On Structured Surfaces with Defects: Geometry,Strain Incompatibility,Stress Field,and Natural Shapes

In this paper we study the stress and deformation fields generated by nonlinear inclusions with finite eigenstrains in anisotropic solids. In particular, we consider finite eigenstrains in transversely isotropic spherical balls and orthotropic cylindrical bars made of both compressible and incompressible solids. We show that the stress field in a spherical inclusion with uniform pure dilatational eigenstrain in a spherical ball made of an incompressible transversely isotropic solid such that the material preferred direction is radial at any point is uniform and hydrostatic. Similarly, the stress in a cylindrical inclusion contained in an incompressible orthotropic cylindrical bar is uniform hydrostatic if the radial and circumferential eigenstrains are equal and the axial stretch is equal to a value determined by the axial eigenstrain. We also prove that for a compressible isotropic spherical ball and a cylindrical bar containing a spherical and a cylindrical inclusion, respectively, with uniform eigenstrains the stress in the inclusion is uniform (and hydrostatic for the spherical inclusion) if the radial and circumferential eigenstrains are equal. For compressible transversely isotropic and orthotropic solids, we show that the stress field in an inclusion with uniform eigenstrain is not uniform, in general. Nevertheless, in some special cases the material can be designed in order to maintain a uniform stress field in the inclusion. As particular examples to investigate such special cases, we consider compressible Mooney-Rivlin and Blatz-Ko reinforced models and find analytical expressions for the stress field in the inclusion.     

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.     

The relation between the Valanis-Landel and classical strain-energy functions

Necessary and sufficient conditions are derived for the strain-energy function of an isotropic elastic solid, regarded as a function of the strain invariants, to be expressible in the Valanis-Landel form, both when the material is compressible and when it is incompressible. In the case when the Valanis-Landel strain-energy function is a polynomial in squares of the principal extension ratios, explicit representations as polynomials in the basic isotropic strain invariants are obtained.     

On the Inviscid Limit for the Compressible Navier–Stokes System in an Impermeable Bounded Domain

In this paper we investigate the issue of the inviscid limit for the compressible Navier–Stokes system in an impermeable fixed bounded domain. We consider two kinds of boundary conditions. The first one is the no-slip condition. In this case we extend the famous conditional result (Kato, T.: Remarks on zero viscosity limit for nonstationary Navier–Stokes flows with boundary. In: Seminar on nonlinear partial differential equations, vol. 2, pp. 85–98. Math. Sci. Res. Inst. Publ., Berkeley 1984) obtained by Kato in the homogeneous incompressible case. Kato proved that if the energy dissipation rate of the viscous flow in a boundary layer of width proportional to the viscosity vanishes then the solutions of the incompressible Navier–Stokes equations converge to some solutions of the incompressible Euler equations in the energy space. We provide here a natural extension of this result to the compressible case. The other case is the Navier condition which encodes that the fluid slips with some friction on the boundary. In this case we show that the convergence to the Euler equations holds true in the energy space, as least when the friction is not too large. In both cases we use in a crucial way some relative energy estimates proved recently by Feireisl et al. in J. Math. Fluid Mech. 14:717–730 (2012).     

Incompressible Elastic Bodies with Non-Convex Energy under Dead-Load Surface Tractions

In this paper we study the equilibrium deformations of an incompressible elastic body with a non-convex strain energy function which is subjected to a homogeneous distribution of dead-load tractions. To determine the stable solutions we consider the mixtures of the phases which minimize the total energy density. In the special case of a trilinear material we discuss the stability of the equilibrium phases in detail. Finally, we show that multiphase solutions are possible when the surface loads correspond to a critical simple shear and we sketch their possible forms. This revised version was published online in July 2006 with corrections to the Cover Date.     

Distortion of Anisotropic Hyperelastic Solids Under Pure Pressure Loading: Compressibility, Incompressibility and Near-Incompressibility

The purpose of this note is to examine distortion during pure pressure loading for anisotropic hyperelastic solids. We contrast the corresponding issues in compressible and incompressible hyperelasticity, and then use these results to examine nearly incompressible materials. An anisotropic compressible hyperelastic solid will generally exhibit both volume change and distortion under hydrostatic pressure loading. In contrast, an incompressible hyperelastic solid—both isotropic and anisotropic—exhibits no change to its current state of deformation as the hydrostatic pressure is varied. Nearly incompressible hyperelastic materials are compressible, but approach an incompressible response in an appropriate limit. We examine this limiting process in the context of transverse isotropy. The issue arises as to how to implement a nearly incompressible version of a given truly incompressible material model. Here we examine how certain implementations eliminate distortion under pure pressure loading and why alternative implementations do not eliminate the distortion.     

Creep transition in torsion

Stresses for a circular cylinder of compressible material subjected to torsion are derived in closed form for steady state creep. It is shown that the asymptotic solution through stress leads from elastic state to plastic and then to creep and through stress difference leads to the creep state. The effect of compressibility is presented graphically. The results indicate that the value of maximum shear stress for a cylinder of compressible material is greater than that for an incompressible material and increases with the increase in a measure index n. For an incompressible material, as a particular case, the results obtained are the same as given by Marin [9].     

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.     

The elastostatic plane strain mode I crack tip stress and displacement fields in a generalized linear neo-Hookean elastomer

A general asymptotic plane strain crack tip stress field is constructed for linear versions of neo-Hookean materials, which spans a wide variety of special cases including incompressible Mooney elastomers, the compressible Blatz–Ko elastomer, several cases of the Ogden constitutive law and a new result for a compressible linear neo-Hookean material. The nominal stress field has dominant terms that have a square root singularity with respect to the distance of material points from the crack tip in the undeformed reference configuration. At second order, there is a uniform tension parallel to the crack. The associated displacement field in plane strain at leading order has dependence proportional to the square root of the same coordinate. The relationship between the amplitude of the crack tip singularity (a stress intensity factor) and the plane strain energy release rate is outlined for the general linear material, with simplified relationships presented for notable special cases.     

On the stress singularities generated by anisotropic eigenstrains and the hydrostatic stress due to annular inhomogeneities

The problems of singularity formation and hydrostatic stress created by an inhomogeneity with eigenstrain in an incompressible isotropic hyperelastic material are considered. For both a spherical ball and a cylindrical bar with a radially symmetric distribution of finite possibly anisotropic eigenstrains, we show that the anisotropy of these eigenstrains at the center (the center of the sphere or the axis of the cylinder) controls the stress singularity. If they are equal at the center no stress singularity develops but if they are not equal then stress always develops a logarithmic singularity. In both cases, the energy density and strains are everywhere finite. As a related problem, we consider annular inclusions for which the eigenstrains vanish in a core around the center. We show that even for an anisotropic distribution of eigenstrains, the stress inside the core is always hydrostatic. We show how these general results are connected to recent claims on similar problems in the limit of small eigenstrains.     

Understanding the need of the compression branch to characterize hyperelastic materials

Soft biological tissues are frequently modeled as hyperelastic materials. Hyperelastic behavior is typically ensured by the assumption of a stored energy function with a pre-determined shape. This function depends on some material parameters which are obtained through an optimization algorithm in order to fit experimental data from different tests. For example, when obtaining the material parameters of isotropic, incompressible models, only the extension part of a uniaxial test is frequently taken into consideration. In contrast, spline-based models do not require material parameters to exactly fit the experimental data, but need the compression branch of the curve. This is not a disadvantage because as we explain herein, to properly characterize hyperelastic materials, the compression branch of the uniaxial tests (or valid alternative tests) is also needed, in general. Then, unless we know beforehand the tendency of the compression branch, a material model should not be characterized only with tensile tests. For simplicity, here we address isotropic, incompressible materials which use the Valanis-Landel decomposition. However, the concepts are also applicable to compressible isotropic materials and are specially relevant to compressible and incompressible anisotropic materials, because in biomechanics, materials are frequently characterized only by tensile tests.     

A theoretical analysis of unsteady incompressible flows for time-dependent constitutive equations based on domain transformations

In this paper, we propose an analysis that allows calculation of kinematic histories in unsteady problems of continuum mechanics, in relation to the use of memory-integral constitutive equations. Such cases particularly concern flow conditions of processing rheology, requiring evaluation of strain or deformation rate tensors, for viscoelastic incompressible fluids as polymers. In two- and three-dimensional cases, we apply concepts of the stream-tube method (STM) initially given for stationary conditions, where unknown local or global mapping functions are defined instead of classic velocity-pressure formulations, leading to consider the flow parameters in domains where the streamlines and trajectories are parallel straight lines. The approach enables us to provide accurate formulae for evaluating the kinematics histories that can be used later for computing the stresses for a given memory-integral model.     

Bulk Cavitation and the Possibility of Localized Interface Deformation due to Surface Layer Swelling

We consider the uniform swelling of a compressible hyperelastic surface layer with finite thickness that is attached to an underlying bulk material composed of a non-swelling incompressible hyperelastic material. In addition to classically smooth solutions, two additional phenomena may occur for sufficiently large swelling. One is the formation of cavities in the interior of the underlying bulk material. The other is the disappearance of smooth solutions in the surface layer while the underlying bulk material remains intact. It is conjectured that the latter may be associated with the concentration of deformation at the swelling interface. Both phenomena are investigated by the consideration of solutions to a boundary value problem for a sphere involving radial deformation with a prescribed swelling field that acts as an effective loading device. Specific material models for both the compressible swollen surface layer and the non-swollen incompressible bulk are invoked so as to permit an analytical treatment. Swelling thresholds are obtained that depend on the thickness of the surface layer for the onset of these separate phenomena.     

Scaling of energy spectra in weakly compressible turbulence

We find an asymptotic expression for the char-acteristic timescales of decorrelation processes in weakly compressible and isothermal turbulence. This result is used in the Eddy-Damped Quasi-Normal Markovian equation to derive the scalings of compressible energy spectra: (1) if the acoustic waves are dominant, the compressible energy spectra exhibit ?7/3 scaling; (2) if local eddy straining is dominant, the compressible energy spectra are scaled as ?3. Meanwhile, the energy spectra of incompressible components display the same scaling of ?5/3 as those in incompressible turbulence. The direct numerical simulations of weakly compressible turbulence are used to examine the scaling.     

On Slightly Compressible Ideal Flow¶in the Half-Plane

We consider the Euler equations of barotropic inviscid compressible fluids in the half-plane. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In 2D (two dimensions) such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial data. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small. We decompose the solution as the sum of the irrotational part, the incompressible part and the remainder, which describes the interaction between the first two components. First we study the life span of smooth irrotational solutions, i.e., the largest time interval T(?) of existence of classical solutions, when the initial data are a small perturbation of size ? from a constant state. Related to this is a decay property for the irrotational part. Then, we study the interaction between the two components and show the existence on any arbitrary time interval, for any Mach number sufficiently small. This yields the existence of smooth compressible flow on any arbitrary time interval. For the proofs we use a combination of energy and decay estimates.     

2D Slightly Compressible Ideal Flow in an Exterior Domain

We consider the Euler equations of barotropic inviscid compressible fluids in the exterior domain. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In dimension 2 such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial data. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small. First we study the life span of smooth irrotational solutions, i.e. the largest time interval of existence of classical solutions, when the initial data are a small perturbation of size from a constant state. Then, we study the nonlinear interaction between the irrotational part and the incompressible part of a general solution. This analysis yields the existence of smooth compressible flow on any arbitrary time interval and with no restriction on the size of the initial velocity, for any Mach number sufficiently small. Finally, the approach is applied to the study of the incompressible limit. For the proofs we use a combination of energy estimates and a decay estimate for the irrotational part.     

A class of exact solutions for finite plane strain deformations of a particular elastic material

Summary In this paper we consider plane deformations of an incompressible elastic material and we show that by a suitable choice of strain energy function we can find the class of deformations with constant local rotation angle. Although the form for the strain energy function is chosen in the first place for mathematical convenience it does correspond to physically reasonable behaviour and such a theory may be regarded as a first order theory. The class of solutions obtained are expressed in a parametric form involving an arbitrary function, simple choices of which correspond to the well known exact solutions of finite elasticity.     

Remarks on the Instability of an Incompressible and Isotropic Hyperelastic, Thick-Walled Cylindrical Tube

The problem of instability of a hyperelastic, thick-walled cylindrical tube was first studied by Wilkes [1] in 1955. The solution was formulated within the framework of the theory of small deformations superimposed on large homogeneous deformations for the general class of incompressible, isotropic materials; and results for axially symmetrical buckling were obtained for the neo-Hookean material. The solution involves a certain quadratic equation whose characteristic roots depend on the material response functions. For the neo-Hookean material these roots always are positive. In fact, here we show for the more general Mooney–Rivlin material that these roots always are positive, provided the empirical inequalities hold. In a recent study [2] of this problem for a class of internally constrained compressible materials, it is observed that these characteristic roots may be real-valued, pure imaginary, or complex-valued. The similarity of the analytical structure of the two problems, however, is most striking; and this similarity leads one to question possible complex-valued solutions for the incompressible case. Some remarks on this issue will be presented and some new results will be reported, including additional results for both the neo-Hookean and Mooney–Rivlin materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

Universal Deformations for a Class of Compressible Isotropic Hyperelastic Materials

Differential conditions are derived for a smooth deformation to be universal for a class of isotropic hyperelastic materials that we regard as a compressible variant (a notion we make precise) of Mooney–Rivlin’s class, and that includes the materials studied originally by Tolotti in 1943 and later, independently, by Blatz. The collection of all universal deformations for an incompressible material class is shown to contain, modulo a uniform dilation, all the universal deformations for its compressible variants. As an application of this result, by searching the known families of universal deformations for all incompressible isotropic materials, a nontrivial universal deformation for Tolotti materials is found. This revised version was published online in August 2006 with corrections to the Cover Date.