## Abstract

Cracking induced by tensile wave at the free surface of an impacted target is an important issue in impact-resistant design. Here, we explore the use of material nonlinearity to undermine the strength of the tensile wave. More specifically, we consider waves in a two-material composite bar subjected to impact loading at one end. Multiple reflections cause a tensile wave being transmitted into the second material. The attention is on analytically and numerically studying the phenomenon that the tensile wave catches the first transmitted compressive wave. It turns out that, depending on the interval of the initial impact, catching-up phenomena can happen in two wave patterns. A general mathematical theory is provided to show the existence of these patterns together with some qualitative information. To gain more insights into such phenomena, asymptotic solutions are also constructed, which provide both qualitative and quantitative results on the requirement of the constitutive relation, the time and place at which the catching takes place, and how the initial impact, material and geometric parameters influence the solutions. Numerical simulations are also performed, confirming the validity of the analytical results. The analysis and results presented here could be useful for designing a composite structure that has a good impact-protection performance.

## 1. Introduction

Critical civil infrastructures such as airports or defence establishments and transport vehicles are vulnerable to manually delivered, small explosive devices. Such structures are increasingly using composite materials, owing to their effective energy-absorption capability (Vinson & Sierakowski 1989; Carruthers *et al.* 1998). However, the structural damage evolution under blast loading is caused by both shock wave and dynamic pressure. Shock-induced microcracking results in macrocracking and final collapse of the whole structure under subsequent dynamic pressure. Hence, there is an urgent need for designing a composite material structure that could reduce effectively the shock-induced microcracking such that the protected structure could survive the subsequent dynamic pressure. One approach for the blast-resistant design is to make use of the relationship between incident, reflected and transmitted waves in a multi-layer composite structure (Achenbach 1984). However, the challenging issue is how to adjust material properties in each layer and the geometrical parameters so that the magnitude of tensile wave could be reduced as much as possible to prevent the composite structure from the formation and evolution of microcracking. One possible way to achieve such a purpose is to use nonlinearly elastic materials in some layers.

Waves in a single nonlinearly elastic material have been studied for a long time. The nonlinearity induced by finite deformations in such a material makes it difficult to deduce the analytical properties of solutions in a boundary/initial value problem. A focus is on finite amplitude waves in an infinite medium. One of the earlier works was carried out by John (1966), which considered plane waves in Hadamard materials and harmonic materials. A review on research in this direction before year 2000 was given by Boulanger & Hayes (2001). In recent years, efforts have been made to take into account other effects for finite amplitude waves in an infinite medium, such as viscoelasticity, dissipation and dispersion (Hayes & Saccomandi 2000; Hacinliyan & Erbay 2004; Haddow & Erbay 2004; Destrade & Saccomandi 2005, 2006). It was found that some exotic waves (e.g. compact-like shear waves) can arise. These works demonstrate that material nonlinearity can induce some interesting wave behaviour. The present study explored the use of material nonlinearity for the purpose of impact protection in a nonlinearly elastic composite bar with two materials.

Nonlinear waves in slender structures, such as bars/rods/beams, made of a single nonlinearly elastic material, have also been studied for a long time. It appears that weakly nonlinear waves in an elastic bar/rod were first studied by Nariboli (1970), who showed that the Korteweg–de Vries equation could be used as the model equation for unidirectional long waves. We refer to the book by Porubov (2003) for reviews on other works on weakly nonlinear waves in elastic rods. One of the authors has also carried out extensive research in this area. For example, Cohen & Dai (1993) proposed two coupled equations for modelling bidirectional waves in a compressible Mooney–Rivlin rod, and the head-on collision of two solitary waves was studied in Dai *et al.* (2000). A new type of nonlinear dispersive equation (similar to the Camassa–Holm equation) was derived (Dai 1998; Dai & Huo 2000) for finite-length and finite-amplitude waves. It turns out that this physical model admits both the peakon solution and compacton solution (Dai 1998; Dai & Huo 2000; Dai *et al.* 2004).

Waves in nonlinearly elastic composite materials have also been extensively studied. For example, Andrianov *et al.* (2011) used an asymptotic homogenization method to study the dynamical properties of layered composite Murnaghan materials. Parnell (2007) considered wave propagation in a pre-stressed nonlinearly elastic composite bar, and the focus was on determining the effective incremental response. Besides analytical studies, many people have also carried out numerical computations on waves in composite materials. For example, Clements *et al.* (1996) used the method of cells to study stress waves in laminated materials, and their results can be used to explain some features of the experimental data. Berezovski *et al.* (2006) used a finite volume method to study waves in layered nonlinear heterogeneous materials due to impact, and their results are in good agreement with the experimental data of Zhuang *et al.* (2003). One focus of these works was how the cell structure of the composite material influences the waves. The present study has a different focus: whether and how the material nonlinearity can be explored to generate the phenomenon that a destructive tensile wave can catch a transmitted compressive wave. This will reduce the strength of the latter.

As will be seen, for such a catch-up phenomenon, we need the stress–strain curve of nonlinearly elastic material to be convex. Some materials indeed exhibit such a behaviour. For example, certain iron-particle-filled magnetorheological elastomers may have such a behaviour (Danas *et al.* 2012). It is also found that certain aluminium foams have such a stress–strain curve (Banhart & Baumeister 1998). Under certain temperatures and humidity some polymers can have a concave-down stress–strain response (Shackelford 2000). Thus, in reality, it is possible to design such a composite material structure with optimized material layer arrangement for resisting blast loads. In this study, an analytical approach is developed to explore how the incident, reflected and transmitted waves could be controlled in a nonlinearly elastic composite bar with two materials.

The rest of study is organized as follows. In §2, we formulate the wave propagation problem. Mathematical theory for two wave patterns for which wave catching-up phenomena can occur in the second material of the composite bar is presented in §3. Some qualitative information is also deduced. There are two parts in §4. The first part is devoted to deriving the asymptotic solutions for the related physical quantities. Those simple analytical expressions provide both qualitative and quantitative information on how the constitutive relation, the strength of the initial impact and the geometrical and material parameters influence the strains and particle velocities. In the second part, we do some numerical experiments on a particular composite material. Comparisons between the numerical and analytical solutions demonstrate the validity of the latter. The paper is concluded in §5 with some remarks.

## 2. Problem formulation

We consider waves in a two-material composite bar due to a compressive impact, with the first layer composed of a linearly elastic material of finite length *h* (called material 1) and the second part composed of a nonlinearly elastic material of infinite length (called material 2). The geometry of the problem is shown in figure 1. The two parts are assumed to remain in perfect contact throughout the time interval of interest. During the motion by the impact, the entire assembly is assumed to undergo uniaxial deformation. To ensure that the deformation is one-to-one, it is natural to suppose the strain *γ*>−1.

In a Lagrangian description, the governing system of equations of motion reads
2.1
where *γ*,*v*,*σ* are smooth functions and denote the strain, particle velocity and nominal stress, respectively. At a moving strain discontinuity *x*=*s*(*t*), the jump conditions are
2.2
where [*f*]=*f*(*t*,*s*(*t*)+0)−*f*(*t*,*s*(*t*)−0), denotes the Lagrangian velocity of the discontinuity. The constant density *ρ* and the stress–strain relation are defined as
2.3
where *E*_{1},*E*_{2} are the Young’s moduli of the materials 1 and 2, respectively.

The composite bar is initially at rest in the reference state. At the initial time *t*=0, an impulsive load described by a rectangular loading function is imposed at the end *x*=0 with the duration *T* and prescribed stress *σ*=*A*, where *A* is a negative constant. So the initial boundary value conditions are given by
2.4

The objective is to investigate the catching-up phenomenon that the tensile wave (arising due to multiple reflections) catches the earlier transmitted compressive waves in the second semi-infinite bar, so that the magnitude of the former (which is destructive) can be reduced.

To this end, throughout the paper, we consider a kind of two-material composite bar, for which the nonlinear function *f*(*γ*) in the stress–strain curve *σ*(*γ*)=*E*_{2}*f*(*γ*) for material 2 is assumed to satisfy the following properties:

(H

_{1})*f*(*γ*) is smooth enough in and*f*(0)=0;(H

_{2}) let*α*_{1},*β*_{1}be assigned constants consistent with , it holds that(H

_{3}) without loss of generality, we may assume*f*′(0)=1.

It will be seen that the convexity of *f*(*γ*) is crucial for the catching-up phenomenon to happen.

## 3. Mathematical theory

In this section, we give a mathematical analysis on the wave propagation in this nonlinearly elastic composite bar. Our attention is mainly focused on the construction of two-wave patterns for the catching-up phenomena, which can happen when the constitutive curve *f*(*γ*) (see (2.3)) is nonlinear and convex.

First, we consider the case that the material 2 is linearly elastic (i.e. *f*(*γ*)=*γ*). Obviously, the reflected and transmitted waves are all linear (figure 2).

By the jump conditions (2.2), it is easy to see that the speeds of the transmitted waves in the second material equal to the sound wave speed *c*_{2}=(*E*_{2}/*ρ*_{2})^{1/2}, which implies that no wave catching-up phenomenon can happen. Through some simple manipulations on the jump conditions, we find that the strains in material 2 are given as
3.1
where *β*=*ρ*_{1}*c*_{1}/*ρ*_{2}*c*_{2} denotes the ratio of the impedances of material 1 and material 2; *c*_{1}=(*E*_{1}/*ρ*_{1})^{1/2} is the sound wave speed in material 1. It is found from (3.1) that the tensile wave (i.e. *γ*^{−}>0) can arise when the impedance ratio of two materials *β* is less than 1.

We now turn to consider the case that the material 2 is nonlinearly elastic and aim to constructing the solutions from (2.1) to (2.4) for the two-wave patterns, which can arise owing to the convexity of *f*(*γ*). It will be seen that the transmitted waves are composed of a rarefaction wave followed by two subsequent shock waves (figure 3). Here, we would like to point out that the rarefaction wave can generically originate from the nonlinearity of the material 2 (here the convexity of the stress–strain relation) and the self-similarity of the problem. By rarefaction, it means that the spatial distributions of the strains and particle velocities evolve with the time and become wider and wider.

We note that the first three moments when the linear waves in material 1 happen to reflect at the interface are given by
3.2
At the same time, we denote the duration time *T*=*θt*_{1} and require
3.3
in order that before the first reflected wave arrives at the end *x*=0, the unloading wave has begun to propagate rightward.

Because of the self-similar structure of our problem (2.1)–(2.4), the strains and particle velocities are piecewise constants or take the forms of a fan (Dai & Kong 2006). Applying the jump conditions (2.2) at the reflected waves in material 1 yields the strains and particle velocities in the following forms:
3.4
3.5
3.6
3.7
and
3.8
where the strains *γ*_{2},*γ*_{5} and *γ*_{8} will be determined through the unknowns and *γ*^{−} in material 2.

It has been assumed that the two parts of the composite bar remain in perfect contact for all times of interest; so the stress must keep continuous across *x*=*h* and therefore
3.9

Now, we consider waves in the second semi-infinite bar. For a compressive impact, it is expected that *γ*^{+}_{1}<0 (see the first equality in (3.1) for the linear case). Thus, *c*(*γ*^{+}_{1})<*c*(0)=*c*_{2} because *c*′(*γ*)=(*c*_{2}/2)[*f*′(*γ*)]^{−1/2}*f*′′(*γ*)>0. As a result, the slope of the characteristic line *t*−*t*_{1}=[1/*c*(*γ*^{+}_{1})](*x*−*h*) in the *xt*-plane is larger than that of the characteristic line *t*−*t*_{1}=[1/*c*(0)](*x*−*h*). The characteristic lines for strains between *γ*^{+}_{1} and 0 form a fan (rarefaction wave). So, the first transmitted wave is rarefactive. For such a wave, the standard argument gives (Dai & Kong 2006)
3.10
where
3.11
Here, the meaning of rarefaction wave is the same as mentioned earlier. It is seen that once the constitutive relation for material 2 is given, the strain can be directly computed from the second equality in (3.10), and the particle velocity will in turn be obtained by integration from the third equality in (3.10).

Using the second equation in (3.4) and the first equations in (3.9) and (3.10) easily yields an equation for ,
3.12
We find that the initial stress *A* is monotonically increasing with respect to , which means that, for any given initial stress *A*<0, there exists a unique such that equation (3.12) holds. Meanwhile, in order the elastic deformation is one-to-one, it is natural to assign a restriction on *A*. If (−1,0)⊂(*α*_{1},*β*_{1}), such a kind of condition appears as
3.13

The second transmitted wave is a shock wave (denoted by *S*_{0} in figure 3*a*,*b*), because (see the argument below (3.18)). The jump conditions across *S*_{0} are
3.14
and
3.15
where *s*_{0}>0 is the speed of *S*_{0}. By using the first equations in (3.6), (3.10) and the second equation in (3.9), equations (3.14) and (3.15) are equivalent to
3.16
and
3.17
Once *γ*^{+}_{1} is solved, we can eliminate *s*_{0}/*c*_{2} from (3.14) and (3.15) and obtain an equation for :
3.18
Furthermore, we can prove that there is a unique solution for (3.18). In fact, replace by *γ* in (3.18) and denote the left-hand side of the resulting equation by *H*(*γ*). We can see that *H*(*γ*) is continuous and
and
The last inequality follows from So there is a solution to the equation (3.18). Moreover, it is easy to verify that *H*(*γ*) is monotonically increasing for *γ*∈(*γ*^{+}_{1},0), which implies that the solution *γ*^{+}_{2} is unique.

Once and are determined, it is easy to derive the shock velocity *s*_{0} by (3.16) or (3.17). Here, we note that the first two transmitted nonlinear waves are all compressive ( and ).

The third transmitted wave is also a shock wave (denoted by *S*_{1} in figure 3*a*,*b*), as the strain *γ*^{−} behind *S*_{1} is less than the strain ahead *S*_{1} (see the argument below (3.20)). By using a similar method as already mentioned, we can deduce the equations for the wave velocity *s*_{1} of *S*_{1} and the strain *γ*^{−}, respectively,
3.19
and
3.20
Denote the left-hand side of (3.20) by *G*(*γ*^{−}). It is found that and *G*(*γ*^{−}) tends to as *γ*^{−} approaches to since the convexity of *f*(*γ*), which means that there exists a solution to (3.20). At present, for a general given stress–response relation *σ*(*γ*)=*E*_{2}*f*(*γ*), it is not so obvious to prove that the strain *γ*^{−} is positive. In §4, we show that by asymptotic analysis, the third transmitted shock wave can be tensile indeed when the amplitude of the impact is small.

Here, it is worth pointing out that, owing to the convexity of the constitutive curve for material 2, the transmitted shock waves exactly satisfy Lax’s entropy condition (Lax 1957) 3.21 which implies the constructed solutions are physically admissible.

From the inequalities shown in (3.21), we see that there are two cases for the wave catching-up phenomena to happen, i.e. case I: the third transmitted shock wave *S*_{1} first captures the second shock wave *S*_{0} (because *s*_{1}>*s*_{0}) and case II: the second transmitted shock wave *S*_{0} first penetrates the rarefaction wave (because ; figure 3).

For case I, the locus of the intersection *A* between the third and second transmitted waves is given by
3.22
Analogously, for case II, the locus of the intersection *B* between the second transmitted shock wave and the rarefaction wave is determined by
3.23
So we define the critical time
3.24
Then, when the duration time *T*≥*T*_{*}, the case I will occur; while at *T*<*T*_{*}, the case II would happen (figure 3).

Now, we give a further analysis on the interaction of the shock *S*_{0} and the first transmitted rarefaction wave in case II. Mathematically, for an initial-value problem such an interaction has been addressed (Chang & Hsiao 1989) in certain aspects. However, for a boundary and initial value problem, as far as the authors are aware of, it has not been considered. Such an interaction will lead to a shock with a varying speed, because the strain and velocity ahead of this shock vary with respect to time (their values are determined by the rarefaction wave). The propagation of the varying speed shock wave is determined by
3.25
In the following, we show that the shock *S*_{0} cannot eliminate the whole rarefaction wave. This is crucial for showing that the third transmitted shock *S*_{1} can capture the varying-speed shock.

From (3.25), one can see that *s*(*γ*^{+}_{2})=*c*(*γ*^{+}_{2}). That is, when the strain value ahead of this shock reaches *γ*^{+}_{2}, the speed of the shock is equal to the speed of the rarefaction wave, and then there is no more interaction. Because the strain value ahead of the varying-speed shock at the intersection *B* is *γ*^{+}_{1}(<*γ*^{+}_{2}), if we can show that *γ* is an increasing function of *t*, then it implies that the interaction stops when *γ* increases to *γ*^{+}_{2}.

The second equation in (3.25) leads to
3.26
Hence, it follows from (3.26) and the first equation in (3.25) that
3.27
It is easy to see that *c*′(*γ*)=(*c*_{2}/2)[*f*′(*γ*)]^{−1/2}*f*′′(*γ*)>0. Thus, to show that *dγ*/*dt*>0 is equivalent to showing *s*(*γ*)>*c*(*γ*). From the first equality in (3.25), by Taylor’s formula, we have
3.28
Thus,
3.29
Because 0<*μ*<1, . Then, because of the convexity of *f*(*γ*) (i.e. *f*′′(*γ*)>0), we have . This completes the proof.

Now, we show that the shock wave *S*_{1} can capture the shock wave with varying speed (the curve *BC* in figure 3*b*, denoted by from now on). It is sufficient to show that *s*_{1}≥*s*(*γ*) for all *γ*. For the shock wave , the strain value increases from *γ*^{+}_{1} as time increases, because it is an increasing function of *t*. The largest possible *γ* value is *γ*^{+}_{2}. Thus, *γ*^{+}_{1}≤*γ*≤*γ*^{+}_{2}. From (3.28), we have
3.30
Therefore, *s*(*γ*) is an increasing function of *γ* and *s*(*γ*)<*s*(*γ*^{+}_{2}). On the other hand, according to the first equation in (3.25) and (3.21), *s*(*γ*^{+}_{2})=*c*(*γ*^{+}_{2})<*s*_{1}. Thus, *s*(*γ*)<*s*_{1}. This shows that the shock wave *S*_{1} will capture the varying-speed shock at some time and interact with it afterwards.

At the end of this section, we would like to remark that the varying-speed shock wave satisfies the following Cauchy problem:
3.31
where *p*=*g*(*q*) is the inverse function of .

## 4. Asymptotic solutions and their comparisons with numerical solutions

### (a) Asymptotic solutions

The mathematical theory presented in §3 shows that, depending on the time duration of the impact, there are two wave patterns in which wave catching-up phenomena can happen, together with some qualitative information. However, how various parameters influence the solutions qualitatively and quantitatively are not determined. In this section, we shall derive the asymptotic solutions for the strains , , *γ*^{−} and other quantities, provided that the amplitude of the impact is small (in comparison with the Young’s modulus of material 2). Then, we can deduce both qualitative and quantitative information and provide analytical insights on the roles played by various parameters.

For simplicity, we mainly give the leading-order terms for the asymptotic expansions, and most higher-order terms are omitted (the interested readers can contact the authors for them).

We introduce a dimensionless quantity
4.1
which is small because the smallness assumption on the initial stress *A*. Assuming having an asymptotic expansion in *ε* and straightforwardly expanding the equation (3.12) with respect to *γ*^{+}_{1} gives
4.2
This simple formula is already informative: the first term is exactly the same as the one given in the first equality in (3.1) for linearly elastic material, and the second term is caused by material nonlinearity through *f*′′(0), which is further influenced by the impedance ratio *β*. By the earlier-mentioned formula, it follows from (3.18) that
4.3
We can see that the material nonlinearity induces a third-order effect for this strain value.

Furthermore, with the aid of (4.2) and (4.3), we can obtain from (3.20) that
4.4
It is seen that indeed *S*_{1} is a tensile shock (*γ*^{−}>0) for small *ε*. Also, the material nonlinearity undermines its magnitude (the second term is negative).

Therefore, it is possible to deduce from the equations (3.16), (3.19) and (3.22)–(3.24) the following asymptotic expansions: 4.5 4.6 4.7 4.8 4.9 4.10 4.11 and 4.12 where .

We see from (4.5) and (4.6) that the material nonlinearity decreases the speed of the shock wave *S*_{0} and increases the speed of the shock wave *S*_{1}, which is crucial for the wave catching-up phenomena. Equation (4.7) tells that the critical time, which separates the two wave patterns, is determined by the impedance ratio and *t*_{1} (which in turn is determined by the thickness of first layer and the wave speed in the first layer, see the first equality in (3.2)).

As for the rarefaction wave, because
we have
4.13
where we have used the formula (4.12). The equation (4.13) means that the term is of *O*(*ε*). Thus, we are able to derive the following expansion from (3.10) and (3.11):
4.14
It is seen that the strain value is independent of the impedance ratio *β* and only determined by the sound speed *c*_{2} and the material nonlinearity through *f*′′(0),*f*′′′(0), etc.

Now, we deal with the varying-speed shock in case II, for which one needs to analyse the orders of the relevant terms carefully. An important observation from (4.13) is that the order of the magnitude of will not be greater than *O*(*ε*). Besides, according to the discussions in §3, the shock wave *S*_{1} cannot penetrate the whole rarefaction wave, and at the end of interaction the strain value *γ*^{+}_{2}=*O*(*ε*^{3}), see (4.3). So the order of the term cannot be smaller than *O*(1)*ε*^{3}. On the basis of this insight and using the formula (4.14), it is not hard to derive from the first equation in (3.25) that
4.15
where is small and satisfies .

Introduce
4.16
and
Then, the equation (4.15) can be written as
4.17
The initial condition *x*(*t*_{B})=*x*_{B} for (4.15) becomes
4.18

Now we are ready to seek the asymptotic expansion for *y*=*y*(*τ*).

Let
4.19
where *y*_{i}(*τ*) (*i*=1,2,3) are functions to be determined later. Substituting (4.19) into (4.17) and equating the coefficients of *ε*,*ε*^{2},*ε*^{3}, respectively, and using the initial condition (4.18) leads to
4.20
and
4.21

Thus, it is found that the asymptotic formula for the shock wave is
4.22
where *y*_{i}(*τ*) (*i*=1,2,3) are given by (4.20), (4.21) and (4.16).

In the sequel, we study the locus (point *C* in figure 3*b*) at which the tensile shock wave *S*_{1} catches the varying-speed compressible shock wave . We note that the mathematical theory presented in §3 does not provide any information on this point, except its existence.

The time and location of *C* are denoted by (*t*_{C},*x*_{C}) and
4.23
and
4.24
where
4.25
It follows from (4.23) and (4.24) that
4.26
Bearing in mind the formulae (4.6), (4.9) and (4.19) for *s*_{1}, *t*_{B} and *y*(*τ*), respectively, we can deduce from (4.26) that
4.27
with
4.28
It is seen that a smaller *ε* causes a longer time for the catching point. We can easily derive from (4.24) the following asymptotic formula for *x*_{C}/*h*:
4.29
We observe that a smaller *ε* causes a farther distance for the catching point. It is found that the parameter *α*=*c*_{1}/*c*_{2} also influences the position of the catching point.

For the particle velocities of the composite bar, it is also easy to obtain their asymptotic expansions. However, for brevity, these formulae are omitted (interested readers can contact the authors for them).

Figures 4 and 5 are the plots of the strains *γ*(*x*,*t*) in the second semi-infinite bar as a function of *x* at several fixed times in case I and case II, respectively. For *t*_{1}<*t*≤*t*_{2} (figures 4*a* and 5*a*), there is only a transmitted rarefaction wave. As time progresses to *t*_{2}<*t*≤*t*_{3} (figures 4*b* and 5*b*), a compressive shock *S*_{0} is further transmitted into the second layer. Then, when *t*_{3}<*t*≤*t*_{A} (figure 4*c*) or *t*_{3}<*t*≤*t*_{B} (figure 5*c*), a tensile shock *S*_{1} propagates in the second semi-infinite bar. In case II, when *t*_{B}<*t*<*t*_{C} (figure 5*d*) a varying-speed shock wave appears owing to the interaction of the shock *S*_{0} and the first transmitted rarefaction wave. It can also be seen that these plots are consistent with the two-wave patterns shown in figure 3.

The analytical expressions obtained in this section reveal clearly the roles played by the constitutive relation, the strength of the initial impact, and the material and geometrical parameters. It is expected that they will be very useful for designing such a composite structure for impact protection purpose.

### (b) Comparisons with numerical solutions

Here, we shall consider the wave catching-up phenomena in a concrete composite bar by selecting a particular constitutive relation for material 2 and then do some numerical simulations to verify the results in the earlier sections.

Because the purpose here is just to examine validity of the asymptotic solutions, we take a simple nonlinear function *f*(*γ*) as follows:
4.30
where *m* is a constant. For simplicity, we assume and then the interval of interest for all strains is . As usual, in order to ensure that the deformation is one-to-one, the initial stress must satisfy
4.31
instead of the assumption (3.13). Equivalently,
4.32

For such a *f*(*γ*), the integrals in (3.12) and (3.18) can be explicitly evaluated. Then (3.12), (3.18) and (3.20) become algebraic equations. By the Newton’s method, the strains and *γ*^{−} can be obtained numerically. Then the speeds of the shock waves *S*_{0},*S*_{1} and the locus of point *A*,*B* can be obtained from (3.16), (3.19), (3.22) and (3.23), respectively.

For the constitutive relation (4.30), it is easy to see from (3.25) that the curve for the varying-speed shock wave can be described in the following dimensionless form:
4.33
It is straightforward to solve the Cauchy problem (4.33) by the Runge–Kutta method. Then, the time and location at which the shock wave *S*_{1} captures the varying-speed shock wave can be determined from (4.23) and (4.24).

We now do some numerical simulations for the particular material (4.30) (tables 1–3). There are five parameters (*m*,*α*,*β*,*θ*,*ε*) needed to assign values. The material constants *m*,*α* are fixed as 0.5 and 2, respectively. We can let *β* be 0.4, according to the fact that the tensile wave can occur when *β*<1. The parameter *θ* measures the initial duration time. When *T*≥*T*_{*}, the case I for the wave catching-up phenomenon will happen, while if *T*<*T*_{*}, the case II then occurs (figure 3). The parameter *ε*=−2*A*/*E*_{2} is dimensionless and assumed to be small. Here it should be pointed out that the analytic results for the strains and other quantities have been calculated by the corresponding asymptotic formulae in §4*a*.

Table 1 shows that when *ε*=0.2 or 0.1 the strain values given by the asymptotic formulae are in very close agreement with the numerical results. Even when *ε*=0.4,0.3 the asymptotic solutions are very good. Comparisons for related quantities of the wave catching-up phenomena in case I and case II are also made (see tables 2 and 3 for details).

Figure 6 shows that the position curve of the varying-speed shock. Once again, the difference between the analytical (asymptotic) solution and the numerical solution is very small, which is also implied by table 3.

### Remark 4.1

In the numerical simulations, we take a relatively large *ε* (from 0.1 to 0.4), because there is the possibility that the second material can be a polymer. If *ε* is smaller (say, 0.001), the asymptotic solutions should be better (owing to the nature of the asymptotic expansions).

## 5. Concluding remarks

For impact waves in a two-layer composite bar, it is found that if the impedance ratio is less than 1, a tensile wave can be transmitted into the second layer. Then, we show that if the constitutive curve of the second material is convex, there are two cases in which it can catch the first transmitted compressive wave. When this catch-up phenomenon happens, the strength of the tensile wave can be reduced, so that such a structure can reduce the chance of the formation and evolution of microcracking. The theory, analytical and numerical solutions presented in this study may provide a new possible way for designing certain structures for impact protection purposes.

We point out that the present study is valid only up to the catching point. To go beyond, some comprehensive numerical simulations are needed. Currently, the discontinuous Galerkin method is used to examine the extent of the magnitude reduction of the tensile wave. The initial results show that a 500 per cent reduction can be achieved, comparing with the linear elastic material. The complete results will be reported elsewhere in the future.

## Acknowledgements

The work was supported by a GRF grant from Hong Kong Research Grants Council (project no. CityU 100911), the 111 Project (no. B08014), the Programme for Changjiang Scholars and Innovative Research Team in University (PCSIRT), the National Key Basic Research Special Foundation of China (2010CB832704) and the NNSF of China (no. 11101001).

- Received May 14, 2012.
- Accepted July 6, 2012.

- This journal is © 2012 The Royal Society