CaltechAUTHORS: Article
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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenThu, 15 Aug 2024 13:08:04 -0700On Saint-Venant's principle and the torsion of solids of revolution
https://resolver.caltech.edu/CaltechAUTHORS:20150811-132215439
Year: 1966
DOI: 10.1007/BF00276511
Although a comprehensive review of the literature on SAINT-VENANT'S principle
in the linearized equilibrium theory of elastic solids would serve a useful
purpose, such a survey is clearly beyond the scope of these introductory remarks.
The principle was originally introduced by SAINT-VENANT in connection with, and with limitation to, the problem of extension, torsion, and flexure of slender cylindrical or prismatic beams. Renewed interest in the intriguing theoretical questions posed by SAINT-VENANT'S principle was stimulated by VON MISES [1] ,(1945), who brought into focus the vagueness of the traditional universal statements of the principle, which go back to BOUSSINESQ [2]. Guided by BOUSSINESQ'S own efforts in support of the principle, and on the basis of two specific examples, YON MISES was led to interpret and amend the conventional statements in terms of assertions concerning the order of magnitude of the stresses at interior points of an elastic body under loads that are confined to several distinct portions of its boundary. The limit process underlying these assertions refers to the contraction of the regions of load-application to fixed points of the boundary and the prevailing order of magnitude of the internal stresses depends upon the nature of the individual load resultants. A mathematical formulation and proof of von Mises' version of SAINT-VENANT'S principle was supplied later by STERNBERG [3]* (1952).https://resolver.caltech.edu/CaltechAUTHORS:20150811-132215439Minimum energy characterizations of Saint-Venant's solution to the relaxed Saint-Venant problem
https://resolver.caltech.edu/CaltechAUTHORS:20150811-131727884
Year: 1966
DOI: 10.1007/BF00266569
The singular importance, in both theory and practice, of SAINT-VENANT'S
celebrated memoirs [1], [2] on what has long since become known as SAINTVENANT'S
problem, requires no emphasis. Indeed, a comprehensive bibliography
of the vast and varied literature to which the work contained in [1], [2] has given
impetus would multiply the length of this rather limited study.
With a view toward describing the aim of the present investigation we recall first that SAINT-VENANT'S problem consists in the determination of the stresses and deformations in an elastic cylinder (or prism), which - in the absence of body forces - is subjected to surface tractions arbitrarily prescribed over its ends and which is free from lateral loading. In this formulation the problem, even within the linearized equilibrium theory of homogeneous and isotropic elastic solids, has remained one of undiminished notoriety. SAINT-VENANT'S treatment of the foregoing problem rests on a relaxed formulation in which the detailed assignment of the terminal tractions is abandoned in favor of prescribing merely the appropriate stress resultants.https://resolver.caltech.edu/CaltechAUTHORS:20150811-131727884On Saint-Venant's principle in the two-dimensional linear theory of elasticity
https://resolver.caltech.edu/CaltechAUTHORS:20150811-130208259
Year: 1966
DOI: 10.1007/BF00253046
This paper presents results which provide a formulation and proof of a version of Saint-Venant's principle appropriate to the plane strain and generalized plane stress solutions of the equations of the linear theory of elastic equilibrium.https://resolver.caltech.edu/CaltechAUTHORS:20150811-130208259A note on elastic surface waves
https://resolver.caltech.edu/CaltechAUTHORS:20141110-095112867
Year: 1966
DOI: 10.1029/JZ071i022p05480
The structure of harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space can be described by proceeding from the following assumptions: (1) the plane boundary is free of surface traction; (2) the Laimé potentials, and consequently all physical quantities, decay exponentially with distance away from the boundary. In the absence of further a priori assumptions, the resulting surface waves need be neither plane nor axially symmetric, and thus the derivation sketched here constitutes a generalization of the ones usually given in the textbook literature [e.g., Love, 1944; Ewing et al., 1957].
With reference to Cartesian coordinates χ_1, χ_2, χ_3, the half-space under consideration occupies the region x_3≥0. The displacement vector u of a typical point has Cartesian components u_j, and the associated components of stress are denoted by τ_(jk). The summation convention is used, Latin and Greek subscripts have the respective ranges 1, 2, 3 and 1, 2, and a subscript preceded by a comma indicates differentiation with respect to the corresponding coordinate.https://resolver.caltech.edu/CaltechAUTHORS:20141110-095112867A Saint-Venant principle for a class of second-order elliptic boundary value problems
https://resolver.caltech.edu/CaltechAUTHORS:20141121-163033508
Year: 1967
DOI: 10.1007/BF01601718
In the present paper we wish to describe a Saint-Venant principle for a special
class of boundary value problems for second order elliptic equations in two independent variables. While the context thus presented is much simpler than that encountered in most problems arising in elasticity, the methods developed here may be of some relevance in more involved situations. Moreover, our estimates are directly applicable to certain problems of physical interest, as we shall point out below; they may therefore be of some interest in themselves.https://resolver.caltech.edu/CaltechAUTHORS:20141121-163033508The Dirichlet problem for a thin rectangle
https://resolver.caltech.edu/CaltechAUTHORS:20151208-154823152
Year: 1967
DOI: 10.1017/S0013091500011974
We consider the Dirichlet problem for Laplace's equation in a rectangle with a view toward determining the asymptotic behaviour of the solution for the case in which the width of the rectangle is small in comparison with its length. Although the construction of an explicit representation of the solution is an elementary matter, the resulting formula is inconvenient for present purposes, and we accordingly proceed along different lines.https://resolver.caltech.edu/CaltechAUTHORS:20151208-154823152On the Dynamic Response of a Beam to a Randomly Moving Load
https://resolver.caltech.edu/CaltechAUTHORS:20151120-111330136
Year: 1968
DOI: 10.1115/1.3601165
The problem considered is that of an infinitely long elastic beam subject to a moving concentrated force whose position is a stochastic function of time, X(t). The expected deflection and expected bending moment are analyzed, with special attention being given to the case of a stationary process X(t) and to the case in which X(t) is a Wiener process.https://resolver.caltech.edu/CaltechAUTHORS:20151120-111330136Propagation of One‐Dimensional Waves from a Source in Random Motion
https://resolver.caltech.edu/CaltechAUTHORS:20141210-083930907
Year: 1968
DOI: 10.1121/1.1910964
This paper is concerned with the transient problem of waves in an infinite one‐dimensional medium owing to a monochromatic source whose position X(t) is a stochastic function of time. Asymptotic results for the farfield are given for the general class of stationary source motions and for the case in which X(t) is a Wiener process, corresponding to Brownian motion.https://resolver.caltech.edu/CaltechAUTHORS:20141210-083930907On the exponential decay of stresses in circular elastic cylinders subject to axisymmetric self-equilibrated end loads
https://resolver.caltech.edu/CaltechAUTHORS:20151207-102452675
Year: 1969
DOI: 10.1016/0020-7683(69)90067-5
Methods involving energy-decay inequalities are applied to the axisymmetric end problem for a circular elastic cylinder. Explicit lower bounds in terms of Poisson's ratio are obtained for the rate of exponential decay of stresses, and these are compared with results of other authors.https://resolver.caltech.edu/CaltechAUTHORS:20151207-102452675Energy inequalities and error estimates for torsion of elastic shells of revolution
https://resolver.caltech.edu/CaltechAUTHORS:20151124-105106068
Year: 1970
DOI: 10.1007/BF01627942
This paper is concerned with the problem of axisymmetric torsion by terminal loads of elastic shells of revolution. Such shells are considered here to be three-dimensional bodies occupying a region of space which includes all points whose distances from a given surface - the midsurface - do not exceed h/2, where h is the shell thickness. The analysis is based on the classical linear theory of elasticity for homogeneous, isotropic materials, and it may be regarded as an extension of the work described in [1] and [2]. The purpose of the present paper is to assess the quality of an approximate solution of the thin shell problem. The case of axisymmetric torsion of thin shells of revolution is perhaps tile simplest one for this purpose, since simple approximate solutions - constructed from two-dimensional shell theories or otherwise - are known [3]. Our results provide estimates, based on the three-dimensional theory of elasticity, for the error involved in a stress analysis in which the 'exact' solution is replaced by an approximate one.https://resolver.caltech.edu/CaltechAUTHORS:20151124-105106068Eigenvalue problems associated with Korn's inequalities
https://resolver.caltech.edu/CaltechAUTHORS:20151124-103729126
Year: 1971
DOI: 10.1007/BF00251798
We shall discuss a class of problems associated with certain inequalities which apparently originated in the work of A. KORN [1, 2] on the linear theory of elasticity. To describe these inequalities we consider a vector field u(x), continuously differentiable on the closure R + B of a bounded domain R with boundary B in two or three dimensions. Let x, be coordinates in a given rectangular Cartesian coordinate system, and denote by u_i the components of u in this system.https://resolver.caltech.edu/CaltechAUTHORS:20151124-103729126A Potential Representation for Two-Dimensional Waves in Elastic Materials of Harmonic Type
https://resolver.caltech.edu/CaltechAUTHORS:20150728-111652292
Year: 1971
In the present note we consider two-dimensional finite dynamical
deformations for the class of homogeneous, isotropic elastic materials introduced by
F. John in [1] and referred to by him as materials of harmonic type. The theory of such
materials, developed in [1] and [2], appears to be simpler in many respects than that of
more general elastic materials, and it may offer the possibility of investigating some
features of nonlinear elastic behavior more explicitly than is possible in general.
For plane motions of such materials, we derive here a representation for the displacements
in terms of two potentials which is analogous to the theorem of Lamé in classical
linear elasticity (see [3]) for the case of plane strain. The two nonlinear differential
equations satisfied by the potentials reduce upon linearization to the wave equations
associated with irrotational and equivoluminai waves in the linear theory.
In the following section we state without derivation the equations governing two-dimensional
waves in an elastic material of harmonic type. The reader is referred to [1]
for details. In Sec. 3 we derive the representation in terms of potentials described briefly
above.https://resolver.caltech.edu/CaltechAUTHORS:20150728-111652292On the spatial decay of solutions of the heat equation
https://resolver.caltech.edu/CaltechAUTHORS:20151124-104815023
Year: 1971
DOI: 10.1007/BF01590873
The spatial decay of solutions of parabolic partial differential equations has been the subject of two recent papers by Edelstein [1] and Sigillito [2]. The results obtained by these authors are in some respects analogous to estimates derived in connection with Saint-Venant's principle in the linear theory of elasticity [3], [4].https://resolver.caltech.edu/CaltechAUTHORS:20151124-104815023On a class of conservation laws in linearized and finite elastostatics
https://resolver.caltech.edu/CaltechAUTHORS:20151124-103510030
Year: 1972
DOI: 10.1007/BF00250778
Several years ago ESHELBY [1] (1956), in a paper devoted to the continuum theory of lattice defects, deduced a surface-integral representation for the "force on an elastic singularity or inhomogeneity", which-in the absence of such
defects-gives rise to a conservation law for regular elastostatic fields appropriate to homogeneous but not necessarily isotropic solids in the presence of infinitesimal deformations. Morevoer, ESHIELBY noted that his result, when suitably interpreted, remains strictly valid for finite deformations of elastic solids.https://resolver.caltech.edu/CaltechAUTHORS:20151124-103510030Finite-deformation analysis of the elastostatic field near the tip of a crack: Reconsideration and higher-order results
https://resolver.caltech.edu/CaltechAUTHORS:20150630-082317785
Year: 1974
DOI: 10.1007/BF00049265
In this paper we return to the asymptotic analysis, within the nonlinear equilibrium theory of compressible elastic solids, of the deformations and stresses near the tip of a traction-free crack in a slab of all-around infinite extent under conditions of plane strain. As before, the loading at infinity is taken to be one of uniform uni-axial tension at right angles to the faces of the crack. We show that once a restrictive assumption introduced at the start of our earlier asymptotic treatment of the problem is relinquished, certain perplexing anomalies encountered in the previous analysis no longer arise. The present reconsideration of the problem leads to modifications in the dominant-order results for the "secondary" deformations and stresses, while those pertaining to the physical quantities of primary interest remain unaffected. Furthermore, this investigation encompasses some higher-order considerations, which supply an essential clarification and improvement of the lowest-order asymptotic solution.https://resolver.caltech.edu/CaltechAUTHORS:20150630-082317785On the ellipticity of the equations of nonlinear elastostatics for a special material
https://resolver.caltech.edu/CaltechAUTHORS:20151109-145649593
Year: 1975
DOI: 10.1007/BF00126996
This paper deals with the ellipticity of the equations of finite elastostatics for a compressible material that corresponds to a special choice of the strain-energy density and has received repeated attention in the literature. The possible failure of ellipticity of the appropriate system of displacement equations of equilibrium at solutions involving large deformations was suggested by certain difficulties encountered in an attempt to determine the deformations and stresses arising in such a material near the tip of a crack. It is shown here that ellipticity prevails only if the principal stretches are suitably restricted and breaks down, in particular, at a local state of uni-axial tension or compression of sufficiently severe intensity.https://resolver.caltech.edu/CaltechAUTHORS:20151109-145649593On the failure of ellipticity of the equations for finite elastostatic plane strain
https://resolver.caltech.edu/CaltechAUTHORS:20151124-104106498
Year: 1976
DOI: 10.1007/BF00279991
In this paper we establish necessary and sufficient conditions, in terms of the local principal stretches, for ordinary and strong ellipticity of the equations governing finite plane equilibrium deformations of a compressible hyperelastic solid. The material under consideration is assumed to be homogeneous and isotropic, but its strain-energy density is otherwise unrestricted. We also determine the directions of the characteristic curves appropriate to plane elastostatic deformations that are accompanied by a failure of ellipticity.https://resolver.caltech.edu/CaltechAUTHORS:20151124-104106498A note on anti-plane shear for compressible materials in finite elastostatics
https://resolver.caltech.edu/CaltechAUTHORS:20151109-153158189
Year: 1977
DOI: 10.1017/S0334270000001399
This note gives a necessary and sufficient condition that a compressible, isotropic elastic material should admit non-trivial states of finite anti-plane shear.https://resolver.caltech.edu/CaltechAUTHORS:20151109-153158189A Note on the Spatial Decay of a Minimal Surface Over a Semi-infinite Strip
https://resolver.caltech.edu/CaltechAUTHORS:20150212-075135517
Year: 1977
DOI: 10.1016/0022-247X(77)90090-7
n/ahttps://resolver.caltech.edu/CaltechAUTHORS:20150212-075135517The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids
https://resolver.caltech.edu/CaltechAUTHORS:20150728-111035202
Year: 1977
DOI: 10.1007/BF00017296
The present paper is concerned with an infinite slab containing a crack and deformed at infinity to a state of finite simple shear. The material of the slab is taken to be homogeneous, isotropic, elastic, and incompressible, and is further assumed to belong to a class of materials which admit nontrivial states of anti-plane shear. The analysis is carried out for the fully nonlinear equilibrium theory of finite elasticity. The stress field near the crack-tips is studied in detail; one of the special materials considered is such that the shear stresses near a crack tip remain bounded, despite the presence of unbounded displacement gradients. An analogy between the crack problem in finite anti-plane shear and the problem of transonic flow of a gas past a flat plate is pointed out and discussed.https://resolver.caltech.edu/CaltechAUTHORS:20150728-111035202On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics
https://resolver.caltech.edu/CaltechAUTHORS:20151106-101307186
Year: 1978
DOI: 10.1007/BF00049187
This investigation concerns equilibrium fields with discontinuous displacement gradients, but continuous displacements, in the theory of finite plane deformations of possibly anisotropic, compressible elastic solids. "Elastostatic shocks" of this kind, which resemble in many respects gas-dynamical shocks associated with steady flows, are shown to exist only if and when the governing field equations of equilibrium suffer a loss of ellipticity. The local structure of such shocks, near a point on the shockline, is studied with particular attention to weak shocks, and an example pertaining to a shock of finite strength is explored in detail. Also, necessary and sufficient conditions for the "dissipativity" of time-dependent equilibrium shocks are established. Finally, the relevance of the analysis carried out here to localized shear failures-such as those involved in the formation of Lüders bands-is discussed.
Since this paper was submitted for publication, Professor James R. Rice has pointed out to us that the dissipation inequality (6.28), which was in essence postulated in the present work, could have been deduced from the thermodynamic requirement of a positive rate of entropy production together with the energy identity (6.23). It is presumed in such a derivation that the underlying quasi-static time dependent equilibrium shock constitutes an isothermal process.https://resolver.caltech.edu/CaltechAUTHORS:20151106-101307186Universal States of Finite Anti-Plane Shear: Ericksen's Problem in Miniature
https://resolver.caltech.edu/CaltechAUTHORS:20151110-152036919
Year: 1979
A solid body which in its undeformed state occupies a cylindrical region is said to undergo a deformation corresponding to anti-plane shear if each particle of the body is displaced parallel to the generators of the cylinder by an amount which is independent of the axial position of the particle. The displacement vector field thus has a nonvanishing component u only in the axial direction, and u is a function of position on a cross-section D of the cylinder. Problems involving such deformations are ordinarily simpler than those in which the displacement vector has a more elaborate character, and for this reason they often serve usefully as pilot problems for the analysis of qualitative effects, especially when nonlinearity is involved. (Examples may be found in [1] and in the references cited in [2].) The present paper is intended to illustrate anti-plane shear in its role as exemplar in the setting of finite elasticity theory and with particular reference to an issue which has come to be called Ericksen' s problem.https://resolver.caltech.edu/CaltechAUTHORS:20151110-152036919On the existence of an elastic potential for a simple material without memory
https://resolver.caltech.edu/CaltechAUTHORS:20151124-103624348
Year: 1979
DOI: 10.1007/BF00276379
It is shown that a compressible elastic body — not necessarily homogeneous or isotropic — is hyperelastic provided the work done by all external forces acting on an arbitrary part of the body vanishes for every sufficiently smooth cyclic motion in which each material point returns to its initial position with a velocity equal to its initial velocity.https://resolver.caltech.edu/CaltechAUTHORS:20151124-103624348On the dissipation associated with equilibrium shocks in finite elasticity
https://resolver.caltech.edu/CaltechAUTHORS:20151109-153157919
Year: 1979
DOI: 10.1007/BF00041322
Equilibrium fields with discontinuous displacement gradients can occur in finite elasticity for certain materials. The presence of such "equilibrium shocks" affects the energy balance in the elastostatic field, and the present paper is concerned with a notion of dissipation associated with this energy balance. A dissipation inequality is proposed for three-dimensional equilibrium shocks for both compressible and incompressible materials. The consequences of this inequality are studied for weak shocks in plane strain for compressible materials and for shocks of arbitrary strength in anti-plane strain for a class of incompressible materials. A thermodynamic argument for the dissipation inequality is also given.https://resolver.caltech.edu/CaltechAUTHORS:20151109-153157919Discontinuous deformation gradients near the tip of a crack in finite anti-plane shear: an example
https://resolver.caltech.edu/CaltechAUTHORS:20141104-081003852
Year: 1980
DOI: 10.1007/BF00043136
This investigation aims at the elastostatic field near the edges (tips) of a plane crack of finite width in an all-round infinite body, which — at infinity — is subjected to a state of simple shear parallel to the crack edges. The analysis is carried out within the fully nonlinear equilibrium theory of homogeneous and isotropic, incompressible elastic solids. Further, the particular constitutive law employed here gives rise to a loss of ellipticity of the governing displacement equation of equilibrium in the presence of sufficiently severe anti-plane shear deformations.
The study reported in this paper is asymptotic in the sense that the actual crack is replaced by a semi-infinite one, while the far field is required to match the elastostatic field predicted near the crack tips by the linearized theory for a crack of finite width. The ensuing global boundary-value problem thus characterizes the local state of affairs in the vicinity of a crack-tip, provided the amount of shear applied at infinity is suitably small.
An explicit exact solution to this problem, which is deduced with the aid of the hodograph method, exhibits finite shear stresses at the tips of the crack, but involves two symmetrically located lines of displacement-gradient and stress discontinuity issuing from each crack-tip.https://resolver.caltech.edu/CaltechAUTHORS:20141104-081003852A nonlinear effect in mode II crack problems
https://resolver.caltech.edu/CaltechAUTHORS:20141031-110333179
Year: 1981
DOI: 10.1016/0013-7944(81)90072-2
A recent result of Stephenson shows that, when finite deformations are taken into account, a crack under Mode II loading conditions in plane strain will open, at least near the crack-tip and at least for certain elastic materials. In this note, the matter is investigated further, and it is shown that, in general, nonlinear effects—even at small loads—lead either to crack-opening or apparent interpenetration of the crack-faces when the loading is of Mode II-type.https://resolver.caltech.edu/CaltechAUTHORS:20141031-110333179Anti-plane shear fields with discontinuous deformation gradients near the tip of a crack in finite elastostatics
https://resolver.caltech.edu/CaltechAUTHORS:20141103-094634483
Year: 1981
DOI: 10.1007/BF00043857
This paper reconsiders the problem of determining the elastostatic field near the tip of a crack in an all-round infinite body deformed by a "Mode III" loading at infinity to a state of anti-plane shear. The problem is treated for a class of incompressible, homogeneous, isotropic elastic materials whose constitutive laws permit a loss of ellipticity in the governing displacement equation of equilibrium at sufficiently severe shearing strains. The analysis represents a generalization of that reported in an earlier study and, as before, is carried out for the "small-scale nonlinear crack problem", in which a crack of finite length is replaced by a semi-infinite one, and the nonlinear field far from the crack-tip is matched to the near field predicted by the linearized theory. The methods employed in the present paper are necessarily largely qualitative, since they apply to all materials in the class considered. The principal feature of the resulting elastic field is the presence of two symmetrically located curves issuing from the crack-tip and bearing discontinuities in displacement gradient and stress.https://resolver.caltech.edu/CaltechAUTHORS:20141103-094634483The effect of nonlinearity on a principle of Saint-Venant type
https://resolver.caltech.edu/CaltechAUTHORS:20141104-103601635
Year: 1981
DOI: 10.1007/BF00041940
This paper establishes a principle of Saint-Venant type associated with finite anti-plane shear of a cylinder whose cross-section is a semi-infinite strip. The long sides of the strip are traction-free, and the short side carries an arbitrarily distributed shear traction. At the infinity in the strip, the deformation is prescribed to be one of simple shear, and the associated shear stress is uniform. The analysis is based on the fully nonlinear theory of finite elastostatics and is carried out for a special class of homogenous, isotropic incompressible materials. It is shown that, along the parallel sides of the strip, the nonvanishing component of shear stress differs from its average value (taken across the strip) by an exponentially decaying function of the distance from the end. A lower bound is given for the rate of decay.https://resolver.caltech.edu/CaltechAUTHORS:20141104-103601635A Note on the Energy Release Rate in Quasi-Static Elastic Crack Propagation
https://resolver.caltech.edu/CaltechAUTHORS:20120718-120448789
Year: 1981
DOI: 10.1137/0141034
This paper considers analytical issues associated with the notion of the energy release rate in quasi-static elastic crack propagation.https://resolver.caltech.edu/CaltechAUTHORS:20120718-120448789Recent Developments Concerning Saint-Venant's Principle
https://resolver.caltech.edu/CaltechAUTHORS:20141104-131016828
Year: 1983
DOI: 10.1016/S0065-2156(08)70244-8
This chapter provides an overview of the recent developments concerning Saint-Venant's principle. The task of determining, within the framework of the linear theory of elasticity, the stresses and displacements in an elastic cylinder in equilibrium, under the action of loads that arise solely from tractions applied to its plane ends has come to be called Saint- Venant's problem. Saint-Venant's construction does not permit the arbitrary preassignment of the point-by-point variation of the end tractions giving rise to these forces and moments; indeed, this variation is essentially determined as a consequence of the special assumptions made in connection with his so-called semi-inverse procedure. The early work of Saint-Venant and Boussinesq furnished the seeds from which grew a large number of more general assertions, most referring to elastic solids of arbitrary shape and many being rather imprecise, concerning the effect on stresses within the body of replacing the tractions acting over a portion of its surface by statically equivalent ones. Such propositions usually went by the name of Saint-Venunt's principle, despite the fact that Saint-Venant's original conjecture was intended to apply only to cylinders. This chapter discusses in detail about flow in a cylinder, a representation for the exact solution, and energy decay for other linear elliptic second-order problem. Linear elastostatic problems are also stated in the chapter.https://resolver.caltech.edu/CaltechAUTHORS:20141104-131016828An Energy Estimate for the Biharmonic Equation and its Application to Saint-Venant's Principle in Plane Elastostatics
https://resolver.caltech.edu/CaltechAUTHORS:20141029-103545954
Year: 1983
A new energy estimate is given for a boundary value problem for the
biharmonic equation. The result is applied to the estimation of stresses in
a plane elasticity problem.https://resolver.caltech.edu/CaltechAUTHORS:20141029-103545954Large deformations near a tip of an interface-crack between two Neo-Hookean sheets
https://resolver.caltech.edu/CaltechAUTHORS:20141104-100901137
Year: 1983
DOI: 10.1007/BF00042997
This paper contains an asymptotic investigation - within the nonlinear theory of elastostatic plane stress - of the deformations and stresses near the tips of a traction-free interface-crack between two dissimilar semi-infinite Neo-Hookean sheets. The results obtained are free of oscillatory singularities of the kind predicted by the linearized theory, which would require the two deformed faces of an interface-crack to overlap in the vicinity of its tips. Instead, the crack is found to open smoothly near its ends, regardless of the specific loading at infinity.https://resolver.caltech.edu/CaltechAUTHORS:20141104-100901137On damped linear dynamical systems
https://resolver.caltech.edu/CaltechAUTHORS:20141104-095911928
Year: 1985
This paper is concerned with damped linear dynamical systems which fail to satisfy the so-called "convenience hypothesis", an assumption that guarantees the existence of classical normal modes. Under suitable conditions, estimates are given for the error committed when such a system is described approximately by a differential equation which does satisfy the convenience hypothesis. One of these estimates is used to derive an upper bound on the frequency response of systems which do not violate the convenience hypothesis too severely.https://resolver.caltech.edu/CaltechAUTHORS:20141104-095911928Remarks on a Question of Ericksen concerning Elastostatic Fields of Saint-Venant Type
https://resolver.caltech.edu/CaltechAUTHORS:20151113-165202758
Year: 1985
DOI: 10.1007/BF00251733
In an effort to understand better the relationship between approximate theories - such as those for thin rods- and the three-dimensional theory of elasticity,
ERICKSEN [1]- [3] has recently suggested a reconsideration of Saint-Yenant's problem for elastic cylinders with traction-free lateral surfaces. Among the various
questions raised in [1]- [3], one concerns the structure and role of the set of all
possible elastostatic fields in an infinitely long cylinder in the absence of lateral
loading and body force, but in the presence of a restriction on the size of a suitable
cross-sectional norm of the associated strain tensor field.
Although the issues which emerge from ERICKSEN's discussion acquire their greatest significance in the setting of finite elasticity, some of them arise in simpler
form in the classical infinitesimal theory, as he points out. In the present paper,
we examine the question mentioned above in a context that permits elementary analysis, and yet is rich enough to illuminate some of the important features:
that of linearized plane strain for homogeneous, isotropic materials. In particular,
the analysis makes especially transparent the exceptional status of the analog in
plane strain of the case of flexure by a transverse force in Saint-Venanfs cylinder
problem.
In the following section, we discuss plane strain for the infinite strip and the
corresponding fields of Saint-Yenant type. One version of ERICKSEN's question
is examined in Section 3 and, in modified form, in Section 4. Section 5 is devoted
to related issues for a strip of finite length.https://resolver.caltech.edu/CaltechAUTHORS:20151113-165202758Path-independent integrals for the direct determination of stress intensity factors in certain classical crack problems
https://resolver.caltech.edu/CaltechAUTHORS:20141103-094347047
Year: 1986
DOI: 10.1007/BF00041766
Stress intensity factors have been determine directly for certain special crack problems with the help of J or other path-independent integrals. Such procedures have not been used successfully in what are perhaps the most classical of all crack problems: those in two dimensions involving a crack of finite length in an infinite medium with loading at infinity of either Mode I, Mode II or Mode III type. We give a new class of path-independent integrals which are suitable for this purpose.https://resolver.caltech.edu/CaltechAUTHORS:20141103-094347047On the Scale of the Nonlinear Effect in a Crack Problem
https://resolver.caltech.edu/CaltechAUTHORS:20141103-084331857
Year: 1986
DOI: 10.1115/1.3171808
When crack problems are analyzed on the basis of nonlinear
theories, such as finite elasticity or deformation theory of
plasticity, it is inevitable that nonlinear effects will
predominate near a crack-tip, even if the loads are small. The
most favorable circumstance in this regard occurs when the
loads are so small that the zone of significant nonlinearity lies
within the region of validity of the near-tip approximation to
the global solution of the associated linearized crack problem.
This situation - called small-scale yielding for crack problems
in plasticity - permits simplifications in analysis which are
often decisive; see, e.g., Knowles (1977) and Rice (1968).
Insofar as we know, there are no analytical estimates
available of the level of load below which nonlinear effects are
guaranteed to be small-scale in the above sense. Indeed, even a
precise version of the question seems to be lacking. In the present
note we formulate and answer such a question for an
especially cooperative crack problem; that corresponding to
finite anti-plane shear of an infinite medium containing a
crack of finite length for an elastic material of Neo-Hookean
type. The associated boundary value problem is a linear one
for Laplace's equation and thus can be solved globally. Nevertheless,
there is a significant nonlinear effect of Kelvin type in
the stress field. We give a condition under which this nonlinear
response occurs on a small scale near the crack tips.https://resolver.caltech.edu/CaltechAUTHORS:20141103-084331857Non-elliptic elastic materials and the modeling of dissipative mechanical behavior: an example
https://resolver.caltech.edu/CaltechAUTHORS:20141029-132515351
Year: 1987
DOI: 10.1007/BF00044195
In the context of a special problem, this paper investigates the possibility of modeling dissipative mechanical response in solids on the basis of the equilibrium theory of finite elasticity for materials that may lose ellipticity at large strains. Quasi-static motions for such materials are in general dissipative if the associated equilibrium fields involve discontinuous displacement gradients. For the problem treated, consideration of such deformations is shown to lead naturally to an internal variable formalism similar to those used to describe macroscopic plastic behavior arising from microstructural effects. For quasi-static motions which are maximally dissipative in a specified sense, this formalism leads to a mechanical response which resembles that associated with the pseudo-elastic effect in shape-memory alloys.https://resolver.caltech.edu/CaltechAUTHORS:20141029-132515351Non-elliptic elastic materials and the modeling of elastic-plastic behavior for finite deformation
https://resolver.caltech.edu/CaltechAUTHORS:20141112-121650858
Year: 1987
DOI: 10.1016/0022-5096(87)90012-3
For illustrative purposes this paper treats a special problem in the theory of finite deformations of elastic materials whose associated displacement equations of equilibrium do not remain elliptic at all strains. The typical deformation arising in this problem possesses a discontinuous gradient, so that quasi-static motions involving such equilibrium states may be dissipative. For a special class of such "non-elliptic" elastic materials, it is shown that the macroscopic response in the problem treated may be precisely of the form associated with elastic—perfectly plastic behavior. The counterparts of yield, plastic strain and plastic strain rate are determined by the underlying elastic strain energy function.https://resolver.caltech.edu/CaltechAUTHORS:20141112-121650858On the dissipative response due to discontinuous strains in bars of unstable elastic material
https://resolver.caltech.edu/CaltechAUTHORS:20141029-093857845
Year: 1988
DOI: 10.1016/0020-7683(88)90105-9
Some elastic materials are capable of sustaining finite equilibrium deformations with discontinuous strains. Boundary-value problems for such "unstable" elastic materials often possess an infinite number of solutions, suggesting that the theory suffers from a constitutive deficiency. In the setting of the one-dimensional theory of bars in tension, the present paper explores the consequences of supplementing the theory with further constitutive information. This additional information pertains to the surface of strain discontinuity and consists of a 'kinetic relation' and a criterion for the "initiation" of such a surface. We show that the quasi-static response of the bar to a prescribed force history is then fully determined. In particular, we observe how unstable clastic materials can he used to model macroscopic behavior similar to that associated with viscoplasticity.https://resolver.caltech.edu/CaltechAUTHORS:20141029-093857845Unstable Elastic Materials and the Viscoelastic Response of Bars in Tension
https://resolver.caltech.edu/CaltechAUTHORS:20141104-135257842
Year: 1988
DOI: 10.1115/1.3173707
Some homogeneous elastic materials are capable of sustaining
finite equilibrium deformations with discontinuous
strains. For materials of this kind, the energetics of isothermal,
quasi-static motions may differ from those conventionally
associated with elastic behavior. When equilibrium states
involving strain jumps occur during such motions, the rate of
increase of stored energy in a portion of the body may no
longer coincide with the rate of work of the external forces
present. In general, energy balance now includes an additional
effect due to the presence of moving strain discontinuities. As
a consequence, the macroscopic response of the body may be
dissipative. This fact makes it possible to model certain types
of inelastic behavior in solids with the help of such "unstable"
elastic materials; see, for example, Abeyaratne and Knowles
(1987a,b,c).https://resolver.caltech.edu/CaltechAUTHORS:20141104-135257842Eli Sternberg Memoriam
https://resolver.caltech.edu/CaltechAUTHORS:20141103-103729493
Year: 1989
DOI: 10.1115/1.3176073
Eli Sternberg, perhaps the best known scholar in the field of
elasticity during most of the past half-century, died suddenly
in Pasadena, California, on October 8, 1988, shortly before
his seventy-first birthday.https://resolver.caltech.edu/CaltechAUTHORS:20141103-103729493On the direct determination of the near-tip stress field for the scattering of SH-waves by a crack
https://resolver.caltech.edu/CaltechAUTHORS:20141103-103118611
Year: 1989
DOI: 10.1007/BF00018860
A procedure is given for the direct determination of the near-tip stress field arising from the scattering of a normally incident SH-wave by a crack in a homogeneous, isotropic elastic medium. The analysis, which circumvents the construction of the global solution of the problem, is based on a conservation law associated with the relevant field equation.https://resolver.caltech.edu/CaltechAUTHORS:20141103-103118611Equilibrium shocks in plane deformations of incompressible elastic materials
https://resolver.caltech.edu/CaltechAUTHORS:20141031-104535753
Year: 1989
DOI: 10.1007/BF00041104
This paper is concerned with piecewise smooth plane deformations in an isotropic, incompressible elastic material. An explicit necessary and sufficient condition for the existence of piecewise homogeneous equilibrium states is established, and the set of all such states is precisely characterized. A particularly simple expression is derived for the "driving traction" on a surface of discontinuity in the deformation gradient.https://resolver.caltech.edu/CaltechAUTHORS:20141031-104535753On the driving traction acting on a surface of strain discontinuity in a continuum
https://resolver.caltech.edu/CaltechAUTHORS:20141021-113953976
Year: 1990
DOI: 10.1016/0022-5096(90)90003-M
The notion of the driving traction on a surface of strain discontinuity in a continuum undergoing a general thermomechanical process is defined and discussed. In addition, the associated constitutive notion of a kinetic relation, in which the normal velocity of propagation of the surface of discontinuity may be a given function of the driving traction and temperature, is introduced for the special case of a thermoelastic material.https://resolver.caltech.edu/CaltechAUTHORS:20141021-113953976Kinetic relations and the propagation of phase boundaries in solids
https://resolver.caltech.edu/CaltechAUTHORS:20141021-141408927
Year: 1991
DOI: 10.1007/BF00375400
This paper treats the dynamics of phase transformations in elastic bars. The specific issue studied is the compatibility of the field equations and jump conditions of the one-dimensional theory of such bars with two additional constitutive requirements: a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition. A special elastic material with a piecewise-linear, non-monotonic stress-strain relation is considered, and the Riemann problem for this material is analyzed. For a large class of initial data, it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities.https://resolver.caltech.edu/CaltechAUTHORS:20141021-141408927A class of compressible elastic materials capable of sustaining finite anti-plane shear
https://resolver.caltech.edu/CaltechAUTHORS:20141029-083432015
Year: 1991
DOI: 10.1007/BF00040926
This paper describes a simple class of homogeneous, isotropic, compressible hyperelastic materials capable of sustaining nontrivial states of finite anti-plane shear.https://resolver.caltech.edu/CaltechAUTHORS:20141029-083432015Implications of Viscosity and Strain-Gradient Effects for the Kinetics of Propagating Phase Boundaries in Solids
https://resolver.caltech.edu/CaltechAUTHORS:20141021-102900385
Year: 1991
DOI: 10.1137/0151061
This paper is concerned with the propagation of phase boundaries in elastic bars. It is known that the Riemann problem for an elastic bar capable of undergoing isothermal phase transitions need not have a unique solution, even in the presence of the requirement that the entropy of any particle cannot decrease upon crossing a phase boundary. For a special class of elastic materials, the authors have shown elsewhere that if all phase boundaries move subsonically with respect to both phases, this lack of uniqueness can be resolved by imposing a nucleation criterion and a kinetic relation for the relevant phase transition. Others have singled out acceptable solutions on the basis of a theory that adds effects due to viscosity and second strain gradient to the elastic part of the stress. It is shown that, for phase boundaries that propagate subsonically, this approach is equivalent to the imposition of a particular kinetic relation at the interface between the phases.https://resolver.caltech.edu/CaltechAUTHORS:20141021-102900385Wave propagation in linear, bilinear and trilinear elastic bars
https://resolver.caltech.edu/CaltechAUTHORS:20141021-103958370
Year: 1992
DOI: 10.1016/0165-2125(92)90006-N
This paper is concerned with the role of supplementary conditions, such as the entropy inequality at shock waves or kinetic relations at phase boundaries, in the selection of physically appropriate solutions to systems of quasi-linear differential equations describing wave propagation. The differences in this respect among various materials are illustrated by constrasting the behaviour of waves in linear, bilinear and trilinear elastic bars.https://resolver.caltech.edu/CaltechAUTHORS:20141021-103958370On a minimization problem associated with linear dynamical systems
https://resolver.caltech.edu/CaltechAUTHORS:20141021-101021488
Year: 1992
DOI: 10.1016/0024-3795(92)90236-4
This paper considers the problem of approximating a given pair of symmetric Cartesian tensors by a pair of symmetric Cartesian tensors that commute.https://resolver.caltech.edu/CaltechAUTHORS:20141021-101021488On the Propagation of Maximally Dissipative Phase Boundaries in Solids
https://resolver.caltech.edu/CaltechAUTHORS:20141024-154040051
Year: 1992
This paper is concerned with the kinetics of propagating phase boundaries
in a bar made of a special nonlinearly elastic material. First, it is shown that
there is a kinetic law of the form f = φ(s) relating the driving traction f at a phase
boundary to the phase boundary velocity s that corresponds to a notion of maximum
dissipation analogous to the concept of maximum plastic work. Second, it is shown
that a modified version of the entropy rate admissibility criterion can be described by
a kinetic relation of the above form, but with a different φ. Both kinetic relations
are applied to the Riemann problem for longitudinal waves in the bar.https://resolver.caltech.edu/CaltechAUTHORS:20141024-154040051A continuum model of a thermoelastic solid capable of undergoing phase transitions
https://resolver.caltech.edu/CaltechAUTHORS:20141021-123729357
Year: 1993
DOI: 10.1016/0022-5096(93)90048-K
We construct explicitly a Helmholtz free energy, a kinetic relation and a nucleation criterion for a one-dimensional thermoelastic solid, capable of undergoing either mechanically- or thermally-induced phase transitions. We study the hysteretic macroscopic response predicted by this model in the case of quasistatic processes involving stress cycling at constant temperature, thermal cycling at constant stress, or a combination of mechanical and thermal loading that gives rise to the shape-memory effect. These predictions are compared qualitatively with experimental results.https://resolver.caltech.edu/CaltechAUTHORS:20141021-123729357Dynamics of propagating phase boundaries: Thermoelastic solids with heat conduction
https://resolver.caltech.edu/CaltechAUTHORS:20141021-154813217
Year: 1994
DOI: 10.1007/BF00375642
This paper is concerned with the incorporation of thermal effects into the continuum modeling of dynamic solid-solid phase transitions. The medium is modeled as a one-dimensional thermoelastic solid characterized by a specific Helmholtz free-energy potential and a specific kinetic relation. Heat conduction and inertia are taken into account. An initial-value problem that gives rise to both shock waves and a propagating phase boundary is analyzed on the basis of this model.https://resolver.caltech.edu/CaltechAUTHORS:20141021-154813217A one-dimensional continuum model for shape-memory alloys
https://resolver.caltech.edu/CaltechAUTHORS:20141021-111150545
Year: 1994
DOI: 10.1016/0020-7683(94)90208-9
In this paper we construct an explicit one-dimensional constitutive model that is capable of describing some aspects of the thermomechanical response of a shape-memory alloy. The model consists of a Helmholtz free-energy function, a kinetic relation and a nucleation criterion. The free-energy is associated with a three-well potential energy function; the kinetic relation is based on thermal activation theory; nucleation is assumed to occur at a critical value of the appropriate energy barrier. The predictions of the model in various quasi-static thermomechanical loadings are examined and compared with experimental observations.https://resolver.caltech.edu/CaltechAUTHORS:20141021-111150545Dynamics of propagating phase boundaries: adiabatic theory for thermoelastic solids
https://resolver.caltech.edu/CaltechAUTHORS:20141024-102313946
Year: 1994
DOI: 10.1016/S0167-2789(05)80008-9
This paper examines adiabatic processes in a thermoelastic material undergoing a solid-solid phase transition. It is shown that an initial-value problem of Riemann type, based on momentum balance, energy balance, kinematic compatibility and the entropy inequality, has a one-parameter family of solutions for a range of given data. Within this adiabatic setting, the notions of driving traction and kinetic relation are discussed. The enforcement of a kinetic relation at phase boundaries is shown to single out a particular solution of this Riemann problem from among the family of available solutions.https://resolver.caltech.edu/CaltechAUTHORS:20141024-102313946A thermoelastic model for an experiment involving a solid-solid phase transition
https://resolver.caltech.edu/CaltechAUTHORS:20141029-133257223
Year: 1995
DOI: 10.1007/BF01175771
An experiment was carried out by Sammis and Dein [1974] to support the notion that large creep strains could be generated in the earth's mantle by solid-solid phase transitions. The predictions of the thermoelastic model given here are in qualitative agreement with the results of their experiment.https://resolver.caltech.edu/CaltechAUTHORS:20141029-133257223On the Representation of the Elasticity Tensor for Isotropic Materials
https://resolver.caltech.edu/CaltechAUTHORS:20141024-141244096
Year: 1995
DOI: 10.1007/BF00043415
This note provides a different proof of the theorem concerning the representation
of the elasticity 4-tensor for isotropic materials. Other relatively recent proofs may
be found in [1]-[3].https://resolver.caltech.edu/CaltechAUTHORS:20141024-141244096Dynamic thermoelastic phase transitions
https://resolver.caltech.edu/CaltechAUTHORS:20141021-120725895
Year: 1995
DOI: 10.1016/0020-7683(94)00292-5
This paper summarizes some recent work carried out jointly with R. Abeyaratne on the continuum modeling of phase transitions in thermoelastic tensile bars. The specific model considered involves a particular Helmholtz free energy potential governing the bulk response of the material as well as a kinetic relation controlling the phase transition. Inertia is taken into account. The discussion here is based on an adiabatic model.https://resolver.caltech.edu/CaltechAUTHORS:20141021-120725895On the decay of end-effects due to self-equilibrated loads
https://resolver.caltech.edu/CaltechAUTHORS:20141020-151253996
Year: 1995
DOI: 10.1007/BF00040763
This note compares the decay of end effects in anti-plane shear for a semi-infinite elastic strip under a self-equilibrated system of concentrated end-loads with the corresponding decay for a smoothly distributed self-equilibrated load.https://resolver.caltech.edu/CaltechAUTHORS:20141020-151253996On the kinetics of an austenite → martensite phase transformation induced by impact in a Cu-Al-Ni shape-memory alloy
https://resolver.caltech.edu/CaltechAUTHORS:20141022-084130812
Year: 1997
DOI: 10.1016/S1359-6454(96)00276-5
The kinetics of a propagating phase boundary in a single crystal of Cu-Al-Ni is determined. As particles cross this moving interface, they transform from austenite to β'_1-martensite; the former phase is cubic, while the latter is monoclinic. The data that are used in this paper were obtained by Escobar and Clifton (Escober, J. C. and Clifton, R. J., On pressure-shear plate impact for studying the kinetics of stress-induced phase transformations. J. Mater. Sci. Engng, 1993, A170, 125–142; Escober, J. C. and Clifton, R. J. Pressure-shear impact-induced phase transitions in Cu-14.4Al-4.19Ni single crystals. SPIE, 1995, 2427, 186–197) and Escobar (Escobar, J. C., Plate impact induced phase transformations is Cu-Al-Ni single crystals. Ph.D. dissertation, Brown University, Providence, RI, 1995) from experiments in which the transformation was induced by impact. These data, together with an analysis based on a certain linearization, are used to determine the values of phase boundary velocity and driving force in each impact experiment. The resulting kinetic law which relates these two quantities is displayed in a figure later in the text.https://resolver.caltech.edu/CaltechAUTHORS:20141022-084130812Impact-induced phase transitions in thermoelastic solids
https://resolver.caltech.edu/CaltechAUTHORS:20141021-102050630
Year: 1997
DOI: 10.1098/rsta.1997.0048
A continuum model of the macroscopic behaviour of solids capable of undergoing
displacive phase transitions is applied to determine the response of such materials
to mechanical loading by impact. The solid is modelled using one-dimensional finite
thermoelasticity, and the model incorporates both a kinetic relation and a nucleation
criterion controlling the evolution and initiation of the phase transition, respectively.https://resolver.caltech.edu/CaltechAUTHORS:20141021-102050630Dynamically Induced Phase Transitions and the Modeling of Comminution in Brittle Solids
https://resolver.caltech.edu/CaltechAUTHORS:20141023-084053866
Year: 1997
DOI: 10.1177/108128659700200201
Some experiments suggest the presence of a sharp interface between comminuted and uncomminuted regimes in a ceramic subject to impact by a penetrator. This in turn suggests that one might model the associated dynamical process with the help of recently developed continuum models of the macroscopic response of solids undergoing phase transitions. A highly idealized phase transition model of such a com- minuted process is analyzed here. The model accounts for the kinetics of the phase transition. If the shear wave speed in the comminuted material is small compared to that in the uncomminuted portion, it is found that the energy reaching the uncomminuted portion of the target is greatly reduced in comparison to its value in the absence of the phase transition.https://resolver.caltech.edu/CaltechAUTHORS:20141023-084053866On an integrodifferential equation arising in a theory of phase transitions in solids
https://resolver.caltech.edu/CaltechAUTHORS:20141024-152623996
Year: 1997
This note is concerned with some properties of an integrodifferential equation a rising in a continuum model of solid-solid phase transitions.https://resolver.caltech.edu/CaltechAUTHORS:20141024-152623996Unstable kinetic relations and the dynamics of solid-solid phase transitions
https://resolver.caltech.edu/CaltechAUTHORS:20141022-084738427
Year: 1997
DOI: 10.1016/S0022-5096(97)00026-4
In recent continuum-mechanical models of phase transitions in solids, the kinetic relation for a transition is usually assumed to be such that the driving force acting on a phase boundary is a monotonically increasing function of phase boundary velocity. The present paper explores the implications of relinquishing this assumption in the dynamics of one-dimensional elastic bars undergoing stress-induced transitions. Among other results, it is found that, for a class of non-monotonic kinetic relations, models of the kind discussed here permit stick-slip motions of a phase boundary, as observed in certain experiments.https://resolver.caltech.edu/CaltechAUTHORS:20141022-084738427On the stability of thermoelastic materials
https://resolver.caltech.edu/CaltechAUTHORS:20141023-155158386
Year: 1998
DOI: 10.1023/A:1007513631783
A notion of material stability is introduced and discussed in the setting of nonlinear thermoelasticity. Necessary and sufficient conditions are established for the stability of a general thermoelastic material. The adiabatic theory and the theory that accounts for heat conduction are considered separately.https://resolver.caltech.edu/CaltechAUTHORS:20141023-155158386Continuum models for irregular phase boundary motion in shape-memory tensile bars
https://resolver.caltech.edu/CaltechAUTHORS:20141021-110039001
Year: 1999
DOI: 10.1016/S0997-7538(99)80001-1
We consider quasi-static displacement-controlled loading through one stress cycle of a shape-memory tensile bar modeled as a one-dimensional, two-phase elastic solid. Our objective is to explore the effect on the associated hysteresis loop of various qualitatively different types of kinetic relations, bearing in mind certain features of such loops that have been observed experimentally. We show that when the model involves a kinetic relation that is 'unstable' in a definite sense, 'stick-slip' motion of the interface between phases and serration of the accompanying stress-elongation curve are both predicted at slow elongation rates. We also show that a 'nonhomogeneous' kinetic relation intended to model the effect of micro-obstacles on interface motion also leads to irregular interface motion and a serrated stress-elongation curve, in this case at all elongation rates.https://resolver.caltech.edu/CaltechAUTHORS:20141021-110039001Stress-induced phase transitions in elastic solids
https://resolver.caltech.edu/CaltechAUTHORS:20141023-090542578
Year: 1999
DOI: 10.1007/s004660050376
This paper describes a continuum model, developed recently by R. Abeyaratne and the author, for the response of elastic solids capable of undergoing stress-induced phase transitions. Models of the kind sketched here and their generalizations are intended to apply to both quasi-static and dynamic experiments for shape memory alloys and to impact-induced phase changes in ceramics. The present discussion is confined to a purely mechanical theory, omitting thermal effects, so that the natural setting is the nonlinear theory of elasticity. The presentation below is limited to one space dimension.https://resolver.caltech.edu/CaltechAUTHORS:20141023-090542578Effect of irreversible phase change on shock-wave propagation
https://resolver.caltech.edu/CaltechAUTHORS:20141021-124357929
Year: 1999
DOI: 10.1016/S0022-5096(98)00090-8
New release adiabat data for vitreous GeO_2 are reported up to ∼25 GPa using the VISAR technique. Numerical modeling of isentropic release wave induced dynamic states achieved from one dimensional strain–stress waves is consistent with a phase change that induce an increase in zero-pressure density from 3.7–6.3 Mg/m^3 starting at ∼8 GPa. The first release adiabat data for SiO_2 ( fused quartz) are presented (obtained with immersed foil technique) . Above 10 GPa, the SiO_2 release isentropes, in analogy with GeO_2, are steeper than the Hugoniot in the volume-pressure space, indicating the presence of an irreversible phase transition (to a stishovite-like phase). We simulate propagation of shock-waves in GeO_2, in spherical and planar symmetries, and predict enhanced attenuation for shock pressures ( p) above the phase change initiation pressure (8 GPa) . The pressure from a spherical source decays with propagation radius r, p ∼ r^x, where x is the decay coefficient. Modeling hysteresis of the phase change gives x = −2.71, whereas without the phase change, x = −1.15. An analytical model is also given.https://resolver.caltech.edu/CaltechAUTHORS:20141021-124357929On a shock-induced martensitic phase transition
https://resolver.caltech.edu/CaltechAUTHORS:20141021-073017252
Year: 2000
DOI: 10.1063/1.371989
A recently developed continuum-mechanical model for stress-induced phase transitions in solids is applied to a transition generated by impact. The role of transition kinetics in determining the macroscopic response to impact is discussed; in addition, the special way that "overdriven" phase boundaries emerge in this model is described. The predictions of the model are compared with experiments involving shock-induced graphite-to-diamond phase transitions.https://resolver.caltech.edu/CaltechAUTHORS:20141021-073017252A phenomenological model for failure waves in glass
https://resolver.caltech.edu/CaltechAUTHORS:20141021-100236468
Year: 2000
DOI: 10.1007/s001930000056
A phenomenological continuum-mechanical model for phase transitions in solids is applied to the description of failure waves in glass. Several predictions of the model are in qualitative agreement with experimental observations.https://resolver.caltech.edu/CaltechAUTHORS:20141021-100236468A note on the driving traction acting on a propagating interface: Adiabatic and non-adiabatic processes of a continuum
https://resolver.caltech.edu/CaltechAUTHORS:20141021-112510693
Year: 2000
DOI: 10.1115/1.1308577
An expression for the driving traction on an interface is derived for an arbitrary continuum undergoing an arbitrary thermomechanical
process which may- or may not be adiabatic.https://resolver.caltech.edu/CaltechAUTHORS:20141021-112510693Hysteresis in the Stress‐Cycling of Bars Undergoing Solid–Solid Phase Transitions
https://resolver.caltech.edu/CaltechAUTHORS:20111110-095056685
Year: 2002
DOI: 10.1093/qjmam/55.1.69
A model is presented for the response to slow cyclic stressing of bars capable of undergoing displacive phase transitions. One‐dimensional nonlinear elasticity is used, and the sharp interface between material phases bears a strain discontinuity. As a result, the theory is dissipative, in the sense that the rate of work supplied to the bar exceeds the rate at which the stored energy increases. An essential part of the theory is a continuum‐mechanical version of a kinetic relation that controls the rate at which the phase transition takes place and therefore determines the ultimate governing dynamical system. A periodic solution of this system is constructed, from which one may determine the hysteresis loop to which the response of the bar is expected to tend after many loading cycles. The energy dissipation associated with this loop is determined for a very general class of kinetic relations. For a particular kinetic relation, the transient cycling problem is also solved explicitly, and it is shown that the response indeed tends to that for the periodic solution as the number of cycles tends to infinity. Several predictions of the model are in qualitative accord with experiments carried out on shape‐memory alloys, suggesting that transition kinetics may provide the mechanism responsible for the hysteretic behaviour observed in such materials.https://resolver.caltech.edu/CaltechAUTHORS:20111110-095056685Elastic Materials with Two Stress-Free Configurations
https://resolver.caltech.edu/CaltechAUTHORS:20141021-100558200
Year: 2002
DOI: 10.1023/A:1022545224008
It is shown that, if an elastic material exhibits two stress-free configurations, it is dynamically unstable in a definite sense.https://resolver.caltech.edu/CaltechAUTHORS:20141021-100558200Impact-induced tensile waves in a rubberlike material
https://resolver.caltech.edu/CaltechAUTHORS:KNOsiamjam02
Year: 2002
DOI: 10.1137/S0036139901388234
This paper concerns the propagation of impact-generated tensile waves in a one-dimensional bar made of a rubberlike material. Because the stress-strain curve changes from concave to convex as the strain increases, the governing quasi-linear system of partial differential equations, though hyperbolic, fails to be genuinely nonlinear so that the standard form of the boundary-initial value problem corresponding to impact is not well-posed at all levels of loading. When the problem fails to be well-posed, it does so by exhibiting a massive loss of uniqueness, even though an entropy-like dissipation inequality is in force. Because the breakdown in uniqueness is reminiscent of a similar phenomenon that occurs in continuum-mechanical models for impact-induced phase transitions, a mathematically suitable, though physically unmotivated, supplementary selection mechanism for determining the solution naturally suggests itself. We describe in detail the solutions determined by two special forms of this selection mechanism, and we show that these two solutions provide bounds on the impact response, regardless of the selection principle used.https://resolver.caltech.edu/CaltechAUTHORS:KNOsiamjam02On the relation between particle velocity and shock wave speed for thermoelastic materials
https://resolver.caltech.edu/CaltechAUTHORS:20141021-093341465
Year: 2002
DOI: 10.1007/s00193-002-0146-1
Results of shock-wave experiments in solids often suggest a nearly-linear relation between the particle velocity behind the shock and the shock wave speed. The present note reconsiders the question of whether thermoelastic material models may be consistent with such observations. Emphasis is placed on the role played by the response of the material in severe compression, as distinguished from its response for small or moderate deformations. The details are illustrated for materials of Mie-Grüneisen type.https://resolver.caltech.edu/CaltechAUTHORS:20141021-093341465Sudden tensile loading of a rubberlike bar
https://resolver.caltech.edu/CaltechAUTHORS:20141023-074114376
Year: 2003
DOI: 10.1016/S0093-6413(03)00070-3
In uniaxial tension, the stress–strain curve for rubber changes curvature from concave to convex as the strain increases. For sudden tensile loading of a bar, a one-dimensional model that reflects this behavior leads to an under-determined problem reminiscent of that arising in materials capable of undergoing phase transitions. In the latter setting, adding the kinetic relation underlying the phase change to the conventional statement of the problem removes the indeterminacy; the same is true when such a relation is used in a formal way in the problem for rubber. This presents a physical question: What is the evolutionary process at the microscale whose kinetics are needed in the dynamics of rubber?https://resolver.caltech.edu/CaltechAUTHORS:20141023-074114376On shock waves in a special class of thermoelastic solids
https://resolver.caltech.edu/CaltechAUTHORS:20141023-074425429
Year: 2005
DOI: 10.1016/j.ijnonlinmec.2004.08.001
This paper describes a thermoelastic model for shock waves in uniaxial strain based on a subclass of the so-called materials of Mie–Grüneisen type. We compare the Hugoniot curve with the isotherms and isentropes for this model, and we construct the shock-wave solution to a simple impact problem.https://resolver.caltech.edu/CaltechAUTHORS:20141023-074425429Preface
https://resolver.caltech.edu/CaltechAUTHORS:20141021-092758564
Year: 2006
DOI: 10.1002/stc.151
The papers making up Issue No. 1 of Volume 13 of Structural Control and Health Monitoring,
aggregated here, are published as a tribute to Thomas K. Caughey, who was Hayman Professor
of Mechanical Engineering, Emeritus, at the California Institute of Technology and Honorary
Associate Editor of this journal prior to his death on 7 December 2004. Professor Caughey was
a major contributor to the general field of dynamics over the past half-century.https://resolver.caltech.edu/CaltechAUTHORS:20141021-092758564On the approximation of damped linear dynamical systems
https://resolver.caltech.edu/CaltechAUTHORS:20141021-091907527
Year: 2006
DOI: 10.1002/stc.130
This paper is concerned with damped linear dynamical systems, a subject of interest to Tom Caughey. Two approximating schemes for such systems are described. One scheme, due to Rayleigh, approximates the damping, while the second approximates both the damping and the stiffness. Both types of approximation are applied to an example.https://resolver.caltech.edu/CaltechAUTHORS:20141021-091907527Driving force and kinetic relations for scalar conservation laws
https://resolver.caltech.edu/CaltechAUTHORS:20141023-145404563
Year: 2007
DOI: 10.1142/S0219891607001057
This paper is concerned with the circumstances under which the dissipative character of a one-dimensional scalar conservation law may be described by a formalism strictly analogous to that arising naturally in the dynamics of nonlinearly elastic materials. It is shown that this occurs if and only if the entropy density, entropy flux pair associated with the conservation law takes a particular form. We compare the admissibility condition associated with this special entropy with other admissibility criteria such as those of Lax, Oleinik and regularization theory. Using the special entropy, we consider the Riemann problem for an example in which genuine nonlinearity fails and a kinetic relation is needed to determine a unique solution.https://resolver.caltech.edu/CaltechAUTHORS:20141023-145404563On shock waves in solids
https://resolver.caltech.edu/CaltechAUTHORS:20100503-101804312
Year: 2007
This paper describes some recent theoretical results pertaining to
the experimentally-observed relation between the speed of a shock wave in a
solid and the particle velocity immediately behind the shock. The new feature
in the present analysis is the assumption that compressive strains are limited
by a materially-determined critical value, and that the internal energy density
characterizing the material is unbounded as this critical strain is approached.
It is shown that, with this assumption in force, the theoretical relation between
shock speed and particle velocity is consistent with many experimental observations in the sense that it is asymptotically linear for strong shocks of the
kind often arising in the laboratory.https://resolver.caltech.edu/CaltechAUTHORS:20100503-101804312On entropy conditions and traffic flow models
https://resolver.caltech.edu/CaltechAUTHORS:20141021-090916484
Year: 2008
DOI: 10.1002/zamm.200700093
This paper is concerned with "entropy conditions" for traffic flow models governed by one-dimensional nonlinear scalar conservation laws. The classical conditions of this type serve as selection principles used to overcome the lack of uniqueness of weak solutions for such laws. These principles select solutions in accord with observations for the simplest traffic flow models, but they seem not to do so when the constitutive law underlying the model is refined so as to be more realistic. The classical conditions are here compared with an analog of the second law of thermodynamics based on an entropy density that crudely measures the disorder of the traffic flow.https://resolver.caltech.edu/CaltechAUTHORS:20141021-090916484On the structure of the Hugoniot relation for a shock-induced martensitic phase transition
https://resolver.caltech.edu/CaltechAUTHORS:20141021-095011345
Year: 2008
DOI: 10.1007/s00193-008-0119-0
The Hugoniot curve relates the pressure and volume behind a shock wave, with the temperature having been eliminated. This paper studies the Hugoniot curve behind a propagating sharp interface between two material phases for a solid in which an impact-induced phase transition has taken place. For a solid capable of existing in only one phase, compressive impact produces a shock wave moving into material, say, at rest in an unstressed state at the ambient temperature. If the specimen can exist in either of two material phases, sufficiently severe impact may produce a disturbance with a two-wave structure: a shock wave in the low-pressure phase of the material, followed by a phase boundary separating the low- and high-pressure phases. We use a theory of phase transitions in thermoelastic materials to construct the Hugoniot curve behind the phase boundary in this two-wave circumstance. The kinetic relation controlling the evolution of the phase transition is an essential ingredient in this process.https://resolver.caltech.edu/CaltechAUTHORS:20141021-095011345On Maximally Dissipative Shock Waves in Nonlinear Elasticity
https://resolver.caltech.edu/CaltechAUTHORS:20100121-141205723
Year: 2010
DOI: 10.1007/s10659-009-9221-5
Shock waves in nonlinearly elastic solids are, in general, dissipative. We study the following question: among all plane shock waves that can propagate with a given speed in a given one-dimensional nonlinearly elastic bar, which one—if any—maximizes the rate of dissipation? We find that the answer to this question depends strongly on the qualitative nature of the stress-strain relation characteristic of the given material. When maximally dissipative shocks do occur, they propagate according to a definite kinetic relation, which we characterize and illustrate with examples.https://resolver.caltech.edu/CaltechAUTHORS:20100121-141205723Discrete and continuous scalar conservation laws
https://resolver.caltech.edu/CaltechAUTHORS:20110621-110113923
Year: 2011
DOI: 10.1177/1081286510382821
Motivated by issues arising for discrete second-order conservation laws and their continuum limits (applicable, for example, to one-dimensional nonlinear spring—mass systems), here we study the corresponding issues in the simpler setting of first-order conservation laws (applicable, for example, to the simplest theory of traffic flow). The discrete model studied here comprises a system of first-order nonlinear differential-difference equations; its continuum limit is a one-dimensional scalar conservation law. Our focus is on issues related to discontinuities — shock waves — in the continuous theory and the corresponding regions of rapid change in the discrete model. In the discrete case, we show that a family of new conservation laws can be deduced from the basic one, while in the continuous case we show that this is true only for smooth solutions. We also examine how well the continuous model approximates rapidly changing solutions of the discrete model, and this leads us to derive an improved continuous model which is of second-order. We also consider the form and implications of the second law of thermodynamics at shock waves in the scalar case.https://resolver.caltech.edu/CaltechAUTHORS:20110621-110113923