Reflection of plane waves at the free surface of a fibre-reinforced elastic half-space

The propagation of plane waves in fibre-reinforced, anisotropic, elastic media is discussed. The expressions for the phase velocity of quasi-P (qP) and quasi-SV (qSV) waves propagating in a plane containing the reinforcement direction are obtained as functions of the angle between the propagation and reinforcement directions. Closed form expressions for the amplitude ratios for qP and qSV waves reflected at the free surface of a fibre-reinforced, anisotropic, homogeneous, elastic half-space are obtained. These expressions are used to study the variation of amplitude ratios with angle of incidence. It is found that reinforcement has a significant effect on the amplitude ratios and critical angle     

Nondestructive Evaluation of Anisotropy in Composite Materials Via Acoustic Birefringence

A linearly polarized ultrasonic shear wave propagating in fiber-reinforced composites produces an elliptical vibration pattern due to birefringence. The acoustic birefringence of shear waves is analyzed and applied for evaluation of in-plane stiffness anisotropy of uni-axial nonfabric, biaxial, and quadra-axial fabric composite materials. The parameters of the elliptical motion are determined by measurements of the amplitude and phase of the transmitted/reflected shear wave as functions of the receiver polarization angle. The strength of birefringence is derived from the measured elliptical pattern and used to quantify the in-plane stiffness asymmetry in glass and carbon fiber-reinforced nonfabric and fabric composites. The technique is shown to be sensitive enough to detect damage induced variation in stiffness anisotropy. The correlation between the asymmetry of shear stiffness and orientation of impact cracking in composites is also discussed.     

Wave Propagation in a Green–Naghdi Thermoelastic Solid with Diffusion

The Green and Naghdi theory of thermoelasticity is applied to study plane-wave propagation in an elastic solid with thermo-diffusion. The governing equations of an elastic solid with generalized thermo-diffusion are solved to show the existence of three coupled longitudinal waves and a shear vertical (SV) wave in a two-dimensional model of the solid with thermo-diffusion. The reflection of plane waves from a thermally insulated stress-free surface of an elastic solid with thermo-diffusion is also studied. A non-homogeneous system of four equations in reflection coefficients is obtained. The speeds of the plane waves are computed numerically and plotted against frequency for a particular range. The complex absolute values of the reflection coefficients of all reflected waves are computed numerically and plotted against the angle of incidence of the striking wave at the free surface. The effects of diffusion parameters are shown graphically for speeds and reflection coefficients of plane waves.     

Initiation and propagation of a penny-shaped crack in a finitely deformed incompressible elastic medium

The effects that the initial lateral stress has on the initiation and the propagation of a penny-shaped crack are investigated on the basis of the theory of small deformations superposed on finite deformation for an incompressible elastic material. Using the methods of the Laplace and Hankel transforms, the crack shape function and the stress distribution with singularities in the crack plane are obtained in closed forms for the crack propagating at a constant speed in the Mooney material. The dynamic stress-intensity factor is obtained as a function of the initial lateral stretch and the ratio of the crack speed to the shear wave speed. For the same crack speed, the value of the dynamic stress-intensity factor increases with increasing lateral stretch, but decreases if the lateral compression increases.The dynamic solutions reduce to the associated static solutions at zero crack speed. For the stationary crack, the stress-intensity factor is shown to be independent of the initial stress. However, the initial lateral stretch increases, but the lateral compression decreases the value of the critical stress required for the initiation of crack growth on the basis of the Griffith theory. The central crack opening displacement is shown to decrease if the lateral stretch increases or the lateral compression decreases.     

Reflection and transmission of waves from a plane interface between two microstretch solid half-spaces

In this paper, we have investigated the wave propagation and their reflection and transmission from a plane interface between two different microstretch elastic solid half-spaces in perfect contact. It is shown that there exist five waves in a linear homogeneous isotropic microstretch elastic solid, one of them travel independently, while other waves are two sets of two coupled waves. It is also shown that these waves travel with different velocities, three of which disappear below a critical frequency. Amplitude ratios and energy ratios of various reflected and transmitted waves are presented when a set of coupled longitudinal waves and a set of coupled transverse waves is made incident. It is found that the amplitude ratios of reflected and transmitted waves are functions of angle of incidence, frequency and are affected by the elastic properties of the media. Some special cases have been reduced from the present formulation.     

A Lagrangian model for hardening behaviour of materials at finite deformation based on the right plastic stretch tensor

In this paper a modified multiplicative decomposition of the right stretch tensor is proposed and used for finite deformation elastoplastic analysis of hardening materials. The total symmetric right stretch tensor is decomposed into a symmetric elastic stretch tensor and a non-symmetric plastic deformation tensor. The plastic deformation tensor is further decomposed into an orthogonal transformation and a symmetric plastic stretch tensor. This plastic stretch tensor and its corresponding Hencky’s plastic strain measure are then used for the evolution of the plastic internal variables. Furthermore, a new evolution equation for the back stress tensor is introduced based on the Hencky plastic strain. The proposed constitutive model is integrated on the Lagrangian axis of the plastic stretch tensor and does not make reference to any objective rate of stress. The classic problem of simple shear is solved using the proposed model. Results obtained for the problem of simple shear are identical to those of the self-consistent Eulerian rate model based on the logarithmic rate of stress. Furthermore, extension of the proposed model to the mixed nonlinear isotropic/kinematic hardening behaviour is presented. The model is used to predict the nonlinear hardening behaviour of SUS 304 stainless steel under fixed end finite torsional loading. Results obtained are in good agreement with the available experimental results reported for this material under fixed end finite torsional loading.     

Fabrics in polycrystalline ice deformed experimentally at −10°C

Blocks of polycrystalline ice were compressed experimentally in an apparatus that restricted the deformation to plane strain at ?10°C, under constant load and at strain rates varying between 10^?8 to 10^?9 s^?1. By inscribing initially circular markers on a face perpendicular to the axis of plane strain, it is possible to measure the strain heterogeneity. The shortening and cumulative strain (γ_oc) measured locally are seen, in general, to be different from the nominal bulk shortening strain. Using these strain markers it has been possible to compare the c-axis preferred-orientation with strain distribution.Variation in ice c-axis fabrics correlate with changes in: (1) mode of deformation from pure shear to combined pure and simple shear, and (2) the magnitude of the finite strain. In pure shear there is a rapid transition from randomly oriented c-axes of the initial ice aggregate to a broad diffuse pattern lying between the 45° and 85° small circles (centred about the shortening axis), and thence to a 60° small circle distribution containing two maxima that can be symmetrically related to the bulk axes of strain. Where there is a component of simple shear, this two-maxima pattern is more asymmetric, having a strong c-axis maximum that can be attributed to the simple shear and a weaker portion of the fabric contributed by the pure shear deformation. These fabric transitions are analogous to those described in natural ice-sheets.     

On the critical angle for ocean waves entering shore fast ice

A theoretical model is derived and presented for sea waves entering shore fast ice at other than normal incidence. The model assumes that the sea ice behaves as a thin elastic plate overlying a deep fluid foundation. Three ice-coupled waves are generated; a propagating wave, and two inhomogeneous plane waves which rapidly decay with penetration. At a certain angle of incidence, the critical angle, these ice-coupled waves coalesce into a wave travelling along the ice edge with an amplitude which varies into the ice cover as the sum of a constant term and an attenuated pattern of corrugations. At angles exceeding the critical angle, the constant term becomes evanescent but the general picture remains the same. A novel break-up mechanism is proposed, based on this modulation of amplitude with distance from the edge, for shore fast ice and ice floes which are large compared to the incident sea's wavelengths.     

Thermodynamic aspects of transverse wave propagation in an incompressible hyperelastic solid

Summary Thermodynamic considerations in the propagation of finite amplitude plane shear waves in an incompressible isotropic hyperelastic solid are examined, in particular the validity of the assumption of zero thermal conductivity which results in the adiabatic approximation. Two problems are considered for a solid withdc/d|F|>0, wherec is the transverse wave speed andF is the shear component of the deformation gradient tensor for simple shear. These problems, which are used to illustrate thermodynamic effects, are the sudden application of a spatially uniform shearing stress to the free surface of an unstressed half space at rest and the sudden removal of the spatially uniform shearing stress on the surface of a half space at rest and in a state of simple shear.With 2 Figures     

Dynamic growth of anti-plane shear cracks in ideally plastic crystals

A near-tip asymptotic analysis is given for the stress and deformation field near the tip of crack propagating dynamically under anti-plane shear in an ideally plastic single crystal. A paricular class of orientations of the crack relative to the crystals is considered so that the yield is so simple diamond shape (relative to directions along perpendicular to the crack) in the plane of the anti-plane shear stresses. The near-tip solution is shown to consists of sectors which carry constant stresses, at yield levels, corresponding to adjacent vertices on the diamond-shaped yield locus, and which are joined along an elastic-plastic shock discontinuity. All plastic flow in the near-tip region occurs in the shock. Plastic strains and particle velocity are finite at the crack tip. The plastic strain is proportional to the elastic strain at onset of yielding and is inversely proportional to the elastic Mach number associated with the speed of crack growth.     

Mechanical behavior of glass/epoxy tubes under combined static loading. Part I: Experimental

A series of biaxial static tests of E-glass/epoxy tubular specimens [±45]₂, subjected to combined torsion and tension/compression were performed to simulate complex stress states encountered in a wind turbine rotor blade. The failure locus in the effective axial-shear stress plane was derived experimentally while in-plane strain tensor components were measured in the tube outer surface. By means of shell theory and strain measurements in the surface of the specimen, the in-plane shear response of the outer ply was obtained, revealing dependence each time to the combined tube loading. The correlation established between the ratio of transverse normal and in-plane shear stress in the principal coordinate ply system and the elastic shear modulus, suggested a strong dependence, warning on the implications for design and certification procedures.     

Asymptotic deformation and stress fields at the tip of a sharp notch in an elastic-plastic material

The singular elastic-plastic stress, strain and the displacement fields at the tip of a sharp notch for both plane stress and plane strain conditions are investigated analytically. The material is assumed to be governed by the deformation theory of plasticity with linear strain hardening characteristic. Since the elastic strain is retained in the analysis, the final strain and displacement fields can be separated into the elastic and the plastic parts. In the case with zero notch angle, the results reduce to the classical crack problem. The relationship of the amplitude of the near crack tip elastic-plastic field to the elastic far field is obtained. Both mode I and mode II cases are investigated. The mixed mode case is also discussed.     

Evolution of ultrasonic velocity and dynamic elastic moduli with shear strain in granular layers

Ultrasonic wave transmission has been used to investigate processes that influence frictional strength, strain localization, fabric development, porosity evolution, and friction constitutive properties in granular materials under a wide range of conditions. We present results from a novel technique using ultrasonic wave propagation to observe the evolution of elastic properties during shear in laboratory experiments conducted at stresses applicable to tectonic faults in Earth’s crust. Elastic properties were measured continuously during loading, compaction, and subsequent shear using piezoelectric transducers fixed within shear forcing blocks in the double-direct-shear configuration. We report high-fidelity measurements of elastic wave properties for normal stresses up to 20 MPa and shear strains up to 500 % in layers of granular quartz, smectite clay, and a quartz-clay mixture. Layers were 0.1–1 cm thick and had nominal contact area of $5 \mathrm{cm} \!\times \! 5 \mathrm{cm}$ . We investigate relationships among frictional strength, granular layer thickness, and ultrasonic wave velocity and amplitude as a function of shear strain and normal stress. For layers of granular quartz, P-wave velocity and amplitude decrease by 20–70 % after a shear strain of 0.5. We find that P-wave velocity increases upon application of shear load for layers of pure clay and for the quartz-clay mixture. The P-wave amplitude of pure clay and quart-clay mixtures first decreases by $\sim $ 50 and 30 %, respectively, and then increases with additional shear strain. Changes in P-wave speed and wave amplitude result from changes in grain contact stiffness, crack density and disruption of granular force chains. Our data indicate that sample dilation and shear localization influence acoustic velocity and amplitude during granular shear.     

Analysis of intrinsic stability criteria for isotropic third-order Green elastic and compressible neo-Hookean solids

Internal stability of isotropic nonlinear elastic materials under homogeneous deformation is studied. Results provide new insight into various intrinsic stability measures, first proposed elsewhere, for generic nonlinear elastic solids. Three intrinsic stability criteria involving three different tangent elastic stiffness matrices are considered, corresponding to respective increments in strain measures conjugate to thermodynamic tension, first Piola–Kirchhoff stress, and Cauchy stress. Primary deformation paths of interest include spherical (i.e., isotropic) deformation, uniaxial strain, and simple shear; unstable modes are not constrained to remain along primary deformation paths. Effects of choices of second- and third-order elastic constants on intrinsic stability are systematically studied for physically realistic ranges of constants. For most cases investigated here, internal stability according to strain increments conjugate to Cauchy stress is found to be the most stringent criterion. When third-order constants vanish, internal stability under large compression tends to decrease as Poisson’s ratio increases. When third-order constants are nonzero, a negative (positive) pressure derivative of the shear modulus often promotes unstable modes in compression (tension). For large shear deformation, larger magnitudes of third-order constants tend to result in more unstable behavior, regardless of the sign of the pressure derivative of the shear modulus. A compressible neo-Hookean model is generally much more intrinsically stable than second- and third-order elastic models when Poisson’s ratio is non-negative.     

Interaction of harmonic plane waves with a mobile elastic phase boundary in anti-plane shear

This paper is concerned with the effect of sustained infinitesimal harmonic plane wave excitation of a phase boundary in a non-linearly elastic material that is subject to anti-plane shear deformation. The phase boundary is capable of motion that is here described by a harmonic travelling waveform. The reflected wave is also a harmonic plane wave, however the transmitted wave may be either in the form of a harmonic plane wave or a harmonic surface wave. The phase boundary motion is determined on the basis of a standard kinetic relation that involves a single mobility parameter. This gives phase boundary motion that is synchronized with the incident wave for the case of a transmitted plane wave, but is not synchronized with the incident plane wave for the case of a transmitted surface wave. A certain fraction of the energy provided by the incident wave is dissipated by phase boundary motion in a fashion that can be explicitly quantified. Special incident angles can suppress the reflected wave, suppress the transmitted wave or cause the dissipation to vanish.     

Some comments on the dynamic properties of anisotropic and strongly anisotropic pre-stressed elastic solids

The effect of pre-stress on the propagation of small amplitude waves in an incompressible, transversely isotropic elastic solid is discussed in respect of the most general appropriate strain energy function. A simple set of sufficient conditions is noted which ensures that two real wave speeds exist for all directions of propagation. In the case of bi-axial primary deformations, and for propagation within each principal plane of the homogeneous primary deformation, the propagation condition is factorised and conditions which are both necessary and sufficient to ensure the existence of two real wave speeds are established. The paper also includes some graphical illustrations of the associated slowness and wave surfaces and discussion of the strongly anisotropic case, for which the extensional modulus along the preferred fibre direction is much larger than other material parameters.     

Analysis of elastoplastic sharp notches

In this paper the elastoplastic solutions with higher-order terms for apex V-notches in power-law hardening materials have been discussed. Two-term expansions of the plane strain and the plane stress solutions have been obtained. It has been shown that the leading-order singularity approaches the value for a crack when the notch angle is not too large. In plane strain cases the elasticity does not enter the second-order solutions when the notch opening angle is too small. For a large notch angle, the two-term expansions of the plane strain near-tip fields are described by a single amplitude parameter. The plane stress solutions generally contain the elasticity terms. The boundary layer formulations based on the small-strain plasticity theory confirm that a dominance zone exists ahead of the notch tip. Finite element results give good agreement to the asymptotic solutions under both plane strain and plane stress conditions. The second-order terms cannot improve the predictions significantly. The near-tip fields are dominated by a single parameter. Finite element calculations under the finite strain J ₂-flow plasticity theory revealed that the finite strains can only affect local characterization of the asymptotic solution. The asymptotic solution has a large dominance zone around the notch tip. For an apex notch bounded to a rigid substrate the leading-order singularity falls with the notch angle significantly more slowly than in the homogeneous material. It vanishes at the notch angle about 135° for all power-hardening exponents. The elasticity effects enter the second-order solutions when the notch angle becomes large enough. The tip fields are characterized by the hydrostatic stress and the shear stress ahead of the notch.     

Biaxial fatigue for proportional and non‐proportional loading paths

In order to study the use of a local approach to predict crack‐initiation life on notches in mechanical components under multiaxial fatigue conditions, the study of the local cyclic elasto‐plastic behaviour and the selection of an appropriate multiaxial fatigue model are essential steps in fatigue‐life prediction. The evolution of stress–strain fields from the initial state to the stabilized state depends on the material type, loading amplitude and loading paths. A series of biaxial tension–compression tests with static or cyclic torsion were carried out on a biaxial servo‐hydraulic testing machine. Specimens were made of an alloy steel 42CrMo4 quenched and tempered. The shear stress relaxations of the cyclic tension–compression with a steady torsion angle were observed for various loading levels. Finite element analyses were used to simulate the cyclic behaviour and good agreement was found. Based on the local stabilized cyclic elastic–plastic stress–strain responses, the strain‐based multiaxial fatigue damage parameters were applied and correlated with the experimentally obtained lives. As a comparison, a stress‐invariant‐based approach with the minimum circumscribed ellipse (MCE) approach for evaluating the effective shear stress amplitude was also applied for fatigue life prediction. The comparison showed that both the equivalent strain range and the stress‐invariant parameter with non‐proportional factors correlated well with the experimental results obtained in this study.     

On finite amplitude waves in heterogeneous elastic solids

The perturbation technique of the ‘stretching of the coordinates’ is used to obtain first and second order perturbation solutions of finite amplitude plane waves which propagate into an elastic half-space whose material property varies in the direction of the propagation. The interaction between nonlinearity and heterogeneity is discussed, and the results are illustrated by means of two examples: the longitudinal waves propagating in an elastic half-space with harmonic heterogeneity; the shear wave in a half-space whose property varies as A[1 + ?(X/L)ⁿ], where A, ? ? 1, L, and n are constants, and X denotes the initial particle position measured normal to the plane boundary.