Objective constitutive relations in elasticity and viscoelasticity

Summary The compatibility between the objectivity principle and affine constitutive equations for the elastic Cauchy and Piola-Kirchhoff stress tensors with non-zero residual stress is examined. It is found that the Cauchy stress is allowed to be only a constant tensor, proportional to the identity tensor, while the Piola-Kirchhoff stress may be a linear function on the deformation gradient thus generalizing previous results by Fosdick and Serrin. The same conclusions are arrived at also by starting from viscoelasticity. Finally, in the case of Maxwell-like materials, the solutions to the objective evolution equations are shown to be objective functionals.

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Sommario Si esamina la compatibilità tra il principio di obiettività ed equazioni costitutive affini per i tensori di stress elastici di Cauchy a Piola-Kirchhoff con stress residuo non nullo. Generalizzando risultati di Fosdick e Serrin si prova che il tensore di Cauchy può essere soltanto un tensore costante, proporzionale al tensore identità, mentre il tensore di Piola-Kirchhoff può essere una funzione lineare del gradiente di deformazione. Alle stesse conclusioni si perviene anche partendo dal funzionale della viscoelasticità. Infine si mostra che, per materiali tipo Maxwell, le soluzioni di equazioni di evoluzione obiettive sono funzionali obiettivi.

     

A gradient elasticity theory for second-grade materials and higher order inertia

Second-grade elastic materials featured by a free energy depending on the strain and the strain gradient, and a kinetic energy depending on the velocity and the velocity gradient, are addressed. An inertial energy balance principle and a virtual work principle for inertial actions are envisioned to enrich the set of traditional theoretical tools of thermodynamics and continuum mechanics. The state variables include the body momentum and the surface momentum, related to the velocity in a nonstandard way, as well as the concomitant mass-accelerations and inertial forces, which do intervene into the motion equations and into the force boundary conditions. The boundary traction is the sum of two parts, i.e. the Cauchy traction and the Gurtin–Murdoch traction, whereas the traction boundary condition exhibits the typical format of the equilibrium equation of a material surface (as known from the principles of surface mechanics) whereby the Gurtin–Murdoch traction (incorporating the inertial surface force) plays the role of applied surfacial force density. The body’s boundary surface constitutes a thin boundary layer which is in global equilibrium under all the external forces applied on it, a feature that makes it possible to exploit the traction Cauchy theorem within second-grade materials. This means that a second-grade material is formed up by two sub-systems, that is, the bulk material operating as a classical Cauchy continuum, and the thin boundary layer operating as a Gurtin–Murdoch material surface. The classical linear and angular momentum theorems are suitably extended for higher order inertia, from which the local motion equations and the moment equilibrium equations (stress symmetry) can be derived. For an isotropic material featured by four constants, i.e. the Lamé constants and two length scale parameters (Aifantis model), the dynamic evolution problem is characterized by a Hamilton-type variational principle and a solution uniqueness theorem. Closed-form solutions of the wave dispersion analysis problem for beam models are presented and compared with known results from the literature. The paper indicates a correct thermodynamically consistent way to take into account higher order inertia effects within continuum mechanics.     

Finite Anticlastic Bending of Hyperelastic Solids and Beams

This paper deals with the equilibrium problem in nonlinear elasticity of hyperelastic solids under anticlastic bending. A three-dimensional kinematic model, where the longitudinal bending is accompanied by the transversal deformation of cross sections, is formulated. Following a semi-inverse approach, the displacement field prescribed by the above kinematic model contains three unknown parameters. A Lagrangian analysis is performed and the compressible Mooney-Rivlin law is assumed for the stored energy function. Once evaluated the Piola-Kirchhoff stresses, the free parameters of the kinematic model are determined by using the equilibrium equations and the boundary conditions. An Eulerian analysis is then accomplished to evaluating stretches and stresses in the deformed configuration. Cauchy stress distributions are investigated and it is shown how, for wide ranges of constitutive parameters, the obtained solution is quite accurate. The whole formulation proposed for the finite anticlastic bending of hyperelastic solids is linearized by introducing the hypothesis of smallness of the displacement and strain fields. With this linearization procedure, the classical solution for the infinitesimal bending of beams is fully recovered.     

Separation phenomena at the interface of a finitely deformed composite sphere

The problem of the finite deformation of a composite sphere subjected to a spherically symmetric dead load traction is revisited focusing on the formation of a cavity at the interface between a hyperelastic, incompressible matrix shell and a rigid inhomogeneity. Separation phenomena are assumed to be governed by a vanishingly thin interfacial cohesive zone characterized by uniform normal and tangential interface force–separation constitutive relations. Spherically symmetric cavity shapes (spheres) are shown to be solutions of an interfacial integral equation depending on the strain energy density of the matrix, the interface force constitutive relation, the dead loading and the volume concentration of inhomogeneity. Spherically symmetric and non-symmetric bifurcations initiating from spherically symmetric equilibrium states are analyzed within the framework of infinitesimal strain superimposed on a given finite deformation. A simple formula for the dead load required to initiate the non-symmetrical rigid body mode is obtained and a detailed examination of a few special cases is provided. Explicit results are presented for the Mooney–Rivlin strain energy density and for an interface force–separation relation which allows for complete decohesion in normal separation.     

Uniqueness and Complementary Energy in Nonlinear Elastostatics

Global uniqueness of the smooth stress and deformation to within the usual rigid-body translation and rotation is established in the null traction boundary value problem of nonlinear homogeneous elasticity on a n-dimensional star-shaped region. A complementary energy is postulated to be a function of the Biot stress and to be para-convex and rank-(n-1) convex, conditions analogous to quasi-convexity and rank-(n-2) of the stored energy function. Uniqueness follows immediately from an identity involving the complementary energy and the Piola-Kirchhoff stress. The interrelationship is discussed between the two conditions imposed on the complementary energy, and between these conditions and those known for uniqueness in the linear elastic traction boundary value problem.     

Linear Constitutive Relations in Isotropic Finite Elasticity

For simple shearing and simple extension deformations of a homogeneous and isotropic elastic body, it is shown that a linear relation between the second Piola-Kirchhoff stress tensor and the Green-St. Venant strain tensor does not predict a physically reasonable response of the body. This constitutive relation implies that the slope of the curve between an appropriate component of the first Piola-Kirchhoff stress tensor and a deformation measure is an increasing functions of the deformation measure. This revised version was published online in August 2006 with corrections to the Cover Date.     

Experimental analysis of the deformation of convex cylindrical surfaces

The described method is based on the moiré technque. By applying a special stripping film with a copied grid to a convex cylindrical surface, it is possible to express the equations of the deformed as well as the undeformed grid. By using the double-exposure photographic technique, a field of moiré fringes will be obtained, which gives information of the component perpendicular to the direction of observation of the deformation vector in the plane perpendicular to the axis of revolution. By observing two different directions α₁ and α₂, the experimental data yield the calculation of the displacement vector. The application of the derived method and its reliability will be demonstrated by several examples. The method does not require highly expensive laboratory equipment and is especially useful for engineers in consulting and structural design.     

Regularized method of fundamental solutions for boundary identification in two-dimensional isotropic linear elasticity

We investigate the stable numerical reconstruction of an unknown portion of the boundary of a two-dimensional domain occupied by an isotropic linear elastic material from a prescribed boundary condition on this part of the boundary and additional displacement and traction measurements (i.e. Cauchy data) on the remaining known portion of the boundary. This inverse geometric problem is approached by combining the method of fundamental solutions (MFS) and the Tikhonov regularization method, whilst the optimal value of the regularization parameter is chosen according to the discrepancy principle. Various geometries are considered, i.e. convex and non-convex domains with a smooth or piecewise smooth boundary, in order to show the numerical stability, convergence, consistency and computational efficiency of the proposed method.     

The Electronic Structure of Smoothly Deformed Crystals: Wannier Functions and the Cauchy–Born Rule

The electronic structure of a smoothly deformed crystal is analyzed for the case when the effective Hamiltonian is a given function of the nuclei by considering the regime when the scale of the deformation is much larger than the lattice parameter. Wannier functions are defined by projecting the Wannier functions for the undeformed crystal to the space spanned by the wave functions of the deformed crystal. The exponential decay of such Wannier functions is proved for the case when the undeformed crystal is an insulator. The celebrated Cauchy–Born rule for crystal lattices is extended to the present situation for electronic structure analysis.     

A contour integral method for dynamic stress intensity factors

The contour integral method previously used to determine static stress intensity factors is applied to dynamic crack problems. The required derivatives of the traction in the reference problem are obtained numerically by the displacement discontinuity method. Stress intensity factors are determined by an integral around a contour which contains a crack tip. If the contour is chosen as the outer boundary of the body, the stress intensity factor is obtained from the boundary values of traction and displacement. The advantage of this path-independent integral is that it yields directly both the opening-mode and sliding-mode stress intensity factors for a straight crack. For dynamic problems, Laplace transforms are used and the dynamic stress intensity factors in the time domain are determined by Durbin's inversion method. An indirect boundary element method, incorporating both displacement discontinuity and fictitious load techniques, is used to determine the boundary or contour values of traction and displacement numerically.     

Equilibrium of an internal transverse crack in a semiinfinite elastic body with thin coating

Static elasticity problems for a half-plane and a strip weakened by a rectilinear transverse crack are studied. In each case, the upper boundary of the body is reinforced by a flexible patch. Various versions of conditions on the lower boundary are considered in the case of the strip. The crack is maintained in the open state by distributed normal forces. The method of generalized integral transforms reduces solving the problem for the equations of equilibriumto solving a singular integral equation of the first kind with the Cauchy kernel with respect to the derivative of the crack opening function. The solutions of the integral equation are constructed by the small parameter and collocation methods for various combinations of the geometric and physical parameters of the problem, and the structure of the solutions is analyzed. The values of the stress intensity factor (SIF) near the crack vertex are obtained.     

On a class of exact solutions in nonlinear elasticity

A class of exact solutions to the equations of nonlinear elasticity that occur at constant pressure on the boundary of the body and null Cauchy deviatoric stress is presented. Stability analysis shows that the solutions in this class are at best neutrally stable.     

A second strain gradient elasticity theory with second velocity gradient inertia – Part I: Constitutive equations and quasi-static behavior

A multi-cell homogenization procedure with four geometrically different groups of cell elements (respectively for the bulk, the boundary surface, the edge lines and the corner points of a body) is envisioned, which is able not only to extract the effective constitutive properties of a material, but also to assess the “surface effects” produced by the boundary surface on the near bulk material. Applied to an unbounded material in combination with the thermodynamics energy balance principles, this procedure leads to an equivalent continuum constitutively characterized by (ordinary, double and triple) generalized stresses and momenta. Also, applying this procedure to a (finite) body suitably modelled as a simple material cell system, in association with the principle of the virtual power (PVP) for quasi-static actions, an equivalent structural system is derived, featured by a (macro-scale) PVP having the typical format as for a second strain gradient material model. Due to the surface effects, the latter model does work as a combination of two subsystems, i.e. the bulk material behaving as a Cauchy continuum, and the boundary surface operating as a membrane-like boundary layer, each subsystem being in (local and global) equilibrium by its own. Further, the applied (ordinary) boundary traction splits into two (response-dependent) parts, i.e. the “Cauchy traction” transmitted to the bulk material and the “Gurtin–Murdoch traction” acting, together with all other boundary tractions, upon the boundary layer. The role of the boundary layer as a two-dimensional manifold enclosing a Cauchy continuum is elucidated, also with the aid of a discrete model. A strain gradient elasticity theory is proposed which includes a minimum total potential energy principle featuring the relevant boundary-value problem for quasi-static loads and its (unique) solution. A simple application is presented. Two appendices are included, one reports the proof of the global equilibrium of the boundary layer, the other is concerned with double and triple stresses. The paper is complemented by a companion Part II one on dynamics. Previous findings by the author [Polizzotto, C., 2012. A gradient elasticity theory for second-grade materials and higher order inertia. Int. J. Solids Struct. 49, 2121–2137] are improved and extended.     

On Molecular Modelling and Continuum Concepts

Continuum concepts and field values are related to local (scale-dependent) spacetime atomistic averages. Spatial averaging is effected by both weighting function and cellular localisation procedures, and resulting forms of linear momentum balance are compared. The former yields a local balance directly, with several candidate interaction stress fields. The latter results in a global balance involving a traction field expressible in terms of an interaction stress tensor field. In both approaches the Cauchy stress incorporates distinct interaction and thermokinetic contributions. Inter alia are addressed physically-distinguished choices of weighting function; the scale-dependence of the boundary of a body, its motion and material points thereof; physical interpretations of various candidate interaction stress tensors; temporal averaging and material systems whose content changes with time; and the possible relevance of the latter to investigating a molecular context for configurational forces.     

Determination of the midsurface of a deformed shell from prescribed surface strains and bendings via the polar decomposition

We show how to determine the midsurface of a deformed thin shell from known geometry of the undeformed midsurface as well as the surface strains and bendings. The latter two fields are assumed to have been found independently and beforehand by solving the so-called intrinsic field equations of the non-linear theory of thin shells. By the polar decomposition theorem the midsurface deformation gradient is represented as composition of the surface stretch and 3D finite rotation fields. Right and left polar decomposition theorems are discussed. For each decomposition the problem is solved in three steps: (a) the stretch field is found by pure algebra, (b) the rotation field is obtained by solving a system of first-order PDEs, and (c) position of the deformed midsurface follows then by quadratures. The integrability conditions for the rotation field are proved to be equivalent to the compatibility conditions of the non-linear theory of thin shells. Along any path on the undeformed shell midsurface the system of PDEs for the rotation field reduces to the system of linear tensor ODEs identical to the one that describes spherical motion of a rigid body about a fixed point. This allows one to use analytical and numerical methods developed in analytical mechanics that in special cases may lead to closed-form solutions.     

Some problems in the antiplane shear deformation of bi-material wedges

The antiplane shear deformation of a bi-material wedge with finite radius is studied in this paper. Depending upon the boundary condition prescribed on the circular segment of the wedge, traction or displacement, two problems are analyzed. In each problem two different cases of boundary conditions on the radial edges of the composite wedge are considered. The radial boundary data are: traction–displacement and traction–traction. The solution of governing differential equations is accomplished by means of finite Mellin transforms. The closed form solutions are obtained for displacement and stress fields in the entire domain. The geometric singularities of stress fields are observed to be dependent on material property, in general. However, in the special case of equal apex angles in the traction–traction problem, this dependency ceases to exist and the geometric singularity shows dependency only upon the apex angle. A result which is in agreement with that cited in the literature for bi-material wedges with infinite radii. In part II of the paper, Antiplane shear deformation of bi-material circular media containing an interfacial edge crack is considered. As a special case of bi-material wedges studied in part I of the paper, explicit expressions are derived for the stress intensity factor at the tip of an edge crack lying at the interface of the bi-material media. It is seen that in general, the stress intensity factor is a function of material property. However, in special cases of traction–traction problem, i.e., similar materials and also equal apex angles, the stress intensity factor becomes independent of material property and the result coincides with the results in the literature.     

Uniqueness in linear elastostatics for problems involving unbounded bodies

The problem of the uniqueness of elastostatic solutions to various boundary value problems involving unbounded two- and three-dimensional bodies is considered. The general boundary value problem is first considered for anisotropic bodies on which the elasticity tensor is uniformly positive definite. The displacement problem is then considered for the cases where the body is homogeneous and isotropic and the elasticity tensor is strongly-elliptic. Finally, the traction problem on homogeneous, isotropic bodies is considered with fairly little restriction placed on the Lamé moduli. In the first two cases no restriction is placed on the geometry of the body other than the normal assumptions allowing for the use of the divergence theorem. For the traction problem, however, it is assumed that one component of the outward normal to the boudary of the body almost never vanishes. In each case it is shown that the desired uniqueness results hold with fairly mild restrictions placed on the displacement or stress fields (depending on the problem) in a neighborhood of infinity. Where possible, the results are shown to hold even though the solutions may possess the sort of discontinuities and singularities commonly found in problems involving cracks, corners, and bonded interfaces.Related results involving the insolvability of certain non-trivial boundary value problems and the behavior of certain elastic states are, where appropriate, developed.     

Flexural Stress Concentration Near a Hyperboloidal Neck in a Transversely Isotropic Piezoceramic Cylinder

The general solution of the electroelastic problem for a transversely isotropic hyperboloid of revolution is used to find the stress concentration near a hyperboloidal neck in a piezoceramic body subjected to bending. The solution is a sum of four partial solutions for the case where the forces and the normal component of the induction vector on the neck surface are equal to zero. Numerical examples are given for specific external loads and properties of the body. The stress components and normal component of the induction vector near the neck vertex are plotted as a function of the external load and neck curvature     

Cauchy Tetrahedron Argument Applied to Higher Contact Interactions

Second gradient theories are nowadays used in many studies in order to describe in detail some transition layers which may occur in micro-structured materials and in which physical properties are sharply varying. Sometimes higher order theories are also evoked. Up to now these models have not been based on a construction of stresses similar to the one due to Cauchy, which has been applied only for simple materials. It has been widely recognized that the fundamental assumption by Cauchy that the traction depends only on the normal of the dividing surface cannot be maintained for higher gradient theories. However, this observation did not urge any author, to our knowledge, to revisit the Cauchy construction in order to adapt it to a more general conceptual framework. This is what we do in this paper for a continuum of grade N (also called N-th gradient continuum). Our construction is very similar to the one due to Cauchy; based on the tetrahedron argument, it does not introduce any argument of a different nature. In particular, we avoid invoking the principle of virtual work. As one should expect, the balance assumption and the regularity hypotheses have to be adapted to the new framework and tensorial computations become more complex.     

Boundary integral equations of unique solutions in elasticity

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