The propagation and reflection of finite amplitude cylindrically symmetric shear waves in a hyperelastic incompressible solid

Summary We consider axially symmetric shear waves propagating in an incompressible hyperelastic thick-walled cylindrical shell, whose strain energy function is expressible as a truncated power series in terms of the basic strain invariants. A continuous pulse is initiated at the interior boundary of the cylinder by surface tractions of finite duration. The pulse propagates away from the interior boundary, then reflects from the outer boundary, and subsequently reflects back and forth between the two boundaries of the cylinder. We analyze shock development of the first incident and first reflected wave. The incident pulse can break before it reaches the outer boundary. Using Whitham's nonlinearization technique, we determine conditions under which the incident wave breaks and which shock waves can subsequently occur. Similar calculations are carried out for the first reflection. The formulas obtained for the incident pulse provide accurate estimates of the breaking distance and time, and the location of the shock paths, for any incident shock waves that occur. Results obtained for the reflected wave cannot be used to make similar estimates, but they do reveal that once the pulse has completely left the outer boundary, the possible shock that can occur is the same as for the incident wave. Our analysis is carried out for axial shear waves. A similar analysis can be done for torsional shear waves, but not for combined axial and torsional shear wave propagation. We illustrate the conclusions of our shock analysis with numerical solutions obtained using a relaxation scheme for systems of conservation laws. Numerical results are obtained for axial shear and for combined axial and torsional shear. These results indicate that the shock behavior indicated by our analysis of axial shear is also valid for combined axial and torsional shear wave propagation.     

Nonlinear axial shear wave propagation in a hyperelastic incompressible solid

Summary This paper is concerned with the propagation of axial shear waves in an incompressible isotropic hyperelastic solid, whose strain energy function is expressible as a power series in (I ₁-3) and (I ₂-3) whereI ₁ andI ₂ are the first and second basic invariants of the left Cauchy-Green tensorB. Numerical solutions are presented for problems of wave propagation produced by a step function application, or a finite duration pulse, of axial shear stress at the surface of a cylindrical cavity in an unbounded medium. A modification of MacCormack's finite difference scheme [1] is proposed and is used to obtain these solutions along with a procedure for the determination of the position of the shock front for the step function application.An estimate of the breaking time of a wave, obtained from a procedure proposed by Whitham [2], is compared with the numerical results. The dissipation of mechanical energy due to shock propagation is considered.With 4 Figures     

On the propagation of waves from spherical and cylindrical cavities in anisotropic materials

The propagation of waves from a spherical or cylindrical cavity in an inhomogeneous anisotropic elastic solid is considered. In the first instance, integral transforms are used to provide solutions to specific boundary value problems involving elastic media exhibiting certain inhomogeneities. It is then noted that the Bergman integral operator method provides a more general analysis. Finally, an asymptotic approach having a wide range of application is discussed and employed to construct wavefront and high-frequency expansions for the solution field in general media.     

Padé extensions to ray series methods and analysis of transients in inhomogeneous viscoelastic media of hereditary type

In this paper we demonstrate that wavefront expansions for the analysis of transient phenomena are far from adequate when numerical information back of the wavefront is required. However, by employing Padé approximants together with ray series methods, we can obtain directly and greatly extend the range of validity of these expansions. The procedure, which is straightforward and requires very little computing time, is here applied to a non-trivial problem involving impact-generated shear transients in inhomogenoues viscoelastic media whose stress-strain laws are given in integral form. For a special combination of the material parameters an exact solution is recovered and used to check the validity of our approximate Padé-extended wavefront solution. We also compare our results from the extended wavefront solution with numerical solutions obtained using Bellman's approximate inversion scheme for Laplace transforms. A further advantage of our approach is that, unlike transform techniques, it does not depend upon being able to find tabulated special function equations for the transformed dependent variables. All numerical results are presented graphically for ease of comparison.     

Non-destructive Detection of a Circular Cavity in a Finite Functionally Graded Material Layer Using Anti-plane Shear Waves

A semi-analytical method is proposed to investigate the non-destructive detection of a circular cavity buried in a functionally graded material layer bonded to homogeneous materials, and the multiple scattering effect of shear waves is described accurately. The image method is used to satisfy the traction free boundary condition at the edge of the functionally graded material layer. The analytical solutions of wave fields are expressed by employing wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions at the edge and around the cavity. The analytical and numerical solutions of dynamic stress concentration factors around the cavity are presented. The effects of the position of the cavity in the material layer, the incident wave number, and the properties of the two phases of materials on the dynamic stress concentration factors are analyzed. Analyses show that when the buried depth of the cavity and the thickness of the layer are relatively small, the properties of the two phases of materials have great effect on the distribution of dynamic stress around the cavity. In the region of higher frequency, the effects of the position of the cavity and the properties of the two phases of materials on the maximum dynamic stress are greater.     

Weak shock waves and shear bands in thermoelastic solids

The linear weak shock wave (acoustic wave) propagation and the existence of shear bands are examined in finitely deformed thermoelastic solids within the framework of the theory of singular surfaces. The jumps of certain field variables across the shock wave front are obtained by using Taylor series expansions of them. The propagation condition of shock waves in a thermoelastic solid is obtained by using the strain–energy function corresponding to Duhamel–Neumann expression. The propagation speeds of weak shock waves are determined for a particular state of deformation, that is, general dilation. The formation of shear bands and the magnitudes of critical stretches are obtained for the deformation states of uniaxial, biaxial extension and for uniform dilation.     

Asymptotic analysis of axisymmetric plane strain problem involving fractional order heat conduction

This work is concerned with the asymptotic solutions of the axisymmetric plane strain problem involving the fractional order heat conduction. The governing equations for the axisymmetric plane strain problem are derived by means of fractional calculus. The Laplace transform technique is used to obtain the general solutions for any set of boundary conditions in the physical domain. The asymptotic solutions for a specific problem of an infinite cylinder with the boundary subjected to a thermal shock is derived by means of the limit theorem of Laplace transform. Utilizing these solutions, the thermoelastic behavior induced by transient thermal shock can be clearly illustrated, and the jumps locating at the position of each wavefront can also be accurately captured. Some comparisons for the predictions of thermoelastic response are conducted to estimate the effect of the fractional order parameter on the thermoelastic behavior.     

Flow feature aligned grid adaptation

A flow feature aligned grid adaptation method is proposed for the solution of Euler and Navier–Stokes equations for compressible flows, motivated by the desire for an efficient grid system for an accurate and robust solution method to best resolve flow features of interest. The method includes extraction of the flow features; generation of the embedded flow feature aligned structured blocks combined with unstructured grid generation for the rest of the flowfield; and adaptation of the hybrid grid for high flow feature resolution. The feature alignment makes it possible to maintain the high resolution property for both shock waves and shear layers of the approximate Riemann solvers and the higher order reconstruction schemes based on one‐dimensional derivation and dimensional splitting. High grid efficiency is obtained with highly anisotropic directional grid corresponding to the feature directions. The computational procedure is described in details in the paper and its application to flow solutions involving shock waves, boundary layers, wakes and shock boundary layer interaction are demonstrated. Its accuracy, efficiency and robustness are discussed in comparison with an anisotropic unstructured grid adaptations for the shock boundary layer interaction case. Copyright © 2006 John Wiley & Sons, Ltd.     

Shear deformation effect in second-order analysis of frames subjected to variable axial loading

In this paper a boundary element method is developed for the second-order analysis of frames consisting of beams of arbitrary simply or multiply connected constant cross section, taking into account shear deformation effect. Each beam is subjected to an arbitrarily concentrated or distributed variable axial loading, while the shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. To account for shear deformations, the concept of shear deformation coefficients is used. Three boundary value problems are formulated with respect to the beam deflection, the axial displacement and to a stress function and solved employing a BEM approach. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress function using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The influence of both the shear deformation effect and the variableness of the axial loading are remarkable.     

斜入射SV波对地铁车站地震响应的影响

在显式有限元法结合黏弹性人工边界的时域波动方法的基础上,建立了三维平面SV波斜入射的输入方法。半空间算例说明了该文方法具有较好的精度,并基于所建立的斜入射方法研究了地震波斜入射对北宫门地铁车站地震响应的影响。该算例结果表明:在地震波斜入射情况下,地铁车站的地震响应规律与垂直入射时的情况具有明显差异;斜入射角度对水平加速度响应并不敏感,但对竖向加速度影响较大;斜入射使得车站柱子构件的剪力和轴力明显改变,柱子轴力随着入射角增加而明显增大;边墙控制点的应力状态也受入射角的影响较大,各控制点的第一、第三主应力都出现了随着入射角度增加而增大的规律。在地铁车站等地下结构抗震研究中,应考虑地震波斜入射的影响。     

CIRCUMFERENTIAL SHEAR HORIZONTAL WAVE AXIAL-CRACK SIZING IN PIPES

Circumferential shear horizontal (SH) waves are used for the inspection and sizing of axial cracks in pipelines. Experiments on two sample pipes having notches with different depths and lengths were carried out utilizing magnetostrictive sensor (MsS) technology for generating circumferential shear horizontal guided waves. A simplified two-dimensional model for crack sizing in pipes was studied through wave-reflection amplitude coefficients. The wave-reflection amplitudes are affected by both defect depths and lengths. To estimate the defect depth, which is critical, length compensation was taken for defects shorter than the beam width. An axial scan was carried out for the defect length estimations and the length compensation. Based on this axial scan, an approximate two-dimensional theory has been developed that has improved the defect depth estimation greatly. Two-dimensional boundary element modeling analysis and normal mode expansion technology are used to study defect sizing theoretically in a pipe-like structure by calculating reflection coefficients. The theoretical results agree with the experiments quite favorably.     

Finite element computation of unsteady viscous transonic flows past stationary airfoils

Results are presented for computations of unsteady viscous transonic flows past a stationary NACA0012 airfoil at various angles of attack. The Reynolds number, based on the chord-length of the airfoil, is 10,000 and the Mach number is 0.85. Stabilized finite-element formulations are employed to solve the compressible Navier-Stokes equations. The equation systems, resulting from the discretization, are solved iteratively by using the preconditioned GMRES technique. Time integration of the governing equations is carried out for large values of the non-dimensional time to understand the unsteady dynamics and long-term behavior of the flows. The results show interesting flow patterns and a complex interaction between the boundary/shear layers, shock/expansion waves and the lateral boundaries of the computational domain. For transonic flow past an airfoil at various angles of attack in a narrow channel/wind-tunnel one can observe solutions that are qualitatively different from each other. At low angles of attack an unsteady wake is observed. At moderate angles of attack the interaction between the shock system and the lateral walls becomes significant and the temporal activity in the wake decreases and eventually disappears. At high angles of attack a reflection shock is formed. Hysteresis is observed at an angle of attack 8^∘. For the flow in a domain with the lateral boundaries located far away, the unsteadiness in the flow increases with an increase in the angle of attack. Computations for a Mach 2, Re 106 flow past an airfoil at 10^∘ angle of attack compare well with numerical and experimental results from other researchers     

Oscillatory and chaotic wakes behind moving boundaries in reaction-diffusion systems

Oscillatory wakes occur in a wide range of reaction-diffusion systems, consisting of either periodic travelling waves or irregular spatiotemporal oscillations, behind a moving transition front. In this paper, the use of a finite boundary moving with an imposed speed to mimic the transition front is considered. For both λ-ω systems and standard predator-prey models, the solutions behind these moving boundaries agree very closely with the behaviour behind transition fronts, provided suitable end conditions are used on the moving boundary. This confirms that the transition front can be regarded as determining the solution, by forcing a particular periodic wave at the boundary of the wake region. In the case of λ-ω systems, a detailed numerical study of solutions on a fixed-length finite domain with a periodic wave solution forced at the boundaries is performed. As the domain length is varied as a parameter, the long-term temporal behaviour undergoes bifurcation sequences that are well known as routes to chaos in ordinary differential equations. This suggests that irregular wakes actually have the form of a perpetual transient in a progression towards chaos. Finally, the way in which the moving boundary results can be used to design an experimental verification of the oscillatory wakes phenomenon in a chemical system is discussed.     

Interface Effect on the Dynamic Stress around an Elliptical Nano-Inhomogeneity Subjected to Anti-Plane Shear Waves

In the design of advanced micro- and nanosized materials and devices containing inclusions, the effects of surfaces/interfaces on the stress concentration become prominent. In this paper, based on the surface/interface elasticity theory, a two-dimensional problem of an elliptical nano-inhomogeneity under anti-plane shear waves is considered. The conformal mapping method is then applied to solve the formulated boundary value problem. The analytical solutions of displacement fields are expressed by employing wave function expansion method, the expanded mode coefficients are determined by satisfying the boundary conditions at the interfaces of the nano-inhomogeneity. Analyses show that the effect of the interfacial properties on the dynamic stress is significantly related to the wave frequency of incident waves, the shear modulus ratio of the nano-inhomogeneity and the matrix, and the dimensions of the elliptical nano-inhomogeneity. Comparison with the previous results is also presented.     

Singular surfaces in linear thermoelasticity

In this paper we give a detailed account, within the framework of the linear theory of thermoelasticity, of the propagation of surfaces of discontinuity in a homogeneous, isotropic elastic solid which is able to conduct heat. The methods used in the investigation are, in large measure, due to T. Y. Thomas. The early sections of the paper contain a derivation of the principal results of Thomas's theory which enables us to determine, from a consideration of the appropriate Cauchy initial-value problem, the characteristic surfaces of the linear thermoelastic equations. The wavefronts associated with these characteristics are found to propagate with one of the constant speeds

, ₀, E_T, v_T being respectively the density, the isothermal Young's modulus and the isothermal Poisson's ratio of the material in its reference state.

A discontinuity surface of order r in the displacement and temperature fields is referred to as a weak thermoelastic wave if r2 and a strong thermoelastic wave if r=0 or 1. Concerning the properties of these waves our main conclusions are as follows. Weak thermoelastic waves and strong waves of order 1 are characteristic and may be described as dilatational or rotational according as their speed of propagation is v_T or v_S. Dilatational strong waves of order 1 are shock waves and rotational waves of this type are propagating vortex sheets. For all thermoelastic waves of order 1 the strength (defined in a natural way) is completely determined by its distribution on an initial configuration of the wavefront. Irrespective of the shape of this initial configuration, the strength of a dilatational wave decays rapidly as the wave propagates on account of thermoelastic dissipation. For rotational waves, however, the variation of strength during propagation depends solely upon the geometrical form of the initial wavefront. A strong thermoelastic wave of order 0 is an absolute singular surface in the temperature field, discontinuities of displacement being excluded from consideration. A wave of this type may be characteristic, in which case its speed of propagation is v_S; or it may be non-characteristic, in which case it is a dilatational shock wave. In neither case is the strength of the wave completely determined by its distribution on an initial wavefront, a situation which leads us to argue that thermoelastie waves of order 0 cannot in practice be created.

In the final section of the paper the properties of singular surfaces in classical elastokinetics are discussed in the light of the foregoing analysis of discontinuous thermoelastic waves.     

Implicit formulation with the boundary element method for nonlinear radiation of water waves

An accurate and efficient numerical method is presented for the two-dimensional nonlinear radiation problem of water waves. The wave motion that occurs on water due to an oscillating body is described under the assumption of ideal fluid flow. The governing Laplace equation is effectively solved by utilizing the GMRES (Generalized Minimal RESidual) algorithm for the boundary element method (BEM) with quadratic approximation. The intersection or corner singularity in the mixed Dirichlet–Neumann problem is resolved by introducing discontinuous elements. The fully implicit trapezoidal rule is used to update solutions at new time-steps, by considering stability and accuracy. Traveling waves generated by the oscillating body are absorbed downstream by the damping zone technique. To avoid the numerical instability caused by the local gathering of grid points, the re-gridding technique is employed, so that all the grids on the free surface may be re-distributed with an equal distance between them. The nonlinear radiation force is evaluated by means of the acceleration potential. For a mixed Dirichlet–Neumann problem in a computational domain with a wavy top boundary, the present BEM yields numerical solutions for the quadratic rate of convergence with respect to the number of boundary elements. It is also demonstrated that the present time-marching and radiation condition work successfully for nonlinear radiation problems of water waves. The results obtained from this study concur reasonably well with other numerical computations.     

Quasi-sudden brittle fracture at both edges of a finite crack

The diffraction of a plane horizontally polarized shear wave by a crack of finite length is analyzed and the extension of both crack edges prior to the arrival of the first diffracted waves, i.e. quasi-sudden fracture, is studied. In light of an energy rate balance criterion it is found that for an incident step-stress pulse, quasi-sudden fracture may occur but always at both crack edges, often initiating at the trailing edge first. For an incident wave whose stress vanishes at the wavefront, however, quasi-sudden fracture may occur only at the leading crack edge, or if at both edges, at the leading edge first. For both waveforms, the rate of crack extension is non-constant and increases rapidly so that crack branching may be expected. Finally instantaneous crack extension at a uniform rate is possible only if the incident wave stress possesses a square-root sinularity at the wavefront. This result agrees with earlier work by Achenbach.     

Elastic waves in a fluid-loaded, semi-infinite axisymmetric rod

In a fluid-loaded, semi-infinite axisymmetric rod, a free shear stress boundary condition on the circular cross-sectional end introduces complicated, nondispersive waves in the solid. They are composed of a pulse wave, which has the same waveform as the transmitted one and travels at speed c1, and different kinds of pulse trains, each of which travels along the rod at the speed of either c1 or square root of 2c2, where c1 and c2 are the propagating speeds of the longitudinal and transversal bulk waves, respectively. Furthermore, one can conclude from the solutions to the boundary conditions that c1 and square root of 2c2 are the only phase speeds of nondispersive waves. Frequency equations associated with these waves are established, and the solutions are solved and discussed analytically and numerically. The acoustic field in the fluid is also fully discussed, and it is more complicated than a single outgoing Hankel function as described for an infinite rod. The acoustic energy coupling between the solid and the fluid and the end reflection and transmission are quantified as well. In the end, experimental examinations of the echo spectra, using an aluminum rod immersed in the water and air, fully confirm the numerical solutions to the frequency equations.     

Shear deformation effect in nonlinear analysis of spatial composite beams subjected to variable axial loading by bem

Summary In this paper a boundary element method is developed for the nonlinear analysis of composite beams of arbitrary doubly symmetric constant cross section, taking into account the shear deformation effect. The composite beam consists of materials in contact, each of which can surround a finite number of inclusions. The materials have different elasticity and shear moduli with same Poisson's ratio and are firmly bonded together. The beam is subjected in an arbitrarily concentrated or distributed variable axial loading, while the shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, the axial displacement and to two stress functions and solved using the Analog Equation Method, a BEM based method. Application of the boundary element technique yields a system of nonlinear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency and the range of applications of the developed method. The influence of both the shear deformation effect and the variableness of the axial loading are remarkable.     

A method for multidimensional wave propagation analysis via component‐wise partition of longitudinal and shear waves

An explicit integration algorithm for computations of discontinuous wave propagation in two‐dimensional and three‐dimensional solids is presented, which is designed to trace extensional and shear waves in accordance with their respective propagation speeds. This has been possible by an orthogonal decomposition of the total displacement into extensional and shear components, leading to two decoupled equations: one for the extensional waves and the other for shear waves. The two decoupled wave equations are integrated with their CFL time step sizes and then reconciled to a common step size by employing a previously developed front‐shock oscillation algorithm that is proven to be effective in mitigating spurious oscillations. Numerical experiments have demonstrated that the proposed algorithm for two‐dimensional and three‐dimensional wave propagation problems traces the stress wave fronts with high‐fidelity compared with existing conventional algorithms. Copyright © 2013 John Wiley & Sons, Ltd.