Separation at the interface of non-linearly elastic spinning tube-rigid shaft subjected to circumferential shear

Separation at the interface of homogeneous, isotropic, compressible, hyperelastic, spinning cylindrical tube-rigid shaft subjected to circumferential shear is investigated within the context of the finite elasticity theory. The compressible, hyperelastic spinning tube with a uniform wall thickness is assumed to be tautly fitted to a rigid shaft along its inner curved surface. The outer surface of the tube is subjected to a constant uniformly distributed circumferential shearing stress while the rigid shaft is assumed to spin with an angular speed. The state when a separation occurs at the interface of the shaft and the tube is investigated. The critical values are given for slightly compressible rubbers and nearly incompressible rubbers.     

On axisymmetric motion of an incompressible elastic medium under impact loading

The method of matched asymptotic expansions was employed to obtain approximate solutions to the one-dimensional boundary-value problems of nonlinear dynamic elasticity theory of impact loading on the surface of a cylindrical cavity of an incompressible medium that causes antiplane motion or torsion of the medium. The expansion of the solution in the near-front region is based on solutions of evolution equations different from the equations for quasi-simple waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 144–151, November–December, 2006.     

Wave propagation in an elastic tube: a numerical study

In this paper, a fluid–wall interaction model, called the elastic tube model, is introduced to investigate wave propagation in an elastic tube and the effects of different parameters. The unsteady flow was assumed to be laminar, Newtonian and incompressible, and the vessel wall to be linear-elastic, isotropic and incompressible. A fluid–wall interaction scheme is constructed using a finite element method. The results demonstrate that the elastic tube plays an important role in wave propagation. It is shown that there is a time delay between the velocity waveforms at two different locations and that the peak velocity increases while the low velocity decreases in the elastic tube model, contrary to the rigid tube model where velocity waveforms overlap each other. Compared with the elastic tube model, the increase of the wall thickness makes wave propagation faster and the time delay cannot be observed clearly, however, the velocity amplitude is reduced slightly due to the decrease of the internal radius. The fluid–wall interaction model simulates wave propagation successfully and can be extended to study other mechanical properties considering complicated geometrical and material factors.     

Fluid flow of incompressible viscous fluid through a non-linear elastic tube

The study of viscous flow in tubes with deformable walls is of specific interest in industry and biomedical technology and in understanding various phenomena in medicine and biology (atherosclerosis, artery replacement by a graft, etc) as well. The present work describes numerically the behavior of a viscous incompressible fluid through a tube with a non-linear elastic membrane insertion. The membrane insertion in the solid tube is composed by non-linear elastic material, following Fung’s (Biomechanics: mechanical properties of living tissue, 2nd edn. Springer, New York, 1993) type strain–energy density function. The fluid is described through a Navier–Stokes code coupled with a system of non linear equations, governing the interaction with the membrane deformation. The objective of this work is the study of the deformation of a non-linear elastic membrane insertion interacting with the fluid flow. The case of the linear elastic material of the membrane is also considered. These two cases are compared and the results are evaluated. The advantages of considering membrane nonlinear elastic material are well established. Finally, the case of an axisymmetric elastic tube with variable stiffness along the tube and membrane sections is studied, trying to substitute the solid tube with a membrane of high stiffness, exhibiting more realistic response.     

Constitutive analysis of thin biological membranes with application to radial stretching of a hollow circular membrane

The constitutive analysis of the mechanical response of thin elastic membranes under inplane deformation is presented by using the multiplicative decomposition of the deformation gradient into its areal and distortional parts. Specific results are obtained for the Evans-Skalak form of the strain energy function. The solution to the problem of radial stretching of a hollow circular membrane obeying this constitutive model is then derived. The stress concentration factor is determined as a function of the relative hole size and the magnitude of the applied tension. The tension boundary is identified above which no compressive stress appears in the membrane. The limit boundary is introduced below which the membrane cannot support the applied loading without unstable wrinkling. For the loading between the tension and the limit boundary, nonuniformly distributed infinitesimal wrinkles appear within the inner portion of the membrane, carrying radial tension but no circumferential stress (tension field). The specific form of the strain energy function is used to describe this behavior, and to calculate the amount of the membrane area absorbed by infinitesimal wrinkles. The wrinkled portion is surrounded by the outer portion of the membrane carrying both radial and circumferential stresses. The limit boundary is reached when wrinkles spread throughout the membrane. It is shown that for a sufficiently large tension at the outer boundary, the wrinkling does not spread throughout the membrane no matter how large the applied tension at the inner boundary of the membrane is, provided that no rupture takes place. The limiting extent of the tension field in such cases is calculated. The linearized version of the analysis is characterized by a closed form solution.     

On the asymptotic behavior of a phase-field model for elastic phase transitions

We study the asymptotic behavior of a one-dimensional, dynamical model of solid-solid elastic transitions in which the phase is determined by an order parameter. The system is composed of two coupled evolution equations, the mechanical equation of elasticity which is hyperbolic and a parabolic equation in the order parameter. Due to the strong coupling and the lack of smoothing in the hyperbolic equation, the asymptotic behavior of solutions is difficult to determine using standard methods of gradient-like systems. However, we show that under suitable assumptions all solutions approach the equilibrium set weakly, while the phase field stabilizes strongly.     

Dynamic stability of externally pressurized elastic rings subjected to high rates of loading

Of interest here is the influence of loading rate on the stability of structures where inertia is taken into account, with particular attention to the comparison between static and dynamic buckling. This work shows the importance of studying stability via perturbations of the initial conditions, since a finite velocity governs the propagation of disturbances. The method of modal analysis that determines the fastest growing wavelength, currently used in the literature to analyze dynamic stability problems, is meaningful only for cases where the velocity of the perfect structure is significantly lower than the associated wave propagation speeds.     

Localized solutions to the nonlinear wave equation for flexural motions of an elastic beam traveling in an air-filled tube

Steady-progressive-wave solutions are sought to the nonlinear wave equation derived previously [J. Fluids Struct. 16 (2002) 597] for flexural motions of an elastic beam traveling in an air-filled tube along its center axis at a subsonic speed. Fluid-structure interactions are taken into account through aerodynamic loading on the lateral surface of the beam subjected to small but finite deflection but end effects and viscous effects are neglected. Linear dispersion characteristics are first examined by exploiting the small ratio of the induced mass to the mass of the beam per unit length. Centered around the traveling speed of the beam, there exists such a narrow range of propagation velocity that the linear steady propagation is prohibited. In this range, it is revealed that some interesting nonlinear solutions exist. The periodic wavetrain is found to exist as the exact solution. Asymptotic analysis is then made by applying the method of multiple scales and the stationary nonlinear Schrödinger equation is derived for a complex amplitude. A monochromatic solution to this equation corresponds to the exact periodic solution. Imposing undisturbed boundary conditions at infinity, it is revealed that the localized solution exists as a result of balance between the linear instability and the nonlinearity. This solution is checked by solving the nonlinear equation numerically. It is further revealed that the amplitude-modulated wavetrain exists not only in the range of the velocity mentioned above but also outside of it.     

不确定性移动载荷激励下弹性简支梁的动态响应边界分析及应用

不确定性移动载荷激励下的弹性梁振动是土木、机械和航空航天等工程领域普遍存在的一类重要问题。在许多实际工程中,不确定移动载荷的样本测试数据有限或测试成本较高,本文引入区间过程模型对此类动态不确定性参数进行描述,提出了一种求解不确定移动载荷激励下弹性梁振动响应边界的非随机振动分析方法。首先,介绍了确定性移动载荷激励下弹性梁的振动微分方程及其解析求解方法;其次,引入区间过程模型,以上下边界函数的形式对不确定性移动载荷进行度量,进而基于模态叠加法发展出弹性梁振动响应边界求解的非随机振动分析方法;最后,将上述非随机振动分析方法应用于车桥耦合振动问题。     

Dynamic response of a reinforced concrete slab subjected to air blast load

Reinforced concrete is the principal material for military engineering and nuclear power plant containment. However, impacts and explosions could completely destroy such structures, causing tremendous casualties and property loss. Hence, this study conducts an analysis on the propagation law of a blast pressure wave and the dynamic response of reinforced concrete structures under explosive pressure wave effects. This study uses proper state material parameters and equations and then applies the nonlinear finite element analysis software LS-DYNA to conduct a numerical simulation of a free-field explosion model. After comparison with the computed results from empirical equations and validating the reliability of the numerical analysis model, the destruction and influencing factors on reinforced concrete slabs, under the effects of a blast pressure wave, are investigated. The results can serve as a reference for future analysis and design.     

Waves in an elastic tube filled with a heterogeneous fluid of variable viscosity

By treating the artery as a prestressed thin elastic tube and the blood as an incompressible heterogeneous fluid with variable viscosity, we studied the propagation of weakly non-linear waves in such a composite medium through the use of reductive perturbation method. By assuming a variable density and a variable viscosity for blood in the radial direction we obtained the perturbed Korteweg-deVries equation as the evolution equation when the viscosity is of order of ε^3/2. We observed that the perturbed character is the combined result of the viscosity and the heterogeneity of the blood. A progressive wave type of solution is presented for the evolution equation and the result is discussed. The numerical results indicate that for a certain value of the density parameter sigma, the wave equation loses its dispersive character and the evolution equation degenerates. It is further shown that, for the perturbed KdV equation both the amplitude and the wave speed decay in the time parameter τ.     

The effect of fiber recruitment on the swelling of a pressurized anisotropic non-linearly elastic tube

We study the effect of fiber recruitment on the mechanical response of a fiber reinforced non-linearly elastic tube that is both swollen and pressurized. Attention is restricted to cylindrically symmetric tube deformation. The constitutive model permits fibers to support tension, but not compression. While many combinations of pressure and swelling cause all of the fibers to be recruited for load support, both large swelling and large deswelling can give rise to fiber derecruitment at certain locations in the tube. This leads to less channel opening than would be the case if the fibers provided support while contracted. The transition between mechanically active and mechanically inactive fibers can be described in terms of the quasi-static motion of a fiber recruiting interface.     

On out-of-plane buckling of anisotropic elastic plate subjected to a simple shear

Out-of-plane buckling of anisotropic elastic plate subjected to a simple shear is investigated. From exact 3-D equilibrium conditions of anisotropic elastic body with a plane of elastic symmetry at critical configuration, the eqution for buckling direction (buckling wave direction) parameter is derived and the shape functions of possible buckling modes are obtained. The traction free boundary conditions which must hold on the upper and lower surfaces of plate lead to a linear eigenvalue problem whose nontrivial solutions are just the possible buckling modes for the plate. The buckling conditions for both flexural and barreling modes are presented. As a particular example of buckling of anisotropic elastic plate, the buckling of an orthotropic elastic plate, which is subjected to simple shear along a direction making an arbitrary angle of θ with respect to an elastic principal axis of materials, is analyzed. The buckling direction varies with θ and the critical amount of shear. The numerical results show that only the flexural mode can indeed exist. Project supported by the National Natural Science Foundation of China (No. 19772032).     

The torsion of a composite,nonlinear-elastic cylinder with an inclusion having initial large strains

This article considers a static problem of torsion of a cylinder composed of incompressible, nonlinear-elastic materials at large deformations. The cylinder contains a central, round, cylindrical inclusion that was initially twisted and stretched (or compressed) along the axis and fastened to a strainless, external, hollow cylinder. The problem statement and solution are based on the theory of superimposed large strains. An accurate analytical solution of this problem based on the universal solution for the incompressible material is obtained for arbitrary nonlinear-elastic isotropic incompressible materials. The detailed investigation of the obtained solution is performed for the case in which the cylinders are composed of Mooney-type materials. The Poynting effect is considered, and it is revealed that composite cylinder torsion can involve both its stretching along the axis and compression in this direction without axial force, depending on the initial deformation.     

On the response of an elastic membrane to a travelling ring load

Exact solutions in closed-form are presented for the dynamic response of an infinite elastic membrane to a suddenly applied, radially expanding ring load.     

A note on incremental equations for a new class of constitutive relations for elastic bodies

Some new classes of constitutive relations for elastic bodies have been proposed in the literature, wherein the stresses and strains are obtained from implicit constitutive relations. A special case of the above relations corresponds to a class of constitutive equations where the linearized strain tensor is given as a nonlinear function of the stresses. For such constitutive equations we consider the problem of decomposing the stresses into two parts: one corresponds to a time-independent solution of the boundary value problem, plus a small (in comparison with the above) time-dependent stress tensor. The effect of this initial time-independent stress in the propagation of a small wave motion is studied for an infinite medium.     

The nonlinear response of Cattaneo-type thermal loading of a laser pulse on a medium using the generalized thermoelastic model

The nonlinear thermoelastic responses of an elastic medium exposed to laser generated shortpulse heating are investigated in this article. The thermal wave propagation of generalized thermoelastic medium under the impact of thermal loading with energy dissipation is the focus of this research. To model the thermal boundary condition(in the form of thermal conduction),generalized Cattaneo model(GCM) is employed. In the reference configuration, a nonlinear coupled Lord-Shulman-type generalized thermoelasticity formulation using finite strain theory(FST) is developed and the temperature dependency of the thermal conductivity is considered to derive the equations. In order to solve the time-dependent and nonlinear equations, Newmark's numerical time integration technique and an updated finite element algorithm is applied and to ensure achieving accurate continuity of the results, the Hermitian elements are used instead of Lagrangian's. The numerical responses for different factors such as input heat flux and nonlinear terms are expressed graphically and their impacts on the system's reaction are discussed in detail.The results of the study are presented for Green–Lindsay model and the findings are compared with Lord-Shulman model especially with regards to heat wave propagation. It is shown that the nature of the laser's thermal shock and its geometry are particularly determinative in the final stage of deformation. The research also concluded that employing FST leads to achieving more accuracy in terms of elastic deformations; however, the thermally nonlinear analysis does not change the results markedly. For this reason, the nonlinear theory of deformation is required in laser related reviews, while it is reasonable to ignore the temperature changes compared to the reference temperature in deriving governing equations.     

Analytical extension of Finite Element solution for computing the nonlinear far field of ultrasonic waves scattered by a closed crack

Nonlinear scattering of ultrasonic waves by closed cracks subject to contact acoustic nonlinearity (CAN) is determined using a 2D Finite Element (FE) coupled with an analytical approach. The FE model, which includes unilateral contact with Coulomb friction to account for contact between crack faces, provides the near-field solution for the interaction between in-plane elastic waves and a crack of different orientations. The numerical solution is then analytically extended in the far-field based on a frequency domain near-to-far field transformation technique, yielding directivity patterns for all linear and nonlinear components of the scattered waves. The proposed method is demonstrated by application to two nonlinear acoustic problems in the case of tone-burst excitations: first, the scattering of higher harmonics resulting from the interaction with a closed crack of various orientations, and second, the scattering of the longitudinal wave resulting from the nonlinear interaction between two shear waves and a closed crack. The analysis of the directivity patterns enables us to identify the characteristics of the nonlinear scattering from a closed crack, which provides essential understanding in order to optimize and apply nonlinear acoustic NDT methods.     

Finite element computation of torsional plastic waves in a thin-walled tube

Summary  A finite element technique is presented for the analysis of one-dimensional torsional plastic waves in a thin-walled tube. Three different nonlinear consitutive relations deduced from elementary mechanical models are used to describe the shear stress–strain characteristics of the tube material at high rates of strain. The resulting incremental equations of torsional motion for the tube are solved by applying a direct numerical integration technique in conjunction with the constitutive relations. The finite element solutions for torsional plastic waves in a long copper tube subjected to an imposed angular velocity at one end are given, and a comparison with available experimental results to assess the accuracy of the constitutive relations considered is conducted. It is demonstrated that the strain-rate dependent solutions show a better agreement with the experimental results than the strain-rate independent solutions. The limitations of the constitutive equations are discussed, and some modifications are suggested. Received 9 February 1999; accepted for publication 28 March 2000     

Numerical study of nonlinear interaction between a crack and elastic waves under an oblique incidence

A Finite Element (FE) model is proposed to study the interaction between in-plane elastic waves and a crack of different orientations. The crack is modeled by an interface of unilateral contact with Coulombs friction. These contact laws are modified to take into account a pre-stress _σ0${\sigma }_0$ that closes the crack. Using the FE model, it is possible to obtain the contact stresses during wave propagation. These contact stresses provide a better understanding of the coupling between the normal and tangential behavior under oblique incidence, and explain the generation of higher harmonics. This new approach is used to analyze the evolution of the higher harmonics obtained as a function of the angle of incidence, and also as a function of the excitation level. The pre-stress condition is a governing parameter that directly changes the nonlinear phenomenon at work at the interface and therefore the harmonic generation. The diffracted fields obtained by the nonlinear and linear models are also compared.