Plastic Wave Propagation Research Articles

Mechanics of Materials: Multiscale Design of Advanced Materials and Structures

Materials can now be designed and architectured like structural components for targeted mechanical and physical properties. Structures and microstructures should not be studied independently and their design will benefit from a multiscale approach combining nonlinear continuum mechanics approaches and physical descriptions of elasticity, viscoplasticity, phase transformations and damage of microstructures, at various scales. The aim of the workshop was to gather outstanding junior and senior researchers in the various branches of mathematics, physics and engineering sciences suited to address the question of design of materials and structures by means of multiscale discrete and continuum approaches to their constitutive behavior. Examples include atomic or macroscopic lattices, random or periodic cellular materials, smart materials like shape memory alloys, 3D woven composites, acoustic and electromagnetic metamaterials, etc. Modern continuum mechanics relies on sophisticated constitutive laws for anisotropic materials exhibiting elastoviscoplastic behavior, still a field of intense research with new mathematical concepts. In particular size-dependent properties are addressed by resorting to generalized continua such as gradient or micromorphic and phase field models. The latter are attractive for the simulation of microstructure evolution coupled with mechanics, due to thermodynamic and metallurgical processes and damage. Scale transition and homogenization methods for continuous and discrete systems are required for the determination of effective material and structural behavior. Metamaterials are architectured materials specifically designed to achieve certain propagation and dispersion properties of elastic and plastic waves. Optimization strategies for the design of optimal architectures are involved in the design process. Target functions for optimization are now based on multicriteria (stiffness, strength, thermal expansion, transport properties, anisotropy etc.).

An analytical model for the penetration of flat-nosed long rods into semi-infinite concrete targets

An analytical model is presented herein on the penetration of a flat-nosed long rod into a semi-infinite concrete target based on the previous theoretical studies and experimental observations. The nose shape of the flat-nosed long rod in deformable penetration state is assumed to be a circular arc and the length of plastic region of the long rod in hydrodynamic penetration is taken into account. The behavior of an erosive penetrator is further divided into two penetration stages (namely semi-hydrodynamic penetration and hydrodynamic penetration) and a new critical impact velocity (i.e. erosive velocity) is derived to characterize the beginning/incipient erosion in accordance with plastic wave propagation theory. According to the new theoretical considerations, the relationship of dimensionless instantaneous mushrooming head radius versus impact velocity is rewritten and the method for predicting semi-hydrodynamic penetration tunnel radius is proposed. It transpires that the present model predictions are in good agreement with available experimental results for the penetration of flat-nosed long rods into semi-infinite concrete targets in terms of penetration depth, penetration modes, penetration tunnel size, residual mass and residual length.

On the development of a constitutive model for steel subjected to fire and explosion

The behavior of steel under thermal and mechanical loading including high transient temperatures and strain rates (e.g., fire, creep or explosion) is extremely complex. To study this behavior, we propose a constitutive model accordance with the laws of thermodynamic, the principle of virtual power and experimental results. The thermodynamic framework considers the Clausius-Duhem inequality and maximum dissipation rate principle, as well as the theory of finite deformations, linear isotropic viscoelasticity, nonlinear isotropic and kinematic hardening, local and non-local viscoplasticity, local and non-local anisotropic viscodamage and parameters associated to material behavior dependent on loadings. The components of the thermodynamic conjugate forces are defined from the Helmholtz free energy function and the rate of energy dissipation, which are postulated considering the creep phenomenon, thermal-mechanical transient phenomenon, cyclic phenomenon and plastic wave propagation phenomenon. The balance of microforces is defined based on the principle of virtual power. A novel rule of non-associated thermo-viscoplastic flow of the Perzyna type and a novel rate dependent local and non-local damage evolution law are introduced from constitutive model. The verification of the model was carried out in two benchmark studies composed of fire tests. The predictions of the model have shown in according with the results of the experiments. A sensitivity study was also performed with a view to analysis the influence of the parameters in achieved results. From this study a simplified constitutive model is presented. Benchmark studies composed of explosion tests are part of future work.

Effects of shear strain on shock response in single crystal iron

With large-scale non-equilibrium molecular dynamics simulations and in situ x-ray diffraction analysis, we conducted a systematic investigation into the effects of pre-existing shear strain (γxy) on the shock response of single crystal iron. Our findings reveal significant effects of γxy on the deformation of the crystal structure during shock loading, leading to noticeable alterations in the propagation of shock waves. Specifically, during the elastic stage, the presence of γxy results in a reduction of shock strength, consequently diminishing the magnitude of elastic lattice strain (εe). In the plastic stage, γxy stimulates the α–ε phase transformation, and structure deformation undergoes a transition from the sequential activity of dislocation-to-transformation to the synchronous activity of dislocation and transformation. This transition inhibits the propagation of plastic waves and consequently broadens the elastic regime. Additionally, the introduction of γxy activates different slip systems, as it alters the corresponding resolved shear stress. Concurrently, the presence of γxy triggers the activation of different high-pressure phase variants. Our investigation sheds light on the fundamental physics of iron under shock compression and the influence of pre-existing shear strain on its behavior.

Research on plastic structural wave behavior and crashworthiness of continuous gradient mesoscopic foam structures under high speed impact

Under high speed impact, the macro and mesoscopic mechanical behavior, the stress wave propagation mechanism, and the relationship between density gradient and crashworthiness of continuous density gradient foam are the key issues to improve and optimize the protection capability of support structures. The numerical simulation of various density gradient foams is carried out, and the meso-discontinuity of foams is overcome by using the local gradient tensor method of least square error, and the plastic structure wavefronts with clear interfaces are obtained. The plastic structure wave propagation mechanism of foam with different density configurations was analyzed and the relationship between density gradient and crashworthiness was further revealed. The results show that inertia effect is the dominant factor of collapse in the early stage of impact. In the middle and late stage of impact, the dominant factor of collapse changes to the microscopic density of foam. The distribution of continuous density gradient significantly affects the impact resistance. Compared with low-high density configuration foam (LH), high-low density configuration gradient foam (HL) has a 140% increase in impact end force, a 24% decrease in support end force, a 3% decrease in maximum compression, and a 41% longer response time.

Effect of Li element on shocking behavior of Fe-Li alloys

Understanding the effect of alloying elements on shock behaviors is significant for constitutive model of alloy materials at under dynamic high pressure. However, an in-depth understanding about the microscopic mechanism of alloying elements effect on plasticity and phase transition is still limited. In this work, applying non-equilibrium molecular dynamic (NEMD) simulations, shock-induced plasticity and phase transition of Fe-Li alloys was investigated in terms of the compositions of Li, crystallographic direction and shock velocity. The results show that yield stress (or phase change pressure) of Fe-Li alloy was reduced doping Li element, and the large internal stresses caused by concentration of Li element effectively activated the nucleation and multiplication of dislocation loops. But with the increase of Li composition, phase transition is inhibited for shock along [111] direction because of stress release caused by multiplication of dislocation. There is a strong dependence between these microscopic processes and the law of plastic wave propagation. Moreover, with the increase of Li composition, the diversity of phase transition variants was inhibited for shock along [110] direction due to the destruction of initial stress and potential energy uniformity.

The problem of propagation of a one-dimensional plastic wave in the environment with linear and polyline unloading

Problems of propagation of plane and spherical waves in a nonlinearly compressible medium with linear and broken line unloading under intense loads are considered. The solutions of the problems are constructed in the opposite way, assuming that the medium at the shock wave front is instantly loaded in a nonlinear manner, and behind the front in the perturbed region, the medium is irreversibly unloaded. For a specific structure of the medium, the results of calculations are presented in the form of graphs of pressure, velocity of the medium at the layer boundary, at the shock wave front and in the disturbed region as a function of time. The influence of the nonlinear properties of the medium on the distribution of the dynamic characteristics of shock-wave processes in it has been studied.

Fast numerical estimation of residual stresses induced by laser shock peening

The aim of this paper is to develop a model allowing a fast first approximate estimation of the elastic–plastic stress wave propagation caused by a laser impact and the resulting residual stress field. We start by modeling the stress wave propagation, adopting a 1D uniaxial modeling, reducing the behavior of the specimen to the axis of the laser impact, excluding any edge effects caused by the edges of the laser spot. The plastic strain field resulting from this propagation can in turn be used to compute the residual stresses, by making use of an analytic modeling in the case of an infinite planar plate. The accuracy of the 1D model is assessed by comparing it to finite elements simulations, acting as a reference solution, for several materials and laser spot diameters. The results show that the stress wave propagation from the 1D model is close to identical to the reference solution. The residual plastic and stress fields from the finite elements model present a uniaxial distribution on a large portion of the laser spot, except for the very edge and spot center. The comparison between the 1D model and the reference solution shows a good match, indicating that the 1D model can be used for a fast approximation the mechanical fields created by a laser impact for laser spot diameters larger than 2 mm.

Time-discontinuous state-based peridynamics for elasto-plastic dynamic fracture problems

A time-discontinuous state-based peridynamic method (SBPD-TD) is developed in this paper to model the transient problems of wave and crack propagation in solids. In this method, the non-local deformation gradient is derived in a consistent way using the least square method, so that the classical elasto-plastic constitutive relationship can be directly adopted in the SBPD based on the correspondence material modeling. Then the displacement field and the velocity field are interpolated independently in the temporal domain respectively, in which the jump terms representing the discontinuities of variables are introduced between the adjacent time steps. After that, an integral weak form in the temporal domain of the spatially discrete governing equations is constructed and the basic formula of SBPD-TD is derived. The above-mentioned characteristics ensure that the SBPD-TD can accurately capture the sharp gradient characteristics in the propagation of elastic and plastic stress waves and effectively control the spurious numerical oscillations. Several representative numerical examples show that the SBPD-TD provides more accurate and satisfactory results when compared with the conventional SBPD, and has broad applications to predict the contact impact fracture behaviors of various materials.

Modelling the penetration of subsonic rigid projectile probes into granular materials using the cavity expansion theory

Typical subsurface investigation tools used on Earth are not applicable for subsurface exploration of the Moon and other extraterrestrial bodies due to payload limitations in space missions. Instead, light and compact subsonic projectile probes can be considered as an alternative subsurface investigation tool in such exploratory missions to overcome a host of challenges. Such probes can be launched from a lunar orbiter or lander to the surface of the Moon to provide the initial effective penetration from the impact. Here, we develop a model based on the spherical cavity expansion theory to predict the deceleration rate and final penetration depth of a rigid projectile probe into geological targets under the perpendicular subsonic impact. Two stress fields are assumed to propagate in the medium, plastic (near field) and elastic (far field), upon the impact. The stresses at the cavity wall are obtained by combining the Mohr–Coulomb failure criterion for the target failure (plastic region) considering two different assumptions for plastic wave propagation. Two field experiments are used to compare and assess the robustness of the proposed solutions in the subsonic range. Base on the simulation results and the experiments, it is concluded that the cavity expansion model considering the locked hydrostat assumption, with the modification here introduced for the volumetric strain, can provide us with a reasonable prediction of the projectile penetration, final penetration depth and the stresses on the probe. Thus, our proposed solution can be used as a benchmark for sophisticated and computationally-expensive numerical calculations.

Nonuniform Deformation of Cell Structures Owing to Plastic Stress Wave Propagation

In cell structures, unlike in dense bodies, nonuniform deformation occurs from the impact end, even at velocities in the order of tens to hundreds of meters per second. In this study, we experimentally examine the nonuniform deformation mechanism of cell structures. They prepared two kinds of specimens: nickel foam (Ni foam) and silicone-rubber-filled nickel foam (Ni/silicone foam). As a dynamic and impact test method (compression velocity of 20 m/s or more), we used a dynamic and impact load-measuring apparatus with opposite load cells to evaluate the loads on both ends of the specimen in one test. At compression velocities of 20 m/s or less, no nonuniform deformations were observed in the Ni foam and the Ni/silicone foam, and the loads on the impact and the fixed ends achieved force equilibrium. The Ni foam showed no change with an increasing strain rate, and the Ni/silicone foam showed a strong strain rate dependence of the flow stress. At a compression velocity of approximately 26 m/s, the loads differed at the two ends of the Ni/silicone foam, and we observed nonuniform deformation from the impact end. The results of the visualization of the load and deformation behavior obtained from both ends of the specimen revealed that the velocity of the plastic stress wave and the length of the specimens are important for nonuniform deformation.

Mathematical simulation of elastoplastic deformation in cubic materials with an account of anisotropic bulk compressibility

It was shown for the first time that when modelling the deformation of materials with cubic symmetry (at full stress), the rotation of the computational axes leads to the identification of anisotropic volumetric compressibility. Loading of the materials with cubic symmetry of properties in the directions not coincided with the main directions (for example, 011) allows one to detect 75% cases of the auxetic single crystals (i.e. with negative Poisson’s ratio). In these cases, the negative volumetric compressibility has anisotropy, in contrast to the volumetric compressibility calculated along the crystallographic axes for cubic materials. Anisotropy of the volumetric compressibility leads to anisotropy of velocities of propagation of body waves. This paper considers several elastoplastic problems for a cubic material with different orientations of a coordinate system about its crystallographic axes. The behaviour of such a material under dynamic loads is modelled with an account of anisotropic bulk compressibility to provide the same anisotropy of bulk wave velocities in the elastic and plastic ranges and a uniform pressure function at the elastic-to-plastic strain transition. For each orientation of the coordinate system and respective planes, different values at the indicatrices of elastic constants are specified, and this specifies different deformation processes in cubic materials. Such an effect is demonstrated by solving three problems in three-dimensional (3D) statements approximating the following processes: (1) one-dimensional elastoplastic deformation in a thin target impacted by a thin plate; (2) uniform compression in a spherical body under pulsed hydrostatic pressure; and (3) 3D elastoplastic deformation in a cylindrical body striking a rigid target in view of anisotropic bulk compressibility. The problems were solved numerically using original programs based on the finite element method modified by GR Johnson for impact problems. Solving problems in a 3D formulation makes it possible to take into account the dependences of the direction of the elastic and plastic characteristics of the material, as well as the velocities of propagation of elastic and plastic waves from that direction. The simulation results suggest that for cubic materials, changing the orientation of two coordinate axes in a plane changes the strains along all three axes, including those perpendicular to this plane. It is concluded that anisotropic bulk compressibility in cubic materials should be allowed for by mathematical models of their elastic and plastic deformation. We demonstrate that the orientation of a computational coordinate system for cubic materials should be in those directions in which their deformation is analysed in each particular case.

Mixed variational one-dimensional dynamic thermo-viscoplasticity for wave propagation

In view of recent developments in advanced manufacturing and nanomaterial technologies, which have enabled realization of novel material architectures at small-scale, a mixed variational temperature-dependent rate-dependent thermo-viscoplastic continuum element is proposed and developed for elastic, plastic, and thermal wave propagation problems. The variational scheme is based on a Hamiltonian approach, while for the temporal and spatial discretization scheme a discrete calculus of variations approach is adopted. Within this framework, an exponential temperature dependence for material properties and second sound effects is included. The developed framework sets the foundations that can lead to new class of temperature dependent models for capturing thermoplastic behavior and wave propagation of different materials. The robustness of the proposed variational approach, and aspects of coupled dynamic temperature-dependent thermoplastic response are demonstrated through several case studies. Applications include elastic, plastic, and thermal wave propagation and reflection in semi-infinite and finite length space at small temporal scales.

Studies of the propagation of elastic and plastic waves in cubic single crystals

The study of the elastic and plastic properties of a single crystal with cubic symmetry of mechanical properties under dynamic loading was carried out. In the work, the method of numerical simulation shows the effect on the output of an elastic precursor and a plastic compression wave of the orientation of the calculated coordinate system relative to the direction of loading. It is shown that the rotation of the calculated coordinate system by 450 around the axis of loading of the projectile and the target in the framework of numerical simulation of the process of shock loading affects the stress state of the target and the velocity profile of its rear surface. This is explained by the fact that in the plane of loading the elastic and plastic properties in a single crystal with cubic symmetry of properties change in accordance with the values of Euler angles.

Effects of underground explosions on soil and structures

Much effort has been dedicated to the study of underground explosions because they pose a major threat to people and structures below or above the ground. In this regard, it is especially important to model the propagation of blast waves in soil and their effects on structures. The main phenomena caused by underground explosive detonation that must be addressed are crater or camouflet formation, shock wave and elastic–plastic wave propagation in soil, and soil-structure interaction. These phenomena can be numerically simulated using hydrocodes, but much care must be taken to obtain reliable results. The objective of this study is to analyze the ability of a hydrocode and simple soil models that do not require much calibration to approximately reproduce experimental and empirical results related to different buried blast events and to provide general guidelines for the simulation of this type of phenomena. In this regard, crater formation, soil ejecta, blast wave propagation in soil, and their effects on structures below and above the ground are numerically simulated using different soil models and parameters; the results are analyzed. The properties of soil have a significant effect on structures, the ejecta, and the propagation of shock waves in soil. Thus, the model of the soil to study these phenomena must be carefully selected. However their effect on the diameter of a crater is insignificant.

Numerical simulation and absorbing boundary conditions for wave propagation in a semi-infinite media with a linear isotropic hardening plastic model

The theory of plastic wave propagation finds its applications in many geotechnical earthquake engineering problems. In this paper, a linear isotropic hardening plastic material is chosen to investigate the plastic wave propagation in a semi-infinite media. The dynamic response of a semi-infinite bar subjected to a triangular load is first derived. The numerical modelling scheme is then proposed and verified by the existing and derived analytical solutions. Novel artificial boundary conditions (ABCs), which are designed for transient dynamic analysis of plastic dilatational wave propagation problem in the media with a linear hardening plastic property, are developed and incorporated into the numerical scheme. Numerical examples illustrate the accuracy of the ABCs in reflecting the complex phenomenon of wave interaction, and reveal good absorption of outgoing stress waves. The study outcome provides reference for developing suitable ABCs for media with complicated nonlinear mechanical properties, and will facilitate the numerical technique in solving soil-structure seismic interaction problems.

Impact compression and plastic wave propagation in closed cell foams

Impact compression and plastic wave propagation in closed cell foams

Plastic deformation of metallic materials during dynamic events

Recently, fully implicit computational capabilities have been developed to predict the plastic behavior of metallic materials (i.e. Ti, Mo) during dynamic events. It is to be noted that within this formulation framework, the equilibrium equations are solved for each time increment. The couplings of the numerical framework to the Cazacu et al. [1] plasticity model that accounts for all the key features of the plastic behavior of airframe materials, i.e. the tension-compression asymmetry and the orthotropic behavior, results in high fidelity prediction of the mechanical behavior during dynamic events. The improved predictive capabilities have been assessed for different strain rate conditions and different metallic materials. Furthermore, validation of the models and FE formulation for Taylor impact conditions through comparisons of experimental deformed profiles of Taylor specimens for Ti and Mo has been done. It is worth noting that for the first time, the extent of the zone of plastic deformation, change in geometry and the transition from transient to quasi-steady plastic wave propagation was captured with great fidelity. Furthermore, the model was used to gain understanding of the dynamic deformation process in terms of time evolution of the pressure, the extent of the plastically deformed zone, distribution of the local plastic strain rates, and when the transition to quasi-steady deformation occurs for different dynamic events. It was thus shown that this model has the potential to be used for virtual testing of complex systems.

A new analytical model for the low-velocity perforation of thin steel plates by hemispherical-nosed projectiles

Abstract Ballistic experiments were conducted on thin steel plates that are normally impacted by hemispherical-nosed projectiles at velocities higher than their ballistic limits. The deformation and failure modes of the thin steel plates were analyzed. A new method was proposed according to the experimental results and the perforation phenomenon of the thin steel plates to determine the radius of the bulging region. In establishing this new method, a dynamic method combined with the plastic wave propagation concept based on the rigid plastic assumption was adopted. The whole perforation process was divided into four consecutive stages, namely, bulging deformation, dishing deformation, ductile hole enlargement, and projectile exit. On the basis of the energy conservation principle, a new model was developed to predict the residual velocities of hemispherical-nosed projectiles that perforate thin steel plates at low velocities. The results obtained from the theoretical calculations by the present model were compared with the experimental results. Theoretical predictions were in good agreement with the experimental results in terms of both the radius of the bulging region and the residual velocity of the projectile when the strain rate effects of the target material during each stage were considered.

Mechanism of shear attenuation near the interface under combined compression and shear impact loading

We investigate the compressional/shear coupling plastic wave propagation characteristics analytically for ideal elastic–plastic materials in both stress and particle velocity spaces, focusing on the shear wave attenuation near the interface occurring in pressure–shear plate impact experiments. The results show that the shear attenuation is strongly associated with the wave propagation characteristics of the coupling waves. In the stress space, as the shear stress increases, an adjustment of the stress components is observed and the final stress state along the wave path is a combined pure shear- and hydrostatic pressure-state. In the particle velocity space, the wave structures with different loading and maximal transverse particle velocity are obtained. The maximal transverse particle velocity varies with the longitudinal velocity and forms a boundary line. Once the loading transverse velocity exceeds this line, a transverse particle velocity discontinuity occurs at the impact interface. If the bonding strength is sufficiently high, there will be a shear band in the target in the extreme vicinity of the interface.