CaltechDATA: Book Chapter
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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenThu, 28 Mar 2024 11:00:44 -0700A Wave Front Approximation Method and its Application to Elastic Stress Waves
https://resolver.caltech.edu/CaltechETD:etd-01152003-152904
Year: 1965
DOI: 10.7907/Y9EG-XT74
This paper presents a new direct method of obtaining wave front approximations for problems involving hyperbolic differential equations. In the problem of a semi-infinite, end-loaded elastic strip (the problem used to illustrate the method), asymptotic solutions are obtained for wave fronts prior to multiple edge interactions. For the special end loading of a step velocity, the results agree with prior results obtained by more complex methods of approximation. Extension of the method to multiple interactions and to other problems of stress wave propagation is briefly discussed.https://resolver.caltech.edu/CaltechETD:etd-01152003-152904Eigenvalue problems associated with Korn's inequalities in the theory of elasticity
https://resolver.caltech.edu/CaltechTHESIS:08062015-145439790
Year: 1970
DOI: 10.7907/3VYK-2B81
<p>Interest in the possible applications of a priori inequalities in
linear elasticity theory motivated the present investigation. Korn's
inequality under various side conditions is considered, with emphasis
on the Korn's constant. In the "second case" of Korn's inequality, a
variational approach leads to an eigenvalue problem; it is shown that,
for simply-connected two-dimensional regions, the problem of determining
the spectrum of this eigenvalue problem is equivalent to finding
the values of Poisson's ratio for which the displacement boundary-value
problem of linear homogeneous isotropic elastostatics has a non-unique solution.</p>
<p>Previous work on the uniqueness and non-uniqueness issue for
the latter problem is examined and the results applied to the spectrum
of the Korn eigenvalue problem. In this way, further information on
the Korn constant for general regions is obtained.</p>
<p>A generalization of the "main case" of Korn's inequality is introduced
and the associated eigenvalue problem is a gain related to the
displacement boundary-value problem of linear elastostatics in two
dimensions.</p>
https://resolver.caltech.edu/CaltechTHESIS:08062015-145439790Singularities and Phase Transitions in Elastic Solids: Numerical Studies and Stability Analysis
https://resolver.caltech.edu/CaltechETD:etd-03082008-083510
Year: 1986
DOI: 10.7907/0ytn-e775
<p>Numerical studies of the deformation near the tip of a crack are presented for a family of incompressible solids in the context of the theory of finite anti-plane shear of an elastic material. The numerical model computes the near-field and far-field solutions simultaneously, enabling observations of both small-scale and large-scale nonlinearity. The computed near-field solution is compared with a lowest-order asymptotic solution. An approximation for the <i>J</i>-integral under conditions of very large loads is discussed and compared with numerical results. The size of the region over which the lowest-order solution applies is observed.</p>
<p>Numerical solutions are presented for the same crack problem with materials for which the equilibrium equation changes in type from elliptic to hyperbolic as a result of deformation. These results show the emergence of surfaces of discontinuity in the displacement field in some cases. In other cases they show a chaotic mixture of elliptic phases near the crack tip.</p>
<p>Analysis of the stability of such coexistent phases is carried out for a specific material, the trilinear material. It is shown that the Maxwell relation, and therefore local stability, cannot in general be satisfied exactly for an arbitrary boundary value problem with this material. However, in those cases where it cannot be satisfied exactly, it may be satisfied in the sense of a limit of a certain sequence of deformations. This sequence produces a progressively chaotic pattern of two coexistent elliptic phases, as was observed numerically. The phases mix over a definite region in a given boundary value problem. This region may be computed using a constitutive relation which characterizes the mixture in the limit of the sequence.</p>https://resolver.caltech.edu/CaltechETD:etd-03082008-083510Some approximate solutions of dynamic problems in the linear theory of thin elastic shells
https://resolver.caltech.edu/CaltechTHESIS:04052013-144005487
Year: 1968
DOI: 10.7907/11GT-4V56
<p>Some aspects of wave propagation in thin elastic shells are considered.
The governing equations are derived by a method which makes their
relationship to the exact equations of linear elasticity quite clear.
Finite wave propagation speeds are ensured by the inclusion of the appropriate
physical effects.</p>
<p>The problem of a constant pressure front moving with constant
velocity along a semi-infinite circular cylindrical shell is studied. The
behavior of the solution immediately under the leading wave is found, as
well as the short time solution behind the characteristic wavefronts. The
main long time disturbance is found to travel with the velocity of very
long longitudinal waves in a bar and an expression for this part of the
solution is given.</p>
<p>When a constant moment is applied to the lip of an open spherical
shell, there is an interesting effect due to the focusing of the waves.
This phenomenon is studied and an expression is derived for the wavefront
behavior for the first passage of the leading wave and its first reflection.</p>
<p>For the two problems mentioned, the method used involves reducing
the governing partial differential equations to ordinary differential equations
by means of a Laplace transform in time. The information sought is
then extracted by doing the appropriate asymptotic expansion with the Laplace
variable as parameter.</p>https://resolver.caltech.edu/CaltechTHESIS:04052013-144005487On the determination of the properties of a medium from its reflection coefficient
https://resolver.caltech.edu/CaltechTHESIS:03272013-095340267
Year: 1971
DOI: 10.7907/fsax-ks83
<p>This thesis demonstrates how the parameters of a slightly
non-homogeneous medium can be derived approximately from the
reflection coefficient.</p>
<p>Two types of media are investigated. The first is described
by the one-dimensional wave equation, the second by the more
complex Timoshenko beam equation. In both cases, the media are
assumed to be infinite in extent, with the media parameters
becoming homogeneous as the space variable approaches positive or
negative infinity.</p>
<p>Much effort is placed in deriving properties of the reflection
coefficient for both cases. The wave equation is considered
primarily to introduce the techniques used to investigate the more
complex Timoshenko equation. In both cases, an approximation is
derived for one of the medium parameters involving the reflection
coefficient.</p>https://resolver.caltech.edu/CaltechTHESIS:03272013-095340267Elastostatic interaction of cracks in the infinite plane
https://resolver.caltech.edu/CaltechTHESIS:09282010-083924490
Year: 1972
DOI: 10.7907/BVJ3-BV94
The stress boundary value problem of an infinite, planar region with embedded rectilinear cracks is investigated from the viewpoint of two-dimensional, static, linear elasticity theory (plane strain or generalized stress). Any finite number of cracks may be considered. Their orientation may be arbitrary, so long as they do not intersect. Boundary loadings may take the form of quite general in-plane tractions along the crack surfaces, together with a bounded in-plane stress field at infinity.
Using Muskhelishvili’s solution for colinear cracks, the problem is reduced to a set of one-dimensional Fredholm integral equations. A simple numerical technique is presented for the approximate solution of these equations. The method is established to possess an extremely high rate of convergence.
Results are presented for a number of two-crack interaction problems. As expected, the interaction of the cracks generally tends to reduce the fracture strength of a material, relative to the strength that would exist with either crack acting independently. However, for certain orientations, it is found that the interaction phenomenon can actually increase the resistance to fracture.https://resolver.caltech.edu/CaltechTHESIS:09282010-083924490A continuum model for phase transformation in thermoelastic solids
https://resolver.caltech.edu/CaltechETD:etd-02232007-155324
Year: 1990
DOI: 10.7907/x4hj-2v63
Under suitable programs of mechanical or thermal loading, many solid materials are capable of undergoing phase transformations from one crystal structure to another. The austenite-martensite transformation that occurs in a variety of metallic alloys, including the so-called shape-memory materials, provides an example. The present paper represents an effort to model coupled thermo-mechanical effects in the macroscopic response of solids that arise from the occurrence of phase transformations. A Helmholtz free energy potential is constructed to describe the thermo-mechanical response of the hypothetical material to be considered here. As a function of strain, the potential is non-convex in a certain range of temperature; this feature is essential for the modeling of phase transformations. Apart from some general preliminary considerations pertaining to finite thermoelasticity, the analysis is carried out in the context of a simple problem, idealized from an experiment, in which an annular cylinder is deformed to a state of radially symmetric, finite anti-plane shear in the presence of differing inner and outer surface temperatures. After constructing all radially symmetric weak solutions involving at most a single surface of discontinuity of strain or temperature gradient, we determine the implications for quasi-static motions of the second law of thermodynamics. The thermo-mechanical phase transformation-induced hysteresis, residual deformation and stress relaxation effects exhibited by this model are discussed and the results concerning creep rate as predicted by the present model are in qualitative agreement with the laboratory observation. Finally, the shape-memory effect as predicted by the present thermo-mechanical model in the setting of finite anti-plane shear is illustrated.https://resolver.caltech.edu/CaltechETD:etd-02232007-155324Thermal Neutron Distributions near Material Discontinuities
https://resolver.caltech.edu/CaltechETD:etd-09132002-091916
Year: 1964
DOI: 10.7907/GSNH-HP15
A method is presented for the approximate calculation of the neutron flux near plane interfaces between different heavy monatomic gaseous media with absorption cross sections inversely proportional to the neutron velocity. Approximate analytic results are obtained for both the diffusion theory and transport theory models. It is found that the flux on each side of the interface can be approximated by the sum of two terms. One term has the same energy dependence that would exist in an infinite medium composed of the heavy monatomic gas that is on that side of the interface. The spatial dependence of this term is determined by diffusion theory. The other term, called a boundary layer correction, makes an appreciable contribution to the flux only near the interface. The procedure presented develops equations and boundary conditions which determine the different terms of the approximate flux. It is found that the approximate flux at the interface, for both diffusion and transport theory, is the average of the two infinite medium fluxes.https://resolver.caltech.edu/CaltechETD:etd-09132002-091916Applications of an Edge-and Corner Layer Technique to Elastic Plates and Shells
https://resolver.caltech.edu/CaltechTHESIS:08192011-135729811
Year: 1962
DOI: 10.7907/W4AN-YY85
This paper contains several problems that can be formulated
mathematically as two-dimensional boundary value problems for partial differential equations containing a parameter. A method is given which leads directly to asymptotic solutions for large values of the parameter without resorting to the exact solutions. The examples discussed involve linear differential equations and are drawn primarily from various problems in the theory of elasticity.
The method involves consideration of what are termed corner-layers in addition to the well known boundary-layers. The need for considering these corner-layers arises from the fact that the problems treated lead to boundary-layer differential equations which contain derivatives, not only with respect to the boundary-layer variable, but also with respect to the remaining independent variable. Thus, the
solution of such boundary-layer equations requires knowledge of boundary conditions in addition to those needed in standard boundary-layer problems.
The applications include: a heat conduction problem, two problems with transverse bending of stretched plates, and two problems from elastic shell theory.
The shell problems concern the bending of both the shallow and the non-shallow helicoidal shell. It is found that these shells have boundary-layers whose characteristic length is proportional to the one-third power of the thickness parameter. This may be contrasted with shells
of revolution, where this characteristic length is proportional to the one-half power of the thickness parameter.
https://resolver.caltech.edu/CaltechTHESIS:08192011-135729811Bending of Thin Elastic Plates Containing Line Discontinuities
https://resolver.caltech.edu/CaltechTHESIS:08182011-105727536
Year: 1962
DOI: 10.7907/9WFR-W284
The purpose of this work is to examine the stress distribution caused by the bending of a thin elastic plate containing a line discontinuity. Specifically, the plate under consideration is of constant thickness and occupies a whole plane exterior to the line discontinuity. The line discontinuity is either a crack or a rigid inclusion.
The loading is applied to the plate at infinity by certain combinations of tractions which leave the plate in equilibirum.
The analysis of the problems considered here is based on
an approximate theory which is more refined than the classical theory ordinarily applied to problems of bending of plates. This is because results based on the classical theory may be incorrect, even in first approximation for thin plates, near a boundary, and it is precisely the region near a boundary (in this case, the line
discontinuity) which is of primary interest in these problems. In fact one of the principal objectives in this work is to compare the stress distributions near the line discontinuity as predicted by the two theories.
The principal techniques used in this work are based on integral equations and the calculus of variations.
Results based on the two theories are found to agree for
thin plates away from the line discontinuity, but differ significantly in the vicinity of the discontinuity, even for very thin plates.
https://resolver.caltech.edu/CaltechTHESIS:08182011-105727536Scattering of a Rayleigh Wave by the Edge of a Thin Surface Layer
https://resolver.caltech.edu/CaltechTHESIS:06062019-152855123
Year: 1975
DOI: 10.7907/TZ1Z-9M12
<p>This investigation treats the problem of the scattering of a Rayleigh wave by the edge of a thin layer which covers half the surface of an elastic half-space. The interaction between the layer and the half-space is described approximately by means of a model in which the effect of the layer is represented by a pair of boundary conditions at the surface of the half-space. Two parameters- one representing mass and the other, stiffness- are found to characterize the layer. The incident Rayleigh wave impinges normally upon the plated region from the unplated side.</p>
<p>In the case where the mass of the layer vanishes, the problem is solved exactly using Fourier transforms and the Wiener-Hop£ technique, and numerical results are obtained for the amplitudes of the reflected and transmitted surface waves. In the more general case of a layer possessing both mass and stiffness, a perturbation procedure leads to a sequence of problems, each of which may be solved using Fourier transforms. The zeroth- and first-order problems are solved and the resulting approximate reflection and transmission coefficients are evaluated numerically for various ratios of layer mass to stiffness.</p>
https://resolver.caltech.edu/CaltechTHESIS:06062019-152855123Load-Absorption and Interaction of Two Filaments in a Fiber-Reinforced Material
https://resolver.caltech.edu/CaltechTHESIS:02062018-115639478
Year: 1973
DOI: 10.7907/GDSK-RJ45
This investigation is concerned with the interaction - as far
as load-absorption is concerned - of a pair of identical and parallel elastic
filaments in a fiber-reinforced composite material. The filaments are
assumed to have uniform circular cross-sections, are taken to be
semi-infinite, and are supposed to be continuously bonded to an all
around infinite matrix of distinct elastic properties. At infinity the
matrix is subjected to uniaxial tension parallel to the filaments. Two
separate but related problems are treated. In the first both filaments
extend to infinity in the same direction and their terminal cross-sections
are coplanar. In the second problem the filaments extend to infinity in opposite
directions and their terminal cross-sections need no longer be coplanar,
the two filaments being permitted to overlap partly. An approximate scheme
based in part on three-dimensional linear elasticity and developed originally
by Muki and Sternberg is employed in the analysis. The problems are ultimately
reduced to Fredholm integral equations which characterize the distribution
of the axial filament force. The integral equations are analyzed asymptotically and
numerically. Results are presented which show the variation of
filament force with position and the effect on this variation of various relevant
geometrical and material properties.
https://resolver.caltech.edu/CaltechTHESIS:02062018-115639478On the Existence and Uniqueness of the Solution to the Small-Scale Nonlinear Anti-Plane Shear Crack Problem in Finite Elastostatics
https://resolver.caltech.edu/CaltechETD:etd-03212008-094413
Year: 1985
DOI: 10.7907/dfcw-j110
<p>This thesis addresses the issue of existence and uniqueness of the solution to the small-scale nonlinear anti-plane shear crack problem in finite elastostatics. The hodograph transformation, commonly used in the theory of compressible fluid flows, plays an essential role. Existence is established by exhibiting an exact closed form solution, constructed via the hodograph transformation. Uniqueness is established by first proving the uniqueness of the solution to a related boundary-value problem, which is linear by virtue of the hodograph transformation, and then examining the implications of this result on the original problem. The possibility of making some of the conditions imposed on the solution to the small-scale nonlinear crack problem less restrictive is then investigated. This leads to several further results, including estimates of the nonvanishing shear stress component of the stress tensor along the crack faces.</p>https://resolver.caltech.edu/CaltechETD:etd-03212008-094413Dynamic and Spectral Features of Semiconductor Lasers
https://resolver.caltech.edu/CaltechETD:etd-02082005-114440
Year: 1985
DOI: 10.7907/H7JA-K512
<p>This thesis is divided into two main subject areas: the fluctuation properties of state of the art semiconductor lasers and the improvement of modulation and fluctuation properties in these devices through a technique called detuned loading.</p>
<p>The discussion of fluctuations in lasers is a topic as old as the device itself, and much of the pioneering work in this field was done in the sixties. Surprisingly, however, several new chapters in this field are being written, because of certain pecularities only recently observed in semiconductor lasers. Chapters 2 and 3 of this thesis will consider these pecularities, which, as it turns out, are quite important in many potential system applications of these devices.</p>
<p>One of the driving forces behind the development of semiconductor lasers has been their application as sources and local oscillators in optical communication systems. In general, such applications require lasers which have low phase and intensity noise, and which can be modulated at high data rates. As is often the case, these requirements are to a certain extent mutually exclusive. Chapter 4 introduces a technique which is an exception to this rule. It relies upon the semiconductor laser physics which produces the fluctuation abnormalities discussed in Chapters 2 and 3. The technique can be used to improve modulation speed while simultaneously reducing noise as compared to the conventional device.</p>https://resolver.caltech.edu/CaltechETD:etd-02082005-114440Dynamics of phase transformations in thermoelastic solids
https://resolver.caltech.edu/CaltechETD:etd-02042008-151022
Year: 1997
DOI: 10.7907/gk0c-8r75
The dynamical aspects of solid-solid phase transformations are studied within the framework of the theory of thermoelasticity. The main purpose is to analyze the role of temperature in the theory of phase transitions. This investigation consists of two parts: first, it is shown that by imposing a kinetic relation and a nucleation criterion it is possible to single out a unique solution to the Riemann problem for an adiabatic process. This extends to the thermomechanical case results previously found in a purely mechanical context. Secondly, based on an admissibility criterion for traveling wave solutions within the context of an augmented theory that includes viscosity, strain gradient and heat conduction effects, a special kinetic relation is derived using singular perturbation techniques.
https://resolver.caltech.edu/CaltechETD:etd-02042008-151022Continuum dynamics of solid-solid phase transitions
https://resolver.caltech.edu/CaltechETD:etd-10222007-135103
Year: 1995
DOI: 10.7907/PW4M-9B73
<p>This work focuses on the applications in dynamics of recently developed continuum-mechanical models of solid-solid phase transitions. The dynamical problems considered here involve only one space coordinate, and attention is limited to hyperelastic materials that involve two phases. This investigation has two purposes. The first is to determine the predictions of the models in complicated situations. Secondly, the present study attempts to develop analytical and numerical approaches to problems that may be relevant to the interpretation and understanding of experiments involving phase transitions under dynamical conditions.</p>
<p>The first problem studied involves the study of a semi-infinite bar initially in an equilibrium state that involves two material phases separated by a phase boundary at a given location. The end of the bar is suddenly subject to a constant impact velocity that persists for a finite time and is then removed. Interaction between the phase boundary and the elastic waves generated by the impact and subsequent reflections are studied in detail, and the trajectory of the phase boundary is determined exactly. The second task addressed involves the development of a Riemann solver to be applied to the numerical solution of Riemann problems for two-phase elastic materials. Riemann problems for such materials involve complications not present in the corresponding problems that arise, for example, in classical gas dynamics. Finally, a finite-difference method of Godunov type is developed for the numerical treatment of boundary-initial-value problems arising in the model of Abeyaratne and Knowles. The method is applied to specific problems.</p>https://resolver.caltech.edu/CaltechETD:etd-10222007-135103Variational Principles and Applications in Finite Elastic Strain Theory
https://resolver.caltech.edu/CaltechETD:etd-09272002-145537
Year: 1964
DOI: 10.7907/PK58-HQ71
The variational principles of finite elastostatic strain theory are presented in a unified manner for both compressible and incompressible bodies. Whereas the principle of stationary potential energy, a restricted case of the general principle of Hu and Washizu, is valid for any elastic deformation, it is found that the principle of stationary complementary energy is valid only for infinitesimal elastic strains. Consequently, Reissner's Theorem is the appropriate stationary principle to use in finite elastic strain theory when the complementary strain energy density is to be the argument function.
The potential energy principle is applied to several problems dealing with the finite straining of a neo-Hookean material. All but one of these problems are concerned with plane strain deformations; the one other problem, in a spherical geometry, involves an unusual stability question. Approximate solutions are obtained for some mixed boundary value problems which are not amenable to the semi-inverse methods of solution frequently used in finite elastic strain theory.
Another plane strain problem, requiring more detailed stress information than can be obtained from the potential energy principle, is studied approximately by means of Reissner's Theorem.https://resolver.caltech.edu/CaltechETD:etd-09272002-145537Propagation of Finite Amplitude Waves in Elastic Solids
https://resolver.caltech.edu/CaltechETD:etd-10032002-104227
Year: 1965
DOI: 10.7907/Q1QQ-VG41
This thesis is devoted to consideration of finite amplitude waves propagating into an elastic half-space in a direction normal to the boundary. Excitation is by means of strains applied at the boundary as step functions of time.
The solutions obtained are combinations of centered simple waves and shock waves. Longitudinal waves may appear alone but waves with transverse displacement components are always accompanied by longitudinal waves. The foregoing solutions are discussed in general and are illustrated by an example problem involving a special nonlinear, compressible, hyperelastic material. A perturbation method, based on the use of characteristic coordinates, which facilitates approximate solution of the problem for arbitrarily prescribed strain boundary conditions is described.https://resolver.caltech.edu/CaltechETD:etd-10032002-104227On the Dynamic Behavior of Thin Elastic Plates
https://resolver.caltech.edu/CaltechETD:etd-10022002-113310
Year: 1968
DOI: 10.7907/2095-RQ32
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
Two wave propagation problems are considered: the propagation of acoustic waves in a fluid slab and the propagation of elastic waves in an elastic slab.
When formulated in terms of nondimensional variables these problems depend explicitly on two small parameters [epsilon] and [delta]. The parameter [epsilon] provides a measure of the thinness of the slabs considered and the parameter [delta] measures the impulsiveness of the applied excitation or loading. Approximation solutions of the problems considered are obtained consisting of several parts, each part having the form of a power series expansion in the parameters [epsilon] and [delta].
The most important result obtained is the development of the approximate theories - the plate wave equation and the Euler-Bernoulli plate equation - directly from the full equations of dynamic elasticity using a rational perturbation expansion technique.https://resolver.caltech.edu/CaltechETD:etd-10022002-113310Two Problems in Plane Finite Elastostatics
https://resolver.caltech.edu/CaltechETD:etd-09182006-085128
Year: 1983
DOI: 10.7907/yhjy-mb48
<p>In this paper the fully nonlinear equilibrium theory of homogeneous and isotropic incompressible elastic solids is used to study the elastostatic fields in plane strain near the point of application of a concentrated force on a deformed half plane and near the vertex of a circular sector whose plane deformed faces are subjected to prescribed tractions.</p>
<p>In the concentrated force problem, restricting only the form of the elastic potential at large extensional deformations, it is shown that, for materials which "harden" in simple shear, the displacement is bounded at the point of application of the load. This is not the case for materials which "soften" in shear. Estimates of the true stress tensor near the singular point are given.</p>
<p>In the sector problem, for a class of the materials mentioned, the deformation and stress field near the vertex of the deformed cross-section are derived and discussed.</p>
https://resolver.caltech.edu/CaltechETD:etd-09182006-085128The conservation laws of three dimensional linearized elasticity theory
https://resolver.caltech.edu/CaltechETD:etd-10122005-085657
Year: 1974
DOI: 10.7907/TC8V-H381
For linearized isotropic elastodynamics and elastostatics, Noether's theorem on invariant variational principles is used to obtain all conservation laws arising from a reasonably general group of infinitesimal transformations. A theorem regarding the completeness of the derived laws is proved, and the conservation laws are then used to derive the wave speed equation for the Rayleigh problem on the surface of an anisotropic half space. An example of additional laws following from the same group but from a more general version of Noether's theorem is given in an appendix devoted to a discussion of limitations on the completeness theorem.https://resolver.caltech.edu/CaltechETD:etd-10122005-085657The Emergence and Propagation of a Phase Boundary in an Elastic Bar
https://resolver.caltech.edu/CaltechETD:etd-09052006-154543
Year: 1983
DOI: 10.7907/85D8-5A19
<p>This dissertation is concerned with the dynamical analysis of an elastic bar whose stress-strain relation is not monotonic. Sufficiently large applied loads then require the strain to jump from one ascending branch of the stress-strain curve to another such branch. For a special class of these materials, a nonlinear initial-boundary value problem in one-dimensional elasticity is considered for a semi-infinite bar whose end is subjected to either a monotonically increasing prescribed traction or a monotonically increasing prescribed displacement. If the stress at the end of the bar exceeds the value of the stress at any local maximum of the stress-strain curve a strain discontinuity or "phase boundary" emerges at the end of the bar and subsequently propagates into the interior. For classically smooth solutions away from the phase boundary, the problem is reducible to a pair of differential-delay equations for two unknown functions of a single variable. The first of these two functions gives the location of the phase boundary, while the second characterizes the dynamical fields in the high-strain phase of the material. In these equations the former function occurs in the argument of the latter, so that the delays in the functional equations are unknown. A short-time analysis of this system provides an asymptotic description of the emergence and initial propagation of the phase boundary. For large-times, a different analysis indicates that the phase boundary velocity approaches a constant which depends on material properties and on the ultimate level reached by the applied load as well. Higher order corrections depend on the detailed way in which the load is applied. Estimates for the various dynamical field quantities are given and a priori conditions are deduced which determine whether the phase boundary eventually becomes the leading disturbance.</p>https://resolver.caltech.edu/CaltechETD:etd-09052006-154543A General Solution Strategy for Large Scale Static and Dynamic Nonlinear Finite Element Problems Employing the Element-by-Element Factorization Concept
https://resolver.caltech.edu/CaltechETD:etd-09012006-080048
Year: 1983
DOI: 10.7907/jqbq-rc43
<p>It is proposed to solve large-scale finite-element equation systems arising in structural and solid mechanics by way of an element-by-element approximate factorization technique which obviates the need for a global coefficient matrix. The procedure has considerable operation count and I/O advantages over direct elimination schemes and it is easily implemented. Numerical results demonstrate the effectiveness of the method and suggest its potential for the analysis of large-scale systems.</p>https://resolver.caltech.edu/CaltechETD:etd-09012006-080048On Two-Dimensional Waves of Finite Amplitude in Elastic Materials of Harmonic Type
https://resolver.caltech.edu/CaltechTHESIS:09272017-154819290
Year: 1971
DOI: 10.7907/C2ZB-4K70
<p>In this thesis, two-dimensional waves of finite amplitude in
elastic materials of harmonic type are considered. After specializing
the basic equations of finite elasticity to these materials,
attention is restricted to plane motions and a new representation
theorem (analogous to the theorem of Lamé in classical linear
elasticity) for the displacements in terms of two potentials is derived.</p>
<p>The two-dimensional problem of the reflection of an obliquely
incident periodic wave from the free surface of a half-space composed
of an elastic material of harmonic type is formulated. The incident
wave is a member of a special class of exact one-dimensional solutions
of the nonlinear equations for elastic materials of harmonic
type, and reduces upon linearization to the classical periodic "shear
wave" of the linear theory.</p>
<p>A perturbation procedure for the construction of an approximate
solution of the reflection problem, for the case where the
incident wave is of small but finite amplitude, is constructed. The
procedure involves series expansions in powers of the ration of the amplitude
to the wavelength of the incident wave and is of the so-called
two-variable type. The perturbation scheme is carried far
enough to determine the second-order corrections to the linearized
theory.</p>
<p>A summary of results for the reflection problem is provided,
in which nonlinear effects on the reflection pattern, on the particle
displacements at the free surface and on the behavior at large depth
in the half-space are detailed.</p>https://resolver.caltech.edu/CaltechTHESIS:09272017-154819290Ellipticity and Deformations with Discontinous Gradients in Finite Elastostatics
https://resolver.caltech.edu/CaltechETD:etd-02122007-094508
Year: 1989
DOI: 10.7907/jn2t-m109
<p>Loss of ellipticity of the equilibrium equations of finite elastostatics is closely related to the possible emergence of elastostatic shocks, i.e., deformations with discontinuous gradients. In certain situations where constitutive response functions are essentially one-dimentional, such as anti-plane shear or bar theories, strong ellipticity is closely related to convexity of the elastic potential and invertibility of certain constitutive response functions.</p>
<p>The present work addresses the analogous issues within the context of three dimensional elastostatics of compressible but not necessarily isotropic hyperelastic materials. A certain direction-dependent resolution of the deformation gradient is introduced and its existence and uniqueness for a given direction are established. The elastic potential is expressed as a function of kinematic variables arising from this resolution. Strong ellipticity is shown to be equivalent to the positive definiteness of the Hessian matrix of this function, thus sufficing for its strict convexity. The underlying variables are interpretable physically as simple shears and extensions. Their work-conjugates define a traction response mapping. It is shown that discontinuous deformation gradients are sustainable if and only if this mapping fails to be invertible. This result is explicit, in the sense that it characterizes the set of all possible piecewise homogeneous deformations given the elastic potential function.</p>https://resolver.caltech.edu/CaltechETD:etd-02122007-094508Continuum modeling of materials that can undergo martensitic phase transformations
https://resolver.caltech.edu/CaltechETD:etd-10192005-160834
Year: 1993
DOI: 10.7907/YB1M-0R23
A continuum model for materials that can undergo martensitic phase transformations is developed and applied to the study of several problems that involve such transformations. One of the several advantages of using this continuum model is that the corresponding boundary value problem is in a form that permits direct linearization, while retaining finite shape deformations for the martensite phases. The continuum model is used to study several problems dealing with which variant of martensite is preferred during the application of a loading. Among these problems is the case of a uniaxial tensile traction applied to a two-phase cylindrical body, and the case of a hydrostatic pressure applied to a two-phase body that has a finite shape deformation with an infinitesimal dilatation. The results that are obtained correspond with those that have been observed from experiments and with those that might be expected from physical considerations. The next problem that is considered involves the temperature at the interface and quasi-static motions of a two-phase thermoelastic bar. The bar is subject to different temperatures at each boundary and to a mechanical end-loading. The last problem that is considered involves the longitudinal free vibrations of a fixed-free, two-phase bar. The main focus in this problem is the damping behavior of the two-phase bar that is due to the motions of the interface during the free vibrations. A finite-difference numerical routine is used to approximate the displacement solutions for this problem. The damping of the bar is studied as the material coefficients are varied, and the values of the material coefficients that produce the maximum damping are investigated.https://resolver.caltech.edu/CaltechETD:etd-10192005-160834Energy inequalities and error estimates for axisymmetric torsion of thin elastic shells of revolution
https://resolver.caltech.edu/CaltechTHESIS:12072015-103241369
Year: 1968
DOI: 10.7907/ARTK-6968
<p>The problem motivating this investigation is that of pure axisymmetric torsion of an elastic shell of revolution. The analysis is carried out within the framework of the three-dimensional linear theory of elastic equilibrium for homogeneous, isotropic solids. The objective is the rigorous estimation of errors involved in the use of approximations based on thin shell theory.</p>
<p>The underlying boundary value problem is one of Neumann type for a second order elliptic operator. A systematic procedure for constructing pointwise estimates for the solution and its first derivatives is given for a general class of second-order elliptic boundary-value problems which includes the torsion problem as a special case.</p>
<p>The method used here rests on the construction of “energy inequalities” and on the subsequent deduction of pointwise estimates from the energy inequalities. This method removes certain drawbacks characteristic of pointwise estimates derived in some investigations of related areas.</p>
<p>Special interest is directed towards thin shells of constant thickness. The method enables us to estimate the error involved in a stress analysis in which the exact solution is replaced by an approximate one, and thus provides us with a means of assessing the quality of approximate solutions for axisymmetric torsion of thin shells. </p>
<p>Finally, the results of the present study are applied to the stress analysis of a circular cylindrical shell, and the quality of stress estimates derived here and those from a previous related publication are discussed. </p>
https://resolver.caltech.edu/CaltechTHESIS:12072015-103241369The Propagation and Arrest of an Edge Crack in an Elastic Half-Space Under Conditions of Anti-Plane Shear: Analytical and Numerical Results
https://resolver.caltech.edu/CaltechETD:etd-09052006-082841
Year: 1983
DOI: 10.7907/jb3j-5460
<p>The motion of an edge crack extending non-uniformly in an elastic half-space under conditions of anti-plane shear is analyzed. An expression for the stress intensity factor at the crack tip is obtained, and an energy balance crack propagation criterion is used to find the equation of motion of the tip. On solving this equation numerically, it is found that crack arrest occurs before the second reflected wave from the boundary reaches the tip.</p>
<p>In the second half of this investigation, a numerical procedure for studying anti-plane shear crack propagation problems using finite differences is developed. To approximate the elastodynamic field as accurately as possible near the moving crack tip, where singular stresses occur, the local asymptotic displacement field near the tip is incorporated into the finite difference scheme. The numerical procedure is applied to the edge crack problem analyzed in the first part of this study, and the numerical and exact results are compared.</p>https://resolver.caltech.edu/CaltechETD:etd-09052006-082841Finite Plane and Anti-Plane Elastostatic Fields with Discontinuous Deformation Gradients Near the Tip of a Crack
https://resolver.caltech.edu/CaltechETD:etd-09122006-153033
Year: 1982
DOI: 10.7907/td0f-kr59
<p>In this paper the fully nonlinear theory of finite deformations of an elastic solid is used to study the elastostatic field near the tip of a crack. The special elastic materials considered are such that the differential equations governing the equilibrium fields may lose ellipticity in the presence of sufficiently severe strains.</p>
<p>The first problem considered involves finite anti-plane shear (Mode III) deformations of a cracked incompressible solid. The analysis is based on a direct asymptotic method, in contrast to earlier approaches which have depended on hodograph procedures.</p>
<p>The second problem treated is that of plane strain of a compressible solid containing a crack under tensile (Mode I) loading conditions. The material is characterized by the so-called Blatz-Ko elastic potential. Again, the analysis involves only direct local considerations.</p>
<p>For both the Mode III and Mode I problems, the loss of equilibrium ellipticity results in the appearance of curves ("elastostatic shocks") issuing from the crack-tip across which displacement gradients and stresses are discontinuous.</p>
https://resolver.caltech.edu/CaltechETD:etd-09122006-153033Impact-Induced Phase Transformations in Elastic Solids: A Continuum Study Including Numerical Simulations for GeO₂
https://resolver.caltech.edu/CaltechETD:etd-02262008-153435
Year: 1999
DOI: 10.7907/4dhf-fj83
<p>This thesis applies recently developed continuum theories of diffusionless phase transformations in solids to the study of impact problems involving materials which can experience such phase changes. Our objective is to compare the theoretical predictions against certain experimental results.</p>
<p>In the experiments of interest, a face-to-face impact occurs between a disk of amorphous germanium dioxide and another material, either tungsten or an aluminum alloy. The GeO₂ is believed to transform to another phase if sufficient compressive stress is achieved.</p>
<p>We model these experiments using one-dimensional finite elasticity. Phase-changing materials are represented by non-convex potential energy functions. This can produce phase boundaries that propagate <i>subsonically</i> or <i>supersonically</i> with respect to the slower longitudinal wave speed of the two phases. When a subsonic phase boundary is possible, it is not uniquely determined by the fundamental field equations and jump conditions. Uniqueness is obtained by invoking a <i>nucleation criterion</i> to control the initiation of the new phase, and a <i>kinetic relation</i> to govern its evolution.</p>
<p>The experiments considered here are sufficiently long in duration (≈ 3 µs) that several reflections and wave interactions occur, and the analysis becomes analytically intractable. Accordingly, a finite-difference method of Godunov type is employed to analyze these experiments numerically. Methods of Godunov type treat adjoining discretized spatial elements as the two sides of a Riemann problem, which is typically solved <i>approximately</i> by linearizing around the initial conditions on each side. Fortuitously, all constitutive models employed in this thesis are such that the required Riemann problems can be solved <i>exactly</i> without too much effort.</p>
<p>Simulations utilizing the numerical method demonstrate that the impact response of a material is sensitive to the kinetic relation that enters the model. It appears the theory may offer a plausible description of the experiments, though the restrictions placed on the constitutive models herein seem too severe to provide a good quantitative match to the experimental results.</p>https://resolver.caltech.edu/CaltechETD:etd-02262008-153435Aspects of the morphological character and stability of two-phase states in non-elliptic solids
https://resolver.caltech.edu/CaltechETD:etd-01302007-160351
Year: 1991
DOI: 10.7907/WJKM-GB39
Part I. This work focuses on the construction of equilibrated two-phase antiplane shear deformations of a non-elliptic isotropic and incompressible hyperelastic material. It is shown that this material can sustain metastable two-phase equilibria which are neither piecewise homogeneous nor axisymmetric, but, rather, involve non-planar interfaces which completely segregate inhomogeneously deformed material in distinct elliptic phases. These results are obtained by studying a constrained boundary value problem involving an interface across which the deformation gradient jumps. The boundary value problem is recast as an integral equation and conditions on the interface sufficient to guarantee the existence of a solution to this equation are obtained. The contraints, which enforce the segregation of material in the two elliptic phases, are then studied. Sufficient conditions for their satisfaction are also secured. These involve additional restrictions on the interface across which the deformation gradient jumps-which, with all restrictions satisfied, constitutes a phase boundary. An uncountably infinite number of such phase boundaries are shown to exist. It is demonstrated that, for each of these, there exists a solution - unique up to an additive constant - for the constrained boundary value problem. As an illustration, approximate solutions which correspond to a particular class of phase boundaries are then constructed. Finally, the kinetics and stability of an arbitrary element within this class of phase boundaries are analyzed in the context of a quasistatic motion.
Part II. This work investigates the linear stability of an antiplane shear motion which involves a planar phase boundary in an arbitrary element of a wide class of non-elliptic generalized neo-Hookean materials which have two distinct elliptic phases. It is shown, via a normal mode analysis, that, in the absence of inertial effects, such a process is linearly unstable with respect to a large class of disturbances if and only if the kinetic response function - a constitutively supplied entity which gives the normal velocity of a phase boundary in terms of the driving traction which acts on it or vice versa - is locally decreasing as a function of the appropriate argument. An alternate analysis, in which the linear stability problem is recast as a functional equation for the interface position, allows the interface to be tracked subsequent to perturbation. A particular choice of the initial disturbance is used to show that, in the case of an unstable response, the morphological character of the phase boundary evolves to qualitatively resemble the plate-like structures which are found in displacive solid-solid phase transformations. In the presence of inertial effects a combination of normal mode and energy analyses are used to show that the condition which is necessary and sufficient for instability with respect to the relevant class of perturbations in the absence of inertia remains necessary for the entire class of perturbations and sufficient for all but a very special, and physically unrealistic, subclass of these perturbations. The linear stability of the relevant process depends, therefore, entirely upon the transformation kinetics intrinsic to the kinetic response function.
Part III. This investigation is directed toward understanding the role of coupled mechanical and thermal effects in the linear stability of an isothermal antiplane shear motion which involves a single planar phase boundary in a non-elliptic thermoelastic material which has multiple elliptic phases. When the relevant process is static - so that the phase boundary does not move prior to the imposition of the disturbance - it is shown to be linearly stable. However, when the process involves a steadily propagating phase boundary it may be linearly unstable. Various conditions sufficient to guarantee the linear instability of the process are obtained. These conditions depend on the monotonicity of the kinetic response function - a constitutively supplied entity which relates the driving traction acting on a phase boundary to the local absolute temperature and the normal velocity of the phase boundary - and, in certain cases, on the spectrum of wave-numbers associated with the perturbation to which the process is subjected. Inertia is found to play an insignificant role in the qualitative features of the aforementioned sufficient conditions. It is shown, in particular, that instability can arise even when the normal velocity of the phase boundary is an increasing function of the driving traction if the temperature dependence in the kinetic response function is of a suitable nature. Significantly, the instability which is present in this setting occurs only in the long waves of the Fourier decomposition of the moving phase boundary, implying that the interface prefers to be highly wrinkled.https://resolver.caltech.edu/CaltechETD:etd-01302007-160351Element-by-Element Solution Procedures for Nonlinear Transient Heat Conduction Analysis
https://resolver.caltech.edu/CaltechETD:etd-01252007-132425
Year: 1984
DOI: 10.7907/G7VB-EV65
<p>Despite continuing advancements in computer technology, there are many problems of engineering interest that exceed the combined capabilities of today's numerical algorithms and computational hardware. The resources required by traditional finite element algorithms tend to grow geometrically as the "problem size" is increased. Thus, for the forseeable future, there will be problems of interest which cannot be adequately modeled using currently available algorithms. For this reason, we have undertaken the development of algorithms whose resource needs grow only linearly with problem size. In addition, these new algorithms will fully exploit the "parallel-processing" capability available in the new generation of multi-processor computers.</p>
<p>The approach taken in the element-by-element solution procedures is to approximate the global implicit operator by a product of lower order operators. This type of "product" approximation originated with ADI techniques and was further refined into the "method of fractional steps." The current effort involves the use of a more natural operator split for finite element analysis based on "element operators." This choice of operator splitting based on element operators has several advantages. First, it fits easily within the architecture of current FE programs. Second, it allows the development of "parallel" algorithms. Finally, the computational expense varies only linearly with the number of elements.</p>
<p>The particular problems considered arise from nonlinear transient heat conduction. The nonlinearity enters through both material temperature dependence and radiation boundary conditions. The latter condition typically introduces a "stiff" component in the resultant matrix ODE's which precludes the use of explicit solution techniques. Implicit solution techniques can be prohibitively expensive. Instead, the matrix equations are solved by combining a modified Newton-Raphson iteration scheme with an element-by-element preconditioned conjugate gradient subiteration procedure. The resultant procedure has proven to be both accurate and reliable in the solution of medium-size problems in this class.</p>
https://resolver.caltech.edu/CaltechETD:etd-01252007-132425Martensitic phase transitions with surface effects
https://resolver.caltech.edu/CaltechETD:etd-11052004-161432
Year: 1992
DOI: 10.7907/NBES-9M76
Continuum treatments of martensitic phase transformations are capable of accounting for a variety of important surface effects attributable to the spatially localized interaction of coexisting material phases. Such phenomena are thought to play a critical role in determining the size, shape, and stability of nucleated embryos as well as to affect the conditions under which nucleation events occur. These issues are examined within a purely mechanical context wherein the special properties are modeled as traction and energy fields defined on a two-dimensional abstraction of the interface region. Materials that undergo martensitic phase changes are modeled as having a hyperelastic character in both the bulk and interface. The characterization of such bodies is examined in detail and a representation theorem is presented for describing the interfaces of isotropic, hyperelastic media. A class of isotropic, nonlinearly hyperelastic bulk material is introduced that is capable of modeling the dilatative component of martensitic phase transformations. Such materials are considered within a noninertial setting referred to as The Cylinder Problem. This problem provides a means of exploring a variety of surface effects, and a criterion for nucleation based on energy is presented towards this end. Here nucleation events are modeled as deterministic, temporal shocks that are global in spatial extent. The fundamental development presented does more than capture the desired surface effects. It shows how they are related to specific assumptions regarding interface and bulk constitution. Four different interface characterizations are presented that serve to illustrate this.
https://resolver.caltech.edu/CaltechETD:etd-11052004-161432Time-dependent monoenergetic neutron transport in two adjacent semi-infinite media
https://resolver.caltech.edu/CaltechTHESIS:10022015-101429428
Year: 1966
DOI: 10.7907/00R6-9A13
<p>An exact solution to the monoenergetic Boltzmann equation is
obtained for the case of a plane isotropic burst of neutrons introduced
at the interface separating two adjacent, dissimilar, semi-infinite
media. The method of solution used is to remove the time dependence
by a Laplace transformation, solve the transformed equation by the
normal mode expansion method, and then invert to recover the time
dependence.</p>
<p>The general result is expressed as a sum of definite, multiple
integrals, one of which contains the uncollided wave of neutrons
originating at the source plane. It is possible to obtain a simplified
form for the solution at the interface, and certain numerical calculations
are made there.</p>
<p>The interface flux in two adjacent moderators is calculated and
plotted as a function of time for several moderator materials. For
each case it is found that the flux decay curve has an asymptotic slope
given accurately by diffusion theory. Furthermore, the interface current
is observed to change directions when the scattering and absorption
cross sections of the two moderator materials are related in a
certain manner. More specifically, the reflection process in two
adjacent moderators appears to depend initially on the scattering
properties and for long times on the absorption properties of the media.</p>
<p>This analysis contains both the single infinite and semi-infinite
medium problems as special cases. The results in these
two special cases provide a check on the accuracy of the general
solution since they agree with solutions of these problems obtained
by separate analyses.</p>
https://resolver.caltech.edu/CaltechTHESIS:10022015-101429428