Showing papers on "Linear elasticity published in 1996"

A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites

Abstract: A variational formulation is employed to derive a micromechanics-based, explicit nonlocal constitutive equation relating the ensemble averages of stress and strain for a class of random linear elastic composite materials. For two-phase composites with any isotropic and statistically uniform distribution of phases (which themselves may have arbitrary shape and anisotropy), we show that the leading-order correction to a macroscopically homogeneous constitutive equation involves a term proportional to the second gradient of the ensemble average of strain. This nonlocal constitutive equation is derived in explicit closed form for isotropic material in the one case in which there exists a well-founded physical and mathematical basis for describing the material's statistics: a matrix reinforced (or weakened) by a random dispersion of nonoverlapping identical spheres. By assessing, when the applied loading is spatially-varying, the magnitude of the nonlocal term in this constitutive equation compared to the portion of the equation that relates ensemble average stresses and strains through a constant “overall” modulus tensor, we derive quantitative estimates for the minimum representative volume element (RVE) size, defined here as that over which the usual macroscopically homogeneous “effective modulus” constitutive models for composites can be expected to apply. Remarkably, for a maximum error of 5% of the constant “overall” modulus term, we show that the minimum RVE size is at most twice the reinforcement diameter for any reinforcement concentration level, for several sets of matrix and reinforcement moduli characterizing large classes of important structural materials. Such estimates seem essential for determining the minimum structural component size that can be treated by macroscopically homogeneous composite material constitutive representations, and also for the development of a fundamentally-based macroscopic fracture mechanics theory for composites. Finally, we relate our nonlocal constitutive equation explicitly to the ensemble average strain energy, and show how it is consistent with the stationary energy principle.

Exact second-order estimates for the effective mechanical properties of nonlinear composite materials

Abstract: Motivated by previous small-contrast perturbation estimates, this paper proposes a new method for estimating the effective behavior of nonlinear composite materials with arbitrary phase contrast. The key idea is to write down a second-order Taylor expansion for the phase potentials, about appropriately defined phase average strains. The resulting estimates, which are exact to second order in the contrast, involve the “tangent” modulus tensors of the nonlinear phase potentials, and reduce the problem for the nonlinear composite to a linear problem for an anisotropic thermoelastic composite. Making use of a well-known result by Levin for two-phase thermoelastic composites, together with estimates of the Hashin-Shtrikman type for linear elastic composites, explicit results are generated for two-phase nonlinear composites with statistically isotropic particulate microstructures. Like the earlier small-contrast asymptotic results, the new estimates are found to depend on the determinant of the strain, but unlike the small-contrast results that diverge for shear loading conditions in the nonhardening limit, the new estimates remain bounded and reduce to the classical lower bound in this limiting case. The general method is applied to composites with power-law constitutive behavior and the results are compared with available bounds and numerical estimates, as well as with other nonlinear homogenization procedures. For the cases considered, the new estimates are found to satisfy the restrictions imposed by the bounds, to improve on the predictions of prior homogenization procedures and to be in excellent agreement with the results of the numerical simulations.

A unified approach to the evaluation of linear elastic stress fields in the neighborhood of cracks and notches

Abstract: The problem of evaluating linear elastic stress fields in the neighborhood of cracks and notches is considered. An analytical solution valid for cracked and notched components is given in general terms, according to Muskhelishvili's method based on complex stress functions. The solution is particularly useful for V-shape notches in wide and finite plates under uniform tensile loading. It will be demonstrated that some remarkable solutions of fracture mechanics and notch analysis already reported in the literature can be considered special cases of this general solution, as soon as appropriate values of the free parameters are adopted.

Strengthening of concrete prisms using the plate-bonding technique

Abstract: This paper presents the use of fracture mechanics for the plate bonding technique. Plates of steel or carbon-fibre reinforced plastic are bonded with an epoxy adhesive to rectangular concrete prisms and loaded in shear up to failure, what is normally known in fracture mechanics as mode II failure. In this special application a linear and a nonlinear approach are presented. The nonlinear equation derived for a realistic shear-deformation curve can only be used for numerical calculations. However, for simplified shear-deformation curves, the derived formula can be solved analytically. Results from tests, which are compared with the theory, are also presented.

A theory of local limiting speed in dynamic fracture

Abstract: A nonlinear continuum analysis is developed to show that stable, steady-state crack motion is limited not only by the macroscopic Rayleigh wave speed as asserted by the established theory of dynamic fracture, but also by a local wave speed governed by the elastic response near the crack tip. The local limiting speed ensures that a subsonic steady-state field can be established in highly nonlinear material regions prior to rupture. A two-dimensional triangular lattice with nearest-neighbor interatomic bonding is studied as a model nonlinear elastic solid that is isotropic under infinitesimal strains, but becomes anisotropic and nonlinear when the lattice is heavily stretched. The local limiting speed is determined by considering the most critical state of deformation on the verge of bond rupture. If the critical state is assumed to be under equibiaxial stressing, the local limiting speed is found to be v 1 = c s σ max μ , where cs is the macroscopic shear wave speed, μ is the shear modulus and σmax is the equibiaxial cohesive strength of the solid (i.e. the maximum equibiaxial tensile stress that a flawless solid can stand without spontaneous rupture). The generality of this result is discussed by relaxing the restrictions in the model problem. It is also shown that lattice dispersion in front of a crack tip can further reduce the speed of bond-breaking stress waves with wavelength on the order of a few atomic spacings. This study lends further support for a viewpoint previously discussed by the author that high speed dynamic fracture involves a competition between a high inertia local crack-tip field and the surrounding low inertia apparent crack field. Motivated by recent molecular dynamics simulations of crack propagation in a 6–12 Lenard-Jones lattice, a variational principle for steady-state deformation is used together with a conjugate gradient minimization algorithm to compute atomistic responses near the tip of a crack moving with constant speed in a similar Lenard-Jones lattice. The computation is performed over a block which moves with the crack and is subjected along the boundary to a low inertia displacement field based on existing solutions for cracks moving in linear elastic solids. The critical velocity at the onset of local crack branching is found to be 0.30cs, in almost exact agreement with the earlier molecular dynamics study. In this case, the local limiting speed is calculated to be v1 = 0.37cs, which is 20% larger than the observed value. This difference can be attributed to the effects of local lattice dispersion. The results are fully supportive of the notion that global-local inertia competition is a key to understanding dynamic fracture instabilities.

Fundamental limitations on the contrast-transfer efficiency in elastography: an analytic study.

Abstract: Elastography is a new ultrasonic imaging technique introduced to produce images of the Young's modulus distribution of compliant tissue. This Young's modulus distribution is derived from the ultrasonically estimated longitudinal internal strains induced by an external compression of the tissue. The displayed two-dimensional images are called elastograms. Recently, contrast-transfer efficiency, defined as the ratio of elasticity contrast as measured from elastogram to the true contrast, was used to illustrate by simulation the fundamental limitation of elastography in displaying the elastic modulus contrast of soft inclusion in a hard background and vice versa. In this paper, using a classical analytic solution of the elasticity equations derived for an infinite medium subjected to a uniaxial compression, we confirm such earlier simulations results. For this purpose we derive an analytic expression predicting the observed contrast in elastograms.

Morphologically representative pattern-based bounding in elasticity

Abstract: A general theory for the homogenization of heterogeneous linear elastic materials that relies on the concept of “morphologically representative pattern” is given. It allows the derivation of rigorous bounds for the effective behaviour of the Voigt-Reuss-type, which apply to any distribution of patterns, or of the Hashin-Shtrikman-type, which are restricted to materials whose pattern distributions are isotropic. Particular anisotropic distributions of patterns can also be considered: Hashin-Shtrikman-type bounds for anisotropic media are then generated. The resolution of the homogenization problem leads to a complex composite inclusion problem with no analytical solution in the general case. Here it is solved by a numerical procedure based on the finite element method. As an example of possible application, this procedure is used to derive new bounds for matrix-inclusion composites with cubic symmetry as well as for transversely isotropic materials.

Cosserat and Cauchy Materials as Continuum Models of Brick Masonry

Abstract: Continuum modeling for masonry-like material accounting for bricks or blocks texture is discussed. The constitutive functions for the contact actions—expressed in terms of size, shape and arrangement of the block assembly-are derived within the framework of the linear elastic Cosserat and Cauchy theories. By varying some important geometrical parameters: the scale factor between the wall and the blocks size, the shape of the bricks and their arrangement, micropolar materials with particular internal constraints are obtained. In a few situations the constrained continuum behaves as a Cauchy continuum. In general, the Cauchy continuum does not provide a proper description of the brick masonry behaviour while the structured continuum model, accounting for the mutual blocks rotation, gives satisfactory results.

Consistent gradient formulation for a stable enhanced strain method for large deformations

Abstract: Considers the problem of stability of the enhanced strain elements in the presence of large deformations. The standard orthogonality condition between the enhanced strains and constant stresses ensures satisfaction of the patch test and convergence of the method in case of linear elasticity. However, this does not hold in the case of large deformations. By analytic derivation of the element eigenvalues in large strain states additional orthogonality conditions can be derived, leading to a stable formulation, regardless of the magnitude of deformations. Proposes a new element based on a consistent formulation of the enhanced gradient with respect to new orthogonality conditions which it retains with four enhanced modes volumetric and shear locking free behaviour of the original formulation and does not exhibit hour‐glassing for large deformations.

Applied Continuum Mechanics

Abstract: 1. Introduction 1.1 General 1.2 Vectors and tensors 1.3 Green-gauss theorem and boundary surfaces 2. Deformations and kinematics 2.1 General 2.2 Coordinate systems 2.3 Non-cartesian 2.4 Strain tensor 2.5 Rate-of-deformation tensor 2.6 Coordinate transformations for strains 2.7 Dilatational and deviatoric properties 2.8 Compatibility equations 3. Equilibrium and kinetics 3.1 General 3.2 Forces and stresses 3.3 Basic balance laws 3.4 Coordinate transformations for stresses 3.5 Deviatoric stress tensor 3.6 Stresses with large strains 3.7 Equations of motion for large strains 4. Elastic solids 4.1 Constitutive equations for linear elastic Solids 4.2 Navier equations 4.3 Energy principles 4.4 Thermodynamics of solids 4.5 Finite elasticity 4.6 Torsional stress 4.7 Fiber components 5. Newtonian fluids 5.1 Introduction 5.2 Constitutive equations 5.3 Navier-Stokes system of equations 5.4 Compressible viscous flow 5.5 Ideal flow 5.6 Rotational flow 5.7 Turbulence 5.8 Boundary layer 5.9 Convective heat transfer 5.10 High-speed aerodynamics 5.11 Acoustics 5.12 Reacting flows References.

Neumann resonances in linear elasticity for an arbitrary body

Abstract: We study resonances (scattering poles) associated to the elasticity operator in the exterior of an arbitrary obstacle inR3 with Neumann boundary conditions. We prove that there exists a sequence of resonances tending rapidly to the real axis.

Application of homogenization FEM analysis to regular and re-entrant honeycomb structures

Abstract: Honeycomb structures are widely used in structural applications because of their high strength per density. Re-entrant honeycomb structures with negative Poisson's ratios may be envisaged to have many potential applications. In this study, an homogenization finite element method (FEM) technique developed for the analysis of spatially periodic materials is applied for the analysis of linear elastic responses of the regular and re-entrant honeycomb structures. Young's modulus of the regular honeycomb increased with volume fraction. Poisson's ratio of the regular honeycomb structure decreased from unity as volume fraction increased. The re-entrant honeycomb structure had a negative Poisson's ratio, its value dependent upon the inverted angle of cell ribs. Young's modulus of the re-entrant honeycomb structure decreased as the inverted angle of cell ribs increased. The results are in good agreement with previous analytical results. This homogenization theory is also applicable to three-dimensional foam materials — conventional and re-entrant.

Pressuring, Shearing, Torsion and Extension of a Circular Tube or Bar of Cylindrically Anisotropic Material

Abstract: One of the novel features of the present paper is that we have written the equation of equilibrium and the stress-strain law of an inhomogeneous anisotropic linear elastic material in a compact form for cylindrical coordinate system using matrix notation. For a two-dimensional deformation the result resembles Stroh’s sextic formalism in a rectangular coordinate system. We then consider the material to be cylindrically anisotropic. It means that the elastic stiffnesses referred to a cylindrical coordinate system are constants. The problem of a circular tube subjected to a uniform normal stress and shearing stresses at the inner and outer surfaces of the tube is studied. Also studied are the axial extension and torsion of the tube. Unlike isotropic materials for which the applied normal stress (or shear stress) induces only the normal (or shear) stress, all three displacement components and most of the six stress components are nonzero for general anisotropic materials. This is particularly interesting for the uniform axial extension of the tube. For an isotropic material the stress σ 33 is the only non-zero and uniform stress inside the tube. For a cylindrically anisotropic material the stresses σ rr , σ θθ , and σ θ3 are also non-zero. Moreover, they depend on r and are not uniform. A solid cylinder or a cylinder with a pin hole is a special case of a tube. It is shown that, for the loads mentioned above including the axial extension, the stress may be unbounded at the pinhole.

A symmetric linear elastic model for helical wire strands under axisymmetric loads

Abstract: Among several mathematical models for predicting the mechanical response of a helical wire strand to axisymmetric tension and torque derived in the literature over five decades, purely tensile wire linear elastic models have the symmetry of a stiffness matrix. Curiously, in those models where wire bending and torsion terms were included there was a lack of symmetry. In this paper the origin of the lack of symmetry in the earlier models has been identified and a symmetric model developed. The correct generalized strains for this purpose were derived using Wempner's theory and verified using Ramsey's theory. The validity of this model has been verified by comparing its results with that of earlier models and experiments available. This linear elastic symmetric model brings forth the much needed agreement between the global (strand) and the local (wire) responses which should help to simplify considerably the analysis of multi-layer strands and multi-strand wire ropes.

First-Order System Least Squares for Velocity-Vorticity-Pressure Form of the Stokes Equations, with Application to Linear Elasticity

Abstract: In this paper, we study the least-squares method for the generalized Stokes equations (including linear elasticity) based on the velocity-vorticity-pressure formulation in d = 2 or 3 dimensions. The least squares functional is defined in terms of the sum of the L(exp 2)- and H(exp -1)-norms of the residual equations, which is weighted appropriately by by the Reynolds number. Our approach for establishing ellipticity of the functional does not use ADN theory, but is founded more on basic principles. We also analyze the case where the H(exp -1)-norm in the functional is replaced by a discrete functional to make the computation feasible. We show that the resulting algebraic equations can be uniformly preconditioned by well-known techniques.

Simple shearing flow of a dry Kelvin soap foam

Abstract: Simple shearing flow of a dry soap foam composed of identical Kelvin cells is analysed. An undeformed Kelvin cell has six planar quadrilateral faces with curved edges and eight non-planar hexagonal faces with zero mean curvature. The elastic-plastic response of the foam is modelled by determining the bubble shape that minimizes total surface area at each value of strain. Computer simulations were performed with the Surface Evolver program developed by Brakke. The foam structure and macroscopic stress are piecewise continuous functions of strain. Each discontinuity corresponds to a topological change (T1) that occurs when the film network is unstable. These instabilities involve shrinking films, but the surface area and edge lengths of a shrinking film do not necessarily vanish smoothly with strain. Each T1 reduces surface energy, results in cell-neighbour switching, and provides a film-level mechanism for plastic yield behaviour during foam flow. The foam structure is determined for all strains by choosing initial foam orientations that lead to strain-periodic behaviour. The average shear stress varies by an order of magnitude for different orientations. A Kelvin foam has cubic symmetry and exhibits anisotropic linear elastic behaviour ; the two shear moduli and their average over all orientations are G min = 0.5706, G max = 0.9646, and G = 0.8070, where stress is scaled by T/V 1/3 , T is surface tension, and V is bubble volume. An approximate solution for the microrheology is also determined by minimizing the total surface area of a Kelvin foam with flat films.

Surgery Simulation Using fast Finite Elements

Abstract: This paper describes our recent work on real-time Surgery Simulation using Fast Finite Element models of linear elasticity [1]. In addition we discuss various improvements in terms of speed and realism.

Mathematical vs. Experimental Stress Analysis of Inhomogeneities in Solids

Abstract: The intent of this paper is to apply the discrete Fourier transform to the computation of eigenstress and eigenstrain fields around heterogeneities in composite materials. To this end the discrete Fourier transform is first briefly reviewed and then used to solve the basic equations of linear elasticity as pertinent to eigenstrained bodies under external loads. The results of this procedure are then used to discuss a few typical geometries such as an hexagonal two-dimensional array of thermally as well as elastically mismatched fibers in a composite matrix and a spherical Zirconia inclusion after a phase transformation. The resulting stress-strain fields are finally compared to experimental observations of eigenstress fields. The experimental techniques considered include photoelastic analyses as well as electron diffusion contrast techniques. It will be shown that the discrete Fourier transform as applied to eigenstress problems is capable of simulating the outcome of such experiments.

Mathematical theory of elastic structures

Abstract: The book covers three main topics: the classical theory of linear elasticity, the mathematical theory of composite elastic structures, as an application of the theory of elliptic equations on composite manifolds developed by the first author, and the finite element method for solving elastic structural problems. The authors treat these topics within the framework of a unified theory. The book carries on a theoretical discussion on the mathematical basis of the principle of minimum potential theory. The emphasis is on the accuracy and completeness of the mathematical formulation of elastic structural problems. The book will be useful to applied mathematicians, engineers and graduate students. It may also serve as a course in elasticity for undergraduate students in applied sciences.

Progressive Failure Analysis of Laminated Composite Beams

Abstract: A progressive failure model for laminated composite beams is formulated using a beam finite element with layer-wise constant shear (BLCS), which permits accurate computation of stresses on each layer. This is the first study to incorporate the stress-prediction accuracy of a layer-wise element for failure prediction of laminates under bending loads. In the present formulation, based on material degradation factors and existing failure criteria, a linear elastic behavior is assumed, and a damaged layer in an element is substituted by a degraded homogeneous layer. Maximum Stress and Tsai-Wu failure criteria are used to assess failure at the Gauss points. The effect of damage accumulation is accounted for by degrading the stiffness properties of failed element-layers in the equilibrium iterations. After equilibrium is satisfied, the load is increased by a constant percentage of first-play-failure load in a load-controlled failure prediction. A displacement-controlled scheme is also implemented. The predictio...

Numerical Simulation of Geosynthetic-Reinforced Flexible Pavements

Abstract: In traditional analyses of flexible pavements the linear elastic material behavior is assumed for pavement materials. However, pavement materials do not behave as linear elastic materials. They can be better modeled by using elasto-plastic constitutive relationships. The consequences of the assumption of linear elasticity in the prediction of the behavior of geosynthetic-reinforced flexible pavements are presented. The effect of the stiffness of geosynthetic reinforcements on pavement behavior is also studied. The behavior of a geosynthetic-reinforced flexible pavement is analyzed by the finite-element method with different constitutive models. The results of six analyses where E is Young's modulus [Case 1, linear elastic models with geosynthetics (Case 1a, E = 1 GPa; Case Ib, E = 100 GPa); Case 2, linear elastic models without geosynthetics; Case 3, elasto-plastic models with geosynthetics (Case 3a, E = 1 GPa; Case 3b, E = 100 GPa); and Case 4, elasto-plastic models without geosynthetics on the same pave...

The settlement of a rigid circular foundation resting on a half-space exhibiting a near surface elastic non-homogeneity

Abstract: The present paper examines the elastostatic problem pertaining to the axisymmetric loading of a rigid circular foundation resting on the surface of a non-homogeneous elastic half-space. The non-homogeneity corresponds to a depth variation in the linear elastic shear modulus according to the exponential form G(z)=G1+G2e-ζz. The equations of elasticity governing this type of non-homogeneity are solved by employing a Hankel transform technique. The mixed boundary value problem associated with the indentation of the half-space by the rigid circular foundation is reduced to a Fredholm integral equation which is solved via a numerical technique. The numerical results presented in the paper illustrate the influence of the near-surface elastic non-homogeneity on the settlement of the foundation.

Effects of curvilinear anisotropy on radially symmetric stresses in anisotropic linearly elastic solids

Abstract: It has been known for some time that certain radial anisotropies in some linear elasticity problems can give rise to stress singularities which are absent in the corresponding isotropic problems. Recently related issues were examined by other authors in the context of plane strain axisymmetric deformations of a hollow circular cylindrically anisotropic linearly elastic cylinder under uniform external pressure, an anisotropic analog of the classic isotropic Lame problem. In the isotropic case, as the external radius increases, the stresses rapidly approach those for a traction-free cavity in an infinite medium under remotely applied uniform compression. However, it has been shown that this does not occur when the cylinder is even slightly anisotropic. In this paper, we provide further elaboration on these issues. For the externally pressurized hollow cylinder (or disk), it is shown that for radially orthotropic materials, the maximum hoop stress occurs always on the inner boundary (as in the isotropic case) but that the stress concentration factor is infinite. For circumferentially orthotropic materials, if the tube is sufficiently thin, the maximum hoop stress always occurs on the inner boundary whereas for sufficiently thick tubes, the maximum hoop stress occurs at the outer boundary. For the case of an internally pressurized tube, the anisotropic problem does not give rise to such radical differences in stress behavior from the isotropic problem. Such differences do, however, arise in the problem of an anisotropic disk, in plane stress, rotating at a constant angular velocity about its center, as well as in the three-dimensional problem governing radially symmetric deformations of anisotropic externally pressurized hollow spheres. The anisotropies of concern here do arise in technological applications such as the processing of fiber composites as well as the casting of metals.