Rapid Sliding Indentation with Friction of a Pre-Stressed Thermoelastic Material

A rigid indentor travels with a constant speed over the surface of an isotropic thermoelastic half-space. Friction exists between the indentor and half-space, and the latter is initially in equilibrium at a uniform temperature under a uniform normal pre-stress. This pre-stress, below but near yield, is assumed to produce deformations that dominate the additional deformations due to indentation. Thus, the process is treated as one of small deformations superposed upon (relatively) large. The governing equations for the superposed deformation are those of nonisotropic coupled thermoelasticity. A steady-state two-dimensional study uses robust asymptotic analytical solutions to reduce the associated mixed boundary value problem to a classical singular integral equation which can be solved analytically. The solutions show that the pre-stress-induced de facto nonisotropy alters the values of the rotational and dilatational wave and Rayleigh speeds in the half-space and, in the case of a compressive pre-stress, generates a second, lower, critical speed. In addition, pre-stress generates noncritical sliding speeds at which the friction-dependent integral equation eigenvalue changes sign. For purposes of illustration, expressions for the half-space surface temperature change and its average over the contact zone, the equations necessary to determine contact zone size and location, the resultant moment on the indentor, and the maximum compressive stress on the contact zone are presented for a parabolic indentor. This revised version was published online in August 2006 with corrections to the Cover Date.     

Wave propagation in a pre-stressed highly elastic body: two exact solutions for Boussinesq problems

Two problems of wave propagation induced by surface loads in a compressible neo-Hookean half-space, initially at rest under a uniform pre-stress, are considered. One problem concerns a plane-strain situation, the other, one of axial symmetry. An accepted general procedure, that of superposing infinitesimal deformations upon the possibly large deformations due to pre-stress, is carried out completely in terms of tractable exact solutions for both the surface behavior and the full field, and analytical expressions for all wave speeds.

The results show that for a tensile pre-stress above a critical value, a negative Poisson effect occurs; for compressive pre-stresses, Rayleigh waves disappear at a critical value. Indeed, all wave speeds in the deformed configuration (effective wave speeds), as well as the solutions, for both problems are clearly sensitive to material properties and to both the magnitude and nature (compressive or tensile) of the pre-stress. In particular, the constraint imposed by plane strain appears to enhance this sensitivity.     

The transient elastodynamic field excited by trans-Rayleigh trains

The elastic wave field due to a surface load in motion over an elastic half-space is investigated. The model serves as a canonical solution for the modelling of high speed ‘trans-Rayleigh’ trains. The analysis presented leads to closed form expressions for the particle displacement, conical waves and Rayleigh waves as separate contributions. The linearized elastodynamic equations are mapped into a proper form in order to apply the Cagniard-de Hoop technique and find closed form time domain solutions for the particle displacement in the subsonic state, transonic state and supersonic state. A special transformation is used that yields closed form space-time domain expressions for the Conical wave as well as the Rayleigh wave contributions. Attention is focussed on surface source speeds in the neighbourhood of the Rayleigh wave speed and speeds that exceed the wave speed of the shear wave. Numerical results for the conical wave field and Rayleigh wave field are presented at observation points just below the surface showing the enormous effects of the Rayleigh wave at source speeds in the near vicinity of the Rayleigh wave speed.     

Two illustrations of rapid crack growth in a pre-stressed compressible neo-Hookean material

An unbounded isotropic compressible neo-Hookean solid is initially in equilibrium under uniform tensile (possibly large) pre-stress. In one case, plane strain conditions generate slit crack growth at a constant sub-critical rate; in the other, axial symmetry produces penny-shaped crack growth. The procedure of superposing infinitesimal deformations upon those that are large is carried out in terms of tractable exact full-field solutions.These solutions are examined apart from a specific fracture mechanics model, nevertheless, they show that pre-stress induces, in addition to the expected anisotropy, a critical value above which a negative Poisson effect occurs. It is also found that dilatational, rotational and Rayleigh wave speeds decrease, and that the decrease is greater for the plane strain state associated with slit crack growth than for the axially symmetric state of the penny-shaped crack.Dynamic stress intensity factors are also extracted, and found to fall below those for a linear isotropic solid at the same pre-stress and crack growth rate. Moreover, the range of growth rates for sub-critical crack propagation is also decreased.     

Rapid sliding motion of a rigid frictionless indentor with a flat base over a thermoelastic half-space

The three-dimensional, rapid sliding indentation of a deformable half-space by a rigid indentor of a flat elliptical base is treated in this paper. The response of the material that fills the half-space is assumed to be governed by coupled thermoelasticity. The indentor translates without friction on the half-space surface at a constant sub-Rayleigh speed and the problem is treated as a steady-state one. An exact solution is obtained that is based on a Green’s function approach, integral equations, and Galin’s theorem. A closed-form expression for the distributed contact pressure under the elliptical base of the indentor is derived. Representative numerical results are given illustrating the effects of the indentor velocity, indentor geometry, and parameters of the thermoelastic solid on the contact displacement. Since there is an analogy between the steady-state theories of thermoelasticity and poroelasticity, the present results carry over to the latter case directly.     

On a 3D moving load problem for an elastic half space

The paper deals with 3D dynamic response of an elastic half-space loaded by a point force moving at a constant speed along a straight line on the surface. The problem is formulated within the framework of the asymptotic hyperbolic–elliptic model developed earlier by two of the authors. The validity of the model is restricted to the range of speeds close to the Rayleigh wave speed. Steady-state near-field solutions are derived in terms of elementary functions. Transient analysis of surface motion illustrates peculiarities of the resonance associated with the Rayleigh wave.     

高速移动荷载下黏弹性半空间体的动力响应

分别以移动荷载和黏弹性半空间体模拟运动列车荷载和地基,分析了地基在运动列车作用下的动力响应.首先采用Green函数法求解黏弹性半空间体在各种移动荷载模式作用下的动力响应的解析解,包括恒常和简谐移动点源、线源和面源荷载.然后采用IFFT算法和自适应数值积分算法计算解析解中的二维积分,得到了包括低音速、跨音速和超音速移动荷载作用下位移的数值结果.最后分析了速度对位移的分布和最大值的影响,发现当速度大于Rayleigh波速时,位移发生显著变化.     

Rapid sliding indentation with friction on a transversely isotropic thermoelastic half-space

A rigid insulated die slides at a constant sub-critical speed on a transversely isotropic half-space in the presence of friction. In a two-dimensional analysis of the dynamic steady-state, the coupled equations of thermoelasticity are invoked. All elements of the Coulomb friction model are strictly enforced, thus giving rise to auxiliary conditions, including two unilateral constraints.Robust asymptotic forms of an exact solution to a related problem with unmixed boundary conditions lead to analytical solutions for the sliding indentation problem. The solution expressions, abetted by calculations for zinc, show the role of frictional heating on the half-space surface. The effects of friction and sliding speed on contact zone size and location and average contact zone temperature are also studied.The analysis is aided by factoring procedures that simplify the complicated forms that arise in anisotropic elasticity. A scheme that renders expressions for roots of certain irrational functions analytic to within a single quadrature also plays a role.     

A new secular equation for slip waves along the interface of two dissimilar anisotropic elastic half-spaces in sliding contact

We consider wave propagation along the interface of two dissimilar anisotropic elastic half-spaces that are in sliding contact. A new secular equation is obtained that covers all special cases in one equation. One special case is when a Rayleigh wave (called the R$R$-wave) can propagate in both half-spaces with the same wave speed. Another special case is when a slip wave (called the S$S$-wave) can propagate in each of the half-spaces with the same wave speed. If a Rayleigh wave and a slip wave can propagate in one of the half-spaces it is called the RS-wave. In this case an interfacial slip wave exists in which the other half-space is at rest unless an RS-wave can also propagate in the other half-space. The results for general anisotropic elastic materials are applied to orthotropic materials.     

Steady-State Response of a Thermoelastic Half-Space to the Rapid Motion of Surface Thermal/Mechanical Loads

Shear and normal tractions and a heat flux are applied to a largely arbitrary area that moves with constant subsonic speed over a half-space surface. The half-space is a coupled thermoelastic solid of the Jeffreys type, so that the governing steady-state equations involve three thermoelastic characteristic lengths and a dimensionless coupling constant. This constant and one of the lengths remain in the limit as the solid reduces to the standard coupled thermoelastic material. The problem is solved exactly in an integral transform space, and asymptotic expressions for the normal displacement and the temperature change induced on the half-space surface are extracted. These are in principle valid for large distances from the loading zone as measured along its line of travel but, because the scaling dimension is of O(10^-14)μ m, they are robust. Exact inversions are performed, and the results show marked dependence on both loading zone speed and thermoelastic parameters. Indeed, the role of the latter is enhanced as the speed is increased. Singular behavior is found, in particular, when the loading zone moves with the effective thermoelastic Rayleigh speed, an exact formula for which is also given. This revised version was published online in August 2006 with corrections to the Cover Date.     

Rayleigh Surface Waves on a Kelvin-Voigt Viscoelastic Half-Space

In this paper we consider the propagation of Rayleigh surface waves in an exponentially graded half-space made of an isotropic Kelvin-Voigt viscoelastic material. Here we take into account the effect of the viscoelastic dissipation energy upon the corresponding wave solutions. As a consequence we introduce the damped in time wave solutions and then we treat the Rayleigh surface wave problem in terms of such solutions. The explicit form of the secular equation is obtained in terms of the wave speed and the viscoelastic inhomogeneous profile. Furthermore, we use numerical methods and computations to solve the secular equation for some special homogeneous materials. The results sustain the idea, existent in literature on the argument, that there is possible to have more than one surface wave for the Rayleigh wave problem.     

Sudden deceleration or acceleration of an intersonic shear crack

Sudden jumps in the crack tip velocity were revealed by numerical simulation (in both continuum/cohesive element and molecular dynamics approaches) and experiments for rapid shear cracking. The cracking velocity may accelerate from a sub-Rayleigh speed to the intersonic range, or from an intersonic speed to a higher one, when the reflected impact wave reloads the crack tip. On the other hand, the cracking velocity may decelerate from an intersonic speed to a lower one or recede to the sub-Rayleigh range when the fracture driving force declines. The velocity change encountered during intersonic cracking plays a different role from that in the acceleration or deceleration of a subsonic crack. A crack propagating at an intersonic speed would leave a shear wave trailing behind. When the crack decelerates or accelerates, the effect of the trailing wave will lead to a transition period from one steady-state solution of crack tip singularity to another. This investigation aims at quantifying these processes. The full field solution of an intersonic mode II crack whose speed changed suddenly from one velocity (intersonic or subsonic) to another (intersonic or subsonic) is given in closed form. The solution is facilitated via superposing a series of propagating crack problems that are loaded by dislocations to seal the unwanted crack-face sliding or by concentrated forces moving at various speeds to negate the crack-face traction. In contrast to the subsonic solution, the results in the intersonic case indicate that the elastic fields around the crack tip depend on the deceleration or acceleration history that is traced back over a long time. Singularity matching dictates the jump that may actually take place.     

Van Leer格式在全速域范围的推广

通过对格式耗散项的修正将Van Leer格式推广至全速域流场求解范围.对格式耗散项的分析表明,在低马赫数流动情况下格式耗散项中不应包含声速项,以此为依据对Van Leer迎风分裂格式提出了耗散项的修正方法.结合对控制方程时间导数项的预处理,修正后的格式能够成功地模拟低速流动问题,同时在其他马赫数范围内也不损失格式的收敛...     

Subsonic and intersonic mode II crack propagation with a rate-dependent cohesive zone

A recent experimental study has demonstrated the attainability of intersonic shear crack growth along weak planes in otherwise homogeneous, isotropic, linear elastic solids subjected to remote loading conditions (Rosakis et al., Science 284 (5418) (1999) 1337). The relevant experimental observations are summarized briefly here and the conditions governing the attainment of intersonic crack speeds are examined. Motivated by experimental observations, subsonic and intersonic mode II crack propagation with a rate-dependent cohesive zone is subsequently analyzed. A cohesive law is assumed, wherein the cohesive shear traction is either a constant or varies linearly with the local sliding rate. Complete decohesion is assumed to occur when the crack tip sliding displacement reaches a material-specific critical value. Closed form expressions are obtained for the near-tip fields. With a cohesive zone of finite size, it is found that the dynamic energy release rate is finite through out the intersonic regime. Crack tip stability issues are addressed and favorable speed regimes are identified. The influence of shear strength of the crack plane and of a rate parameter on crack propagation behavior is also investigated. The isochromatic fringe patterns predicted by the analytical solution are compared with the experimental observations of Rosakis et al. (1999) and comments are made on the validity of the proposed model.     

Existence of Surface Waves and Band Gaps in Periodic Heterogeneous Half-spaces

We find a sufficient condition for the existence of surface (Rayleigh) waves based on the Rayleigh-Ritz variational method. When specialized to a homogeneous half-space, the sufficient condition recovers the known criterion for the existence of subsonic surface waves. A simple existence criterion in terms of material properties is obtained for periodic half-spaces of general anisotropic materials. Further, we numerically compute the dispersion relation of the surface waves for a half-space of periodic laminates of two materials and demonstrate the existence of surface wave band gaps.     

Finite-amplitude Love waves in a pre-stressed compressible elastic half-space with a double surface layer

The dispersive behavior of finite-amplitude time-harmonic Love waves propagating in a pre-stressed compressible elastic half-space overlaid with two compressible elastic surface layers of finite thickness is investigated. The half-space and layers are made of different pre-stressed compressible neo-Hookean materials. The dispersion relation which relates wave speed and wavenumber is obtained in explicit form. Results for the energy density and energy flux of the waves are also presented. The special case where the interfaces between the layers and the half-space are principal planes of the left Cauchy–Green deformation tensor is also investigated. Numerical results are presented showing the variation of the Love wave speed with the pre-stress and the propagation angle.     

A knife-edge load traveling on the surface of an elastic halfspace

A load moving on the surface of an elastic halfspace forms a basic problem that is related to different fields of engineering, such as the subsoil response due to vehicle motion or the ultrasound field due to an angle beam transducer. Many numerical techniques have been developed to solve this problem, but these do not provide the fundamental physical insights that are offered by closed form solutions, which are very rare in comparison. This paper describes the development and analysis of the closed form space-time domain solution for a knife-edge load, i.e. a line segment of normal traction, moving at a constant speed on the surface of an elastic halfspace. The various contributions making up the exact solution, obtained with the Cagniard-De Hoop method, produce all the complicated wave patterns from this distributed type of loading. Examples are the transient wave field at the starting position of the load, angled conical and plane waves propagating into the solid, Rayleigh waves propagating along the surface, and head waves spreading and attenuating in specific directions from the loading path. The influence of the load speed on the wave field is investigated by considering the singularities in the relevant complex domains, for each sonic range relative to the bulk wave velocities. The characteristic wave fronts and wave patterns as exhibited by the particle displacements are evaluated for subsonic, transonic and supersonic load speeds.     

Wave propagation along a non-principal direction in a compressible pre-stressed elastic layer

The dispersive behavior of small amplitude waves propagating along a non-principal direction in a pre-stressed, compressible elastic layer is considered. One of the principal axes of stretch is normal to the elastic layer and the direction of propagation makes an angle θ with one of the in-plane principal axes. The dispersion relations which relate wave speed and wavenumber are obtained for both symmetric and anti-symmetric motions by formulating the incremental boundary value problem for a general strain energy function. The behavior of the dispersion curves for symmetric waves is for the most part similar to that of the anti-symmetric waves at the low and high wavenumber limits. At the low wavenumber limit, depending on the pre-stress and propagation angle, it may be possible for both the fundamental mode and the next lowest mode to have finite phase speeds, while other higher modes have an infinite phase speed. At the high wavenumber limit, the phase speeds of the fundamental mode and the higher modes tend to the Rayleigh surface wave speed and the limiting wave speeds of the layer, respectively. Numerical results are presented for a Blatz–Ko material and the effect of the propagation angle is clearly illustrated.     

各向异性弹性体光滑接触界面上的滑移脉冲波

利用Stroh公式,Fourier分析和奇异积分方程技术研究了两各向异性弹性半空间光滑接触可分离界面上滑移脉冲波的存在及其传播特性。结果表明,如果至少能在一种介质中存在Rayleigh波,且其波速小于两种介质中的最小极限速度,则滑移脉冲波就可以存在。这种脉冲波传播速度不确定,可在最小极限波速与较低的Rayleigh波速之间取值,而该取值范围又取决于无界面分离情况下的第一、第二滑移波的解。分离区大小取决于扰动的强度,界面法向力和质点速度在分离区两端有 1 /2奇异性。