disturbed region behind the shock (0≤*x*<*st*) with *v*=*V* and an undisturbed region ahead of the shock (*x*>*st*) with *u*=*v*=0, *a*=1. Assuming conservation of momentum across the discontinuity, we have by (2.4) the jump condition
2.11and the second jump condition requires continuity of mass, namely
2.12Inserting (2.10) into jump conditions (2.11) and (2.12) results in two relations for the field variables in the disturbed range
2.13implying
2.14Simple algebraic considerations together with shock condition (2.9) show that the Lagrangian shock wave velocity is smaller then the characteristic velocity, namely *s*<*C*.

### (c) The continuous field

To analyse the continuous field, where the wave will gradually disperse, we generalize the von Karman & Duwez [2] small-strain similarity solution. That study has used the assumption *C*=*x*/*t*, which solves the small-strain wave equation identically. Thus, we assume that the stretch *a* is a function of *x*/*t*, namely
2.15and upon differentiation of (2.1), we obtain
2.16Likewise, the second time derivative of the displacement is given, again via equation (2.1), by
2.17here the change in the order of integration and differentiation is consistent owing to the smoothness of the integrand. By inserting the second derivatives (2.16) and (2.17) back into the wave equation (2.5) we find the simple solution
2.18which is identical to the early result by von Karman & Duwez [2]. Equation (2.18) defines the profile of the stretch as *a*=*a*(*ξ*) and hence that of the stress as *σ*=*σ*(*ξ*).

If the shock condition (2.9) does not hold then the continuous self-similar field will evolve. Furthermore, if the stress–stretch curve changes from convex to concave, or vice versa, then the nature of the field will change accordingly with a continuous transition.

A purely continuous field can be divided into three zones, as suggested by Knowles [8] for uniaxial tension,
2.19once *a*(*ξ*) is determined by (2.18) the only remaining unknowns are the constant *a*_{1} and the velocity profile *v*(*ξ*). Note that *ξ*_{2}, which is determined by the equality *a*(*ξ*_{2})=1, is the velocity of propagation of the disturbance. To obtain the velocity profile we now return to equation (2.3) in the self-similar form
2.20which is integrated with the aid of (2.19) to obtain
2.21Specifying (2.21) at *ξ*=*ξ*_{1}, where the velocity is equal to the imposed velocity, provides
2.22which is a closed relation for *a*_{1}. Recalling that by (2.1), it is possible to obtain the displacement by integration, using via (2.15) the substitution d*x*=*t* d*ξ*, namely
Integration by parts then yields
Now, replacing the integral term with *v*(*ξ*) from equation (2.21) and inserting *ξ*=*x*/*t* we have
2.23which is identical with relation (2.10). The above analysis holds for both tension (*V* <0) and compression (*V* >0) as long as the field is continuous, according to (2.9).

### (d) Energy considerations

As discussed by Courant & Freidrichs [16], shock discontinuities are necessarily associated with entropy rise manifested by a jump in temperature across the shock. Therefore, once a shock discontinuity emerges, the power input is higher than the power consumed by the deformation field. As shock intensity increases the thermodynamic effect may become pronounced and the influence of thermomechanical coupling is more significant. The power input (per unit area) equals the rate of accumulation of strain energy, kinetic energy and rate of shock dissipation *D*_{s} (scaled by elastic modulus *E*)
Realizing that in this formulation only the discontinuity contributes to *D*_{s}, it is possible to write the rate of dissipation across a given discontinuity in the form
2.24

Considering a pure shock response, so that velocity and stress are constant in the disturbed region, (2.24) becomes
2.25Now, inserting relations (2.1) into (2.25) we find
2.26where the *driving force* (per unit area) *F*=*D*_{s}/*s*, defined by Knowles [8], represents the thermodynamic force required to ‘move’ the discontinuity. A non-dissipative response necessarily implies *F*=0, or
2.27Thus, depending on the constitutive response, a critical applied velocity for which no shock dissipation occurs could exist. We notice that the constitutive relation *σ*=*k*(*a*−1), where *k* is a constant, satisfies (2.27) identically. Therefore, for this material, the response is non-dissipative at any applied velocity. This constitutive relation coincides with the linear Hookean curve in uniaxial strain even beyond the small-strain regime. The pure shock response implies a purely concave or convex material response hence there is no other constitutive relation which satisfies (2.27).

To evaluate the relative role of shock dissipation in the field response it is instructive to examine the *shock dissipation efficiency* defined by
2.28which is the fractional power input consumed by the dissipation at the shock discontinuity (here *σ* represents the applied stress at *x*=0). The term *shock dissipation efficiency*is justified upon considering the proposed piston as a mechanism for maximal energy consumption under extreme loading conditions. Such mechanisms are often found in protective structures, as, for example, the bumper of a car. In that case, the appearance of plastic discontinuity may enhance the energy consumption and therefore increase the protection efficiency. Inserting relations (2.13), (2.15) and (2.24) into (2.28) it can be shown that *η*_{s}<1/2, as long as the accumulation of strain energy is finite.

## 3. The hardening elastoplastic Mises solid

While §2 deals with the general formulation of the field induced by imposing the velocity at the surface of a semi-infinite solid, in this section we examine specific characteristics of this field: hyperbolicity condition, shock condition, characteristic velocity and transition points for elastoplastic solids with arbitrary hardening characteristics.

Material response is modelled by the Mises flow theory of plasticity which centres on the (non-dimensional) Mises effective stress defined by
3.1where (*σ*_{1},*σ*_{2},*σ*_{3}) denote the principal stresses. In the uniaxial strain field considered here, we identify *σ*_{1} with the longitudinal stress *σ* and, by transverse isotropy, with standing for the (non-dimensional) transverse stress. Thus, from (3.1)
3.2

Within the framework of finite-strain-associated Mises flow theory, the total strain rate is the sum of a Hookean elastic part and a plastic part derived from (3.1) as the plastic potential along with the principle of plastic power equivalence. For the present orthogonal deformation pattern, the three-dimensional tensorial constitutive equation of Mises flow theory (e.g. [19–21]) is reduced to just two scalar relations. Indeed, with total strain rates decomposed on the principal axes of strain (coaxial with principal axes of stress) we have the two relations
3.3and
3.4where *ν* is the elastic Poisson ratio and *ε*_{p} is the effective (logarithmic) plastic strain and a known function of the effective stress *σ*_{e} determined by the standard tension test.

Relations (3.3)–(3.4) can be integrated over the loading history to obtain the Mises deformation theory relations
3.5and
3.6or, put differently,
3.7and
3.8where *β*=1−2*ν* is the compressibility parameter and
3.9is the total effective strain. In passing, note that *m*, defined in (3.2), is by (3.8) positive (+1) in tension and negative (−1) in compression.

Comparison between the Mises (*J*_{2}) flow and deformation theories for several basic problems is given in Durban & Stronge [19], Durban [20], Masri *et al.* [21]. The stress path in the case considered here is proportional in the sense that both stress deviator components (*σ* and ) increase in constant proportion to effective stress (3.2); hence flow theory relations (3.3)–(3.4) integrate exactly to deformation theory relations (3.7)–(3.8) thus providing implicitly the stress–stretch connection *σ*=*σ*(*a*).

If the material exhibits a definite yield stress *σ*_{e}=*σ*_{y} then in the pre-yield zone (*ε*_{p}=0) we have the stress components
3.10The characteristic velocity, as obtained via relation (2.6), is then
3.11Hyperbolicity requirement (2.7) is therefore fulfilled identically throughout the field in tension and compression. In the small-strain regime, with *a*≈1, characteristic velocity (3.11) reduces to
3.12with *C*_{L} denoting the longitudinal elastic wave velocity. The second derivative of the longitudinal stress (3.10), with respect to stretch, is negative throughout the field with no transitions; hence in view of the shock condition (2.9), the compression field will be discontinuous while the tension field will be continuous.

Longitudinal shock waves have been studied in a recent paper by Howell *et al.* [5], employing an elastic viscous constitutive equation. That model has a natural time constant, unlike the material model used here. The elastic branch of the material investigated by Howell *et al.* [5] is derived from a strain energy function which, in the absence of viscoplastic effects, generates stress components similar to (3.10) only with replaced by (*a*−1). Thus (2.27) will be satisfied and no dissipation will take place. The rate-dependent branch in that study is activated in the post-yield regime.

Proceeding with the elastoplastic range beyond yield, in uniaxial deformation, the first derivative of the longitudinal stress *σ*′, deduced from (3.5)–(3.6), is substituted into relation (2.6) to obtain the elastoplastic characteristic velocity
3.13where
is the non-dimensional hardening (tangent) modulus of the stress–strain curve.

Characteristic velocity (3.13) reduces to the small-strain plastic wave velocity
3.14derived by Hopkins [22], with the earlier reference to Luntz [23], in the context of spherical cavity expansion. Cohen *et al.* [24] generalized this result to include finite-strains in the dynamic expansion of a spherical cavity, resulting in a characteristic velocity identical to (3.13). More recently, Cohen & Durban [25] examined the effect of initial porosity on the material response in the context of hypervelocity cavity expansion.

We limit our analysis to materials with real characteristic slopes to maintain hyperbolicity of the equation, thus from (3.13) we have the bounds
This limitation holds for practically all materials with realistic hardening/softening characteristics. The second derivative of the longitudinal stress is now examined towards further investigation of field properties,
3.15where *h**=d*h*/d*ε*_{e}. Under the constraint of the hyperbolicity condition we find that the shock condition (2.9) translates to
3.16

This formulation accounts for arbitrary hardening/softening characteristics; however, for the investigation of constitutive sensitivity, we employ the Ludwik hardening law
3.17which covers a broad range of plastic response. Here, *σ*_{y} denotes the non-dimensional yield stress and *n* is the hardening index. For such materials *h* is positive and decreases as straining progresses (*h**<0). It is clear from condition (3.16) that a transition from convex to concave response will occur for the Ludwik hardening solid under compression while in tension no shock will appear.

The power-law relation in (3.17) has been experimentally verified for a wide family of materials with hardening index in the range of 0<*n*<1. The perfectly plastic solid is obtained at the limit of . Note that while (3.17) is determined by the uniaxial tension test, for uniaxial strain the effective stress is given by (3.2).

An example of the (squared) characteristic velocity profile and variation of the stretch *a* with *mσ*_{e}/*σ*_{y} is shown in figure 2. Different zones are separated by dotted lines and labelled by Roman numerals. The elastic zone is bounded by *σ*_{e}/*σ*_{y}<1 and a jump in the characteristic velocity is observed at yield (*σ*_{e}=*σ*_{y}, *a*=*a*_{y}) owing to a jump in the tangent modulus (*h*). As deduced from (3.16), the shock condition holds only in compression within the elastic range. In tension (*m*=1), even beyond the elastic limit, the shock condition is not fulfilled, implying that the material response is purely continuous and the solution is as in §2*c*. In zone I (*a*_{o}<*a*≤*a*_{y}), the material is compressed with *σ*′′>0 (as marked on figure 2). Under these circumstances, the field will consist of a continuous self-similar plastic zone. A transition from concave to convex behaviour occurs at *a*=*a*_{o}, where *σ*′′=0, therefore, in zone II (*a*_{ep}<*a*≤*a*_{o}) a plastic shock will appear. However, the characteristic velocity is smaller than that in the elastic field, implying that co-existence of the two wave solutions is feasible. Specific definitions of stretches *a*_{ep} and *a*_{p} are given in the subsequent discussion.

Overall, it is understood that the characteristic velocity, as dictated by the stress–stretch curve, has a profound effect on material response. It would be instructive therefore to examine the sensitivity of the characteristic velocity to material parameters. Figure 3 displays characteristic curves and the corresponding stretch for various values of material parameters (*σ*_{y},*n*). It is observed that increasing both yield stress and hardening delays the appearance of plastic shock when *σ*′′=0. At the limit, for a perfectly plastic solid with , shock condition (3.16) implies a purely discontinuous field. It is also noted that though the effective stress levels at the transition point (*σ*′′=0) are sensitive to both yield stress and hardening index, the stretch value (*a*_{o}) and the corresponding characteristic velocity are hardly affected. It will be shown in the subsequent analysis of zone II (of figure 2) that the minimal plastic shock velocity is equal to the characteristic velocity at the transition point and is thus hardly sensitive to variations in yield stress and strain hardening.

Figure 4 illustrates the sensitivity to Poisson ratio revealing that, as expected, material compressibility has an appreciable influence on the characteristic velocity. For an incompressible solid with *β*=0, (*ν*=1/2) the characteristic velocity is infinite and, by (3.15), *σ*′′<0 throughout the field hence no transition point (*a*=*a*_{o}) exists. The relation between the effective stress and the stretch is hardly influenced by the elastic compressibility (3.7).

Sample calculations over a range of parameters (*n*,*σ*_{y},*ν*) show similar results, implying that figures 2–4 can serve as a prototype of field behaviour. The axial stress level (*σ*) can be much higher than the yield stress (*σ*_{y}) according to (3.8), even though the effective stress (*σ*_{e}) is just several times the yield stress. See also the analysis of constitutive sensitivity of spherical shock waves in Cohen *et al.* [24,25]. In the next subsections, we will analyse the field which develops in each zone of figure 2 and the corresponding constant imposed velocity.

### (a) Zone I—elastic shock followed by continuous plastic field

In zone I, the elastoplastic field has a transition point at *σ*_{e}=*σ*_{y}, where the stress–stretch curve changes from concave to convex. Therefore, that field can be divided into four distinct regions
3.18An elastic shock, with Lagrangian (referential) velocity *s*_{E}=*ξ*_{3}, propagates into the undisturbed material. Behind that shock, for *ξ*_{2}≥*ξ*>*ξ*_{3}, the stretch and the velocity are constant with the elastic limit values so that the effective stress is *σ*_{e}=*σ*_{y} (*a*_{y} is the yield stretch as shown on figure 2). A continuous plastic wave then follows in the range of *ξ*_{1}≥*ξ*≥*ξ*_{2} behind which a constant region develops with a velocity that equals the imposed velocity. Transitions at *ξ*_{1} and *ξ*_{2} are continuous.

The Lagrangian shock velocity *s*_{E}=*ξ*_{3} and the transition velocity, which is equal to *ξ*_{2}, follow from the similarity solution (2.18) at *a*=*a*_{y}. While the characteristic velocity has two distinct values at that point owing to the jump in the tangent modulus *h*, the shock velocity *ξ*_{3} assumes the pre-yield value and *ξ*_{2} takes the post-yield value. The particle velocity behind the elastic shock at *ξ*_{2}≥*ξ*>*ξ*_{3} is obtained from the Hugoniot jump conditions, as in §2 for the discontinuous field, so that
3.19where *a*_{y} and *σ*(*a*_{y}) are determined by inserting *σ*_{e}=*σ*_{y} in (3.5) and (3.6), respectively, resulting in
3.20Therefore, upon substitution of relations (3.20) into (3.19) we have
3.21The elastic shock velocity (*s*_{E}) is then given, as in (2.14), by
3.22Proceeding with the continuous range where similarity solution (2.18) applies, integration of (2.20) over [*ξ*_{1},*ξ*_{2}] yields
3.23where field continuity implies that *a*(*ξ*_{2})=*a*_{y} and *v*(*ξ*_{2})=*v*_{y}.

We are particularly interested in the limit value *V* _{o}=*V* (*a*_{1}=*a*_{o}) beyond which (*V* >*V* _{o}) a plastic shock will appear. It is conceivable that the appearance of a shock wave can cause material destruction, implying that *V* _{o} is an important factor in design of protective structures. Sensitivity of *V* _{o} to material parameters is thus illustrated in figure 5, where a representative elastic wave velocity in metals (*C*_{E}=5000 m s^{−1}) is taken for the evaluation of a dimensional result. Sensitivity is notable. Inversion between the curves and sudden change in tendency is observed when approaching the incompressibility limit . This behaviour is because of the disappearance of zone I, implying that, beyond a definite value of *ν*, close to the incompressibility limit, zone I does not exist and *V* _{o}=*v*_{y}.

### (b) Zone II—two-shock solution

If the imposed velocity *V* is high enough to induce straining in zone II then a solution which consists of a plastic shock wave propagating behind the continuous regime is suggested. That field can be divided into four distinct regions as in (3.18), yet continuity at *ξ*_{1} does not hold and the jump conditions (2.11) and (2.12) must be fulfilled. The velocity at (the superposed (+) represents values in front of the shock and accordingly a superposed (−) represents values behind the shock) is obtained from integral relation (3.23)
3.24Jump conditions (2.11)–(2.12) imply
3.25and
3.26where *s*_{P}=*ξ*_{1} denotes the plastic shock referential velocity and follows from the similarity solution (2.18) in the continuous range. Extracting from (3.26) and inserting into (3.25) yields an implicit relation between *V* and with the aid of (3.24), (3.5) and (3.6). Combining (3.25) and (3.26) we arrive at an expression for the plastic shock velocity
3.27which for a weak shock with *a*_{1}∼*a*_{o} and implies that , namely the velocity of the weak shock is equal to the characteristic velocity (also shown on figure 2). Note that *s*_{P} should not be confused with the small-strain plastic wave velocity *C*_{P} defined in (3.14).

At the limit where and (as shown on figure 2) the continuous range is overdriven (*ξ*_{1}=*ξ*_{2}) and a purely discontinuous field develops. Values of *V* _{ep}=*V* (*a*_{ep}) are shown in figure 6. It is observed that while the existence of the continuous region is highly sensitive to material hardening (*n*), sensitivity to yield stress (*σ*_{y}) is low with inverted range near *ν*=1/2; thus, increasing yield stress lowers *V* _{ep}. As shown in figure 5, the disappearance of zones I and II is noticed when as reflected by a sudden change in curve behaviour where *V* _{ep}=*v*_{y}.

### (c) Zone III—purely discontinuous solution

In the absence of the continuous regime, discussed previously, the present field is divided into three subregimes by
3.28The elastic shock remains unchanged as discussed in the previous subsection and is now followed by a plastic shock. Continuity of mass and momentum across the jump implies, by (2.11) and (2.12)
3.29and
3.30Combining equations (3.29) and (3.30) to eliminate *s*_{P}, and writing *a*_{1} and *σ*_{1}=*σ*(*a*_{1}) as functions of the effective stress (*σ*_{e}) with the aid of (3.5)–(3.6), along with the hardening law relating *σ*_{e} and *ε*_{e}, we arrive at an explicit relation for the imposed velocity as a function of the effective stress
3.31

At the zone boundary, where *a*_{1}=*a*_{p} (see figure 2), the plastic wave velocity overtakes the elastic shock so that *s*_{P}>*s*_{E} and the field consists of a single plastic shock with the same solution as in §2 for the discontinuous field. Figure 7 reveals very little sensitivity of *V* _{p}=*V* (*a*_{p}) to yield stress (*σ*_{y}) and hardening index (*n*), while the sensitivity to Poisson ratio (*ν*) is considerable. The purely discontinuous field with one plastic shock (zone IV) is expected to develop in the deep plastic zone, thus it is possible to explain this behaviour by returning to equations (3.5) and (3.6) with *σ*_{e}≪*ε*_{e}, implying the explicit relation
3.32For example, in the deep plastic zone the axial stress–stretch relation is practically unaffected by yield stress and material hardening. Sensitivity is noticed for a limited range near *ν*=1/2, where zones I–III do not exist. The variation of the plastic shock velocity along with effective stress is shown in figure 2.

### (d) Dissipation effect

The *shock dissipation efficiency* (*η*_{s}), as defined in (2.28), serves as a measure of the thermodynamic effect on the material response in the presence of a shock discontinuity. As the dissipation effect intensifies, the present solution becomes inaccurate. According to (2.25), the rate of dissipation across the elastic shock is
3.33where *v*_{y} and *s*_{E} are given in (3.19) and (3.22), respectively, and the longitudinal stress *σ*(*a*) is given for the Hookean elastic field by (3.10). Inserting these relations in (3.33) and performing integration yields the elastic shock dissipation efficiency (2.28).
3.34which increases with the yield stress to first-order *η*_{E}≈*Y*/(24*G*), where *Y* =*σ*_{y}*E* is the dimensional yield stress and *G* the elastic shear modulus. For *σ*_{y}=0.01 and *ν*=0.3, we have *η*_{E}∼0.001 implying that in the elastic field the dissipation effect can be neglected.

The rate of dissipation across the plastic shock is deduced from relation (2.24), separately for each zone, by numerical integration. The explicit stress–stretch relation in the deep plastic range (3.32) allows analytical evaluation of the plastic shock dissipation efficiency
3.35which is accurate for *a*>*a*_{p}. At the limit the induced velocity is infinite while the strain energy remains finite; thus *η*_{P}=1/2 is the highest possible shock dissipation efficiency.

The plastic shock dissipation efficiency *η*_{P}, as a function of the non-dimensional effective stress, is displayed in figures 8 and 9. While the dissipation effects in zone IV may be significant, as deduced from (3.35), in zone II the maximum of 2.2% of the power input is consumed by dissipation effects and less than 10% in zone III. Sensitivity to yield stress and hardening index is displayed in figure 8. Though sensitivity is observed it is found that the shock dissipation efficiency at the transition between zones is little affected by *n* and *σ*_{y}. Sensitivity of the dissipation efficiency, as a function of effective stress, to *ν* is hardly noted in figure 9, where curves (for different levels of *ν*) coincide.

## 4. The effect of geometry

Most of the available literature on analytical investigation of shock wave behaviour in solids examines longitudinal fields. Such fields may vary in conditions. For example, the classical result [9] for linear elastic solids is for a purely one-dimensional field with no transverse stress or strain . Knowles [8] also studied the uniaxial field but allowed changes in cross-sectional area with no transverse stress , as in von Karman & Duwez [2]. In this work, an analogous shock tube is modelled with active transverse stresses but no transverse straining permitted . We show that the geometry in which the deformation is activated may have an appreciable effect on the material response.

The formulation in §2 accounts for shocks in all of the aforementioned longitudinal deformation patterns. The constitutive relations (3.5)–(3.6) are generalized to include both longitudinal and transverse (logarithmic) strains by
4.1and
4.2In the uniaxial stress state of a Hookean elastic field, with active longitudinal and transverse strains, the stress–stretch relation can be derived from equations (4.1)–(4.2) with *ε*_{p}=0 to obtain
The longitudinal characteristic velocity can now be written via relation (2.6) as
4.3This characteristic velocity can be up to three times larger than that obtained from (3.11) for the piston field. If changes in cross-sectional area of the bar are neglected, then only constitutive relation (4.1) is active and , therefore the characteristic velocity reduces to
4.4which agrees with the small-strain result [9] for *a*∼1. As the characteristic velocity in (4.4) is independent of material compressibility, it can differ considerably between (3.11) and (4.3).

Turning to the elastoplastic bar with active longitudinal and transverse logarithmic strains , plastic incompressibility implies a characteristic velocity identical to that of the Hookean elastic solid (4.3). Therefore in this geometry, the appearance of shock waves is independent of hardening and yield characteristics. It should also be noted that the appearance of a shock in such a field is necessarily involved with an unrealistic jump in cross-sectional area of the bar. Further reduction to the one-dimensional field with no changes in cross-sectional area implies and by (2.6) we obtain the characteristic velocity
4.5which is proportional to tangent modulus and independent of material compressibility. Also, to maintain hyperbolicity of the one-dimensional field (2.7), strain softening (*h*<0) is prohibited.

Characteristic velocities for a hardening elastoplastic solid with *σ*_{y}=0.001, *n*=0.3, and *ν*=0.3 for the three different geometries (3.13), (4.3) and (4.5), are shown in figure 10 for comparison. It is seen that each geometry exhibits a specific and distinct behaviour. While the piston pattern response is complicated, with a variety of zones of continuity and discontinuity, the uniaxial stress field, studied by von Karman & Duwez [2], is purely discontinuous with convex stress–stretch curve in compression and continuous in tension. Further constraining of the uniaxial stress field makes the stress–stretch curve concave, and thus the response is purely continuous in the plastic range for both tension and compression.

Considering the sensitivity of the response to geometrical particularities we compare the present results with those obtained for dynamic spherical cavity expansion by Cohen & Durban [24]. In that study, the steady-state, self-similar, expansion of a spherical cavity embedded in an infinite medium is induced by constant applied pressure at the cavity wall. A remote continuous elastic wave propagates followed by a continuous elastoplastic field up to a critical value of constant expansion velocity where appearance of a plastic shock is observed. Behind the plastic shock, the field is continuous up to the cavity wall where singularity induces infinite level of the effective strain. Values of the critical velocity at the appearance of a plastic shock with Poisson ratio 0≤*ν*<1/2, hardening index 0.1≤*n*≤0.3 and yield stress 0.001≤*σ*_{y}≤0.01, are in the range of 2050≤*V* _{c}≤3300 [m s^{−1}] with *C*_{E}=5000 [m s^{−1}]. Comparison with the present uniaxial field velocities, figures 5–7, reveals that overall *V* _{ep}≤*V* _{c}≤*V* _{p} implying that the appearance of shock waves is delayed in the spherical geometry.

## 5. Concluding remarks

We have examined uniaxial strain states that include shock waves in the unidirectional motion of elastoplastic solids. Formulation is within the framework of large-strain rate-independent Mises theory, accounting for strain hardening and elastic compressibility. Problem setting centres on the piston shock analogue, commonly employed in fluid mechanics, as the representative elemental unit of the half-space field in longitudinal motion.

Derivation draws on earlier work by von Karman & Duwez [2], and on the recent work by Knowles [8], both dealing with uniaxial stress patterns in tension. Shock conditions are formulated, for both compression and tension, including the rate of dissipation, regardless of constitutive particularities.

A detailed analysis and sectors mapping is given for the associated Mises flow theory which coincides with the deformation theory for all cases considered here. Sensitivity of critical velocities for the appearance of a shock wave to material stress-strain curve is studied and illustrated. A limited comparison with axial stress fields reveals the influence of geometry on the material response.

Most of available theoretical studies on plastic wave propagation are unidirectional with analysis limited to uniaxial stress or uniaxial strain. Even the cylindrical and spherical wave patterns consider motion in radial direction only. Nevertheless, the essential nature of plastic shock waves in uniaxial strain and continuous plastic loading, with isotropic hardening, has been exposed, paving the way for study of more complicated geometries in future studies.

## Footnotes

↵† This work is based on part of a Ph.D Thesis at the Technion.

- Received January 30, 2013.
- Accepted January 7, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.