# Shock Wave Formation and Cloaking in Hyperelastic Rods
## 1 Introduction  
### An Overview  
(A) Shock waves in fluid and gaseous media. The first works on the formation of acoustic shock wave fronts in gaseous media descend to [PERSON] [1] and [PERSON] [2]. It was found that at a moving surface of a weak discontinuity, known also as the weak shock wave front, the thermodynamic equations should be taken into consideration to account for the heat production. Later, a complete theory for the formation and propagation of shock wave fronts in gaseous and liquid media was developed [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16]. Several numerical methods for analyzing the propagation of shock wave fronts were developed: (i) the shock-fitting method [17], and (ii) the non-linearization techniques [18; 19]. Both methods, along with (iii) the finite difference methods [20; 21; 22; 23], (iv) the finite element methods [24; 25; 26], and (v) various types of the discrete element approaches [27; 28; 29], are widely used in the study of acceleration waves. Note also that according to [30], the shock waves in gaseous and liquid media propagate with faster velocities than the local sound velocity. (B) Shock waves in solids. The first theoretical works on the formation and propagation of shock waves in elastic-plastic solids are mainly due to [PERSON], who gave an explicit representation for the displacement of the initial part of the propagating pulse in a semi-infinite rod [31]:  
\[u(x,t)=c_{0}(t+x/c_{\text{shock}})\] (1a) where \[c_{\text{shock}}\] is the unknown shock wave velocity; \[t\] is the time; and \[x\] is the distance from the rod edge. \[u(x,t)\big{|}_{x=0}=c_{0}t \tag{1b}\]  
The last equation defines \(c_{0}\) as the velocity of the impact applied at the rod edge. It was found that (i) the \(c_{\text{shock}}\) velocity depends upon \(c_{0}\); (ii) it depends upon actual strain; and (iii) the shock wave velocity in an elastic-plastic target slab may be larger than the impact velocity [32; 33]:  
\[c_{\text{shock}}\geq c_{0} \tag{1c}\]  
It should be mentioned that in physical experiments [32; 33], inequality Equation (1c) was obtained by indirect measurement of the particle velocities at the rear face of theslab. In this respect, see previous studies [34; 35; 36; 37; 38; 39; 40]. Most of these works are concerned with elastic-plastic solids of the Prandtl-Reuss type. However, despite a significant achievement in finding the shock wave velocity, the problem of mechanical energy dissipation and heat production remained unresolved; meanwhile, physical experiments revealed a substantial temperature rise [33]. [PERSON] [41] proposed, for the first time, a comprehensive theory of shock waves in hypoelastic solids satisfying the following equation of state:  
\[\dot{\mathbf{\tau}}=\mathbf{C}(\mathbf{\varepsilon})\cdot\dot{\mathbf{\varepsilon}} \tag{1d}\]  
where \(\dot{\mathbf{\tau}}\) is the Kirchhoff stress rate; \(\mathbf{C}\) is the fourth-order elasticity tensor, assumed to be positive definite; and \(\dot{\mathbf{\varepsilon}}\) is the infinitesimal strain rate. For such a hypoelastic material, the shock wave front velocity could be explicitly constructed [34; 35; 42]. \[c_{\text{shock}}=\frac{[\mathbf{v}]\cdot\mathbf{n}}{[\mathbf{\varepsilon}]\cdot\mathbf{n} \otimes\mathbf{n}} \tag{1e}\]  
where \([\mathbf{v}]\) is a jump in the velocity field; and \([\mathbf{\varepsilon}]\) is a jump in the corresponding strain field. Note that for shock waves in a 1D rod, Equation (1e) reduces to:  
\[c_{\text{shock}}=[\mathbf{v}]/[\mathbf{\varepsilon}] \tag{1f}\]  
According to Equation (1e), the shock wave velocity depends upon the jump in velocity and strain and the inclination of the jump with respect to the unit normal, ensuring \(c_{\text{shock}}\in(0;\,\infty)\)[43].