Academic literature on the topic 'Linear elastic solids'

The word “viscoelastic” is used to describe the mechanical response of materials exhibiting both the springiness associated with elastic solids and viscous flow characteristics associated with fluids. A familiar example of a material called viscoelastic is Silly PuttyTM. If a blob of Silly Putty is rolled into a ball and then dropped onto a hard surface, it will bounce like an elastic ball. If the ball is placed on a hard surface, its own weight will cause it to flow into a puddle. This behavior indicates that time is an intrinsic parameter in discussing viscoelastic response of materials. The elastic response is associated with a contact force of very short duration. The flow into a puddle occurs when forces act for a long period of time.Viscoelastic response occurs in materials such as soils, concrete, cartilage, biological tissue, and polymers. Soils and cartilage can be thought of as porous solids filled with fluid. Viscous response is due to the flow of the fluid in the pores; elastic response is due to the distortion of the porous solid.

Solutions are obtained for the stress and pore pressure due to sudden introduction of plane strain dislocations in a linear elastic, fluid-infiltrated, Biot, solid. Previous solutions have required that the pore fluid pressure and its gradient be continuous. Consequently, the antisymmetry (symmetry) of the pore pressure p about y = 0 requires that this plane be permeable (p = 0) for a shear dislocation and impermeable (∂p/∂y = 0) for an opening dislocation. Here Fourier and Laplace transforms are used to obtain the stress and pore pressure due to sudden introduction of a shear dislocation on an impermeable plane and an opening dislocation on a permeable plane. The pore pressure is discontinuous on y = 0 for the shear dislocation and its gradient is discontinuous on y = 0 for the opening dislocation. The time-dependence of the traction induced on y = 0 is identical for shear and opening dislocations on an impermeable plane, but differs significantly from that for dislocations on a permeable plane. More specifically, the traction on an impermeable plane does not decay monotonically from its short-time (undrained) value as it does on a permeable plane; instead, it first increases to a peak in excess of the short-time value by about 20 percent of the difference between the short and long time values. Differences also occur in the distribution of stresses and pore pressure depending on whether the dislocations are emplaced on permeable or impermeable planes.

Clay, rocks, concrete and other composite solids show evidence of a hierarchical structure. A fractal tree of nested viscoelastic boxes is proposed to describe the elastic after-effects in these composite solids. A generalized fractal transmission line approach is developed to relate the strain and stress responses. Power law for strain, under an applied stress step, is derived. The exponent in the power law is obtained as a well-defined function of the branching numbers and scaling parameters of the viscoelastic hierarchy. Then, a composite solid with both instantaneous (linear) elastic strain response and power law type (linear) elastic after-effect for an applied stress step, is considered. The stretched exponential stress relaxation to an applied strain step is derived as an approximation. For the same composite solid, the stretch parameter of the stretched exponential and the exponent of the power law result to be equal to each other.

Finite-element analyses for elastoplastic cone indentations were conducted in which the effect of linear strain hardening on indentation behavior was intensively examined in relation to the influences of the frictional coefficient (μ) at the indenter/material contact interface and of the inclined face angle (β) of the cone indenter. A novel procedure of “graphical superposition” was proposed to determine the representative yield stress YR. It was confirmed that the concept of YR applied to elastic-perfectlyplastic solids is sufficient enough for describing the indentation behavior of strainhardening elastoplastic solids. The representative plastic strain of εR (plastic) ≈ 0.22 tan β, at which YR is prescribed, is applicable to the strain-hardening elastoplastic solids, affording a quantitative relationship of YR = Y + ε;R (plastic) × EP in terms of the strain-hardening modulus EP. The true hardness H as a measure for plasticity is estimated from the Meyer hardness HM and then successfully related to the yield stress Y as H = C(β,μ) × Y for elastic-perfectly-plastic solids and as H = C(β,μ) × YR for strain-hardening solids, by the use of a β- and μ-dependent constraint factor C(β,μ) ranging from 2.6 to 3.2.

A linear elastic no-tension material model is implemented in this contribution to cope with the analysis of masonry-like solids in case of either elastic or inelastic settlements. Instead of implementing an incremental non-linear approach, an energy-based method is adopted to address the elastic no-tension equilibrium. Under a prescribed set of compatible loads, and possible enforced displacements, a solution is found by distributing an equivalent orthotropic material having negligible stiffness in tension, such that the overall strain energy is minimized and the stress tensor is negative semi-definite all over the domain. A preliminary implementation of the proposed method is given by adopting a heuristic approach to turn the multi-constrained minimization problem into an unconstrained one. Numerical simulations focus on a wall with an opening subjected to either inelastic settlement or standing on elastic soil.

Expressions for the inner and bare components of the elastic constants of crystalline solids are derived. The inner elastic constants are complex functions of the force constants and vanish only for centrosymmetric solids. Using a linear-chain model, the force-constant dependence of inner, bare, and total elastic constants is studied. The linear-chain model is also utilized in derivation of composition-dependent elastic constant equations. Single-parameter and two-parameter theoretical calculations are compared with the experimental composition-dependent Young's moduli of a number of metal–metalloid glasses.

L’objectiu d’aquesta tesis és aplicar mètodes de teoria de camps per entendre la resposta elàstica no-lineal (ENL) dels sòlids. La resposta ENL conté un gran número de quantitats observables, que no sempre són fàcils de derivar de la composició microscòpica del material. Un actor esencial en la resposta elàstica dels sòlids són els fonons, que poden ser descrits com els bosons de Goldstone d’una ruptura espontània de les simetries de l’espai-temps. Com a tals, la seva dinàmica a baixes energies (incloent no-linealitats) pot ser capturada sistemàticament per mètodes estàndard de Teoria de Camps Efectiva (TCE). Això ofereix naturalment una manera nova de lidiar amb la fenomenologia ENL. Una conclusió principal es que, efectivament, aquests mètodes de baixes energies donen informació no trivial, com relacions entre differents observables ENL. Il·lustrem aquest fet obtenint límits en la màxima deformació que un material pot tolerar, que pot ser expressat en funció d’altres observables ENL. Un cas especial són els sòlids invariants d’escala (IE). Això inclou dos sub-casos diferents, ja que l’EI pot realitzar-se de manera manifesta o com una simetria trencada espontàniament. El primer cas correspon a un punt fix no trivial i requereix l’ús de metodes hologràfics (AdS/CFT). El segon cas pot ser descrit utilitzant mètodes TCE. Comparem els resultats obtinguts als dos casos i trobem que els límits difereixen significativament als dos sub-casos.
El objetivo de esta tesis es aplicar métodos de teoría de campos para entender la respuesta elástica no-lineal (ENL) de los sólidos. La respuesta ENL contiene un gran número de cantidades observables, que no siempre son fáciles de derivar de la composición microscópica del material. Un actor esencial en la respuesta elástica de los solidos son los fonones, que pueden ser descritos como bosones de Goldstone de una ruptura espontánea de las simetrías del espacio-tiempo. Como tales, su dinàmica a bajas energías (incluyendo no linealidades) puede ser capturado sistemáticamente con métodos estándar de Teoría de Campos Efectiva (TCE) a bajas energías. Esto ofrece naturalmente una manera nueva de tratar la fenomenología ENL. Una conclusión principal es que, efectivamente, los métodos de baja energía TCE ofrecen información no trivial, como relaciones entre diferentes observables ENL. Ilustramos esto obteniendo límites en la máxima deformation que un material puede tolerar, lo cual puede ser expresado en función de otros observables ENL. Un caso de especial interés son los sólidos invariantes de escala (IE). Esto incluye dos sub-casos distintos, puesto que la IE puede ser realizada de manera manifiesta o como una simetría rota espontáneamente. El primer caso corresponde a un punto fijo no trivial y requiere el uso de métodos holográficos (AdS/CFT). El segundo caso puede ser descrito con métodos TCE estándar. Comparamos los resultados obtenidos en ambos casos y encontramos que los límites elásticos difieren significativamente en los dos sub-casos.
The goal of this thesis is to apply modern field theory methods to understand the nonlinear elastic (NLE) response of solids. The NLE response contains a large number of low-energy observable quantities, not always easy to derive from the microscopic composition of the material. An essential actor in the elastic response are the phonons, which can be described as the Goldstone bosons of the spontaneously broken spacetime symmetries. As such, their low energy dynamics (including non-linearities) can be captured systematically by standard low energy Effective Field Theory (EFT) methods. This offers naturally a novel approach to tackle NLE phenomenology. One main conclusion is that indeed the low energy effective methods can provide non-trivial information, as relations among various different NLE observables. We illustrate this by obtaining bounds on the maximum deformation that a material can tolerate, which can be expressed in function of other NLE observables. A case of special interest is that of scale invariant (SI) solids. This includes two distinct sub-cases, since SI can be realized either as a manifest symmetry or a spontaneously broken symmetry. The former case corresponds to a nontrivial fixed point and requires the use of holographic (AdS/CFT) techniques. The latter case instead can be described with more standard EFT methods. We compare the results obtained in the two cases, and find that the obtained elasticity bounds differ significantly in the two sub-cases.

The study of local and global instability and bifurcation phenomena is crucial for many engineering applications in the field of solid mechanics. In particular, interfaces within solid bodies are of great importance in the bifurcation analysis, as they constitute localized zones in which discontinuities or jumps in displacement, strain or stress may occur. Different instability phenomena, heavily conditioned by the presence of interfaces, were analyzed in the present thesis. The first phenomenon that has been considered is the propagation of a shear band, which is a localized shear deformation developing in a ductile material. This shear band, assumed to be already present inside of a ductile matrix material (obeying von Mises plasticity with linear hardening), is modelled as a discontinuity interface following two different approaches. In the first approach, the conditions describing the behavior of a layer of material in which localized strain develop are introduced and implemented in a finite element computer code. A shear deformation is simulated by imposing appropriate displacement conditions on the boundaries of the matrix material, in which the shear band is present and modelled through an imperfect interface, having null thickness. The second approach is based on a perturbative technique, developed for a J2-deformation theory material, in which the shear band is modeled as the emergence of a discontinuity surface for displacements at a certain stage of a uniform deformation process, restricted to plane strain conditions. Both the approaches concur in showing that shear bands (differently from cracks) propagate rectilinearly under shear loading and that a strong stress concentration is expected to be present at the tip of the shear band, two key features in the understanding of failure mechanisms of ductile materials [results of this study have been reported in (Bordignon et al. 2015)]. The second type of interface analyzed in the present thesis is a perfectly frictionless sliding interface, subject to large deformations and assumed to be present within a uniformly strained nonlinear elastic solid. This type of interface may model lubricated sliding contact between soft solids, a topic of interest in biomechanics and for the design of small-scale engineering devices. The analyzed problem is posed as follows. Two elastic nonlinear solids are considered jointed through a frictionless and bilateral surface, so that continuity of the normal component of the Cauchy traction holds across the surface, but the tangential component is null. Moreover, the displacement can develop only in a way that the bodies in contact do neither detach, nor overlap. Surprisingly, this finite strain problem has not been correctly formulated until now, so that this formulation has been developed in the thesis. The incremental equations are shown to be non-trivial and different from previously (and erroneously) employed conditions. In particular, an exclusion condition for bifurcation is derived to show that previous formulations based on frictionless contact or ‘spring-type’ interfacial conditions are not able to predict bifurcations in tension, while experiments (one of which, ad hoc designed, is reported) show that these bifurcations are a reality and can be predicted when the correct sliding interface model is used. Therefore, the presented approach introduces a methodology for the determination of bifurcations and instabilities occurring during lubricated sliding between soft bodies in contact [results of this study have been reported in (Bigoni et al. 2018)].

The study of local and global instability and bifurcation phenomena is crucial for many engineering applications in the field of solid mechanics. In particular, interfaces within solid bodies are of great importance in the bifurcation analysis, as they constitute localized zones in which discontinuities or jumps in displacement, strain or stress may occur. Different instability phenomena, heavily conditioned by the presence of interfaces, were analyzed in the present thesis. The first phenomenon that has been considered is the propagation of a shear band, which is a localized shear deformation developing in a ductile material. This shear band, assumed to be already present inside of a ductile matrix material (obeying von Mises plasticity with linear hardening), is modelled as a discontinuity interface following two different approaches. In the first approach, the conditions describing the behavior of a layer of material in which localized strain develop are introduced and implemented in a finite element computer code. A shear deformation is simulated by imposing appropriate displacement conditions on the boundaries of the matrix material, in which the shear band is present and modelled through an imperfect interface, having null thickness. The second approach is based on a perturbative technique, developed for a J2-deformation theory material, in which the shear band is modeled as the emergence of a discontinuity surface for displacements at a certain stage of a uniform deformation process, restricted to plane strain conditions. Both the approaches concur in showing that shear bands (differently from cracks) propagate rectilinearly under shear loading and that a strong stress concentration is expected to be present at the tip of the shear band, two key features in the understanding of failure mechanisms of ductile materials [results of this study have been reported in (Bordignon et al. 2015)]. The second type of interface analyzed in the present thesis is a perfectly frictionless sliding interface, subject to large deformations and assumed to be present within a uniformly strained nonlinear elastic solid. This type of interface may model lubricated sliding contact between soft solids, a topic of interest in biomechanics and for the design of small-scale engineering devices. The analyzed problem is posed as follows. Two elastic nonlinear solids are considered jointed through a frictionless and bilateral surface, so that continuity of the normal component of the Cauchy traction holds across the surface, but the tangential component is null. Moreover, the displacement can develop only in a way that the bodies in contact do neither detach, nor overlap. Surprisingly, this finite strain problem has not been correctly formulated until now, so that this formulation has been developed in the thesis. The incremental equations are shown to be non-trivial and different from previously (and erroneously) employed conditions. In particular, an exclusion condition for bifurcation is derived to show that previous formulations based on frictionless contact or ‘spring-type’ interfacial conditions are not able to predict bifurcations in tension, while experiments (one of which, ad hoc designed, is reported) show that these bifurcations are a reality and can be predicted when the correct sliding interface model is used. Therefore, the presented approach introduces a methodology for the determination of bifurcations and instabilities occurring during lubricated sliding between soft bodies in contact [results of this study have been reported in (Bigoni et al. 2018)].

Thesis (S.M.)--Massachusetts Institute of Technology, Department of Civil and Environmental Engineering, 2013.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 167-169).
Fracture mechanics is a field of continuum mechanics with the objective to predict how cracks initiate and propagate in solids. It has a wide domain of application. While aerospace engineers want to make sure a defect in a structure will not grow and possibly lead to failure, petroleum engineers try to increase the permeability of gas shale rocks by fracturing it. In this context, we introduce some elements of linear elastic fracture mechanics in anisotropic solids. Notably, a special attention is paid to transverse isotropy, often used to model rocks but also some piezoelectric materials or fiber-reinforced composites. We focus on brittle materials, that is, we consider only elastic deformations; we thus ignore dissipative phenomena other than the one associated with the creation of crack surface. This thesis aims at understanding and predicting how pressurized cracks propagate in anisotropic brittle solids, in the framework of linear elastic fracture mechanics. The elastic coefficients relevant to the study of a pressurized crack in such materials are identified. Interestingly, they are directly related to quantities easily measured in a lab at the macroscopic scale through indentation tests and acoustic measurements. As an application, the fluid-driven crack problem is addressed. It is shown that the classical tools of the isotropic fluid-driven crack model remain valid in anisotropy, provided the appropriate elastic constants are used. We introduce the concept of crack-shape adaptability: the ability of three-dimensional cracks to shape with the elastic content. This ability could be ruled by three criteria herein introduced. The first one is based on the maximum dissipation principle. The second one is based on Irwin's theory of fracture and the concept of stress intensity factors. As for the third one, it is based on Griffith's energetic theory. While the first criterion predicts that circular cracks are more favorable, the others predict that elliptical shapes are more likely to be seen. This thesis could be valuable in the context of the stimulation of unconventional oil and gas from organic-rich shale.
by Hadrien Hyacinthe Laubie.
S.M.

This thesis provides a study of different aspects of the mechanical and vibrational properties of disordered and amorphous solids. Resorting to the theoretical framework of non-affine lattice dynamics the attention is focused on the analysis of disordered networks and lattices which serve as tractable model systems for real materials. Firstly, we discuss the static elastic response and the vibrational spectra of defective fcc crystals. The connection to different types of microstructural disorder in the form of bond-depletion and vacancies is described within the context of the inversion symmetry breaking of the local particle configurations. We identify the fluctuations of the local inversion symmetry breaking, which is directly linked to the non-affinity of the disordered solid, as the source of different scalings behaviours of the position of the boson peak. Furthermore, we describe the elastic heterogeneities occurring in a bond-depleted two- dimensional lattice with long-range interactions. The dependence of the concomitant correlations of the local elastic moduli are studied in detail in terms of the interaction range and the degree of disorder. An analytical scaling relation is derived for the radial part of the elastic correlations in the affine limit. Subsequently, we provide an argument for the change of the angular symmetry of the elastic correlation function which was observed in simulations and experiments on glasses and colloids, respectively. Moving to the dynamical behaviour of disordered solids, a framework is developed based on the kernel polynomial method for the approximate computation of the non- affine correlator of displacement fields which is the key requirement to describe the linear viscoelastic response of the system within the quasi-static non-affine formalism. This approach is then extended to the case of multicomponent polymer melts and validated against molecular dynamics simulations at low non-zero temperatures. We also consider the dynamical behaviour of metallic glasses in terms of its shear elasticity and viscosity. A theoretical scheme is suggested which links the repulsive strength of the interatomic potential to the viscoelasticity and fragility in metallic glasses in the quasi-affine limit.

[EN] Phononic crystals are artificial materials formed by a periodic arrangement of inclusions embedded into a host medium, where each of them can be solid or fluid. By controlling the geometry and the impedance contrast of its constituent materials, one can control the dispersive properties of waves, giving rise to a huge variety of interesting and fundamental phenomena in the context of wave propagation. When a propagating wave encounters a medium with different physical properties it can be transmitted and reflected in lossless media, but also absorbed if dissipation is taken into account. These fundamental phenomena have been classically explained in the context of homogeneous media, but it has been a subject of increasing interest in the context of periodic structures in recent years as well. This thesis is devoted to the study of different effects found in sonic and phononic crystals associated with transmission, reflection and absorption of waves, as well as the development of a technique for the characterization of its dispersive properties, described by the band structure. We start discussing the control of wave propagation in transmission in conservative systems. Specifically, our interest is to show how sonic crystals can modify the spatial dispersion of propagating waves leading to control the diffractive broadening of sound beams. Making use of the spatial dispersion curves extracted from the analysis of the band structure, we first predict zero and negative diffraction of waves at frequencies close to the band-edge, resulting in collimation and focusing of sound beams in and behind a 3D sonic crystal, and later demonstrate it through experimental measurements. The focusing efficiency of a 3D sonic crystal is limited due to the strong scattering inside the crystal, characteristic of the diffraction regime. To overcome this limitation we consider axisymmetric structures working in the long wavelength regime, as a gradient index lens. In this regime, the scattering is strongly reduced and, in an axisymmetric configuration, the symmetry matching with acoustic sources radiating sound beams increase its efficiency dramatically. Moreover, the homogenization theory can be used to model the structure as an effective medium with effective physical properties, allowing the study of the wave front profile in terms of refraction. We will show the model, design and characterization of an efficient focusing device based on these concepts. Consider now a periodic structure in which one of the parameters of the lattice, such as the lattice constant or the filling fraction, gradually changes along the propagation direction. Chirped crystals represent this concept and are used here to demonstrate a novel mechanism of sound wave enhancement based on a phenomenon known as "soft" reflection. The enhancement is related to a progressive slowing down of the wave as it propagates along the material, which is associated with the group velocity of the local dispersion relation at the planes of the crystal. A model based on the coupled mode theory is proposed to predict and interpret this effect. Two different phenomena are observed here when dealing with dissipation in periodic structures. On one hand, when considering the propagation of in-plane sound waves in a periodic array of absorbing layers, an anomalous decrease in the absorption, combined with a simultaneous increase of reflection and transmission at Bragg frequencies is observed, in contrast to the usual decrease of transmission, characteristic in conservative periodic systems at these frequencies. For a similar layered media, backed now by a rigid reflector, out-of-plane waves impinging the structure from a homogeneous medium will increase dramatically the interaction strength. In other words, the time delay of sound waves inside the periodic system will be considerably increased resulting in an enhanced absorption, for a broadband spectral range.
[ES] Los cristales fonónicos son materiales artificiales formados por una disposición periódica de inclusiones en un medio, pudiendo ambos ser de carácter sólido o fluido. Controlando la geometría y el contraste de impedancias entre los materiales constituyentes se pueden controlar las propiedades dispersivas de las ondas. Cuando una onda propagante se encuentra un medio con diferentes propiedades físicas puede ser transmitida y reflejada, en medios sin pérdidas, pero también absorbida, si la disipación es tenida en cuenta. La presente tesis está dedicada al estudio de diferentes efectos presentes en cristales sónicos y fonónicos relacionados con la transmisión, reflexión y absorción de ondas, así como el desarrollo de una técnica para la caracterización de sus propiedades dispersivas, descritas por la estructura de bandas. En primer lugar, se estudia el control de la propagación de ondas en transmisión en sistemas conservativos. Específicamente, nuestro interés se centra en mostrar cómo los cristales sónicos son capaces de modificar la dispersión espacial de las ondas propagantes, dando lugar al control del ensanchamiento de haces de sonido. Haciendo uso de las curvas de dispersión espacial extraídas del análisis de la estructura de bandas, se predice primero la difracción nula y negativa de ondas a frecuencias cercanas al borde de la banda, resultando en la colimación y focalización de haces acústicos en el interior y detrás de un cristal sónico 3D, y posteriormente se demuestra mediante medidas experimentales. La eficiencia de focalización de un cristal sónico 3D está limitada debido a las múltiples reflexiones existentes en el interior del cristal. Para superar esta limitación se consideran estructuras axisimétricas trabajando en el régimen de longitud de onda larga, como lentes de gradiente de índice. En este régimen, las reflexiones internas se reducen fuertemente y, en configuración axisimétrica, la adaptación de simetría con fuentes acústicas radiando haces de sonido incrementa la eficiencia drásticamente. Además, la teoría de homogenización puede ser empleada para modelar la estructura como un medio efectivo con propiedades físicas efectivas, permitiendo el estudio del frente de ondas en términos refractivos. Se mostrará el modelado, diseño y caracterización de un dispositivo de focalización eficiente basado en los conceptos anteriores. Considérese ahora una estructura periódica en la que uno de los parámetros de la red, sea el paso de red o el factor de llenado, cambia gradualmente a lo largo de la dirección de propagación. Los cristales chirp representan este concepto y son empleados aquí para demostrar un mecanismo novedoso de incremento de la intensidad de la onda sonora basado en un fenómeno conocido como reflexión "suave". Este incremento está relacionado con una ralentización progresiva de la onda conforme se propaga a través del material, asociado con la velocidad de grupo de la relación de dispersión local en los planos del cristal. Un modelo basado en la teoría de modos acoplados es propuesto para predecir e interpretar este efecto. Se observan dos fenómenos diferentes al considerar pérdidas en estructuras periódicas. Por un lado, si se considera la propagación de ondas sonoras en un array periódico de capas absorbentes, cuyo frente de ondas es paralelo a los planos del cristal, se produce una reducción anómala en la absorción combinada con un incremento simultáneo de la reflexión y transmisión a las frecuencias de Bragg, de forma contraria a la habitual reducción de la transmisión, característica de sistemas periódicos conservativos a estas frecuencias. En el caso de la misma estructura laminada en la que se cubre uno de sus lados mediante un reflector rígido, la incidencia de ondas sonoras desde un medio homogéneo, cuyo frente de ondas es perpendicular a los planos del cristal, produce un gran incremento de la fuerza de
[CAT] Els cristalls fonònics són materials artificials formats per una disposició d'inclusions en un medi, ambdós poden ser sòlids o fluids. Controlant la geometría i el contrast d'impedàncies dels seus materials constituents, és poden controlar les propietats dispersives de les ondes, permetent una gran varietatde fenòmens fonamentals interessants en el context de la propagació d'ones. Quan una ona propagant troba un medi amb pèrdues amb propietats físiques diferents es pot transmetre i reflectir, però també absorbida si la dissipació es té en compte. Aquests fenòmens fonamentals s'han explicat clàssicament en el context de medis homogenis, però també ha sigut un tema de creixent interés en el context d'estructures periòdiques en els últims anys. Aquesta tesi doctoral tracta de l'estudi de diferents efectes en cristalls fonònics i sònics lligats a la transmissió, reflexió i absorció d'ones, així com del desenvolupament d'una tècnica de caracterització de les propietats dispersives, descrites mitjançant la estructura de bandes. En primer lloc, s'estudia el control de la propagació ondulatori en transmissió en sistemes conservatius. Més específicament, el nostre interés és mostrar com els cristalls sonors poden modificar la dispersió espacial d'ones propagants donant lloc al control de l'amplària per difracció dels feixos sonors. Mitjançant les corbes dispersió espacial obtingudes de l'anàlisi de l'estructura de bandes, es prediu, en primer lloc, la difracció d'ones zero i negativa a freqüències próximes al final de banda. El resultat és la collimació i focalització de feixos sonors dins i darrere de cristalls de so. Després es mostra amb mesures experimentals. L'eficiència de focalització d'un cristall de so 3D està limitada per la gran dispersió d'ones dins del cristall, que és característic del règim difractiu. Per a superar aquesta limitació, estructures axisimètriques que treballen en el règim de llargues longituds d'ona, i es comporten com a lents de gradient d'índex. En aquest règim, la dispersió es redueix enormement i, en una configuració axisimètrica, a causa de l'acoblament de la simetría amb les fonts acústiques que radien feixos sonors, l'eficiència de radiació s'incrementa significativament. D'altra banda, la teoria d'homogeneïtzació es pot utilitzar per a modelar, dissenyar i caracteritzar un dispositiu eficient de focalització basat en aquests conceptes. Considerem ara una estructura periòdica en la qual un dels seus paràmetres de xarxa, com ara la constant de xarxa o el factor d'ompliment canvia gradualment al llarg de la direcció de propagació. Els cristalls chirped representen aquest concepte i s'utilitzen ací per a demostrar un mecanisme nou d'intensificació d'ones sonores basat en el fenòmen conegut com a reflexió "suau". La intensificació està relacionada amb la alentiment progressiva de l'ona conforme propaga al llarg del material, que està associada amb la velocitat de grup de la relació de dispersió local en els diferents plànols del cristall. Es proposa un model basat en la teoria de modes acoblats per a predir i interpretar este efecte. Dos fenòmens diferents cal destacar quan es tracta d'estructures periòdiques amb dissipació. Per un costat, al considerar la propagació d'ones sonores en el plànol en un array periòdic de capes absorbents, s'observa una disminució anòmala de l'absorció i es combina amb un augment simultani de reflexió i transmissió en les freqüències de Bragg que contrasta amb la usual disminució de transmissió, característica dels sistemes conservatius a eixes freqüències. Per a un medi similar de capes, amb un reflector rígid darrere, les ones fora del pla incidint l'estructura des de un medi homogeni, augmentaran considerablement la interacció. En altres paraules, el retràs temporal de les ones sonores dins del sistema periòdic augmentarà significativament produint un augmen
Cebrecos Ruiz, A. (2015). Transmission, reflection and absorption in Sonic and Phononic Crystals [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/56463
TESIS
Premiado

This thesis dwells upon several aspects of continuum mechanics and thermodynamics to model elastic and inelastic response of solids. Broadly, the work presented may be categorized into two parts{ one focusing on the development of generalized continuum description in the context of elastic materials and the other on the thermodynamics of dissipative processes whilst considering, in some detail, the physics of deformation too. The first part begins with the proposal for a state-based micropolar peridynamic theory for linear elastic solids. The main motivation is to introduce additional micro-rotational degrees of freedom to each material point and thus naturally bring in the physically relevant material length scale parameters into peridynamics. Non-ordinary type modelling via constitutive correspondence is adopted here to de ne the micropolar peridynamic material. Along with a general three-dimensional model, homogenized one dimensional Timoshenko type beam models for both the proposed micropolar and the standard non-polar peridynamic variants are derived. The efficacy of the proposed models in analyzing continua with length scale effects is established via numerical simulations of a few beam and plane-stress problems. Continuing with our e ort in developing homogenized reduced dimensional models, a state-based peridynamic formulation for linear elastic shells is presented next. The emphasis is on introducing, perhaps for the first time, a general surface based peridynamic model to represent the deformation characteristics of structures that have one geometric dimension much smaller than the other two. A new notion of curved bonds is exploited to model force transfer between the peridynamic particles describing the shell. Starting with the three dimensional force and deformation states, appropriate surface based force, moment and several deformation states are arrived at. Upon application on the curved bonds, such states yield the necessary force and deformation vectors governing the motion of the shell. The peridynamic shell theory is numerically assessed against simulations on static deformation of spherical and cylindrical shells and those on at plates. As a transition to the second part of the thesis, our next work shares features of the first part (micropolarity and homogenization) as well as the second (equation with viscous force, i.e., dissipative process). Starting with a micropolar formulation, known to account for nonlocal microstructural effects at the continuum level, a generalized Langevin equation (GLE) for a particle, describing the predominant motion of a localized region through a single displacement degree-of-freedom, is derived. The GLE features a memory dependent multiplicative or internal noise, which appears upon recognising that the micro-rotation variables possess randomness owing to an uncertainty principle. Unlike its classical version, the new GLE qualitatively reproduces the experimentally measured fluctuations in the steady-state mean square displacement of scattering centers in a polyvinyl alcohol slab. In the second part of the thesis, a series of physically motivated models for dislocation mediated thermoviscoplastic deformation and micro-void mediated ductile damage in metals are proposed. The methodology of modelling brittle damage also constitutes another part of discussion. The models are, in essence, posited in the framework of internal-variables theory of thermodynamics, wherein effective dislocation densities, void volume fractions etc., which assume the role of internal variables, track permanent changes in the internal structure of metals undergoing plastic deformation and damage. The thermodynamic formulation involves a two-temperature description of viscoplasticity and damage that appears naturally if one considers the thermodynamic system to be composed of two weakly interacting subsystems, namely, a kinetic- vibrational subsystem of the vibrating atomic lattices and a con gurational subsystem of the slower degrees-of-freedom of defect motion. While most of the models are proposed satisfying the thermodynamic requirements asserted by the second law, one specific interest, however, has been to explore the possible application of a fluctuation relation that subsumes the second law of thermodynamics en route to deriving the evolution equations for the internal variables. Full- edged three-dimensional continuum formulations, valid for the finite deformation regime, are also set forth. Several numerical exercises, including impact dynamic simulations, are carried out and validated against experimental data.

This monograph describes various methods for solving deformation problems of particulate solids, taking the reader from analytical to computational methods. The book is the first to present the topic of linear elasticity in mathematical terms that will be familiar to anyone with a grounding in fluid mechanics. It incorporates the latest advances in computational algorithms for elliptic partial differential equations, and provides the groundwork for simulations on high performance parallel computers. Numerous exercises complement the theoretical discussions, and a related set of self-documented programs is available to readers with Internet access. The work will be of interest to advanced students and practicing researchers in mechanical engineering, chemical engineering, applied physics, computational methods, and developers of numerical modeling software.

Abstract If a solid is initially at rest and equal and opposing forces are applied to that object, Newton’s Second Law guarantees that the object will remain at rest because the net force on the sample is zero. If that object is an elastic solid, then those forces will cause the solid to deform by an amount that is directly proportional to those applied forces. When the forces are removed, the sample will return to its original shape and size. That reversibility is the characteristic that is required if we say the behavior of the solid is “elastic.” This chapter will quantify the elastic behavior of solids by introducing the concepts of stress and strain and expressing their linear relationship through the definition of elastic moduli that depend only upon the material and the nature of the deformation and not upon the shape of the object. Those concepts allow us to generalize Hooke’s law. As before, the combination of a linear equation of state with Newton’s Second Law will now describe wave motion in solids. The introduction of a relaxation time, through the Maxwell model, will let these results be generalized to viscoelastic materials and then be applied to rubber vibration isolators.

Abstract Elastohydrodynamic lubrication (EHL) is the name given to hydrodynamic lubrication when it is applied to solid surfaces of low geometric conformity (counterformal contacts) that are capable of, and are subject to, elastic deformation. In bearings relying on EHL principles, the residence time of the fluid is less than 1 ms, the pressures are up to 4 GP, the film is thin, down to 0.1 μm, and shear rates are up to 108 s−1 — under such conditions lubricants exhibit material behavior that is distinctly different from their behavior in bulk at normal temperature and pressure. In fact, without taking into account the viscosity-pressure characteristics of the liquid lubricant and the elastic deformation of the bounding solids, hydrodynamic theory is unable to explain the existence of continuous lubricant films in highly loaded gears and rolling contact bearings.

Abstract This paper presents the development of a computational model for the generalized topology optimization problem, using a material distribution approach, of 2-D linear elastic solids subjected to thermal loads, with compliance objective function and an isoperimetric constraint on volume. The model relies on homogenization asymptotic methods to characterize the influence of the material periodic microstructure and a finite element displacement formulation is used to approximate the homogenized equilibrium equations obtained. The computational model developed is tested in several examples considering different finite element approximations and the influence of the design variables (material density and orientation) is analyzed.

The paper reexamines the extraction of material properties using nanoindentation for linearly elastic and elastic-plastic materials. The paper considers indentation performed using a rigid conical indenter, as follows. Linearly elastic solids: The reduction of nanoindentation test data of elastic solids is usually processed using Sneddon’s relation [1], which assumes a linearly elastic infinite half space and an infinitely sharp indenter tip. These assumptions are violated in practical indentation experiments. Since most of the research on the extraction of material properties relies heavily on numerical simulations, we used them to investigate the specimen dimensions required for it to qualify as an infinite body, and the indentation conditions for finite tip radius effect to be negligible. The outcome of this part is firstly, the definition of a “converged” 2D geometry so that additional magnification of the numerical model does not influence the load-displacement curve, and secondly, an explicit relationship between the measured load and displacement that takes into account the finite tip radius. Elastic-plastic solids: Here, the main data reduction technique was proposed by Pharr et al. [2], assuming elastic unloading of a plastic nanoindentation. We investigated the effects of finite tip radius in elastic-plastic indentations and found that the accuracy of the prediction is currently limited by the accurate determination of the projected contact area. This point will be discussed and a new experimental technique to measure the projected contact area will be proposed. The Poisson’s ratio effect in elastic-plastic indentations is found to be different from the linearly elastic case. This leads to the discussion on the applicability of the correction factor (for Poisson’s ratio effect) derived in linear elastic indentations, on elastic-plastic indentations. Finally, a technique to obtain an upper bound estimate of the yield stress for the indented elastic-plastic material (which is an exact estimation for non-hardening materials), will be presented.

Abstract A technique for solving boundary value problems for inelastic solids based on their elastic solution is presented. For this purpose, a constitutive equation for inelastic materials is proposed in terms of variable parameters, similar in form to linear elastic equation. A method of obtaining these parameters based on energy consideration is presented. The technique is applied to predict the elastic-plastic analysis of a flat plate with circular notch subjected to uniform all around tension. It is also used to obtain the residual stress field induced by cold work expansion of fastener holes. Results of this method are compared to those obtained from finite element analysis and experimental data.

This paper presents an enriched meshless method based on an improved moving least-square approximation (IMLS) method for fracture analysis of cracks in homogeneous, isotropic, linear-elastic, two-dimensional bimaterial solids, subject to mixed-mode loading conditions. The method involves an element-free Galerkin formulation in conjunction with IMLS and a new enriched basis functions to capture the singularity field in linear-elastic bi-material fracture mechanics. In the IMLS method, the orthogonal function system with a weight function is used as the basis function. The IMLS has higher computational efficiency and precision than the MLS, and will not lead to an ill-conditioned system of equations. The proposed enriched basis function can be viewed as a generalized enriched basis function, which degenerates to a linear-elastic basis function when the bimaterial constant is zero. Numerical examples are presented to illustrate the computational efficiency and accuracy of the proposed method.

A number of experimental studies [1–3] revealed that the normal displacement in a contact of rough surfaces due to asperities presence is a nonlinear function of local pressure and it can be approximated by a power function of pressure. Originally, a linear mathematical model accounting for surface roughness of elastic solids in contact was introduced by I. Shtaerman [4]. He assumed that the effect of asperities present in a contact of elastic solids can be essentially replaced by the presence of a thin coating simulated by an additional normal displacement of solids’ surfaces proportional to a local pressure. Later, a similar but nonlinear problem formulation that accounted for the above mentioned experimental fact was proposed by L. Galin. In a series of papers this problem was studied by numerical and asymptotic methods [5–9]. The present paper has a dual purpose: to analyze the problem analytically and to provide some asymptotic and numerical solutions. The results presented below provide an overview of the results obtained on the topic and published by the author earlier in the journals hardly accessible to the international tribological community (such as Russian and mathematical journals) and, therefore, mostly unknown by tribologists. A number of recent publications on contacts of rough elastic solids supports the view that these results are still of value to the specialists involved in nanotribology. The existence and uniqueness of a solution of a contact problem for elastic bodies with rough (coated) surfaces is established based on the variational inequalities approach. Four different equivalent formulations of the problem including three variational ones were considered. A comparative analysis of solutions of the contact problem for different values of initial parameters (such as the indenter shape, parameters characterizing roughness, elastic parameters of the substrate material) is done with the help of calculus of variations and the Zaremba-Giraud principle of maximum for harmonic functions [10,11]. The results include the relations between the pressure and displacement distributions for rough and smooth solids as well as the relationships for solutions of the problems for rough solids with fixed and free contact boundaries. For plane and axially symmetric cases some asymptotic and numerical solutions are presented.

This paper presents an enriched meshless method for fracture analysis of cracks in homogeneous, isotropic, nonlinear-elastic, two-dimensional solids, subject to mode-I loading conditions. The method involves an element-free Galerkin formulation and two new enriched basis functions (Type I and Type II) to capture the Hutchinson-Rice-Rosengren singularity field in nonlinear fracture mechanics. The Type I enriched basis function can be viewed as a generalized enriched basis function, which degenerates to the linear-elastic basis function when the material hardening exponent is unity. The Type II enriched basis function entails further improvements of the Type I basis function by adding trigonometric functions. Three numerical examples are presented to illustrate the proposed method. The boundary layer analysis indicates that the crack-tip field predicted by using the proposed basis functions matches with the theoretical solution very well in the whole region considered, whether for the near-tip asymptotic field or for the far-tip elastic field. Numerical analyses of standard fracture specimens by the proposed meshless method also yield accurate estimates of the J-integral for the applied load intensities and material properties considered. Furthermore, the meshless results show excellent agreement with the experimental measurements, indicating that the new basis functions are also capable of capturing elastic-plastic deformations at a stress concentration effectively.

Elastomeric materials have a capability to withstand large deformations and still be able to fully recover their original dimensions. Natural and synthetic elastomers and their derivatives can reach strains as high as 500–1000%. Engineering materials, such as crystalline metals are classified as linear elastic solids, whereas elastomeric materials are considered as nonlinear elastic solids. Elastomers present a very complicated mechanical behavior that exceed the linear elastic theory and contain large deformations, plastic and viscoelastic properties. Finite element (FE) is a powerful tool to analyze such elastomers. Design of elastomeric systems in an industrial scenario generally requires (i) reliability and (ii) a minimum cycle time. This paper starts with a review of the hyperelastic theory, followed by a detailed discussion on the process involved in the material characterization of hyperelastic material like DuPont™ Viton® fluoroelastomer and polyacrylic elastomer in industrial application point of view. The paper also discusses guidelines to be followed in the various stages of material characterization such as testing, sampling and finite element simulation. Numerical stability issues associated with elastomeric modeling in finite element context and a set of guidelines to be followed in finite element analysis of elastomers are illustrated through a DuPont™ Viton® fluoroelastomer and polyacrylic elastomer pad vibration isolation systems. The above technique has been applied for designing vibration isolation systems for generators.

Viscoelastic analysis for numerical modeling of IC assembly processes are generally non-linear and require extensive time and computational resources when compared to a linear elastic analysis. Experimental identification/approximation of the viscoelastic properties (in terms of the Prony series) of any polymeric material is also an exhaustive effort. These drawbacks in experimental procedures and modeling activities have forced us to explore the possibility of approximating viscoelastic response of a polymer with an appropriate linear-elastic model. This paper discusses the impact of different approximation methodologies for a viscoelastic material most commonly used in electronic packaging — the underfill material in flip-chip technology. The modeling methodologies discussed here include the use of long term modulus, short term modulus and other “effective” stiffness measures to approximate the response of underfills during assembly processes. The package response, from each of these models, has been compared to the package response from a linear viscoelastic analysis assuming underfill materials to behave as Maxwell solids. It has been observed that the short term modulus consistently over predicts the ILD (interlayer dielectric in the silicon backend) peel stress for various underfill materials. In addition, the paper also explores the existence of an “effective” stiffness that can be used in lieu of a full fledged viscoelastic analysis. The ability to estimate this “effective” stiffness using available temperature dependent modulus data (from DMA tests) is also discussed. The applicability of this effective metric to different underfill materials has also been explored. In conclusion, this study highlights the need for accurate characterization of polymeric materials so that numerical predictions can provide realistic risk assessments for future packaging technologies.

The paper addresses one of the key problems in structural health monitoring — how to separate defects, both microscopic and macroscopic, accumulated in structural members from damage in the joints and boundaries. It is suggested to use a simultaneous set of measurements — nonlinear vibration frequencies and electrical conductivities — to distinguish between three distinct types of defects. Nonlinearities occur as a result of breathing internal cracks and slapping in damaged joints. The use of proper orthogonal decomposition and a local equivalent linear stiffness method allows for model updating to account for the damage. The theory is presented for full three-dimensional elastic solids and illustrated via the problem of a beam damaged both internally as well as at the boundaries.

One major difficulty in developing fitness-for-purpose based flaw acceptance criteria for pipeline branch connections is the calculation of the stress distributions in the vicinity of the welds. Even with the latest computer-aided modelling technologies, finite element modeling of branch connections using three-dimensional (3-D) solid elements is the only way to accurately determine the stress distributions local to the weld toes of branch connections. This report describes work to access the applicability of linear elastic fracture mechanics (LEFM) principles to a saddle-pad reinforced branch connection modelled using 3-D elements, The effort concentrated on determining the amount and location of plastic zones resulting from different combinations of service loadings. Loadings considered include pressurization to 72% of the specified minimum yield strength with the addition of main pipe end tension or compression. The effect of contact and different material models were also quantified.

Recent concerns regarding global warming and energy security have accelerated research and developmental efforts to produce biofuels from agricultural and forestry residues, and energy crops. Anaerobic digestion is a promising process for producing biogas-biofuel from biomass feedstocks. However, there is a need for new reactor designs and operating considerations to process fibrous biomass feedstocks. In this research project, the multiphase flow behavior of biomass particles was investigated. The objective was accomplished through both simulation and experimentation. The simulations included both particle-level and bulk flow simulations. Successful computational fluid dynamics (CFD) simulation of multiphase flow in the digester is dependent on the accuracy of constitutive models which describe (1) the particle phase stress due to particle interactions, (2) the particle phase dissipation due to inelastic interactions between particles and (3) the drag force between the fibres and the digester fluid. Discrete Element Method (DEM) simulations of Homogeneous Cooling Systems (HCS) were used to develop a particle phase dissipation rate model for non-spherical particle systems that was incorporated in a two-fluid CFDmultiphase flow model framework. Two types of frictionless, elongated particle models were compared in the HCS simulations: glued-sphere and true cylinder. A new model for drag for elongated fibres was developed which depends on Reynolds number, solids fraction, and fibre aspect ratio. Schulze shear test results could be used to calibrate particle-particle friction for DEM simulations. Several experimental measurements were taken for biomass particles like olive pulp, orange peels, wheat straw, semolina, and wheat grains. Using a compression tester, the breakage force, breakage energy, yield force, elastic stiffness and Young’s modulus were measured. Measurements were made in a shear tester to determine unconfined yield stress, major principal stress, effective angle of internal friction and internal friction angle. A liquid fludized bed system was used to determine critical velocity of fluidization for these materials. Transport measurements for pneumatic conveying were also assessed. Anaerobic digestion experiments were conducted using orange peel waste, olive pulp and wheat straw. Orange peel waste and olive pulp could be anaerobically digested to produce high methane yields. Wheat straw was not digestible. In a packed bed reactor, anaerobic digestion was not initiated above bulk densities of 100 kg/m³ for peel waste and 75 kg/m³ for olive pulp. Interestingly, after the digestion has been initiated and balanced methanogenesis established, the decomposing biomass could be packed to higher densities and successfully digested. These observations provided useful insights for high throughput reactor designs. Another outcome from this project was the development of low cost devices to measure methane content of biogas for off-line (US$37), field (US$50), and online (US$107) applications.