Showing papers on "Linear elasticity published in 2006"

Elasticity: Theory, Applications, and Numerics

Abstract: Elasticity: Theory, Applications, and Numerics, Third Edition, continues its market-leading tradition of concisely presenting and developing the linear theory of elasticity, moving from solution methodologies, formulations, and strategies into applications of contemporary interest, such as fracture mechanics, anisotropic and composite materials, micromechanics, nonhomogeneous graded materials, and computational methods. Developed for a one- or two-semester graduate elasticity course, this new edition has been revised with new worked examples and exercises, and new or expanded coverage of areas such as spherical anisotropy, stress contours, isochromatics, isoclinics, and stress trajectories. Using MATLAB software, numerical activities in the text are integrated with analytical problem solutions. These numerics aid in particular calculations, graphically present stress and displacement solutions to problems of interest, and conduct simple finite element calculations, enabling comparisons with previously studied analytical solutions. Online ancillary support materials for instructors include a solutions manual, image bank, and a set of PowerPoint lecture slides. * Thorough yet concise introduction to linear elasticity theory and applications* Only text providing detailed solutions to problems of nonhomogeneous/graded materials* New material on stress contours/lines, contact stresses, curvilinear anisotropy applications* Further and new integration of MATLAB software* Addition of many new exercises* Comparison of elasticity solutions with elementary theory, experimental data, and numerical simulations* Online solutions manual and downloadable MATLAB code

Fundamentals of Micromechanics of Solids

Abstract: Preface. 1 Introduction. 1.1 Background and Motivation. 1.2 Objectives. 1.3 Organization of Book. 1.4 Notation Conventions. References. 2 Basic Equations of Continuum Mechanics. 2.1 Displacement and Deformation. 2.2 Stresses and Equilibrium. 2.3 Energy, Work, and Thermodynamic Potentials. 2.4 Constitutive Laws. 2.5 Boundary Value Problems for Small-Strain Linear Elasticity. 2.6 Integral Representations of Elasticity Solutions. Problems. Appendix 2.A. Appendix 2.B. Appendix 2.C. References. Suggested Readings. 3 Eigenstrains. 3.1 Definition of Eigenstrains. 3.2 Some Examples of Eigenstrains. 3.3 General Solutions of Eigenstrain Problems. 3.4 Examples. Problems. Appendix 3.A. Appendix 3.B. References. Suggested Readings. 4 Inclusions and Inhomogeneities. 4.1 Definitions of Inclusions and Inhomogeneities. 4.2 Interface Conditions. 4.3 Ellipsoidal Inclusion with Uniform Eigenstrains (Eshelby Solution). 4.4 Ellipsoidal Inhomogeneities. 4.5 Inhomogeneous Inhomogeneities. Problems. Appendix 4.A. Appendix 4.B. Suggested Readings. 5 Definitions of Effective Moduli of Heterogeneous Materials. 5.1 Heterogeneity and Length Scales. 5.2 Representative Volume Element. 5.3 Random Media. 5.4 Macroscopic Averages. 5.5 Hill's Lemma. 5.6 Definitions of Effective Modulus of Heterogeneous Media. 5.7 Concentration Tensors and Effective Properties. Problems. Suggested Readings. 6 Bounds for Effective Moduli. 6.1 Classical Variational Theorems in Linear Elasticity. 6.2 Voigt Upper Bound and Reuss Lower Bound. 6.3 Extensions of Classical Variational Principles. 6.4 Hashin-Shtrikman Bounds. Problems. Appendix 6.A. References. Suggested Readings. 7 Determination of Effective Moduli. 7.1 Basic Ideas of Micromechanics for Effective Properties. 7.2 Eshelby Method. 7.3 Mori-Tanaka Method. 7.4 Self-Consistent Methods for Composite Materials. 7.5 Self-Consistent Methods for Polycrystalline Materials. 7.6 Differential Schemes. 7.7 Comparison of Different Methods. Problems. Suggested Readings. 8 Determination of the Effective Moduli-Multiinclusion Approaches. 8.1 Composite-Sphere Model. 8.2 Three-Phase Model. 8.3 Four-Phase Model. 8.4 Multicoated Inclusion Problem. Problems. Appendix 8.A. Appendix 8.B. Appendix 8.C. References. Suggested Readings. 9 Effective Properties of Fiber-Reinforced Composite Laminates. 9.1 Unidirectional Fiber-Reinforced Composites. 9.2 Effective Properties of Multilayer Composites. 9.3 Effective Properties of a Lamina. 9.4 Effective Properties of a Laminated Composite Plate. Problems. Appendix 9.A. References. Suggested Readings. 10 Brittle Damage and Failure of Engineering Composites. 10.1 Imperfect Interfaces. 10.2 Fiber Bridging. 10.3 Transverse Matrix Cracks. Problems. Appendix 10.A. References. Suggested Readings. 11 Mean Field Theory for Nonlinear Behavior. 11.1 Eshelby's Solution and Kro..ner's Model. 11.2 Applications. 11.3 Time-Dependent Behavior of Polycrystalline Materials: Secant Approach. Problems. References. 12 Nonlinear Properties of Composites Materials: Thermodynamic Approaches. 12.1 Nonlinear Behavior of Constituents. 12.2 Effective Potentials. 12.3 The Secant Approach. Problems. Suggested Readings. 13 Micromechanics of Martensitic Transformation in Solids. 13.1 Phase Transformation Mechanisms at Different Scales. 13.2 Application: Thermodynamic Forces and Constitutive Equations for Single Crystals. 13.3 Overall Behavior of Polycrystalline Materials with Phase Transformation. Problems. References. Suggested Readings. Index.

Dynamical fracture instabilities due to local hyperelasticity at crack tips

Abstract: As the speed of a crack propagating through a brittle material increases, a dynamical instability leads to an increased roughening of the fracture surface. Cracks moving at low speeds create atomically flat mirror-like surfaces; at higher speeds, rougher, less reflective ('mist') and finally very rough, irregularly faceted ('hackle') surfaces are formed. The behaviour is observed in many different brittle materials, but the underlying physical principles, though extensively debated, remain unresolved. Most existing theories of fracture assume a linear elastic stress-strain law. However, the relation between stress and strain in real solids is strongly nonlinear due to large deformations near a moving crack tip, a phenomenon referred to as hyperelasticity. Here we use massively parallel large-scale atomistic simulations--employing a simple atomistic material model that allows a systematic transition from linear elastic to strongly nonlinear behaviour--to show that hyperelasticity plays a governing role in the onset of the instability. We report a generalized model that describes the onset of instability as a competition between different mechanisms controlled by the local stress field and local energy flow near the crack tip. Our results indicate that such instabilities are intrinsic to dynamical fracture and they help to explain a range of controversial experimental and computational results.

A Generalized Description of the Elastic Properties of Nanowires

Abstract: We report a model of nanowire (NW) mechanics that describes force vs displacement curves over the entire elastic range for diverse wire systems. Due to the clamped-wire measurement configuration, the force response in the linear elastic regime can be linear or nonlinear, depending on the system and the wire displacement. For Au NWs the response is essentially linear since yielding occurs prior to the onset of the inherent nonlinearity, while for Si NWs the force response is highly nonlinear, followed by brittle fracture. Since the method describes the entire range of elastic deformation, it unequivocally identifies the yield points in both of these materials.

Non-Hertzian approach to analyzing mechanical properties of endothelial cells probed by atomic force microscopy.

Abstract: Detailed measurements of cell material properties are required for understanding how cells respond to their mechanical environment. Atomic force microscopy (AFM) is an increasingly popular measurement technique that uniquely combines subcellular mechanical testing with high-resolution imaging. However, the standard method of analyzing AFM indentation data is based on a simplified "Hertz" theory that requires unrealistic assumptions about cell indentation experiments. The objective of this study was to utilize an alternative "pointwise modulus" approach, that relaxes several of these assumptions, to examine subcellular mechanics of cultured human aortic endothelial cells (HAECs). Data from indentations in 2- to 5-microm square regions of cytoplasm reveal at least two mechanically distinct populations of cellular material. Indentations colocalized with prominent linear structures in AFM images exhibited depth-dependent variation of the apparent pointwise elastic modulus that was not observed at adjacent locations devoid of such structures. The average pointwise modulus at an arbitrary indentation depth of 200 nm was 5.6+/-3.5 kPa and 1.5+/-0.76 kPa (mean+/-SD, n=7) for these two material populations, respectively. The linear structures in AFM images were identified by fluorescence microscopy as bundles of f-actin, or stress fibers. After treatment with 4 microM cytochalasin B, HAECs behaved like a homogeneous linear elastic material with an apparent modulus of 0.89+/-0.46 kPa. These findings reveal complex mechanical behavior specifically associated with actin stress fibers that is not accurately described using the standard Hertz analysis, and may impact how HAECs interact with their mechanical environment.

A composites-based hyperelastic constitutive model for soft tissue with application to the human annulus fibrosus

Abstract: This paper presents a composites-based hyperelastic constitutive model for soft tissue. Well organized soft tissue is treated as a composite in which the matrix material is embedded with a single family of aligned fibers. The fiber is modeled as a generalized neo-Hookean material in which the stiffness depends on fiber stretch. The deformation gradient is decomposed multiplicatively into two parts: a uniaxial deformation along the fiber direction and a subsequent shear deformation. This permits the fiber–matrix interaction caused by inhomogeneous deformation to be estimated by using effective properties from conventional composites theory based on small strain linear elasticity and suitably generalized to the present large deformation case. A transversely isotropic hyperelastic model is proposed to describe the mechanical behavior of fiber-reinforced soft tissue. This model is then applied to the human annulus fibrosus. Because of the layered anatomical structure of the annulus fibrosus, an orthotropic hyperelastic model of the annulus fibrosus is developed. Simulations show that the model reproduces the stress–strain response of the human annulus fibrosus accurately. We also show that the expression for the fiber–matrix shear interaction energy used in a previous phenomenological model is compatible with that derived in the present paper.

Buckling of swelling gels

Abstract: The patterns arising from the differential swelling of gels are investigated experimentally and theoretically as a model for the differential growth of living tissues. Two geometries are considered: a thin strip of soft gel clamped to a stiff gel, and a thin corona of soft gel clamped to a disk of stiff gel. When the structure is immersed in water, the soft gel swells and bends out of plane leading to a wavy periodic pattern whose wavelength is measured. The linear stability of the flat state is studied in the framework of linear elasticity using the equations for thin plates. The flat state is shown to become unstable to oscillations above a critical swelling rate and the computed wavelengths are in quantitative agreement with the experiment.

The Cosserat couple modulus for continuous solids is zero viz the linearized Cauchy-stress tensor is symmetric

Abstract: We investigate weaker than usual constitutive assumptions in linear Cosserat theory still providing for existence and uniqueness and show a continuous dependence result for Cosserat couple modulus µc → 0. This result is needed when using Cosserat elasticity not as a physical model but as a numerical regularization device. Thereafter it is shown that the usually adopted material restrictions of uniform positivity for a linear Cosserat model cannot be consistent with experimental findings for continuous solids: the analytical solutions for both the torsion and the bending problem in general predict an unbounded stiffness for ever thinner samples. This unphysical behaviour can only be avoided for specific choices of parameters in the curvature energy expression. However, these choices do not satisfy the usual constitutive restrictions. We show that the possibly remaining linear elastic Cosserat problem is nevertheless well-posed but that it is impossible to determine the appearing curvature modulus independent of boundary conditions. This puts a doubt on the use of the linear elastic Cosserat model (or the geometrically exact model with µc > 0) for the physically consistent description of continuous solids like polycrystals in the framework of elasto-plasticity. The problem is avoided in geometrically exact Cosserat models if the Cosserat couple modulus µc is set to zero.

Failure criteria for linear elastic materials with U-notches

Abstract: This paper provides a simple, albeit accurate, criterion for prediction of the rupture loads of brittle, or quasi-brittle, U-notched samples, where linear elastic fracture mechanics is not applicable because blunted notches do not exhibit stress singularities. Good agreement is found between numerical predictions and experimental results. The results of fracture tests from 18 different ceramic materials and a polymer (at − 60°C) are summarized and are used as a reference for checking the fracture criterion. Seven fracture criteria are reviewed and it is shown that all can be recast into the proposed criterion.

On the Size of RVE in Finite Elasticity of Random Composites

Abstract: This paper presents a quantitative study of the size of representative volume element (RVE) of random matrix-inclusion composites based on a scale-dependent homogenization method. In particular, mesoscale bounds defined under essential or natural boundary conditions are computed for several nonlinear elastic, planar composites, in which the matrix and inclusions differ not only in their material parameters but also in their strain energy function representations. Various combinations of matrix and inclusion phases described by either neo-Hookean or Ogden function are examined, and these are compared to those of linear elastic types.

Behavior of Corrugated Web I-Girders under In-Plane Loads

Abstract: A theoretical formulation of the linear elastic in-plane and torsional behavior of corrugated web I-girders under in-plane loads is presented. A typical corrugated web steel I-girder consists of two steel flanges welded to a corrugated steel web. Under a set of simplifying assumptions, the equilibrium of an infinitesimal length of a corrugated web I-girder is studied, and the cross-sectional stresses and stress resultants due to primary bending moment and shear are deduced. The analysis shows that a corrugated web I-girder will twist out-of-plane simultaneously as it deflects in-plane under the action of in-plane loads. In the paper, the in-plane bending behavior is analyzed using conventional beam theory, whereas the out-of-plane torsional behavior is analyzed as a flange transverse bending problem. The results for a simply supported span subjected to a uniformly distributed load are presented. Finally, finite element analysis results are presented and compared to the theoretical results for validation.

On the plane strain and plane stress solutions of functionally graded rotating solid shaft and solid disk problems

Abstract: Closed form solutions to functionally graded rotating solid shaft and rotating solid disk problems are obtained under generalized plane strain and plane stress assumptions, respectively. The nonhomogeneity in the material arises from the fact that the modulus of elasticity of the material varies radially according to two different continuously nonlinear forms: exponential and parabolic. Both forms contain two material parameters and lead to finite values of the modulus of elasticity at the center. Analytical expressions for the stresses at the center are determined. These limiting expressions indicate that at the center of shaft/disk: (i) the stresses are finite, (ii) the radial and the circumferential stress components are equal, and (iii) the values of the stresses are independent of the variation of the modulus of elasticity. It is also shown mathematically that the nonhomogeneous solutions presented here reduce to those of homogeneous ones by an appropriate choice of the material parameters describing the variation of the modulus of elasticity.

Defferential Complexes and Stability of Finite Element Methods II: The Elasticity Complex

Abstract: A close connection between the ordinary de Rham complex and a corresponding elasticity complex is utilized to derive new mixed finite element methods for linear elasticity. For a formulation with weakly imposed symmetry, this approach leads to methods which are simpler than those previously obtained. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field. We also discuss how the strongly symmetric methods proposed in [8] can be derived in the present framework. The method of construction works in both two and three space dimensions, but for simplicity the discussion here is limited to the two dimensional case.

Experimental identification of a nonlinear model for composites using the grid technique coupled to the virtual fields method

Abstract: The present paper shows some experimental results of the identification of the full set of parameters driving a nonlinear model for the in-plane behaviour of a unidirectional composite laminate. The strain field at the surface of a rectangular coupon submitted to a shear/bending loading is measured using the grid technique. The strain fields are then processed by the virtual fields method to retrieve the six constitutive parameters: the linear elastic orthotropic in-plane stiffnesses Q xx , Q yy , Q xy , Q ss and softening parameters K and e s 0 driving the shear nonlinearity. It is shown that the shear response is correctly identified, with coefficients of variation similar to the ones of standard tests. The other parameters are identified with larger errors and higher coefficients of variation because their identification is affected by the lower normal strain levels in the specimen. It is therefore necessary to design new tests providing balanced strain levels for improving the identification procedure.

Locking-free adaptive discontinuous Galerkin FEM for linear elasticity problems

Abstract: An adaptive discontinuous Galerkin finite element method for linear elasticity problems is presented. We develop an a posteriori error estimate and prove its robustness with respect to nearly incompressible materials (absence of volume locking). Furthermore, we present some numerical experiments which illustrate the performance of the scheme on adaptively refined meshes.

Application of polygonal finite elements in linear elasticity

Abstract: In this paper, a conforming polygonal finite element method is applied to problems in linear elasticity. Meshfree natural neighbor (Laplace) shape functions are used to construct conforming interpolating functions on any convex polygon. This provides greater flexibility to solve partial differential equations on complicated geometries. Closed-form expressions for Laplace shape functions on pentagonal, hexagonal, heptagonal, and octagonal reference elements are derived. Numerical examples are presented to demonstrate the accuracy of the method in two-dimensional elastostatics.

Analysis of interfacial shear stresses in beams strengthened with bonded prestressed laminates

Abstract: In this paper, the problem of interfacial shear stresses in beams strengthened with bonded prestressed composite laminates is analyzed using linear elastic theory. The analysis provided a closed-form formula for calculating the critical maximum shear stress at the end of the laminate for a beam with arbitrary cross-section and material. A demonstration study on strengthening an existing steel bridge using this technique has also been conducted using FE-analysis. The results from both analyses agreed very well. Also, a parametric study is performed in order to identify the effects of various geometrical and material properties on the magnitude of interfacial shear stresses. The results show that there exists high concentration of shear stresses at the ends of the laminate, which might result in a premature failure of the strengthening scheme at these locations. Material properties such as laminate and adhesive stiffness and the dimensions of the laminate where all found to have a marked effect on the magnitude of maximum shear stress in the composite member.

Elastic and viscoelastic indentation of flat surfaces by pyramid indentors

Abstract: The present work presents the results of frictionless and adhesionless contact of flat surfaces by pyramid indentors. The materials of the contacting solids were modelled as homogeneous and isotropic, linear elastic, as well as linear viscoelastic. The theoretical analysis is complemented by experiments and numerical calculations. The results include explicit relations between the normal applied load and the depth of penetration, details of the contact area shapes, the surface stresses and the contact pressure distributions. The standard shapes of the Vickers, Berkovich and Knoop pyramids were examined in particular. Certain aspects of geometrical imperfections, transverse isotropy and adhesion were considered. When elasticity or viscoelasticity provide adequate models of material behavior, micro- or nano-indentation by sharp pyramid indentors can be very useful and perhaps the only possible test in probing mechanical properties of small volumes of materials. The results can be particularly useful in using instrumented indentation for assessing mechanical properties of materials at cryogenic temperatures, of bio-materials and of micro-electro-mechanical components.

Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb{RT}_0$-$\mathbb{P}_1$-$\mathbb{P}_0$ approximations

Abstract: We present a new stabilized mixed finite element method for the linear elasticity problem in R 2 . The approach is based on the introduction of Galerkin least-squares terms arising from the constitutive and equilibrium equations, and from the relation defining the rotation in terms of the dis- placement. We show that the resulting augmented variational formulation and the associated Galerkin scheme are well posed, and that the latter becomes locking-free and asymptotically locking-free for Dirichlet and mixed boundary conditions, respectively. In particular, the discrete scheme allows the utilization of Raviart-Thomas spaces of lowest order for the stress tensor, piecewise linear elements for the displacement, and piecewise constants for the rotation. In the case of mixed boundary con- ditions, the essential one (Neumann) is imposed weakly, which yields the introduction of the trace of the displacement as a suitable Lagrange multiplier. This trace is then approximated by piecewise linear elements on an independent partition of the Neumann boundary whose mesh size needs to satisfy a compatibility condition with the mesh size associated to the triangulation of the domain. Several numerical results illustrating the good performance of the augmented mixed finite element scheme in the case of Dirichlet boundary conditions are also reported.