Abstract
Rock-like materials are composites that can be regarded as a mixture composed of elastic, plastic, and viscous components. They exhibit viscoelastic-plastic behavior under a high-strain-rate loading according to element model theory. This paper presents an analytical solution for stress wave propagation in viscoelastic-plastic rock-like materials under a high-strain-rate loading and verifies the solution through an experimental test. A constitutive equation of viscoelastic-plastic rock-like materials was first established, and then kinematic and kinetic equations were then solved to derive the analytic solution for stress wave propagation in viscoelastic-plastic rock-like materials. An experimental test using the SHPB (Split Hopkinson Pressure Bar) for a concrete specimen was conducted to obtain a stress-strain curve under a high-strain-rate loading. Inverse analysis based on differential evolution was conducted to estimate undetermined variables for constitutive equations. Finally, the relationship between the attenuation factor and the strain rate in viscoelastic-plastic rock-like materials was investigated. According to the results, the frequency of the stress wave, viscosity coefficient, modulus of elasticity, and density play dominant roles in the attenuation of the stress wave. The attenuation decreases with increasing strain rate, demonstrating strongly strain-dependent attenuation in viscoelastic-plastic rock-like materials.
Highlights
Previous studies have examined the propagation of the stress wave in composite materials, such as rock and concrete under a loading for the seismic design of various infrastructure systems [1,2,3].The propagation of the stress wave is governed mainly by the inherent physical properties of the material and characteristics of the stress wave
From Equations (20a) and (22a), the analytical solution for stress wave propagation in viscoelastic-plastic rock-like materials suggests that the attenuation factor α can be determined by the viscous coefficient of the material, the elastic modulus, density, and the frequency of the stress wave
The estimated stress-strain curve from the analytic solution was identical to the stress-strain curve from the experimental test regardless of the strain rate, and the mean square error between them was less than 10% of the strain range of the intersections
Summary
Previous studies have examined the propagation of the stress wave in composite materials, such as rock and concrete under a loading for the seismic design of various infrastructure systems [1,2,3]. According to element model theory [11,12,13], which is widely accepted in the constitutive equation of rock-like materials, the constitutive model of viscoelasticplastic rock-like materials can be established This model is composed of three main components: an elastomer, a Newtonian body, and a Materials 2016, 9, 377. Analytical solution for stressrelationship wave propagation viscoelastic materialsequation is used widely in polymer materials based on the viscosity effects according to the elastic responses [8]. An analytical σ = ε solution for stress wave propagation in viscoplastic materials has been developed by considering the where E1 effects is the initial of the elastic εE and σEfor arestress the strain stress of viscosity basedelastic on themodulus plastic responses [9]. Analytic Solution for Stress Wave Propagation in Viscoelastic-Plastic Rock-Like Materials (2) Developed based on the responses and the constitutive equation can be defined as: 2. Analytic Solution for Stress Wave Propagation in Viscoelastic-Plastic Rock-Like Materials (2)
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