The plane-strain straightening deformation of certain elastic annular cylindrical sectors is considered in conjunction with the closely related deformation of uniaxial extension of elastic rectangular strips. For the case of mixed boundary conditions of place and traction, sufficient and necessary conditions for the infinitesimal stability of these deformations are provided and several illustrative examples are analysed.

The combined axial shearing, extension, and straightening of an annular cylindrical sector is a deformation that, following Truesdell & Noll (1965, The Non-Linear Field Theories of Mechanics, Encyclopedia of Physics III/3) and Hill (1973, Z. Angew. Math. Phys., 24, 609-618), we describe in terms of two prescribed constants and two unknown functions that depend only on the radial material coordinate. Under the assumption that the material is elastic, compressible, and isotropic we show that for equilibrium in the absence of body forces the unknown functions must satisfy a system of first-order nonlinear ordinary differential equations. The system of differential equations can be decoupled for certain material classes, one of which is the class of Hadamard-Green materials. Thus, several new exact solutions are obtained and, under the assumption that the annular cylindrical sector is composed of a Hadamard-Green material that is strongly elliptic, the existence and uniqueness of solutions for two types of boundary conditions is established.

The paper is concerned with solvability of a mixed boundary value problem of place and traction that describes the straightening of annular cylindrical sectors composed of unconstrained isotropic hyperelastic solids. The existence and uniqueness of solutions is discussed and several examples are exhibited.

A principle of Saint-Venant type is established for the theory of linear micropolar elastodynamics, and the connection that exists between this principle and the domain of influence theorems, uniqueness theorems, and continuous dependence theorems is discussed. The body, which is assumed to be of arbitrary regular shape and is subjected to loadings that possess a bounded support DT for the time interval [0, T], can be bounded or unbounded. According to this principle, there exists a constant c > 0 such that a certain energetic measure of the displacement vanishes for r > ct and decays to zero for r < ct, where r is the distance from a generic point to the support DT, and t is any time in the interval [0, T]. The decay rate is controlled by the factor 1 -r/(ct).

The authors consider the plane-strain straightening of annular cylindrical sectors composed of isotropic, compressible, nonlinearly elastic solids. For zero-body forces and boundary conditions of place, the existence and uniqueness of solutions are established under the assumption that the material satisfies the tension-extension condition. The result is exemplified by considering a compressible neo-Hookean material and a generalized Blatz-Ko material for which closed-form solutions are displayed. Also discussed are certain cases of nonexistence and nonuniqueness of solutions.

. The main stability result established in [15] (according to which a plane strain equilibrium solution of a harmonic material which satisfies the tension-extension condition is globally stable for boundary conditions of place and zero body forces if and only if a weakened form of the Baker-Ericksen inequality is satisfied at this particular solution) is extended in two different directions. Firstly, it is shown that this stability result holds not only for plane strain but also for three-dimensional deformations and secondly, for the case when the sum of principal stretches is constant, it is shown (following [17] where such a result was obtained within the context of a discussion regarding the plane strain radial inflation of hollow cylinders) that if for boundary conditions of place and zero body forces a certain equilibrium solution is stable relative to the harmonic materials, then it is also stable relative to a certain generalization of these materials.

It is shown in this paper that for certain classes of unconstrained isotropic elastic solids, predictions can be made as to whether the overall volume changes which accompany two nonlinear elastic deformations (namely the radial deformation of cylindrical tubes and the bending of rectangular blocks into annular cylindrical sectors) are expansive or contractive. It is emphasised that such predictions may be viewed as universal relations for the material classes concerned.

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 C1 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 C1 and (ii) to obtain explicit lower bounds for the total energy associated with any plane deformation.

The conditions for the strong ellipticity of the equilibrium equations of compressible, isotropic, nonlinearly elastic solids (established by Simpson and Spector [1]) are expressed in terms of the stored-energy function regarded as a function of the principal stretches. The applicability of this reformulation is illustrated with the help of two specific examples.

On certain loading paths the changes in the total energy associated with the deformations with constant modified stretches are due entirely to volume changes. We illustrate this situation by considering in plane-strain framework the deformations with constant modified stretches that describe the bending of rectangular blocks subject to prescribed boundary displacements. In particular we discuss certain conditions which imply that the total energy, regarded as a function of the deformed volume, V, attains a minimum at a certain value V-0 of V. A compressible Varga material and a compressible neo-Hookean material are then used to exemplify our results.

Under consideration is the problem of flexure of compressible nonlinearly elastic rectangular blocks. The discussion is confined to deformations describing the bending of a rectangular block into a sector of a circular cylindrical tube. The predictions based upon the well-known semilinear material model are investigated. Addressed, in particular, are some problems concerning the existence, uniqueness and stability of solutions to specific boundary value problems.

We consider deformations of unconstrained, isotropic hyperelastic solids which satisfy the condition that the determinant of the deformation gradient is constant. In the absence of body forces, it is shown (i) that a certain deformation in this class (which describes the bending of rectangular blocks into annular cylindrical sectors) is not possible in any of the considered materials, (ii) that in the case when the body fills the whole space, it is composed of a compressible neo-Hookean material and it is subjected to relatively moderate loads, these deformations are necessarily homogeneous and (iii) that for boundary conditions of place and relative to a certain sub-class of the class of considered materials, these deformations are globally stable, in the sense that they are minimizers for the total energy with respect to smooth variations that are compatible with the boundary conditions.

Upper and lower bounds for the maximum shear stress in a configuration corresponding to a purely distortional deformation originating from a given undistorted (ground) state are obtained in the framework of plane, isotropic, nonlinear elasticity. The bounds are shown to be expressible in terms of the deformation and the boundary traction that is required for maintaining the purely dilatational deformation in the ground state.

Two inequalities which for certain deformation classes may be viewed as universal relations are shown to hold for two distinct subclasses of unconstrained rubberlike solids. The inequalities express the fact that the mean stress corresponding to any purely distortional deformation originating from a given ground state is dominated by (and, respectively, dominates) the mean stress in that ground state. Also discussed is a case in which the deformations involved are necessarily homogeneous.

A surface measure of the displacement is employed to obtain decay results of Saint-Venant's type in cylinders composed of a physically nonlinear micropolar elastic solid. Finite length cylinders as well as cylinders which are infinite in one direction are considered.

The boundary value problem of place in nonlinear hyperelastostatics is considered. Integral estimates for the displacement and the strain energy are obtained for the weak (or classical) solution in terms of the given body force. These estimates then imply continuous dependence of the solution on the given data in the appropriate norm.