Showing papers on "Linear elasticity published in 2009"

Applied Mechanics of Solids

Abstract: 1 Overview of Solid Mechanics DEFINING A PROBLEM IN SOLID MECHANICS 2 Governing Equations MATHEMATICAL DESCRIPTION OF SHAPE CHANGES IN SOLIDS MATHEMATICAL DESCRIPTION OF INTERNAL FORCES IN SOLIDS EQUATIONS OF MOTION AND EQUILIBRIUM FOR DEFORMABLE SOLIDS WORK DONE BY STRESSES: PRINCIPLE OF VIRTUAL WORK 3 Constitutive Models: Relations between Stress and Strain GENERAL REQUIREMENTS FOR CONSTITUTIVE EQUATIONS LINEAR ELASTIC MATERIAL BEHAVIORSY HYPOELASTICITY: ELASTIC MATERIALS WITH A NONLINEAR STRESS-STRAIN RELATION UNDER SMALL DEFORMATION GENERALIZED HOOKE'S LAW: ELASTIC MATERIALS SUBJECTED TO SMALL STRETCHES BUT LARGE ROTATIONS HYPERELASTICITY: TIME-INDEPENDENT BEHAVIOR OF RUBBERS AND FOAMS SUBJECTED TO LARGE STRAINS LINEAR VISCOELASTIC MATERIALS: TIME-DEPENDENT BEHAVIOR OF POLYMERS AT SMALL STRAINS SMALL STRAIN, RATE-INDEPENDENT PLASTICITY: METALS LOADED BEYOND YIELD SMALL-STRAIN VISCOPLASTICITY: CREEP AND HIGH STRAIN RATE DEFORMATION OF CRYSTALLINE SOLIDS LARGE STRAIN, RATE-DEPENDENT PLASTICITY LARGE STRAIN VISCOELASTICITY CRITICAL STATE MODELS FOR SOILS CONSTITUTIVE MODELS FOR METAL SINGLE CRYSTALS CONSTITUTIVE MODELS FOR CONTACTING SURFACES AND INTERFACES IN SOLIDS 4 Solutions to Simple Boundary and Initial Value Problems AXIALLY AND SPHERICALLY SYMMETRIC SOLUTIONS TO QUASI-STATIC LINEAR ELASTIC PROBLEMS AXIALLY AND SPHERICALLY SYMMETRIC SOLUTIONS TO QUASI-STATIC ELASTIC-PLASTIC PROBLEMS SPHERICALLY SYMMETRIC SOLUTION TO QUASI-STATIC LARGE STRAIN ELASTICITY PROBLEMS SIMPLE DYNAMIC SOLUTIONS FOR LINEAR ELASTIC MATERIALS 5 Solutions for Linear Elastic Solids GENERAL PRINCIPLES AIRY FUNCTION SOLUTION TO PLANE STRESS AND STRAIN STATIC LINEAR ELASTIC PROBLEMS COMPLEX VARIABLE SOLUTION TO PLANE STRAIN STATIC LINEAR ELASTIC PROBLEMS SOLUTIONS TO 3D STATIC PROBLEMS IN LINEAR ELASTICITY SOLUTIONS TO GENERALIZED PLANE PROBLEMS FOR ANISOTROPIC LINEAR ELASTIC SOLIDS SOLUTIONS TO DYNAMIC PROBLEMS FOR ISOTROPIC LINEAR ELASTIC SOLIDS ENERGY METHODS FOR SOLVING STATIC LINEAR ELASTICITY PROBLEMS THE RECIPROCAL THEOREM AND APPLICATIONS ENERGETICS OF DISLOCATIONS IN ELASTIC SOLIDS RAYLEIGH-RITZ METHOD FOR ESTIMATING NATURAL FREQUENCY OF AN ELASTIC SOLID 6 Solutions for Plastic Solids SLIP-LINE FIELD THEORY BOUNDING THEOREMS IN PLASTICITY AND THEIR APPLICATIONS 7 Finite Element Analysis: An Introduction A GUIDE TO USING FINITE ELEMENT SOFTWARE A SIMPLE FINITE ELEMENT PROGRAM 8 Finite Element Analysis: Theory and Implementation GENERALIZED FEM FOR STATIC LINEAR ELASTICITY THE FEM FOR DYNAMIC LINEAR ELASTICITY FEM FOR NONLINEAR (HYPOELASTIC) MATERIALS FEM FOR LARGE DEFORMATIONS: HYPERELASTIC MATERIALS THE FEM FOR VISCOPLASTICITY ADVANCED ELEMENT FORMULATIONS: INCOMPATIBLE MODES, REDUCED INTEGRATION, AND HYBRID ELEMENTS LIST OF EXAMPLE FEA PROGRAMS AND INPUT FILES 9 Modeling Material Failure SUMMARY OF MECHANISMS OF FRACTURE AND FATIGUE UNDER STATIC AND CYCLIC LOADING STRESS- AND STRAIN-BASED FRACTURE AND FATIGUE CRITERIA MODELING FAILURE BY CRACK GROWTH: LINEAR ELASTIC FRACTURE MECHANICS ENERGY METHODS IN FRACTURE MECHANICS PLASTIC FRACTURE MECHANICS LINEAR ELASTIC FRACTURE MECHANICS OF INTERFACES 10 Solutions for Rods, Beams, Membranes, Plates, and Shells PRELIMINARIES: DYADIC NOTATION FOR VECTORS AND TENSORS MOTION AND DEFORMATION OF SLENDER RODS SIMPLIFIED VERSIONS OF THE GENERAL THEORY OF DEFORMABLE ROD EXACT SOLUTIONS TO SIMPLE PROBLEMS INVOLVING ELASTIC RODS MOTION AND DEFORMATION OF THIN SHELLS: GENERAL THEORY SIMPLIFIED VERSIONS OF GENERAL SHELL THEORY: FLAT PLATES AND MEMBRANES SOLUTIONS TO SIMPLE PROBLEMS INVOLVING MEMBRANES, PLATES, AND SHELLS Appendix A: Review of Vectors and Matrices A.1. VECTORS A.2. VECTOR FIELDS AND VECTOR CALCULUS A.3. MATRICES Appendix B: Introduction to Tensors and Their Properties B.1. BASIC PROPERTIES OF TENSORS B.2. OPERATIONS ON SECOND-ORDER TENSORS B.3. SPECIAL TENSORS Appendix C: Index Notation for Vector and Tensor Operations C.1. VECTOR AND TENSOR COMPONENTS C.2. CONVENTIONS AND SPECIAL SYMBOLS FOR INDEX NOTATION C.3. RULES OF INDEX NOTATION C.4. VECTOR OPERATIONS EXPRESSED USING INDEX NOTATION C.5. TENSOR OPERATIONS EXPRESSED USING INDEX NOTATION C.6. CALCULUS USING INDEX NOTATION C.7. EXAMPLES OF ALGEBRAIC MANIPULATIONS USING INDEX NOTATION Appendix D: Vectors and Tensor Operations in Polar Coordinates D.1. SPHERICAL-POLAR COORDINATES D.2. CYLINDRICAL-POLAR COORDINATES Appendix E: Miscellaneous Derivations E.1. RELATION BETWEEN THE AREAS OF THE FACES OF A TETRAHEDRON E.2. RELATION BETWEEN AREA ELEMENTS BEFORE AND AFTER DEFORMATION E.3. TIME DERIVATIVES OF INTEGRALS OVER VOLUMES WITHIN A DEFORMING SOLID E.4. TIME DERIVATIVES OF THE CURVATURE VECTOR FOR A DEFORMING ROD References

Tight-binding approach to uniaxial strain in graphene

Abstract: We analyze the effect of tensional strain in the electronic structure of graphene. In the absence of electron-electron interactions, within linear elasticity theory, and a tight-binding approach, we observe that strain can generate a bulk spectral gap. However, this gap is critical, requiring threshold deformations in excess of 20% and only along preferred directions with respect to the underlying lattice. The gapless Dirac spectrum is robust for small and moderate deformations and the gap appears as a consequence of the merging of the two inequivalent Dirac points only under considerable deformations of the lattice. We discuss how strain-induced anisotropy and local deformations can be used as a means to affect transport characteristics and pinch off current flow in graphene devices.

Effective elastic mechanical properties of single layer graphene sheets.

Abstract: The elastic moduli of single layer graphene sheet (SLGS) have been a subject of intensive research in recent years. Calculations of these effective properties range from molecular dynamic simulations to use of structural mechanical models. On the basis of mathematical models and calculation methods, several different results have been obtained and these are available in the literature. Existing mechanical models employ Euler-Bernoulli beams rigidly jointed to the lattice atoms. In this paper we propose truss-type analytical models and an approach based on cellular material mechanics theory to describe the in-plane linear elastic properties of the single layer graphene sheets. In the cellular material model, the C-C bonds are represented by equivalent mechanical beams having full stretching, hinging, bending and deep shear beam deformation mechanisms. Closed form expressions for Young's modulus, the shear modulus and Poisson's ratio for the graphene sheets are derived in terms of the equivalent mechanical C-C bond properties. The models presented provide not only quantitative information about the mechanical properties of SLGS, but also insight into the equivalent mechanical deformation mechanisms when the SLGS undergoes small strain uniaxial and pure shear loading. The analytical and numerical results from finite element simulations show good agreement with existing numerical values in the open literature. A peculiar marked auxetic behaviour for the C-C bonds is identified for single graphene sheets under pure shear loading.

Elastic theory of unconstrained non-Euclidean plates

Abstract: Non-Euclidean plates are a subset of the class of elastic bodies having no stress-free configuration. Such bodies exhibit residual stress when relaxed from all external constraints, and may assume complicated equilibrium shapes even in the absence of external forces. In this work we present a mathematical framework for such bodies in terms of a covariant theory of linear elasticity, valid for large displacements. We propose the concept of non-Euclidean plates to approximate many naturally formed thin elastic structures. We derive a thin plate theory, which is a generalization of existing linear plate theories, valid for large displacements but small strains, and arbitrary intrinsic geometry. We study a particular example of a hemispherical plate. We show the occurrence of a spontaneous buckling transition from a stretching dominated configuration to bending dominated configurations, under variation of the plate thickness.

Local elasticity map and plasticity in a model Lennard-Jones glass.

Abstract: In this work we calculate the local elastic moduli in a weakly polydispersed two-dimensional Lennard-Jones glass undergoing a quasistatic shear deformation at zero temperature. The numerical method uses coarse-grained microscopic expressions for the strain, displacement, and stress fields. This method allows us to calculate the local elasticity tensor and to quantify the deviation from linear elasticity (local Hooke's law) at different coarse-graining scales. From the results a clear picture emerges of an amorphous material with strongly spatially heterogeneous elastic moduli that simultaneously satisfies Hooke's law at scales larger than a characteristic length scale of the order of five interatomic distances. At this scale, the glass appears as a composite material composed of a rigid scaffolding and of soft zones. Only recently calculated in nonhomogeneous materials, the local elastic structure plays a crucial role in the elastoplastic response of the amorphous material. For a small macroscopic shear strain, the structures associated with the nonaffine displacement field appear directly related to the spatial structure of the elastic moduli. Moreover, for a larger macroscopic shear strain we show that zones of low shear modulus concentrate most of the strain in the form of plastic rearrangements. The spatiotemporal evolution of this local elasticity map and its connection with long term dynamical heterogeneity as well as with the plasticity in the material is quantified. The possibility to use this local parameter as a predictor of subsequent local plastic activity is also discussed.

Spherical indentation of soft matter beyond the Hertzian regime: numerical and experimental validation of hyperelastic models

Abstract: The lack of practicable nonlinear elastic contact models frequently compels the inappropriate use of Hertzian models in analyzing indentation data and likely contributes to inconsistencies associated with the results of biological atomic force microscopy measurements. We derived and validated with the aid of the finite element method force-indentation relations based on a number of hyperelastic strain energy functions. The models were applied to existing data from indentation, using microspheres as indenters, of synthetic rubber-like gels, native mouse cartilage tissue, and engineered cartilage. For the biological tissues, the Fung and single-term Ogden models achieved the best fits of the data while all tested hyperelastic models produced good fits for the synthetic gels. The Hertz model proved to be acceptable for the synthetic gels at small deformations (strain < 0.05 for the samples tested), but not for the biological tissues. Although this finding supports the generally accepted view that many soft materials can be assumed to be linear elastic at small deformations, the nonlinear models facilitate analysis of intrinsically nonlinear tissues and large-strain indentation behavior.

Free vibration analysis of functionally graded panels and shells of revolution

Abstract: The aim of this paper is to study the dynamic behaviour of functionally graded parabolic and circular panels and shells of revolution. The First-order Shear Deformation Theory (FSDT) is used to study these moderately thick structural elements. The treatment is developed within the theory of linear elasticity, when the materials are assumed to be isotropic and inhomogeneous through the thickness direction. The two-constituent functionally graded shell consists of ceramic and metal that are graded through the thickness, from one surface of the shell to the other. Two different power-law distributions are considered for the ceramic volume fraction. For the first power-law distribution, the bottom surface of the structure is ceramic rich, whereas the top surface is metal rich and on the contrary for the second one.

Mechanical properties of graphene nanoribbons.

Abstract: Herein, we investigate the structural, electronic and mechanical properties of zigzag graphene nanoribbons in the presence of stress by applying density functional theory within the GGA-PBE (generalized gradient approximation-Perdew–Burke–Ernzerhof) approximation. The uniaxial stress is applied along the periodic direction, allowing a unitary deformation in the range of ± 0.02%. The mechanical properties show a linear response within that range while a nonlinear dependence is found for higher strain. The most relevant results indicate that Young's modulus is considerable higher than those determined for graphene and carbon nanotubes. The geometrical reconstruction of the C–C bonds at the edges hardens the nanostructure. The features of the electronic structure are not sensitive to strain in this linear elastic regime, suggesting the potential for using carbon nanostructures in nano-electronic devices in the near future.

Non-Linear Mechanics of Materials

Abstract: Preface by Jean Lemaitre Chapter 1 Introduction 1.1. Model construction 1.2. Applications to models Chapter 2 General concepts 2.1. Formulation of the constitutive equations 2.2. Principle of virtual power 2.3. Thermodyna~nicso f irreversible processes 2.4. Main class of constitutive equations 2.5. Yield criteria 2.6. Numerical methods for nonlinear equations 2.7. Numerical solution of differential equations 2.8. Finite element Chapter 3 Plasticity and 3D viscoplasticity 3.1. Generality 3.2. Formulation of the constitutive equations 3.3. Flow direction associated to the classical criteria 3.4. Expression of some particular constitutive equations in plasticity 3.5. Flow under prescribed strain rate 3.6. Non-associated plasticity 3.7. Nonlinear hardening 3.8. Some classical extensions 3.9. Hardening and recovery in viscoplasticity 3.10. Multimechanism models 3.1 1. Behaviour of porous materials Chapter 4 Introduction to damage mechanics 4.1. Introduction 4.2. Notions and general concepts 4.3. Damage variables and state laws 4.4. State and dissipative couplings 4.5. Damage deactivation 4.6. Damage evolution laws 4.7. Examples of damage models in brittle materials Chapter 5 Microstructural mechanics 5.1. Characteristic lengths and scales in microstructural mechanics 5.2. Some homogenization techniques 5.3. Application to linear elastic heterogeneous materials 5.4. Some examples. applications and extensions 5.5. Homogenization in thermoelasticity 5.6. Nonlinear homogenization 5.7. Computation of RVE 5.8. Homogenization of coarse grain structures Chapter 6 Finite deformations 6.1. Geometry and kinematics of continuum 6.2. Sthenics and statics of the continuum 6.3. Constitutive laws 6.4. Application: Simple glide 6.5. Finite deformations of generalized continua Chapter 7 Nonlinear structural analysis 7.1. The material object 7.2. Examples of implementations of particular models 7.3. Specificities related to finite elements Chapter 8 Strain localization 8.1. Bifurcation modes in elastoplasticity 8.2. Regularization methods Appendix Notation used A.1. Tensors A.2. Vectors, Matrices A.3. Voigt notation

On planar biaxial tests for anisotropic nonlinearly elastic solids. A continuum mechanical framework

Abstract: The mechanical testing of anisotropic nonlinearly elastic solids is a topic of considerable and increasing interest. The results of such testing are important, in particular, for the characterization of the material properties and the development of constitutive laws that can be used for predictive purposes. However, the literature on this topic in the context of soft tissue biomechanics, in particular, includes some papers that are misleading since they contain errors and false statements. Claims that planar biaxial testing can fully characterize the three-dimensional anisotropic elastic properties of soft tissues are incorrect. There is therefore a need to clarify the extent to which biaxial testing can be used for determining the elastic properties of these materials. In this paper this is explained on the basis of the equations of finite deformation transversely isotropic elasticity, and general planar anisotropic elasticity. It is shown that it is theoretically impossible to fully characterize the properties of anisotropic elastic materials using such tests unless some assumption is made that enables a suitable subclass of models to be preselected. Moreover, it is shown that certain assumptions underlying the analysis of planar biaxial tests are inconsistent with the classical linear theory of orthotropic elasticity. Possible sets of independent tests required for full material characterization are then enumerated.

A simple approach to nonlinear tensile stiffness for accurate cloth simulation

Abstract: Recent mechanical models for cloth simulation have evolved toward accurate representation of elastic stiffness based on continuum mechanics, converging to formulations that are largely analogous to fast finite element methods. In the context of tensile deformations, these formulations usually involve the linearization of tensors, so as to express linear elasticity in a simple way. However, this approach needs significant adaptations and approximations for dealing with the nonlinearities resulting from large cloth deformations. Toward our objective of accurately simulating the nonlinear properties of cloth, we show that this linearization can indeed be avoided and replaced by adapted strain-stress laws that precisely describe the nonlinear behavior of the material. This leads to highly streamlined computations that are particularly efficient for simulating the nonlinear anisotropic tensile elasticity of highly deformable surfaces. We demonstrate the efficiency of this method with examples related to accurate garment simulation from experimental tensile curves measured on actual materials.

Pulse propagation in a linear and nonlinear diatomic periodic chain: effects of acoustic frequency band-gap

Abstract: One-dimensional nonlinear phononic crystals have been assembled from periodic diatomic chains of stainless steel cylinders alternated with Polytetrafluoroethylene spheres. This system allows dramatic changes of behavior (from linear to strongly nonlinear) by application of compressive forces practically without changes of geometry of the system. The relevance of classical acoustic band-gap, characteristic for chain with linear interaction forces and derived from the dispersion relation of the linearized system, on the transformation of single and multiple pulses in linear, nonlinear and strongly nonlinear regimes are investigated with numerical calculations and experiments. The limiting frequencies of the acoustic band-gap for investigated system with given precompression force are within the audible frequency range (20–20,000 Hz) and can be tuned by varying the particle’s material properties, mass and initial compression. In the linear elastic chain the presence of the acoustic band-gap was apparent through fast transformation of incoming pulses within very short distances from the chain entrance. It is interesting that pulses with relatively large amplitude (nonlinear elastic chain) exhibit qualitatively similar behavior indicating relevance of the acoustic band gap also for transformation of nonlinear signals. The effects of an in situ band-gap created by a mean dynamic compression are observed in the strongly nonlinear wave regime.

Fine Tuning the Adhesive Properties of a Soft Nanostructured Adhesive with Rheological Measurements

Abstract: The major objective of this article is to present recent advances in the methodology to fine tune the adhesive performance of a PSA. In addition to the so-called Dahlquist criterion requiring a low modulus, we propose two additional rheological predictors of the adhesive properties. The first one is derived from the description of the detachment of a linear elastic layer from a rigid substrate. We made an approximate extension of this analysis to the viscoelastic regime and showed that the transition from interfacial cracks to cavitation and fibrillation can be quantitatively predicted from the easily measurable ratio tan(δ)/G′(ω). If a fibrillar structure is formed, the nonlinear large strain properties become important. We showed that the ability of the fibrils to be stretched before final debonding can be predicted from the analysis of simple tensile tests. The softening, which occurs at intermediate strains, and, more importantly, the hardening which occurs at large strains, can be used to predict the...

Physically-Based Approach to the Mechanics of Strong Non-Local Linear Elasticity Theory

Abstract: In this paper the physically-based approach to non-local elasticity theory is introduced. It is formulated by reverting the continuum to an ensemble of interacting volume elements. Interactions between adjacent elements are classical contact forces while long-range interactions between non-adjacent elements are modelled as distance-decaying central body forces. The latter are proportional to the relative displacements rather than to the strain field as in the Eringen model and subsequent developments. At the limit the displacement field is found to be governed by an integro-differential equation, solved by a simple discretization procedure suggested by the underlying mechanical model itself, with corresponding static boundary conditions enforced in a quite simple form. It is then shown that the constitutive law of the proposed model coalesces with the Eringen constitutive law for an unbounded domain under suitable assumptions, whereas it remains substantially different for a bounded domain. Thermodynamic consistency of the model also has been investigated in detail and some numerical applications are presented for different parameters and different functional forms for the decay of the long range forces. For simplicity, the problem is formulated for a 1D continuum while the general formulation for a 3D elastic solid has been reported in the appendix.

A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy.

Abstract: This is an essay on a linear Cosserat model with weakest possible constitutive assumptions on the curvature energy still providing for existence, uniqueness and stability. The assumed curvature energy is the conformally invariant expression L 2 k dev symr axlAk 2 , where axlA is the axial vector of the skewsymmetric microrotation A 2 so(3), dev is the orthogonal projection on the Lie-algebra sl(3) of trace free matrices and sym is the orthogonal projection onto symmetric matrices. It is observed that unphysical singular stiening for small samples is avoided in torsion and bending while size eects are still present. The number of Cosserat parameters is reduced from six to four: in addition to the (size-independent) classical linear elastic Lam e moduli and only one Cosserat coupling constant c > 0 and one length scale parameter Lc > 0 need to be determined. We investigate those deformations not leading to moment stresses for dierent curvature

Flexible microfluidic device for mechanical property characterization of soft viscoelastic solids such as bacterial biofilms

Abstract: We introduce a flexible microfluidic device to characterize the mechanical properties of soft viscoelastic solids such as bacterial biofilms. In the device, stress is imposed on a test specimen by the application of a fixed pressure to a thin, flexible poly(dimethyl siloxane) (PDMS) membrane that is in contact with the specimen. The stress is applied by pressurizing a microfabricated air channel located above the test area. The strain resulting from the applied stress is quantified by measuring the membrane deflection with a confocal laser scanning microscope. The deflection is governed by the viscoelastic properties of the PDMS membrane and of the test specimen. The relative contributions of the membrane and test material to the measured deformation are quantified by comparing a finite element analysis with an independent (control) measurement of the PDMS membrane mechanical properties. The flexible microfluidic rheometer was used to characterize both the steady-state elastic modulus and the transient strain recoil of two soft materials: gellan gums and bacterial biofilms. The measured linear elastic moduli and viscoelastic relaxation times of gellan gum solutions were in good agreement with the results of conventional mechanical rheometry. The linear Young's moduli of biofilms of Staphylococcus epidermidis and Klebsiella pneumoniae, which could not be measured using conventional methods, were found to be 3.2 and 1.1 kPa, respectively, and the relaxation time of the S. epidermidis biofilm was 13.8 s. Additionally, strain hardening was observed in all the biofilms studied. Finally, design parameters and detection limits of the method show that the device is capable of characterizing soft viscoelastic solids with elastic moduli in the range of 102-105 Pa. The flexible microfluidic rheometer addresses the need for mechanical property characterization of soft viscoelastic solids common in fields such as biomaterials, food, and consumer products. It requires only 200 pL of the test specimen.

A hybridizable discontinuous Galerkin method for linear elasticity

Abstract: This paper describes the application of the so-called hybridizable discontinuous Galerkin (HDG) method to linear elasticity problems. The method has three significant features. The first is that the only globally coupled degrees of freedom are those of an approximation of the displacement defined solely on the faces of the elements. The corresponding stiffness matrix is symmetric, positive definite, and possesses a block-wise sparse structure that allows for a very efficient implementation of the method. The second feature is that, when polynomials of degree k are used to approximate the displacement and the stress, both variables converge with the optimal order of k+1 for any k⩾0. The third feature is that, by using an element-by-element post-processing, a new approximate displacement can be obtained that converges at the order of k+2, whenever k⩾2. Numerical experiments are provided to compare the performance of the HDG method with that of the continuous Galerkin (CG) method for problems with smooth solutions, and to assess its performance in situations where the CG method is not adequate, that is, when the material is nearly incompressible and when there is a crack. Copyright © 2009 John Wiley & Sons, Ltd.

Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach

Abstract: In this paper, we examine heuristically the reasons for locking in poroelasticity. As a first step, we first reexamine the problem of locking in linear elasticity. From this, we discover how the problem arises in the poroelasticity setting and how the problem might be overcome.

Nonlinear mesoscopic elasticity : the complex behaviour of granular media including rocks and soil

Abstract: Preface . Acknowledgements . 1 Introduction . 1.1 Systems . 1.2 Examples of Phenomena . 1.3 The Domain of Exploration . 1.4 Outline . References . 2 Microscopic/Macroscopic Formulation of the Traditional Theory of Linear and Nonlinear Elasticity . 2.1 Prefatory Remarks . 2.2 From Microscopic to Continuum . 2.3 Continuum Elasticity and Macroscopic Phenomenology . 2.4 Thermodynamics . 2.5 Energy Scales . References . 3 Traditional Theory of Nonlinear Elasticity, Results . 3.1 Quasistatic Response Linear and Nonlinear . 3.2 Dynamic Response Linear . 3.3 Quasistatic/Dynamic Response Nonlinear . 3.4 Dynamic Response Nonlinear . 3.5 Exotic Response Nonlinear . 3.6 Green Functions . References . 4 Mesoscopic Elastic Elements and Macroscopic Equations of State . 4.1 Background . 4.2 Elastic Elements . 4.3 Effective Medium Theory . 4.4 Equations of State Examples . References . 5 Auxiliary Fields . 5.1 Temperature . 5.2 Saturation . 5.3 The Conditioning Field, X . References . 6 Hysteretic Elastic Elements . 6.1 Finite Displacement Elastic Elements Quasistatic Response . 6.2 Finite Displacement Elastic Elements: Inversion . 6.3 Finite Displacement Elastic Elements: Dynamic Response . 6.4 Models with Hysteresis . 6.5 Summary . 6.6 Models with Hysteresis, Detail . References . 7 The Dynamics of Elastic Systems Fast and Slow . 7.1 Fast/Slow Linear Dynamics . 7.2 Fast Nonlinear Dynamics . 7.3 Auxiliary Fields and Slow Dynamics . 7.4 Summary . References . 8 Q and Issues of Data Modeling/Analysis . 8.1 Attenuation in Linear Elastic Systems . 8.2 Nonlinear Attenuation . 8.3 Why Measure Q ? 8.4 How to Measure Q. 8.5 Resonant Bar Revisited . References . 9 Elastic State Spectroscopies and Elastic State Tomographies . 9.1 Spectroscopies . 9.2 Tomographies, Linear, Inhomogeneous . 9.3 Tomographies, Nonlinear, Inhomogeneous . References . 10 Quasistatic Measurements . 10.1 Some Basic Observations . 10.2 Quasistatic Stress-Strain Data Hysteresis . 10.3 Coupling to Auxiliary Fields . 10.4 Inversion . References . 11 Dynamic Measurements . 11.1 Quasistatic-Dynamic . 11.2 Dynamic-Dynamic . 11.3 Examples of Systems . References . 12 Field Observations . 12.1 Active Probes . 12.2 Passive Probes . References . 13 Nonlinear Elasticity and Nondestructive Evaluation and Imaging . 13.1 Overview . 13.2 Historical Context . 13.3 Simple Conceptual Model of a Crack in an Otherwise Elastically Linear Solid . 13.4 Nonlinear Elastic Wave Spectroscopy in Nondestructive Evaluation (NEWS) . 13.5 Progressive Mechanical Damage Probed by NEWS Techniques . 13.6 Mechanical Damage Location and Imaging . 13.7 Other Methods for Extracting the Elastic Nonlinearity . 13.8 Summary . References . Color Plates . Index .

Connections between Elastic and Conductive Properties of Heterogeneous Materials

Abstract: We discuss cross-property connections that interrelate effective linear elastic and conductive properties of heterogeneous materials. More precisely, they relate changes in the properties, as compared to the ones of the bulk material, caused by various inhomogeneities (cracks, pores, inclusions). They may also be developed for microstructures formed by multiple contacts, such as rough surfaces pressed against each other. Such connections are especially useful if one property (say, electrical conductivity) is easier to measure than the other (anisotropic elastic constants). For the properties governed by mathematically similar laws (for example, electrical and thermal conductivities), the connections are straightforward. However, for the elasticity–conductivity connections – the main focus of the present work – their very existence is nontrivial: not only the governing equations are different but even the ranks of tensors characterizing the properties are different (fourth-rank tensor of elastic constants versus second-rank conductivity tensor). We overview various approaches to the problem and then advance the approach rooted in similarity of the microstructural parameters that control the given pair of properties. This similarity leads to connections that, albeit approximate, have explicit closed form. They have been experimentally verified on several heterogeneous materials (metal foams, short fiber reinforced composites, metals with fatigue microcracks, sprayed coatings). Moreover, for properties controlled by entirely essentially different parameters (such as permeability or fracture of a microcracked material and its elasticity), the correlations may hold only qualitatively, at best.

Robust Mesh Deformation using the Linear Elasticity Equations

Abstract: A modification is proposed to the equations of linear elasticity as used to deform Euler and Navier-Stokes meshes. In particular it is seen that the equations do not admit rigid body rotations as solutions, and it is shown how these solutions may be recovered by modifying the constitutive law. The result is significantly more robust to general deformations, and combined with incremental application generates valid meshes well beyond the point at which remeshing is required.

Static and dynamic moduli of a weak sandstone

Abstract: Static moduli derived from the slope of a stress-strain curve and dynamic moduli derived from the velocity of elastic waves are significantly different for rocks, even though they should be equal according to the theory of linear elasticity. Proper knowledge about this difference might be useful because dynamic measurements are often the only information available about a rock. In tests on a dry sandstone, static and dynamic moduli are always different, except immediately after the direction of loading has been reversed. The results support the assumption that the difference between static and dynamic moduli can be ascribed to the difference in strain amplitude between static and dynamic measurements. At low stress levels, static and dynamic moduli increase with increasing stress during initial loading. In uniaxial compaction tests, the static compaction modulus decreases with increasing stress at higher stress levels, revealing a sensitivity to the location of the failure envelope. However, the correspon...