On a representation of the solution of the linear elasticity equations

A new representation of the stress tensor in the linear theory of elasticity is proposed. The representation satisfies the equilibrium equations and the compatibility conditions for strains. In this representation, the stress tensor is expressed in terms of a harmonic vector. The second boundary-value problem for an elastic half-space and elastic layer is considered as an example.Translated from Prikladnaya Mekhanika, Vol. 40, No. 11, pp. 85–91, November 2004.This revised version was published online in April 2005 with a corrected cover date.     

Representation Theorems for Hemitropic and Tranversely Isotropic Tensor Functions

A representation theorem for transversely isotropic tensor-valued functions of a symmetric tensor variable is proved. The theorem holds in any finite dimension. The proof is based on the decomposition of a symmetric tensor of dimension N into a scalar, a vector, and a symmetric tensor of dimension N-1, and on the fact that the transverse isotropy of the original function is equivalent to the hemitropy of three functions, one scalar-valued, one vector-valued, and one tensor-valued, of the last two terms in the decomposition. Representation theorems for the three functions are obtained as generalizations of two theorems of W. Noll on isotropic functions. The proofs make use of an appropriate algebraic structure based on alternating forms. The three-dimensional case, as well as those of linear and of hyperelastic functions, are treated as special cases. This revised version was published online in August 2006 with corrections to the Cover Date.     

Relaxed Energy for Transversely Isotropic Two-Phase Materials

The paper gives a simple derivation of the relaxed energy W ^qc for the quadratic double-well material with equal elastic moduli and analyzes W ^qc in the transversely isotropic case. We observe that the energy W is a sum of a degenerate quadratic quasiconvex function and a function that depends on the strain only through a scalar variable. For such a W, the relaxation reduces to a one-dimensional convexification. W ^qc depends on a constant g defined by a three-dimensional maximum problem. It is shown that in the transversely isotropic case the problem reduces to a maximization of a fraction of two quadratic polynomials over [0,1]. The maximization reveals several regimes and explicit formulas are given in the case of a transversely isotropic, positive definite displacement of the wells. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the Spectral Representation of Stretch and Rotation

By means of Sylvester's spectral representation of tensor-valued functions, a representation is derived for the finite stretch and rotation associated with the polar decomposition of a given deformation gradient. This revised version was published online in August 2006 with corrections to the Cover Date.     

Remarks on the Behavior of Simple Directionally Reinforced Incompressible Nonlinearly Elastic Solids

The effect of directional reinforcing in generating qualitative changes in the mechanical response of a base neo-Hookean material is examined in the context of homogenous deformation. Single axis reinforcing giving transverse isotropy is the major focus, in which case a standard reinforcing model is characterized by a single constitutive reinforcing parameter. Various qualitative changes in the mechanical response ensue as the reinforcing parameter increases from the zero-value associated with neo-Hookean response. These include (in order): the existence of a limiting contractive stretch for transverse-axis tensile load; loss of monotonicity in off-axis simple shear; loss of monotonicity in on-axis compression; loss of positivity in the stress-shear product in off-axis simple shear; and loss of monotonicity for plane strain in on-axis compression. The qualitative changes in the simple shear response are associated with stretch relaxation in the reinforcing direction due to finite rotation. This revised version was published online in August 2006 with corrections to the Cover Date.     

Analytical solution of lipid membrane morphology subjected to boundary forces on the edges of rectangular membranes

We develop a complete analytical solution predicting the deformation of rectangular lipid membranes resulting from boundary forces acting on the perimeter of the membrane. The shape equation describing the equilibrium state of a lipid membrane is taken from the classical Helfrich model. A linearized version of the shape equation describing membrane morphology (within the Monge representation) is obtained via a limit of superposed incremental deformations. We obtain a complete analytical solution by reducing the corresponding problem to a single partial differential equation and by using Fourier series representations for various types of boundary forces. The solution obtained predicts smooth morphological transition over the domain of interest. Finally, we note that the methods used in our analysis are not restricted to the particular type of boundary conditions considered here and can accommodate a wide class of practical and important edge conditions.     

On the Representation of Symmetric Isotropic 4-Tensors

This note provides short proof of the representation of a symmetric isotropic 4-tensor in an n-dimensional real Euclidean space. This revised version was published online in August 2006 with corrections to the Cover Date.     

Lighthill transpiration velocity revisited: An exact formulation

The Lighthill transpiration-velocity correction is commonly used in the analysis of viscous flows of aeronautical interest, in order to take into account the perturbation to the potential flow caused by the presence of the vorticity in the boundary layer. This correction consists of considering the boundary as a permeable surface from which the fluid flows through the boundary surface, with a velocity (named the transpiration velocity) determined on the basis of the local boundary-layer characteristics. Here, we use a new potential-vorticity decomposition in order to derive an exact representation of the effects of the vorticity on the external flow. The relationship between approximate transpiration-velocity representation and the exact one presented here is analyzed: it is shown that, under typical boundary-layer assumptions, the new representation reduces to that by Lighthill, except for a corrective field term. Finally, in order to quantify, in a simple case, the contribution of the corrective terms which arise in the new formulation, we examine, as a numerical test case, the problem of an attached boundary-layer flow over a flat plate: the numerical results indicate that the corrective term is negligible for Reynolds numbers above 10⁴.A preliminary version of this paper was presented at VI Convegno Italiano di Meccanica Computazionale, Brescia, Italy, October 1991.     

The concept of material forces in phase transition problems within the level-set framework

The dynamics of a phase transition front in solids using the level set method is examined in this paper. Introducing an implicit representation of singular surfaces, a regularized version of the sharp interface model arises. The interface transforms into a thin transition layer of nonzero thickness where all quantities take inhomogeneous expressions within the body. It is proved that the existence of an inhomogeneous energy of the material predicts inhomogeneity forces that drive the singularity. The driving force is a material force entering the canonical momentum equation (pseudo-momentum) in a natural way. The evolution problem requires a kinetic relation that determines the velocity of the phase transition as a function of the driving force. Here, the kinetic relation is produced by invoking relations that can be considered as the regularized versions of the Rankine–Hugoniot jump conditions. The effectiveness of the method is illustrated in a shape memory alloy bar.     

Contact Problem for Porous Elastic Half-Plane

The paper is concerned with a static contact problem about a rigid punch on the free surface of a linear porous elastic half-plane. With the use of the Fourier transform the problem is reduced to a singular integral equation holding over the contact zone. This integral representation permits consideration of the Flamant problem (a line load on the half-plane) to be explicitly reduced to some quadratures. It is shown that in the classical linear elasticity limit the main integral equation has a Cauchy-type kernel, so distribution of the contact pressure is like in the Sadowsky punch-problem. For arbitrary porosity a numerical co-location technique is applied that allows one to analyze in detail the distribution of the contact pressure versus porosity. Both in the Flamant and Sadowsky problems we demonstrate a higher compliance of the porous foundation, with respect to the classical linear elastic results. This revised version was published online in July 2006 with corrections to the Cover Date.     

On Constitutive Equations of Cauchy Elastic Solids: All Kinds of Crystals and Quasicrystals

Our concern is with the problem of determining general reduced forms of constitutive equations of Cauchy elastic solids under all kinds of material symmetries, including well-known crystal classes and newly discovered quasicrystal classes. By means of Tschebysheff polynomials we present in unified forms simple irreducible representations for elastic constitutive equations under the infinitely many subgroup classes C _2m+1, C _2m+2, D _2m+1 and D _2m+2 for all m = 1, 2,... Moreover, we provide a simple representation for elastic constitutive equations under the most complicated point group, i.e. the icosahedral group. Each presented representation is expressed in terms of not more than nine polynomial tensor generators only. This revised version was published online in August 2006 with corrections to the Cover Date.     

Determination of the Rotation Tensor in the Polar Decomposition

An explicit representation for the rotation tensor which contains the lower powers of deformation gradient is proposed and used to evaluate the angle and axis of the rotation tensor. Some related equations about the rotation tensor are established. Through the approximate analysis, the relation between the S-R decomposition and the polar decomposition is examined. This revised version was published online in July 2006 with corrections to the Cover Date.     

The non-uniqueness of the atomistic stress tensor and its relationship to the generalized Beltrami representation

The non-uniqueness of the atomistic stress tensor is a well-known issue when defining continuum fields for atomistic systems. In this paper, we study the non-uniqueness of the atomistic stress tensor stemming from the non-uniqueness of the potential energy representation. In particular, we show using rigidity theory that the distribution associated with the potential part of the atomistic stress tensor can be decomposed into an irrotational part that is independent of the potential energy representation, and a traction-free solenoidal part. Therefore, we have identified for the atomistic stress tensor a discrete analog of the continuum generalized Beltrami representation (a version of the vector Helmholtz decomposition for symmetric tensors). We demonstrate the validity of these analogies using a numerical test. A program for performing the decomposition of the atomistic stress tensor called MDStressLab is available online at http://mdstresslab.org.     

On the stress field singularity in an incompressible half-space weakened by two near-surface wedge-like cracks

We consider the equilibrium problem for an elastic incompressible half-space weakened by two near-surface wedge-like cracks, whose lie in the same plane perpendicular to the half-space surface and have a common vertex. We use the Papkovich-Neuber representation to reduce the problem to finding two harmonic functions satisfying the mixed boundary conditions. These functions are constructed in spherical coordinates by using a Mehler-Fock type integral representation in Legendre functions. The analytic solution thus obtained permits finding the character of the stress distribution near the common tip of the cracks.     

Symmetrical Representation of Stresses in the Stroh Formalism and its Application to a Dislocation and a Dislocation Dipole in an Anissotropic Elastic Medium

Symmetrical stress representation in the Stroh formalism for anisotropic elastic bodies is introduced and the range of its applicability is analysed. By making use of this stress representation new formulae for influence functions giving stresses in an infinite anisotropic medium subjected to a straight dislocation and a straight dislocation dipole are derived. The advantage of the new formulae is that they explicitly show the symmetrical structure of these influence functions not referred to previously. Relations of these influence functions to influence functions giving stresses and Airy stress function due to a straight wedge disclination, whose explicit expressions are also introduced, are derived. Application of these results in computation of stresses by the hypersingular and regularized Somigliana stress identities is discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Elastic Anisotropic Material with Purely Longitudinal and Transverse Waves

The simplest form of the matrix of elasticity moduli of an anisotropic material conducting purely longitudinal and transverse waves with an arbitrary direction of the wave normal is obtained. A generic solution of equations in displacements is represented in terms of three functions satisfying independent wave equations. In the case of planar deformation, this solution yields a complex representation coinciding with the Kolosov–Muskhelishvili formulas for an isotropic material. The formulas in the present work also determine an anisotropic material with Young's modulus identical for all directions, as in an isotropic medium.     

The Representation Theorem for Linear, Isotropic Tensor Functions in Even Dimensions

In this note we give a proof of the representation theorem for linear, isotropic, tensor functions, which only assumes invariance under proper orthogonal tensors. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the explosion of linearly distributed charge with curvilinear shape on the ground surface

We use the momentum version proposed by M. A. Lavrent'yev [1, 2] to treat the two-dimensional problem of the explosion of a linearly distributed charge of curvilinear shape on the ground surface. The problem of the explosion of a straight charge was solved for the first time in [2] in this version. The ground is assumed to be an ideal incompressible liquid at velocities exceeding some critical velocity which remains constant along the crater; beyond this boundary, the medium is fixed. The potential of the velocity is assumed to be constant on the charge and vanishing on the ground surface.     

The Lagrange Multiplier in Incompressible Elasticity Theory

The first variation condition for the potential energy in nonlinear elasticity for incompressible materials provides a linear functional which vanishes on an appropriately constrained set of variations. We prove a representation theorem for such linear functionals which forms the basis for the existence of a constraint reaction (Lagrange multiplier) field. This revised version was published online in August 2006 with corrections to the Cover Date.