Plane strain dynamics of elastic solids with intrinsic boundary elasticity, with application to surface wave propagation

In this paper, in a development of the static theory derived by Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853), we establish the equations of motion for a non-linearly elastic body in plane strain with an elastic surface coating on part or all of its boundary. The equations of (linearized) incremental motions superposed on a finite static deformation are then obtained and applied to the problem of (time-harmonic) surface wave propagation on a pre-stressed incompressible isotropic elastic half-space with a thin coating on its plane boundary. The secular equation for (dispersive) wave speeds is then obtained in respect of a general form of incompressible isotropic elastic strain-energy function for the bulk material and a general energy function for the coating material. Specialization of the form of strain-energy function enables the secular equation to be cast as a quartic equation and we therefore focus on this for illustrative purposes. An explicit form for the secular equation is thereby obtained. This involves a number of material parameters, including residual stress and moment in the properties of the coating. It is shown how this equation relates to previous work on waves in a half-space with an overlying thin layer set in the classical theory of isotropic elasticity and, in particular, the significant effect of omission of the rotatory inertia term, even at small wave numbers, is emphasized. Corresponding results for a membrane-type coating, for which the bending moment, inertia and residual moment terms are absent, are also obtained. Asymptotic formulas for the wave speed at large wave number (high frequency) are derived and it is shown how these results influence the character of the wave speed throughout the range of wave number values. A bifurcation criterion is obtained from the secular equation by setting the wave speed to zero, thereby generalizing the bifurcation results of Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853) to the situation in which residual stress and moment are present in the coating. Numerical results which show the dependence of the wave speed on the various material parameters and the finite deformation are then described graphically. In particular, features which differ from those arising in the classical theory are highlighted.     

Post-bifurcation of perfect and imperfect spherical elastic membranes

When a spherical elastic membrane is inflated it is well known that it may bifurcate into an aspherical mode after the pressure maximum is reached. Upon further inflation the spherical configuration is regained. Here we follow the developing aspherical solution path, for specific forms of strain-energy function, using a simple numerical method. For a realistic strain-energy function it is shown that the post-bifurcation solution curve connects the two bifurcation points. We also consider the inflation of imperfect spherical membranes and show that bifurcation may still occur. For the class of Ogden materials we investigate the asymptotic shape of arbitrary axisymmetric membranes.     

Simple Torsion of Isotropic, Hyperelastic, Incompressible Materials with Limiting Chain Extensibility

The purpose of this research is to investigate the simple torsion problem for a solid circular cylinder composed of isotropic hyperelastic incompressible materials with limiting chain extensibility. Three popular models that account for hardening at large deformations are examined. These models involve a strain-energy density which depends only on the first invariant of the Cauchy–Green tensor. In the limit as a polymeric chain extensibility tends to infinity, all of these models reduce to the classical neo-Hookean form. The main mechanical quantities of interest in the torsion problem are obtained in closed form. In this way, it is shown that the torsional response of all three materials is similar. While the predictions of the models agree qualitatively with experimental data, the quantitative agreement is poor as is the case for the neo-Hookean material. In fact, by using a global universal relation, it is shown that the experimental data cannot be predicted quantitatively by any strain-energy density which depends solely on the first invariant. It is shown that a modification of the strain energies to include a term linear in the second invariant can be used to remedy this defect. Whether the modified strain-energies, which reflect material hardening, are a feasible alternative to the classic Mooney–Rivlin model remains an open question which can be resolved only by large strain experiments. This revised version was published online in August 2006 with corrections to the Cover Date.     

Thermal softening induced plastic instability in rate-dependent materials

A linear perturbation analysis is performed for a class of rate-dependent materials, such as the Johnson-Cook model, in which the rate contribution to the stress can be separated from that of the plastic strain and temperature and in which the temperature rises adiabatically. The analysis is facilitated by perturbing both the rate of momentum equation and the momentum equation. An identical material stability/instability criterion is deduced from the characteristic spectral equations for one-dimensional deformation, one-dimensional shearing, and general three-dimensional field equations, and thus shows that the instability derived here is a material constitutive instability.The criteria indicate that the materials become unstable once the thermal softening overcomes the strain hardening, regardless of the strain rate. The strain rate enters the criteria through its effects on the accumulated temperature and the current stress. Based on the criterion, the three-dimensional instability surface is established in the space of plastic strain, plastic strain rate, and temperature. Instability surface is shown as a material property and independent of deformation histories or modes. Both necking and shear banding are simulated to validate the excellent predictive capability of the criterion.     

Nonlinear viscoelasticity of polymer melts in different types of flow

The nonlinear viscoelastic properties of a fairly large class of polymeric fluids can be described with the factorable single integral constitutive equation. For this class of fluids, a connection between the rheological behaviour in different flow geometries can be defined if the strain tensor (or the damping function) is expressed as a function of the invariants of a tensor which describes the macroscopic strain, such as the Finger tensor. A number of these expressions, proposed in the literature, are tested on the basis of the measuring data for a low-density polyethylene melt. In the factorable BKZ constitutive equation the strain-energy function must be expressed as a function of the invariants of the Finger tensor. The paper demonstrates that the strain-energy function can be calculated from the simple shear and simple elongation strain measures, if it is assumed to be of the shape proposed by Valanis and Landel. The measuring data for the LDPE melt indicate that the Valanis-Landel hypothesis concerning the shape of the strainenergy function is probably not valid for polymer melts.     

On SH waves in a pre-stressed layered half-space for an incompressible elastic material

The propagation of Love waves along the boundary between a half-space and a layer of different pre-stressed material is examined for incompressible isotropic elastic materials. The secular equation is obtained for a general strain-energy function and analysed for particular deformations and materials. For the neo-Hookean strain-energy function, numerical results are obtained to illustrate the dependence of the wavespeed on the wave number and on the deformation.     

Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation

In a recent paper we examined the loss of ellipticity and its interpretation in terms of fiber kinking and other instability phenomena in respect of a fiber-reinforced incompressible elastic material. Here we provide a corresponding analysis for fiber-reinforced compressible elastic materials. The analysis concerns a material model which consists of an isotropic base material augmented by a reinforcement dependent on the fiber direction. The assessment of loss of ellipticity can be cast in terms of the eigenvalues of the acoustic tensors associated with the isotropic and anisotropic parts of the strain-energy function. For the anisotropic part, two different reinforcing models are examined and it is shown that, depending on the choice of model and whether the fiber is under compression or extension, loss of ellipticity may be associated with, in particular, a weak surface of discontinuity normal to or parallel to the deformed fiber direction or at an intermediate angle. Under compression the associated failure interpretations include fiber kinking and fiber splitting, while under extension fiber de-bonding and matrix failure are included.     

Surfactant effects on the Rayleigh instability in capillary tubes – non-ideal systems

In a previous work, the instability of a liquid film deposited on the inner walls of a capillary under the presence of insoluble surfactant was analyzed; for that purpose the surface tension was related to the interfacial concentration of surfactant by a linear equation. In general, that assumption is valid when just trace amounts of surfactant are present. The present work extends previous analysis by considering a non-linear surface equation of state derived from the Frumkin adsorption isotherm. This equation of state account not only for the existing quantities of surfactant but also for non-ideal interactions between adsorbed molecules. Except for the equation of state, both the model and the numerical technique employed do not differ from those used in the preceding work. The new predictions here presented show that a linear surface equation of state gives reasonable results for strong surfactants. However, the action of weaker surfactants strongly depends on other parameters: the initial concentration and the type and strength of interaction between adsorbed molecules. Thus, the use of a linear equation of state in these circumstances might give erroneous results.     

The Valanis-Landel Strain-Energy Function

A constitutive equation is derived for the Cauchy stress matrix for arbitrary deformations of an isotropic elastic solid characterized by a Valanis-Landel strain-energy function. A simple example is given of the way in which results for controllable deformations of an incompressible elastic solid, with a Valanis-Landel strain-energy function, can be obtained from the known results for the more general strain-energy function employed by Rivlin.     

On the Normal Stresses in Simple Shearing of Fiber-Reinforced Nonlinearly Elastic Materials

We are concerned with a particular aspect of the simple shear problem within the framework of nonlinear elasticity for a class of incompressible transversely-isotropic fiber-reinforced materials. It is well known that, for isotropic hyperelastic materials, the normal stress effect characteristic of nonlinear elasticity is crucial in order to maintain a homogeneous deformation state in the bulk of the specimen. For the fiber-reinforced materials of concern here, we show that the confining traction that needs to be applied to the top and bottom faces of a block in order to maintain simple shear can be compressive or tensile depending on the degree of anisotropy and on the angle of orientation of the fibers. Inclusion of the second invariant in the isotropic part of the strain-energy used is shown to be of crucial importance in assessing the nature of the confining traction. In the absence of such an applied traction, an unconfined sample tends to bulge outwards or contract inwards perpendicular to the direction of shear. The character of the normal component of traction on the inclined faces is also investigated. The results are relevant to the development of accurate shear test protocols for the determination of constitutive properties of fiber-reinforced rubber-like materials and fibrous biological soft tissues.     

Elastic instabilities for strain-stiffening rubber-like spherical and cylindrical thin shells under inflation

This paper is concerned with investigation of the effects of strain-stiffening on the classical limit point instability that is well-known to occur in the inflation of internally pressurized rubber-like spherical thin shells (balloons) and circular cylindrical thin tubes composed of incompressible isotropic non-linearly elastic materials. For a variety of specific strain-energy densities that give rise to strain-stiffening in the stress-stretch response, the inflation pressure versus stretch relations are given explicitly and the non-monotonic character of the inflation curves is examined. While such results are known for constitutive models that exhibit a gradual stiffening (e.g. exponential and power-law models), our primary focus is on materials that undergo severe strain-stiffening in the stress-stretch response. In particular, we consider two phenomenological constitutive models that reflect limiting chain extensibility at the molecular level. It is shown that for materials with sufficiently low extensibility no limit point instability occurs and so stable inflation is then predicted for such materials. Potential applications of the results to the biomechanics of soft tissues are indicated.     

Strain-energy density function for rubberlike materials

The strain-energy density function surface for the rubber tested by L. R.G. Treloak (1944a) is determined from bis stress-strain data. The data were given for the three pure homogeneous strain paths of simple elongation, pure shear, and equi-biaxial extension of a thin sheet. The surface is formed by plotting calculated points of the strain-energy function above a plane having the first and second strain invariants as rectangular cartesian coordinates. The strain-energy function is expressed as a double power series in the invariants expanded about the zero energy state which is the origin of coordinates. An analysis of this experimentally derived surface provides the information required for the rational selection of terms and the determination of the coefficients in the series expansion, thus defining a function within the Rivlin-type formulation. The function so determined is tested by employing it in the appropriate constitutive formulae to compute stresses for comparison with experimental values. Another test is made by utilizing the function to predict shapes of an inflated membrane for comparison with experimentally observed shapes. Excellent agreement between prediction and experiment is found. A second demonstration is given for another rubber tested by D.F. Jones and L.R.G. Treloar (1975). Again, excellent results are obtained.     

Some New Implications of Certain Constitutive Inequalities in Plane Isotropic Elasticity

In plane isotropic elasticity a strengthened form of the Ordered–Forces inequality is shown to imply that the restriction of the strain-energy function to the class of deformation gradients which share the same average of the principal stretches is bounded from below by the strain energy corresponding to the conformal deformations in this class. For boundary conditions of place, this property (together with a certain version of the Pressure–Compression inequality) is then used (i) to show that the plane radial conformal deformations are stable with respect to all radial variations of class C ¹ and (ii) to obtain explicit lower bounds for the total energy associated with arbitrary plane radial deformations. For the same type of boundary conditions and together with a different version of the Pressure–Compression inequality, an analogous property in plane isotropic elasticity (established in [3] under the assumption that the material satisfies a strengthened form of the Baker–Ericksen inequality and according to which the restriction of the strain-energy function to the class of deformation gradients which share the same determinant is bounded from below by the strain energy corresponding to the conformal deformations in that class) is used (i) to show that the plane radial conformal deformations are stable with respect to all variations of class C ¹ and (ii) to obtain explicit lower bounds for the total energy associated with any plane deformation.     

竖直振动无黏液滴的法拉第不稳定性分析

由于外部周期性的振动而在液滴表面产生的Faraday不稳定效应广泛存在于超声雾化、喷涂加工等应用中,对Faraday不稳定性进行分析对研究振动液滴的表面动力学有着重要意义.本文将Faraday不稳定性问题从径向振动拓展到竖直振动,研究了竖直振动无黏液滴表面波的不稳定性.竖直方向的振动使得液滴动量方程为含有空间相关项和时...     

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

In the conventional theory of finite deformations of fibre-reinforced elastic solids it is assumed that the strain-energy is an isotropic invariant function of the deformation and a unit vector A that defines the fibre direction and is convected with the material. This leads to a constitutive equation that involves no natural length. To incorporate fibre bending stiffness into a continuum theory, we make the more general assumption that the strain-energy depends on deformation, fibre direction, and the gradients of the fibre direction in the deformed configuration. The resulting extended theory requires, in general, a non-symmetric stress and the couple-stress. The constitutive equations for stress and couple-stress are formulated in a general way, and specialized to the case in which dependence on the fibre direction gradients is restricted to dependence on their directional derivatives in the fibre direction. This is further specialized to the case of plane strain, and finite pure bending of a thick plate is solved as an example. We also formulate and develop the linearized theory in which the stress and couple-stress are linear functions of the first and second spacial derivatives of the displacement. In this case for the symmetric part of the stress we recover the standard equations of transversely isotropic linear elasticity, with five elastic moduli, and find that, in the most general case, a further seven moduli are required to characterize the couple-stress.     

Modelling the slight compressibility of anisotropic soft tissue

In order to avoid the numerical difficulties in locally enforcing the incompressibility constraint using the displacement formulation of the Finite Element Method, slight compressibility is typically assumed when simulating transversely isotropic, soft tissue. The current standard method of accounting for slight compressibility of hyperelastic soft tissue assumes an additive decomposition of the strain-energy function into a volumetric and a deviatoric part. This has been shown, however, to be inconsistent with the linear theory for anisotropic materials. It is further shown here that, under hydrostatic tension or compression, a transversely isotropic cube modelled using this additive split is simply deformed into another cube regardless of the size of the deformation, in contravention of the physics of the problem. A remedy for these defects is proposed here: the trace of the Cauchy stress is assumed linear in both volume change and fibre stretch. The general form of the strain-energy function consistent with this model is obtained and is shown to be a generalisation of the current standard model. A specific example is used to clearly demonstrate the differences in behaviour between the two models in hydrostatic tension and compression.     

Cavitation in finite elasticity with surface energy effects

Cavity formation in incompressible as well as compressible isotropic hyperelastic materials under spherically symmetric loading is examined by accounting for the effect of surface energy. Equilibrium solutions describing cavity formation in an initially intact sphere are obtained explicitly for incompressible as well as slightly compressible neo-Hookean solids. The cavitating response is shown to depend on the asymptotic value of surface energy at unbounded cavity surface stretch. The energetically favorable equilibrium is identified for an incompressible neo-Hookean sphere in the case of prescribed dead-load traction, and for a slightly compressible neo-Hookean sphere in the case of prescribed surface displacement as well as prescribed dead-load traction. In the presence of surface energy effects, it becomes possible that the energetically favorable equilibrium jumps from an intact state to a cavitated state with a finite cavity radius, as the prescribed loading parameter passes a critical level. Such discontinuous cavitation characteristics are found to be highly dependent on the relative magnitude of the surface energy to the bulk strain energy.     

Constraints, Constitutive Limits, and Instability in Finite Thermoelasticity

This paper examines all the possible types of thermomechanical constraints in finite-deformational elasticity. By a thermomechanical constraint we mean a functional relationship between a mechanical variable, either the deformation gradient or the stress, and a thermal variable, temperature, entropy or one of the energy potentials; internal energy, Helmholtz free energy, Gibbs free energy or enthalpy. It is shown that for the temperature-deformation, entropy-stress, enthalpy-deformation, and Helmholtz free energy-stress constraints equilibrium states are unstable, in the sense that certain perturbations of the equilibrium state grow exponentially. By considering the constrained materials as constitutive limits of unconstrained materials, it is shown that the instability is associated with the violation of the Legendre–Hadamard condition on the internal energy. The entropy-deformation, temperature-stress, internal energy-stress, and Gibbs free energy-deformation constraints do not exhibit this instability. It is proposed that stability of the rest state (or equivalently convexity of internal energy) is a necessary requirement for a model to be physically valid, and hence entropy-deformation, temperature-stress, internal energy-stress, and Gibbs free energy-deformation constraints are physical, whereas temperature-deformation constraints (including the customary notion of thermal expansion that density is a function of temperature only), entropy-stress constraints, enthalpy-deformation constraints, and Helmholtz free energy-stress constraints are not.     

Simple shear is not so simple

For homogeneous, isotropic, non-linearly elastic materials, the form of the homogeneous deformation consistent with the application of a Cauchy shear stress is derived here for both compressible and incompressible materials. It is shown that this deformation is not simple shear, in contrast to the situation in linear elasticity. Instead, it consists of a triaxial stretch superposed on a classical simple shear deformation, for which the amount of shear cannot be greater than 1. In other words, the faces of a cubic block cannot be slanted by an angle greater than 45° by the application of a pure shear stress alone. The results are illustrated for those materials for which the strain-energy function does not depend on the principal second invariant of strain. For the case of a block deformed into a parallelepiped, the tractions on the inclined faces necessary to maintain the derived deformation are calculated.