Stress Wave Propagation in Elastic-elastic and Elastic-viscoelastic Bilaminates

In the present study normal plate-impact experiments are conducted to investigate the propagation of acceleration waves in 2-D layered material systems. The objective of these experiments is to understand the role of material architecture and material inelasticity in controlling precursor decay and late-time dispersion of acceleration waves in elastic-elastic and elasticviscoelastic laminates. The experiments are conducted using the 82.5 mm single-stage gas-gun facility at CWRU. The history of the free-surface particle velocity at the rear surface of the target plate is measured by employing a VALYN VISAR. In order to understand the effects of layer architecture, experiments are conducted on elastic-elastic bilaminates fabricated with different layer thicknesses and impedance mismatch. Moreover, in order to understand the effects of material inelasticity, experiments on elastic-viscoelastic bilaminates are utilized. The results of the study indicate that the structure of acceleration waves is strongly influenced by impedance mismatch of the layers constituting the bilaminates, density of interfaces, distance of wave propagation, and the material inelasticity. INTRODUCTION The understanding of dynamic behavior of materials is vital to many areas of both civil and military applications. Better understanding of dynamic response has important practical implications connected with impact and blast mitigation, design of lightweight armor, and optimal design of engineered structures with potential danger of shock loading. A number of material systems ranging from metal, ceramics, and polymers in both monolithic and composite forms are currently being used to achieve a combination of characteristics to meet the desired goals. Some of the recent examples highlighting the success of these systems include, layered materials, woven composites and functionally graded materials. These material systems promise light weight armors which are structurally robust, and are being contemplated for use in future combat vehicles and other defense applications. A large body of knowledge currently exists in the literature on the propagation of acceleration waves and finite amplitude shock waves in heterogeneous materials. For such systems, scattering, dispersion and attenuation play a critical role in determining the thermo-mechanical response of the media. This phenomenon can be attributed to a number of non-linearities arising from loading conditions, material heterogeneity at various length scales, and material inelasticity and failure. However, the phenomenon of wave scattering and dispersion during propagation of shock waves in heterogeneous material systems continues to be poorly understood. In most homogeneous materials shock waves in the absence of phase transitions are understood to have a one-wave structure. However, upon dynamic loading of bilaminates, a two-wave structure is obtained-a leading shock front followed by a complex pattern that varies with time. This complex pattern is generated by a continuous interaction of compression and rarefaction waves due to the presence of inter-laminar interfaces. To date, only a limited number of experiments have been conducted that concern the propagation of finite amplitude shock waves in heterogeneous materials. Barker et al. [1] conducted experiments on periodic laminates, and found that below certain critical input amplitude, the stress waves decayed exponentially with distance. However, above the critical amplitude a structured shock-wave was obtained. Lundergan and Drumhellar [2] and Oved et al. [3] conducted shock-wave experiments on layered stacks. These experiments showed that the oscillatory nature of the stress waves due to layering. Nesterenko et al. [4-6] observed an anomaly in the precursor decay for the case of propagation of strong shock waves in periodic bilaminates with a relatively small cell size. They noted that for bilaminates with a relatively small layer thickness the jump in particle velocity at the wave front is essentially higher than the jump obtained with the larger thickness layer at the same distance of propagation. Comparison of experimental results and computer simulations indicated that this effect is primarily due to interactions of secondary compression waves with the leading shock front. More recently, Zhuang [7] has conducted a c combined experimental and computational investigation to study the effects of interface scattering on shock wave propagation in heterogeneous material systems. In the study they investigated high velocity plate-impact experiments on bilaminates to investigate the effect of impedance mismatch and the density of interfaces on the structure of strong shock waves. The focus of the present research is to better understand wave scattering and dispersion at material interfaces and the role of material inelasticity in determining the structure of acceleration waves in 2-D layered material systems. The 2-D layered systems offer a unique opportunity for designing interpretable experiments as-well-as for providing insights into wave propagation in much more complicated microstructures, e.g. fiber reinforced, particulate and woven composites. In the present study wave propagation in both elastic-elastic and elastic-viscoelastic bilaminates is analyzed. The analysis makes use of the Laplace transform and Floquet theory for ordinary differential equations with periodic coefficients. In addition, normal plate impact experiments are conducted using the 82.5 mm single-stage gas-gun facility at CWRU. In these experiments the particle velocity profile at the free surface of the target plate is measured by using a multi-beam VALYN VISAR and compared with the predictions of the analytical solutions. These comparisons are used to understand the effects of layer thickness, impedance mismatch, and material inelasticity on precursor decay and late-time dispersion, WAVE PROPAGATION IN ELASTIC-VISCOELASTIC BILAMINATES Consider bilaminates consisting of alternating elastic and viscoelastic layers of uniform thickness and infinite lateral extent. The elastic layers occupy odd numbered layers, i.e. n , and the viscoelastic layers occupy even numbered layers, i.e. . Consider the individual layers to be homogeneous and isotropic, and the layer thickness of the both constituents to be the same, l l , where l and l are the thickness of the elastic and viscoelastic layers, respectively, and d is the thickness of a typical bilaminate. 1, 3, 5.. = 1 2 2, 4, 6.. n = 1 2 0.5 = = d For infinitesimal deformation, longitudinal waves propagating in the x-direction are governed by the balance of linear momentum and continuity for elastic and viscoelastic laminae ( ) ( ) ( ) ( 1 1 1 1 1 , , 0 and , u u x t x t x t x t t x t x σ ε ρ ∂ ∂ ∂ ∂ − = = ∂ ∂ ∂ ∂ ) , (1) ( ) ( ) ( ) ( 2 2 2 2 2 , , 0 and , u u x t x t x t x t t x t x σ ε ρ ∂ ∂ ∂ ∂ − = = ∂ ∂ ∂ ∂ ) , (2) and, the constitutive equations for elastic and viscoelastic laminae ( ) ( ) 1 1 , , x t E x t ε = σ , and σ . (3) ( ) ( ) 2 , t x t G t d = − ∫ −∞ 2 τ ε ) In Eqs. (1) to (3), σ and are the longitudinal components of the stress tensor in the elastic and viscoelastic laminae, u and are the longitudinal component of the particle velocity in the elastic and viscoelastic laminae, ε and are the longitudinal component of the strain tensor in the elastic and viscoelastic laminae, E and G t represent the elastic and the viscoelastic modulus, respectively, and ρ and ρ are the mass density of the elastic and the viscoelastic laminae. The relaxation function for the viscoelastic material behavior is assumed to be described by an exponential function of the type 1 2 σ 1 2 u 1 2 ε ( )