Effective wave propagation in a prestressed nonlinear elastic composite bar

The problem of determining the effective incremental responseof nonlinearly elastic composite materials given some initialprestress is of interest in numerous application areas. In particular,the case when small-amplitude elastic waves pass through a prestressedinhomogeneous structure is of great importance. Of specificinterest is how the initial finite deformation affects the microstructureand thus the subsequent response of the composite. Modellingthis effect is in general extremely difficult. In this article,we consider the simplest problem of this type where the materialis a one dimensional composite bar consisting of two distinctphases periodically distributed. Neglecting lateral contractions,the initial deformation is thus piecewise homogeneous and wecan therefore determine the incremental behaviour semi-analytically,given the constitutive behaviour (strain energy function) ofthe phases in question. We apply asymptotic homogenization theoryin the deformed configuration in order to find the effectiveresponse of the deformed material in the low-frequency limitwhere the wavelength of the propagating waves is much longerthan the characteristic length scale of the microstructure.We close by considering the arbitrary frequency case and illustratehow the initial deformation affects the location of stop bandsand pass bands of the material. Work is under way to confirmthese results experimentally.     

Stability of post-fertilization traveling waves

This paper studies the stability of a family of traveling wave solutions to the system proposed by Lane et al. [D.C. Lane, J.D. Murray, V.S. Manoranjan, Analysis of wave phenomena in a morphogenetic mechanochemical model and an application to post-fertilization waves on eggs, IMA J. Math. Appl. Med. Biol. 4 (4) (1987) 309-331], to model a pair of mechanochemical phenomena known as post-fertilization waves on eggs. The waves consist of an elastic deformation pulse on the egg's surface, and a free calcium concentration front. The family is indexed by a coupling parameter measuring contraction stress effects on the calcium concentration. This work establishes the spectral, linear and nonlinear orbital stability of these post-fertilization waves for small values of the coupling parameter. The usual methods for the spectral and evolution equations cannot be applied because of the presence of mixed partial derivatives in the elastic equation. Nonetheless, exponential decay of the directly constructed semigroup on the complement of the zero eigenspace is established. We show that small perturbations of the waves yield solutions to the nonlinear equations decaying exponentially to a phase-modulated traveling wave.     

Differaction of plane deformation waves on a cylindrical cavity in an anisotropic bulk composed of a monoclinic system

An analytic-numerical method based on perturbations in the elastic moduli is proposed for solving problems involving the diffraction of two-dimensional longitudinal-shear harmonic deformation waves on a cavity with a smooth curvilinear cross section in an elastic medium with an anisotropy corresponding to a monoclinic system in the dynamic deformation plane. Results from a numerical analysis of the distribution of the dynamic stresses in the near diffraction field are presented for the case of longitudinal waves scattered on a dome-shaped cavity as the direction of the incident wave and the degree of anisotropy of the medium are varied. Donetsk State University. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 137–147, 1999.     

About the Microstructural Effects of Polycrystalline Materials and their Macroscopic Representation at Finite Deformation

In the sheet bulk metal forming field, the strict geometrical requirements of the workpieces result in a need of a precise prediction of the material behaviour. The simulation of such forming processes requires a valid material model, performing well for a huge variety of different geometrical characteristics and finite deformation. Because of the crystalline nature of metals, anisotropies have to be taken into account. Macroscopically observable plastic deformation is traced back to dislocations within considered slip systems in the crystals causing plastic anisotropy on the microscopic and the macroscopic level. A finite crystal plasticity model is used to model polycrystalline materials in representative volume elements (RVEs) of the microstructure. A multiplicative decomposition of the deformation gradient into elastic and plastic parts is performed, as well as a volumetric-deviatoric split of the elastic contribution. In order to circumvent singularities stemming from the linear dependency of the slip system vectors, a viscoplastic power-law is introduced providing the evolution of the plastic slips and slip resistances. The model is validated with experimental microstructural data under deformation. The validation on the macroscopic scale is performed through the reproduction of the experimentally calculated initial yield surface. Additionally, homogenised stress-strain curves from the microstructure build the outcome for a suitable effective material model. Through optimisation techniques, effective material parameters can be determined and compared to results from real forming processes. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Amplitude modulation of nonlinear waves in a thick elastic tube filled with an inviscid fluid

In the present work, employing the nonlinear equations of an incompressible, isotropic and elastic thick tube and the approximate equations of an incompressible inviscid fluid, and then utilizing the reductive perturbation technique the amplitude modulation of weakly nonlinear waves is examined. It is shown that the amplitude modulation of these waves is governed by a nonlinear Schrödinger(NLS) equation. The range of modulational instability of the monochromatic wave solution with the initial deformation, material and geometrical characteristics is discussed for some elastic materials.     

A crystal plasticity framework based on the Continuum Dislocation Dynamics theory: relaxation of an idealized micropillar

The motion and interaction of dislocation lines are the physical basis of the plastic deformation of metals. Although ‘discrete dislocation dynamic’ (DDD) simulations are able to predict the kinematics of dislocation microstructure (i.e. the motion of dislocations in a given velocity field) and therefore the plastic behavior of crystals in small length scales, the computational cost makes DDD less feasible for systems larger than a few micro meters. To overcome this problem, the Continuum Dislocation Dynamics (CDD) theory was developed. CDD describes the kinematics of dislocation microstructure based on statistical averages of internal properties of dislocation systems. In this paper we present a crystal plasticity framework based on the CDD theory. It consists of two separate parts: a classical 3D elastic boundary value problem and the evolution of dislocation microstructure within slip planes according to the CDD constitutional equations. We demonstrate the evolution of dislocation density in a micropillar with a single slip plane. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

不可压缩弹性固体中的二维应力波分析

本文研究不可压缩弹性固体中的二维应力波.首先对一般的应变能函数给出了分析简单波和激波的基本方程,然后求出了波速和相应的本征向量,证明在一般情况下有两组简单波和两组激波,最后举了平面变形和反平面变形两个例子.在平面变形的情况下,平面激波的斜反射问题一般无解.     

Space-time ray method for waves of small deformation in a nonlinear elastic medium

Space-time ray solutions for longitudinal waves of small deformation in a non-linear elastic medium are constructed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 61–72, 1984.     

The diffraction of plane elastic waves by a delaminated rigid inclusion when there is smooth contact in the delamination region

A solution of the problem of the diffraction of harmonic elastic waves by a thin rigid strip-like delaminated inclusion in an unbounded elastic medium, in which the conditions for plane deformation are satisfied, is proposed. We mean by a delaminated inclusion an inclusion, one side of which is completely bonded to the elastic medium, while the second does not interact in any way with it, or this interaction is partial. It is assumed that the conditions for smooth contact are satisfied in the delamination region. The method of solution is based on the use of previously constructed discontinuous solutions of the equations describing the vibrations of an elastic medium under plane deformation conditions. The problem therefore reduces to solving a system of three singular integral equations in the unknown stress and strain jumps at the inclusion. An approximate solution of the latter enabled formulae to be obtained that are convenient for numerical realization when investigating the stressed state in the region of the inclusion and its displacements when acted upon by incident waves.     

Effect of gravity on subject-specific human lung deformation

A biomechanical model of human lung is developed and used to investigate the effect of gravity on lung deformation. The lung is assumed to behave as a poro-elastic medium with spatially dependent elastic property. Finite element analysis is performed on a three-dimensional (3D) lung geometry reconstructed from a four-dimensional Computed Tomography (4DCT) scan dataset of human patient. The spatially dependent Young’s modulus (YM) values are estimated using inverse analysis from a linear elastic deformation model. The predicted deformation of selected landmarks is monitored with and without gravity, and compared with data obtained from 4DCT registration. The results show that gravity indeed significantly affects the magnitude and distribution of lung deformation with the maximum displacement enhanced by 54% in the direction of gravity, for the conditions investigated. In summary, the accuracy of predicted deformation is improved through incorporation of gravity in the biomechanical model of lung.     

The harmonic oscillations of a viscoelastic rod of triangular cross-section

Two exact solutions of the plane strain problem of the harmonic oscillations of a viscoelastic rod, the cross-section of which is a right triangle, are proposed. Either the normal displacement and the shear stress or the shear displacement and the normal stress of the side surface of the rod are given. Six dimensionless parameters which affect the dynamic deformation process are derived. Two parameters characterize the contribution of the viscous properties with respect to the elastic properties, two others define the logarithmic decrement of the longitudinal and shear harmonic waves, and two other parameters affect the wavelength of the corresponding wave and the velocity of motion of the wave front of these waves. The velocities of both types of waves and their wavelengths turn out to be greater than the velocities and wavelengths of the corresponding elastic waves. It is shown that, for certain values of the viscosity and the oscillation frequency, pseudo-resonance frequencies are possible which are higher than the resonance frequencies for an elastic medium.     

研究金属中激波构造与衰减的一个物理模型

本文给出了研究金属中激波构造与衰减的一个物理模型.为了建立高速形变下材料的本构方程和研究激波过渡带的构造,需要考虑二个独立的理论方面.首先,将比内能分解成弹性压缩能和弹性形变能,而将形变能作为弹性应变和熵的函数展开到三阶项,其中考虑了热与机械能的耦合效应.其次,从位错动力学角度建议了一个塑性松弛函数以便描述高温、高压下塑性流动的特性.另外,本文给出了一个常微分方程组用以计算定态激波过渡带中各状态变量的分布以及激波的厚度.倘若假定在激波上熵的跳跃可以忽略,并用Hugoniot压缩模量代替等熵压缩摸量,可以获得一个分析解.最后,本文还提出了求解平板对称碰撞中激波波头衰减的一个近似方法。     

非局部非对称弹性固体理论

本文基于非局部连续统场论和非线性连续体力学理论,建立了非局部非对称弹性固体的非线性理论.它完善和发展了Eringen等人所建立的非局部弹性场论.将文献[1]中所建立的非局部非对称弹性力学的线性理论推广到有限变形.证明了在非局部弹性固体中存在着非局部体力矩.非局部体力矩引起应力的非对称性,而非局部体力矩则由原子晶格相互作用形成的共价键所产生的.并应用本文建立的理论合理地解释了平面横波和纵波色散系关的不相似性.     

Love waves of SH type in an inhomogeneous transversely isotropic elastic medium

The asymptotics of high-frequency Love waves, which are analogous to transverse surface SH waves, is considered for a special type of anisotropy (transverse isotropy) of elastic media. The wave field is represented as a sum of the space-time (ST) caustic expansion and two additional ST ray series for faster (relative to the transverse surface wave) body waves, decaying exponentially with depth. Near the surface, the coefficients of the ST caustic and ray series, as well as the eikonals of waves, are determined in the form of expansions in a small parameter, which characterizes the proximity of the caustic of the ray field to the surface. With regard for the specific structure of the elasticity tensor of a transversely isotropic medium, the surface is treated as a plane. Interrelations between the parameters of elasticity, which are consistent with the conditions of the positivity of the elastic deformation energy and provide for the origination of the surface waves considered, are obtained.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 243–262.This work was supported by the Russian Foundation for Basic Research under grant No. 96-01-00666.     

Structurally damped elastic waves in 2D

The goal of the paper is to study structurally damped elastic waves in 2D. A suitable diagonalization procedure allows to derive WKB representations for solutions to the Cauchy problems. As consequences, we obtain results on energy decay with and without additional regularity for the data, Gevrey smoothing, and propagation of singularities for the visco‐elastic damped elastic wave model. Copyright © 2016 John Wiley & Sons, Ltd.     

Harmonic decomposition analysis of contact mechanics of bonded layered elastic solids

The paper presents an iterative method for obtaining footprint, pressure distribution, local deformation and sub-surface stress field for the contact between a rigid cylindrical indenter and an elastic flat substrate. The methodology is applicable for semi-infinite, as well as for thin or thick bonded elastic layered solids with high or low elastic moduli. All findings are in accord with the observed behaviour of hard wear resistant and soft solid lubricating coatings. It is shown that the decomposed contact pressure distribution into a series of harmonic waves induces sub-surface stress fields that decay into the depth of the solid according to their wavelengths. Consequently, conditions vis-à-vis fatigue spalling and adhesion performance may be predicted for given thickness of layered bonded elastic solids.     

Implementation and computational aspects of a 3D elastic wave modelling by PUFEM

This paper presents an enriched finite element model for three dimensional elastic wave problems, in the frequency domain, capable of containing many wavelengths per nodal spacing. This is achieved by applying the plane wave basis decomposition to the three-dimensional (3D) elastic wave equation and expressing the displacement field as a sum of both pressure (P) and shear (S) plane waves. The implementation of this model in 3D presents a number of issues in comparison to its 2D counterpart, especially regarding how S-waves are used in the basis at each node and how to choose the balance between P and S-waves in the approximation space. Various proposed techniques that could be used for the selection of wave directions in 3D are also summarised and used. The developed elements allow us to relax the traditional requirement which consists to consider many nodal points per wavelength, used with low order polynomial based finite elements, and therefore solve elastic wave problems without refining the mesh of the computational domain at each frequency. The effectiveness of the proposed technique is determined by comparing solutions for selected problems with available analytical models or to high resolution numerical results using conventional finite elements, by considering the effect of the mesh size and the number of enriching 3D plane waves. Both balanced and unbalanced choices of plane wave directions in space on structured mesh grids are investigated for assessing the accuracy and conditioning of this 3D PUFEM model for elastic waves.     

Stress softening, strain localization and permanent set in the circumferential shear of an incompressible elastomeric cylinder

Corresponding author, email huntley{at}umich.edu A constitutive theory for elastomeric materials has recentlybeen developed according to which stress is generated by differentmicromechanisms at different levels of deformation. When thedeformation is small, the stress is given by the usual theoryof rubber elasticity. As the deformation increases, some junctionsof the macromolecular microstructure rupture. Junctions thenre-form to generate a new microstructure. The constitutive equationallows for continuous scission of the original junctions andformation of new ones as deformation increases. The macromolecularscission causes stress reduction. The formation of new microstructuresresults in permanent set on release of external load. The present work considers a hollow circular cylinder composedof such a material, also assumed to be incompressible and isotropic.The cylinder is fixed rigidly at its inner surface and undergoesaxisymmetric deformation due to a uniform axial moment appliedat the outer surface. There develops an outer zone of materialwith the original microstructure and an inner zone of materialhaving undergone macromolecular scission, separated by a cylindricalinterface, the radius of which increases with the rotation ofthe outer surface. The shear deformation distribution, moment-rotationresponse and permanent set on release of moment are determined.It is found that microstructural scission can lead to higherlevels of shear deformation near the inner surface of the cylinderthan in the case of purely elastic response. It is also seenthat a residual state of high shear deformation can arise ina thin layer of material at the inner boundary of the cylinder.     

Computational Homogenisation of Polycrystalline Elastoplastic Microstructures at Finite Deformation

During sheet bulk metal forming processes both, flat geometries and three-dimensional structures change their shape significantly while undergoing large plastic deformations. As for forming processes, FE-simulations are often done before in situ experiments, a very accurate material model is required, performing well for a huge variety of different geometrical characteristics. Because of the crystalline nature of metals, anisotropies have to be taken into account. Macroscopically observable plastic deformation is traced back to dislocations within considered slip systems in the crystals causing plastic anisotropy on the microscopic and the macroscopic level. A finite crystal plasticity model is used to model the behaviour of polycrystalline materials in representative volume elements (RVEs) of the microstructure. A multiplicative decomposition of the deformation gradient into elastic and plastic parts is performed, as well as a volumetric-deviatoric split of the elastic contribution. In order to circumvent singularities stemming from the linear dependency of the slip system vectors, a viscoplastic power-law is introduced providing the evolution of the plastic slips and slip resistances. The model is validated with experimental microstructural data under deformation. Through homogenisation and optimisation techniques, effective stress-strain curves are determined and can be compared to results from real forming processes leading to a suitable effective material model. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Propagation of weak perturbations in cracked porous media

A three-velocity, three-pressure mathematical model is proposed which enables one to study wave processes in the case of a double porosity, deformable, fluid-saturated medium. This model takes account of the differences in the velocities and pressures in pore systems of different characteristic scales of the pores, fluid exchange between these pore systems and the unsteady forces due to interphase interactions. It is established that a single transverse and three longitudinal waves: one deformation wave and two filtration waves, propagate in such a medium. The existence of two filtration waves is associated with the two different characteristic scales of the pores and the difference in the velocities and pressures of the fluid in these pore systems. The filtration waves decay considerably more rapidly than the deformation and transverse waves. The velocities of the deformation and transverse waves are mainly determined by the elastic moduli of the skeleton. The velocity and decay of the first filtration wave depend strongly on the intensity of the interphase interaction force while the velocity of the second filtration wave depends strongly on the rate of mass exchange between the pores and the cracks. The rate of decay of the second filtration wave is significantly higher than that of the first filtration wave.