One-dimensional wave propagation in an initially deformed incompressible medium with different behaviour in tension and compression

The propagation of 1-dimensional waves in an initially deformed incompressible medium with different moduli in tension and compression is investigated. Depending on the sign of the initial strains, various possibilities of propagation are shown to exist. The governing equations are nonlinear and a hardening or softening behaviour in shear may be present. Simple wave analytical solutions are given in a semi-infinite incompressible half-space. It is shown that in some situations, a shear pulse applied to the surface of an initially deformed half-space propagates linearly up to a specific value of the shear deformation and nonlinearly after that point.     

Study on cavitated bifurcation problems for spheres composed of hyper-elastic materials

In this paper, spherical cavitated bifurcation problems are examined for incompressible hyper-elastic materials and compressible hyper-elastic materials, respectively. For incompressible hyper-elastic materials, a cavitated bifurcation equation that describes cavity formation and growth for a solid sphere, composed of a class of transversely isotropic incompressible hyper-elastic materials, is obtained. Some qualitative properties of the solutions of the cavitated bifurcation equation are discussed in the different regions of the plane partitioned by material parameters indicating the degree of radial anisotropy in detail. It is shown that the cavitated bifurcation equation is equivalent, by use of singularity theory, to a class of normal forms with single-sided constraint conditions at the critical point. Stability and catastrophe of the solutions of the cavitated bifurcation equation are discussed by using the minimal potential-energy principle. For compressible hyper-elastic materials, a group of parameter-type solutions for the cavitated deformation for a solid sphere, composed of a class of isotropic compressible hyper-elastic materials, is obtained. Stability of the solutions is also discussed.     

Analysis of the dispersion relation for an incompressible transversely isotropic elastic plate

Summary The dispersion relation associated with harmonic wave propagation in an incompressible, transversely isotropic elastic plate is derived. Such a material is characterized by only three material constants, contrasting with five in the corresponding compressible case. Motivated by a numerical investigation, asymptotic expansions, giving phase speed and frequency as functions of wave number, are derived in both the long and short wave regimes. These approximations, which owing to the constitutive simplifications are readily available, are shown to provide excellent agreement with the corresponding numerical solution. It is envisaged that the detailed investigation carried out in this paper will aid numerical inversion of the transform solutions often used in impact problems. Additionally, the asymptotic investigation provides the necessary basis for future studies to derive asymptotically approximate models to describe long and short wave motion.     

Directed fluid sheets and gravity waves in compressible and incompressible fluids

This paper extends the range of applicability of the theory of directed fluid sheets [1] to the propagation of fairly long gravity waves in a compressible inviscid fluid and in a uniform shear flow of an incompressible inviscid fluid over a stream of constant initial depth. Applications are also made to nonhomogeneous immiscible fluid layers and the effect of compressibility on wave propagation over a stream with a level bottom is examined.     

Meshfree particle simulation of micro channel flows with surface tension

This paper presents a study of micro channel flows using a meshfree particle approach. The approach is based on smoothed particle hydrodynamics (SPH) and its variant, adaptive smoothed particle hydrodynamics (ASPH). The incompressible flow in the micro channels is modeled as an artificially compressible flow. The surface tension is incorporated into the equations of motion. The classic Poiseuille flow and a practical micro channel flow problem of flip-chip underfill encapsulation process are investigated. It is found that the adaptive kernel can well match the computational geometry with long channels and can greatly save computational time. The simulation results are in close agreement with the analytical solutions.     

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.     

The simple tension problem at large volumetric strains computed from finite hyperelastic material laws

Summary In this paper, different formulations of finite isotropic hyperelastic material laws for compressible solids are considered. Material laws with an additive split of the hyperelastic strain energy function into isochoric parts and volumetric parts are often used in the numerical treatment of nearly incompressible solids. It will be shown that this formulation leads to unphysical results in the simple tension problem when we do not restrict ourselves to nearly incompressible materials.     

Nonlinear wave modulation in fluid filled distensible tubes

Summary The present work considers one dimensional wave propagation in an infinitely long, straight and homogeneous nonlinear viscoelastic or elastic tube filled with an incompressible, inviscid fluid. Using the reductive perturbation technique, and assuming the weakness of dissipative effects, the amplitude modulation of weakly nonlinear waves is examined. It is shown that the amplitude modulation of these waves is governed by a dissipative nonlinear Schrödinger equation (NLS). In the absence of dissipative effects, this equation reduces to the classical NLS equation. The examination of the coefficients of the dissipative and classical NLS equations reveals the significance of the tube wall inertia to obtain a balance between nonlinearity and dispersion. Some special solutions of the NLS equation are given and the modulational instability of the plane wave solution is discussed for various incompressible hyperelastic materials.     

An extension of Key's principle to nonlinear elasticity

A variational principle for finite isothermal deformations of anisotropic compressible and nearly incompressible hyperelastic materials is presented. It is equivalent to the nonlinear elastic field (Lagrangian) equations expressed in terms of the displacement field and a scalar function associated with the hydrostatic mean stress. The formulation for incompressible materials is recovered from the compressible one simply as a limit. The principle is particularly useful in the development of finite element analysis of nearly incompressible and of incompressible materials and is general in the sense that it uses a general form of constitutive equation. It can be considered as an extension of Key's principle to nonlinear elasticity. Various numerical implementations are used to illustrate the efficiency of the proposed formulation and to show the convergence behaviour for different types of elements. These numerical tests suggest that the formulation gives results which change smoothly as the material varies from compressible to incompressible.     

Symmetry Transformations and Exact Solutions of a Generalized Hyperelastic Rod Equation

In this paper, a nonlinear wave equation with variable coefficients is studied, interestingly, this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities. With the aid of Lou’s direct method1, the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained. The corresponding numerical examples of exact solutions are presented by using different coefficients. Particularly, while the variable coefficients are taken as some special constants, the nonlinear wave equation with variable coefficients reduces to the one with constant coefficients, which can be used to describe the propagation of the travelling waves in general cylindrical rods composed of generally hyperelastic materials. Using the same method to solve the nonlinear wave equation, the validity and rationality of this method are verified.     

Dynamical problem for an incompressible laminated cylinder with helical anisotropy. Part 1. Theory

Exact analytical solutions have been obtained of a one-dimensional dynamical problem for an incompressible elastic radially inhomogeneous helically orthotropic thick-walled cylinder under plane strain conditions, loaded with nonsteady internal or external pressure. Necessary and sufficient conditions for the existence, uniqueness and physical adequacy of the solutions have been established. Convergence of wave solutions for slightly compressible cylinders to the analytical dependences obtained for incompressible cylinders has been analytically proved. __________ Translated from Problemy Prochnosti, No. 2, pp. 114–123, March–April, 2006.     

Smoothed finite element method for topology optimization involving incompressible materials

It is well known that the finite element method (FEM) suffers severely from the volumetric locking problem for incompressible materials in topology optimization owing to its numerical ‘overly stiff’ property. In this article, two typical smoothed FEMs with a certain softened effect, namely the node-based smoothed finite element method (NS-FEM) and the cell-based smoothed finite element method, are formulated to model the compressible and incompressible materials for topology optimization. Numerical examples have demonstrated that the NS-FEM with an ‘overly soft’ property is fairly effective in tackling the volumetric locking problem in topology optimization when both compressible and incompressible materials are involved.     

Performance of an analogy-based all-speed procedure without any explicit damping

 The performance of a numerical method which solves flow at all speeds, and does not use any explicit artificial viscosity or damping mechanism whatsoever, is investigated by testing a number of selected cases in compressible and incompressible flows. Contrary to existing methods, the momentum components are chosen as the dependent variables instead of the velocity components in order to provide a number of advantages. Among the motivations for this change is a flow analogy which permits incompressible methods to be used to solve compressible flows. The method is formulated within a control-volume-based finite-element approach using a collocated grid arrangement. The definition of two types of mass flux components at the control volume surfaces removes the possibility of velocity-pressure decoupling in the incompressible or Euler limits. In the absence of any dissipation mechanisms, the main concern of this work is to evaluate the performance of the method and the analogy for solving high speed compressible flows with shocks. The results and performance of the present work are compared with the exact and benchmark solutions and the results of other workers who use dissipation mechanisms to solve flow at all speeds.     

Evolution equations for nonlinear waves in a tapered elastic tube filled with a viscous fluid

In this work, employing the reductive perturbation method and treating the arteries as a tapered, thin walled, long and circularly conical prestressed elastic tube, the propagation of weakly nonlinear waves is investigated in such a fluid-filled elastic tube. By considering the blood as an incompressible viscous fluid, depending on the viscosity and perturbation parameters we obtained various evolution equations as the extended Korteweg-de Vries (KdV), extended KdV Burgers and extended perturbed KdV equations. Progressive wave solutions to these evolution equations are obtained and it is observed that the wave speeds increase with the distance for negative tapering while they decrease for positive tapering.     

Aspects of energy propagation in highly anisotropic elastic solids

Summary Anomalies in the theory of wave propagation in constrained materials may be reconciled with the standard theory of wave propagation in unconstrained materials by relaxing the constraint slightly and then taking the limit as the constraint is obeyed exactly. In this paper the same method is employed in an attempt to reconcile anomalies in the propagation of energy in a constrained material with the known propagation propertics for unconstrained materials. On relaxing the constraint in a singly constrained material, it is found that the energetics associated with two of the three propagating waves tend to the appropriate known forms for the corresponding constrained material in the limit where the constraint holds exactly. The third wave has no counterpart in the constrained theory and it is conjectured that both the total energy density and the energy flux vector tend to zero as the constrained limit is approached. This conjecture is shown to be true for two simple boundary value problems involving incompressible, and inextensible, elastic half-spaces.     

Radial propagation of rotary shear waves in a finitely deformed elastic solid

A study is made of the radial propagation of rotary shear waves in an incompressible elastic solid under finite radial deformations. Basic equations are derived on the basis of Biot's mechanics of incremental deformations, and analysis is made by specializing the initial deformations to two cases: (a) an infinite solid with a cylindrical bore is inflated by all-around tension, and (b) a cuboid is rounded into a ring and its ends are bonded to each other. The influence of the inhomogeneities of such deformations upon the laws of shear wave propagation is presented in the form of curves.     

Einige Überlegungen zur Unified Strength Theory von Mao-Hong Yu

The equivalent stress concept is well-established, since it is rather useful for formulation of phenomenological material models. However, there is no connection to any physical principles or the microstructure known, which is a disadvantage of the concept. Despite of it, numerous hypotheses based on the equivalent stress are known and used in applications more or less successfully. As a basis for systematization of different models the Unified Stress Theory (UST) was proposed by Yu. It can be used in order to describe incompressible inelastic material behavior as well as compressible materials with different behavior at tension and compression. In this paper Yu’s theory is analyzed, compared to the existing models and approximated by simpler ones. For the UST and other models the Poisson’s ratio and the hydrostatic stress are computed. Using the Poisson’s ratio at tension as well as simple stress relations different models are compared to each other. Constraints for parameters of the model based on the latter values can be provided. In order to reduce the number of the models an alternative systematization based on the geometrical properties of the equivalent stresses is suggested. In this systematization the model of Yu plays a quite important role.     

On the unsteady inviscid force on cylinders and spheres in subcritical compressible flow

The unsteady inviscid force on cylinders and spheres in subcritical compressible flow is investigated. In the limit of incompressible flow, the unsteady inviscid force on a cylinder or sphere is the so-called added-mass force that is proportional to the product of the mass displaced by the body and the instantaneous acceleration. In compressible flow, the finite acoustic propagation speed means that the unsteady inviscid force arising from an instantaneously applied constant acceleration develops gradually and reaches steady values only for non-dimensional times c(infinity)t/R approximately >10, where c(infinity) is the freestream speed of sound and R is the radius of the cylinder or sphere. In this limit, an effective added-mass coefficient may be defined. The main conclusion of our study is that the freestream Mach number has a pronounced effect on both the peak value of the unsteady force and the effective added-mass coefficient. At a freestream Mach number of 0.5, the effective added-mass coefficient is about twice as large as the incompressible value for the sphere. Coupled with an impulsive acceleration, the unsteady inviscid force in compressible flow can be more than four times larger than that predicted from incompressible theory. Furthermore, the effect of the ratio of specific heats on the unsteady force becomes more pronounced as the Mach number increases.     

Compressibility effects on outflows in a two-fluid system. 1. Line source in cylindrical geometry

A two-dimensional line source outflow is considered, in which the evolution of a sharp interface separating an incompressible fluid from a bounding weakly compressible gas is analysed. Linear theory is applied, assuming that anisotropies in the source outflow are small, to develop an approximate solution for the interfacial evolution. The simplest solutions to the governing linearised equations require the presence of a high-order velocity singularity at the location of the line source. A spectral method is also developed to capture the nonlinear behaviour of the flow; after some finite time, curvature singularities are found to develop on the interface. Comparisons are made between the stability of the interface and its analogue which separates two incompressible fluids. It is found that when the bounding fluid is weakly compressible rather than incompressible, the stability of the interface is significantly increased.     

Surface wave propagation in a fluid-saturated incompressible porous medium

A study of surface wave propagation in a fluid-saturated incompressible porous half-space lying under a uniform layer of liquid is presented. The dispersion relation connecting the phase velocity with wave number is derived. The variation of phase velocity and attenuation coefficients with wave number is presented graphically and discussed. As a particular case, the propagation of Rayleigh type surface waves at the free surface of an incompressible porous half-space is also deduced and discussed.