Radial deformation of a plane sheet containing a circular hole or inclusion

Seth's generalized measure of strain is used in the determination of the deformation field produced by radial forces acting on a thin plane sheet of incompressible isotropic elastic material, containing a circular hole or circular rigid inclusion, at its outer edge. Results calculated are comparable with the experimental results obtained by Rivlin and Thomas, employing a vulcanized natural rubber.     

Growth of longitudinal and torsional waves through a rod for second order elasticity

A boundary value problem connected with the propagation and growth of wave through a rod of second order elastic materials is studied. Two one-dimensional equations of motions are derived from the exact three dimensional equations which govern the torsional and longitudinal wave motions. The torsional wave does not grow at all while there is a distinct possibility for a compressive wave to grow into a shock. For Seth's stress strain relations the compressive wave grows into a shock while a tension wave decays.     

Derivation of strain measures from strand convection models for polymer melts

Constitutive equations for polymer melts often fail to describe realistically experimental data because they do not describe adequately the melt's elastic behaviour; i.e., the instantaneous response of the melt to the imposed deformation. Elastic behaviour is represented in differential constitutive equations by the convected time derivative and in BKZ integral models by the elastic potential function or strain measure. Here we propose a general molecular model to describe strand convection; that is, the instantaneous response of a polymer strand to an imposed deformation. The model includes the affine, Gordon-Schowalter, and apparently the Seth models as special cases. It is found that by requiring the convection to be such that the principal axes of stress and of strain rotate together, the strain measure becomes a function of strain history, and the Lodge-Meissner relationship is satisfied, in accord with experimental observations.     

Nonlinear characterization of sand-asphalt concrete by means of permanent-memory norms

Employing a constitutive equation developed by Farris and Fitzgerald which accounts for the maximum strain ever imposed upon a material as well as a weighted average of the strain history, the family of P^th order Lebesgue norms, the applicability to a sand-asphalt concrete is demonstrated. The inadequacy of linear viscoelasticity theory under repeated or decreasing loadings for these materials is also demonstrated. Practical laboratory determination of the material parameters is described.     

A new approach to the analysis of shells,plates and membranes with finite deflections

This paper presents a new perturbation method of analysis applicable to a class of geometrically non-linear problems of shells, plates, and membranes with translationally restrained edges. The perturbation parameter is a linear function of Poisson's ratio. The convergence of successive perturbations (i.e., approximations) is independent of the magnitudes of deflections. The method also offers a rational explanation of the efficacy of Berger's approximate equations, thus placing Berger's method on a firmer foundation while at the same time weakening his hypothesis of vanishing second membrane strain invariant in the strain energy integral. Several solutions and results are obtained for the purposes of illustration and discussion. Whenever possible, calculated values are compared with results obtained by other means.     

Il'iushin's postulate and resulting thermodynamic conditions on elasto-plastic coupling

Il'iushin's postulate is restated within a general thermodynamic strain space formulation of rate independent plasticity by means of plastic internal variables. This yields a general expression in terms of appropriate thermodynamic potentials. A combination of a thermodynamic condition, derived from the general development, with the results of Il'iushin's postulate, furnishes explicit conditions on elasto-plastic coupling. A specific example is presented, with the plastic work being the only plastic internal variable. Necessary and sufficient consitbns on the elastic moduli and their change with plastic deformation are derived, for the thermodynamic condition to be satisfied.     

Decay Estimates for the Quadratic Tilt-Excess of Integral Varifolds

This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space with their first variation given by either a Radon measure or a function in some Lebesgue space. Pointwise decay results for the quadratic tiltexcess are established for those varifolds. The results are optimal in terms of the dimension of the varifold and the exponent of the Lebesgue space in most cases, for example if the varifold is not two-dimensional.     

On the determination of elastodynamic crack tip stress fields

A linear elastic body in plane strain which contains a stationary crack and which is initially at rest and stress free is considered. It is shown that if the elastodynamic displacement field and stress intensity factor are known, as functions of crack length, for any symmetrical distribution of time-varying forces which acts on the body, subsequent to t=0, then the stress intensity factor due to any other symmetrical load system whatsoever which acts on the same body may be directly determined. The other load system may be of arbitrary spatial distribution and time variation. Further, that part of the elastodynamic displacement field due to the other load system, which arises from the presence of the crack, may also be directly determined. The results are obtained by extension of Rice's mode of derivation of the corresponding Bueckner-Rice elastostatic results to Laplace-transformed elastodynamic variables. Likewise, the existence of a universal elastodynamic “weight function” for any given cracked body is demonstrated. As an application, Freund's recent result for the stress intensity factor due to suddenly applied concentrated forces on the crack surfaces is derived directly by our method, from de Hoop's earlier solution for suddenly applied uniform pressures.     

Moderate deformations in extension-torsion of incompressible isotropic elastic materials

It has been previously shown by anand (1979) that the classical strain energy function of infinitesimal isotropic elasticity is in good agreement with experiment for a wide class of materials for moderately large deformations, provided the infinitesimal strain measure occurring in the strain energy function is replaced by the Hencky or logarithmic measure of finite strain. The basis in Anand's paper for relating Hencky's strain energy function to experiment was data from experiments on metals and rubbers in uniaxial strain, simple tension and compression, and pure shear. Here, to test further the validity of this strain energy function for moderate deformations, its predictions for the twisting moment and the axial force in simple torsion and combined extension-torsion of solid cylinders of incompressible materials are calculated and shown to be in good agreement with data from the classical experiments of Rivlin and Saunders (1951) on vulcanized natural rubber. Indeed, the predictions from Hencky's strain energy function are in better accord with experiment than the predictions from the widely used Mooney (or Mooney-Rivlin) strain energy function.     

Pesin’s Formula for Random Dynamical Systems on \mathbf {R}^d

Pesin’s formula relates the entropy of a dynamical system with its positive Lyapunov exponents. It is well known, that this formula holds true for random dynamical systems on a compact Riemannian manifold with invariant probability measure which is absolutely continuous with respect to the Lebesgue measure. We will show that this formula remains true for random dynamical systems on $\mathbf {R}^d$ which have an invariant probability measure absolutely continuous to the Lebesgue measure on $\mathbf {R}^d$ . Finally we will show that a broad class of stochastic flows on $\mathbf {R}^{d}$ of a Kunita type satisfies Pesin’s formula.     

Impedance at a rough waveguide boundary

The impedance at the randomly rough upper boundary of an ocean waveguide is derived. The sound speed of the waveguide is an arbitary function of depth. The boundary surface height is assumed to be a statistically homogeneous Gaussian process. Integral equations for the Green's function and its normal derivative on the boundary are derived. These are solved to second order in the surface interaction. The result is a rational approximation to the impedance in terms of the wave guide Green's function and the statistical properties of the surface. The special cases of small and large roughness as well as that of a constant sound speed profile are presented. For simplicity we restrict the analysis to a semi-infinite waveguide where the waveguide Green's function vanishes at the surface (Dirichlet problem).     

Application of a discrete polycrystal model to the analysis of cyclic straining in copper

A discrete polycrystal model, designed to simulate a metal aggregate macro-element, is applied to the study of cyclic straining in copper. The numerical method of solution (an adaptation of the “finite element method”) incorporates a convergent discrete Green's function within the constrained minimum principle which governs the (crystallographic) plastic shear increments at each load step. Isothermal elastic moduli of copper crystals and Taylor's hardening rule with constant hardening modulus are used in the calculations. Numerical results are obtained for macroscopic elastic properties, cyclic stress-strain curves (which indicate the contribution of aggregate heterogeneity to macroscopic hardening), macroscopic plastic work, and residual (latent) strain energy through four loading cycles between fixed macrostrain limits. Other estimates for elastic properties also are included, and all results are compared, both qualitatively and quantitatively, with published experiments. The predictions of the model are in general satisfactory.     

Viscosity,first normal-stress coefficient,and molecular stretching in dilute polymer solutions

Warner's numerical method for finitely extensible nonlinear elastic (“FENE”) dumbbells in a dilute suspension undergoing steady-state shear flow has been improved by assuming a form of the distribution function that removes the singularity at R = 0 and improves the behavior of the weight function in the Galerkin expression for large b. The comparison of the results of the present method to those of Christiansen and Bird's extrapolation and Warner's numerical solution indicates the success of this method. The material functions, the dumbbell elongation, and the distribution function for steady-state shear flow are given. In addition the newly obtained results are used to assess the accuracy of two approximate methods referred to as the FENE-P model and the FENE-P-B model.     

On thermoelastic dielectrics

Constitutive equations for a linear thermoelastic dielectric are derived from the energy balance equation assuming dependence of the stored energy function on the strain tensor, the polarization vector, the polarization gradient tensor and entropy. A method is indicated for constructing a hierarchy of constitutive equations for materials with arbitrary symmetry by introducing various thermodynamic potentials. Maxwell's relations are constructed for the thermodynamic potential W^L. The entropy inequality is used to obtain stability conditions for an elastic dielectric in equilibrium under prescribed boundary constraints. Frequencies are explicitly determined for a plane wave propagating along the x₁-axis in an infinite centro-symmetric isotropic thermoelastic dielectric.     

Plastic deformation of aluminium alloy under abruptly-changing loading or strain paths

The effect of deformation history on the plastic behaviour of thin-walled tubular specimens of aluminium alloy 5056 was examined for three types of abruptly-changing loading or strain paths: namely, reverse loading after pre-loading, orthogonal straining after pre-strain, and orthogonal reloading after pre-loading and then perfect unloading, by applying combined loadings of axial force and torque. The experimental results revealed the following trends. The relation between the magnitudes of stress and strain after the comer is expressed for every pre-strain by a single curve parallel to the extension of the pre-loading curve, when the strain after the corner exceeds 1.2 per cent. Moreover, the relations between the stress reduced by pre-stress and strain after the corner for various values of the same type of pre-strain agree with each other; however, the relation for the tensile pre-strain differs clearly from that for the torsional pre-strain. Thus, A.A. Il'yushin's postulate of isotropy does not hold accurately for the above loading histories for the aluminium alloy even if the effect of the third invariant of stress deviator is eliminated.     

A simplified nonlinear theory of the generalized plane strain

A finite deformation theory of plane strain is formulated for transversely isotropic, homogeneous bodies with nonlinear stress-strain law. A new set of simplified field equations, which is valid in the case of some deviations from Hooke's law, is derived systematically with the help of the method of order estimation. For illustration purposes, a circular hole in a body under generalized plane strain is considered, together with the solution of an example problem by perturbation techniques.     

Weighted estimates for strongly singular integral operators with rough kernels

The Fourier transform and the Littlewood-Paley theory are used to give the weighted boundedness of a strongly singular integral operator defined in this paper. The paper shows that the strongly singular integral operator is bounded from the Sobolev space to the Lebesgue space.     

Forced axisymmetric motions of viscoelastic cylindrical shells

The isothermal response of a viscoelastic cylindrical shell, of finite length, to arbitary axisymmetric surface forces, initial conditions, and boundary conditions is considered within the linear theory of thin shells. The problem is formulated with the effects of shear deformation and rotatory inertia included; the viscoelastic properties are assumed to be isotropic and homogeneous. The response is first found formally in terms of a causal Green's function. It is then shown that when Poisson's ratio is constant, the causal Green's function can be expanded in a series of orthonormal spatial eigenfunctions of an associated elastic shell eigenvalue problem. The resulting solution for the general problem is an eigenfunction series with Laplace transformed time-dependent coefficients. The general solution is applied to predicting the motion of a uniform, simply-supported cylindrical shell, initially quiescent, which is subjected to a step pressure moving with constant velocity. For this example, the relaxation function of the shell material in uniaxial extension is taken to be that of a standard linear solid. The motions predicted by simpler shell models, namely, shells with bending only and without bending, are also considered for comparison. Here, the absolute values of the Fourier coefficients in the shell displacement series go to zero faster than the inverse of the first or second power of positive integers when bending is excluded or included, respectively. Numerical results are presented for a moderately long and relatively thick, nearly elastic, cylindrical shell.     

An existence theory for nonlinear elasticity that allows for cavitation

In this paper the existence of minimizers in nonlinear elasticity is established under assumptions on the stored energy that permit the formation of new holes in the body. Such cavities have been observed in experiments on elastomers, and a mathematical theory for radially symmetric cavities has been developed by Ball. Here the full three-dimensional problem is considered and an additional, physically motivated, energy term that is proportional to the area of the boundary of the deformed body is included. The minimizers lie in a subclass of those maps in W ^{1, p} , 2<p<3, that are one-to-one almost everywhere and preserve orientation. Roughly speaking, this subclass consists of those maps in which cavities in one part of the body are not filled by material from other parts of the body. Such maps are shown to be much more regular than expected. In particular, some ideas of verák are used to show that each map in this subclass has a representative which is continuous outside a set of Hausdorff dimension 3 — p and that this representative also satisfies Lusin's condition (N), i.e., it maps Lebesgue null sets onto such sets. It is also shown that the distributional Jacobian of such a map is a measure which is the sum of a measure that is absolutely continuous with respect to Lebesgue measure and (at most) a countable number of Dirac measures.     

Dynamical properties of 2-torus parabolic maps

In this paper, a class of linear maps on the 2-torus are discussed. Discussions are focused on the case that the maps are parabolic. It is shown that the maximal invariant set for a 2-torus parabolic map is indeed invariant, and is almost closed, and the Lebesgue measure restricted to a maximal invariant set is invariant. Under this invariant measure, all Lyapunov exponents of a parabolic map are zero. In certain simple cases, the Lebesgue measure of the maximal invariant sets are computed and estimated. For the case the maps are invertible, it is shown that the inverse of a non-horocyclic parabolic map is no longer a parabolic map. Interesting properties of the conjugation of invertible parabolic maps by automorphisms of the torus are characterized, and a conjugation invariant for such maps are obtained. And it is proven that all these maps can be reduced to a family of one parameter rigid rotations. Mathematics Subject Classification: 37C15, 37D50