scispace - formally typeset
Search or ask a question

Showing papers on "Linear elasticity published in 1997"


Journal ArticleDOI
TL;DR: It is shown that transformations constrained by quadratic regularization methods such as the Laplacian, biharmonic, and linear elasticity models, do not ensure that the transformation maintains topology and, therefore, must only be used for coarse global registration.
Abstract: Presents diffeomorphic transformations of three-dimensional (3-D) anatomical image data of the macaque occipital lobe and whole brain cryosection imagery and of deep brain structures in human brains as imaged via magnetic resonance imagery. These transformations are generated in a hierarchical manner, accommodating both global and local anatomical detail. The initial low-dimensional registration is accomplished by constraining the transformation to be in a low-dimensional basis. The basis is defined by the Green's function of the elasticity operator placed at predefined locations in the anatomy and the eigenfunctions of the elasticity operator. The high-dimensional large deformations are vector fields generated via the mismatch between the template and target-image volumes constrained to be the solution of a Navier-Stokes fluid model. As part of this procedure, the Jacobian of the transformation is tracked, insuring the generation of diffeomorphisms. It is shown that transformations constrained by quadratic regularization methods such as the Laplacian, biharmonic, and linear elasticity models, do not ensure that the transformation maintains topology and, therefore, must only be used for coarse global registration.

543 citations


Journal ArticleDOI
TL;DR: In this article, a device and procedure for measuring elastic properties of gelatin for elasticity imaging (elastography) was described. And the measured compression forces were comparable to results obtained from finite element analysis when linear elastic media are assumed.
Abstract: Acoustic and mechanical properties are reported for gelatin materials used to construct tissue-like phantoms for elasticity imaging (elastography). A device and procedure for measuring elastic properties are described. The measured compression forces were comparable to results obtained from finite element analysis when linear elastic media are assumed. Also measured were the stress relaxation, temporal stability, and melting point of the materials. Aldehyde concentration was used to increase the stiffness of the gelatin by controlling the amount of collagen cross-linking. A broad range of tissue-like elastic properties was achieved with these materials, although gels continued to stiffen for several weeks. The precision for elastic modulus measurements ranged from less than 0.1% for 100 kPa samples to 8.9% for soft (<10 kPa), sticky samples.

511 citations


Journal ArticleDOI
TL;DR: In this paper, a micromechanical analysis for the linear elastic behavior of a low-density foam with open cells is presented, where the foam structure is based on the geometry of Kelvin soap froth with flat faces.
Abstract: A micromechanical analysis for the linear elastic behavior of a low-density foam with open cells is presented. The foam structure is based on the geometry of Kelvin soap froth with flat faces: 14-sided polyhedral cells contain six squares and eight hexagons. Four struts meet at every joint in the perfectly ordered, spatially periodic, open-cell structure. All of the struts and joints have identical shape. Strut-level force-displacement relations are expressed by compliances for stretching, bending, and twisting. We consider arbitrary homogeneous deformations of the foam and present analytic results for the force, moment, and displacement at each strut midpoint and the rotation at each joint. The effective stress-strain relations for the foam, which has cubic symmetry, are represented by three elastic constants, a bulk modulus, and two shear moduli, that depend on the strut compliances. When these compliances are evaluated for specific strut geometries, the shear moduli are nearly equal and therefore the elastic response is nearly isotropic. The variational results of Hashin and Shtrikman are used to calculate the effective isotropic shear modulus of a polycrystal that contain grains of Kelvin foam.

308 citations


Journal ArticleDOI
TL;DR: In this article, a linear elastic contact between a rigid plane and a halfspace whose surface height is described by a bandwidth-limited Fourier series is considered, and the surface normal displacements and contact pressures are found by a numerical technique that exploits the structure of the Fast Fourier Transform (FFT) and an exact result in linear elasticity.
Abstract: Elastic contact between a rigid plane and a halfspace whose surface height is described by a bandwidth-limited Fourier series is considered The surface normal displacements and contact pressures are found by a numerical technique that exploits the structure of the Fast Fourier Transform (FFT) and an exact result in linear elasticity The multiscale nature of rough surface contact is implicit to the method, and features such as contact agglomeration and asperity interaction-a source of difficulty for asperity-based models-evolve naturally Both two-dimensional (2-D) and three-dimensional ( 3-D ) contact are handled with equal ease Finally, the implementation is simple, compact, and fast

265 citations


Journal ArticleDOI
TL;DR: In this paper, the effective properties of a linear elastic medium reinforced periodically with thin parallel fibers made up of a much stronger elastic medium were studied and it was shown that the effective material is a second gradient material, i.e., a material whose energy depends on the second gradient of the displacement.
Abstract: Homogenization may change fundamentally the constitutive laws of materials. We show how a heterogeneous Cauchy continuum may lead to a non Cauchy continuum. We study the effective properties of a linear elastic medium reinforced periodically with thin parallel fibers made up of a much stronger linear elastic medium and we prove that, when the Lame coefficients in the fibers and the radius of the fibers have appropriate order of magnitude, the effective material is a second gradient material, i.e. a material whose energy depends on the second gradient of the displacement.

233 citations


Journal ArticleDOI
TL;DR: In this article, the authors compare and contrast plane strain predictions of ground movement for both single and twin-tunnel excavations in stiff clay modelled as (a) isotropic linear elastic perfectly plastic (b) anisotropic linear elastic perfect plastic (c) isotropy non-linear elastic perfectly plastic with shear stiffness dependent on deviatoric strain and mean effective stress, and bulk modulus dependent on volumetric strain and means effective stress.
Abstract: The use of the finite-element method to analyse tunnels is becoming more widespread, but any prediction is dependent (among other things) on the model adopted for the pre-failure soil behaviour. This paper compares and contrasts plane strain predictions of ground movement for both single- and twin-tunnel excavations in stiff clay modelled as (a) isotropic linear elastic perfectly plastic (b) anisotropic linear elastic perfectly plastic (c) isotropic non-linear elastic perfectly plastic with shear stiffness dependent on deviatoric strain and mean effective stress, and bulk modulus dependent on volumetric strain and mean effective stress (d) anisotropic non-linear elastic perfectly plastic employing the model in (c) above (e) isotropic non-linear elastic perfectly plastic with shear and bulk stiffness dependent on deviatoric strain level, mean effective stress, and loading reversals. The analyses model the geometry of the twin Jubilee Line Extension Project tunnels beneath St James's Park (London, UK), and ...

228 citations


Journal ArticleDOI
TL;DR: In this paper, a critical value of K n r δ (δ K n ) was determined for notched mode I three-point flexure specimens using a combination of the Williams (Williams, M. L. (1952) asymptotic method, dimensional considerations, and detailed finite element analysis.

220 citations


Journal ArticleDOI
TL;DR: In this paper, a two-dimensional model composite with elastic reinforcements in a crystalline matrix subject to macroscopic shear is analyzed using both discrete dislocation plasticity and conventional continuum slip crystal plasticity.

212 citations


Journal ArticleDOI
TL;DR: In this paper, a least-squares functional for the generalized Stokes equations was developed by adding a pressure term in the continuity equation, which yields optimal discretization error estimates for finite element spaces in an H1 product norm appropriately weighted by the Reynolds number.
Abstract: Following our earlier work on general second-order scalar equations, here we develop a least-squares functional for the two- and three-dimensional Stokes equations, generalized slightly by allowing a pressure term in the continuity equation. By introducing a velocity flux variable and associated curl and trace equations, we are able to establish ellipticity in an H1 product norm appropriately weighted by the Reynolds number. This immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems. Both estimates are naturally uniform in the Reynolds number. Moreover, our pressure-perturbed form of the generalized Stokes equations allows us to develop an analogous result for the Dirichlet problem for linear elasticity, where we obtain the more substantive result that the estimates are uniform in the Poisson ratio.

183 citations


Journal ArticleDOI
TL;DR: The main components needed for an adaptivehp-version finite element algorithm are discussed: an adaptive hp-refinement strategy, effective methods for constructing conforming hp-approximations, and, efficient solvers for the large, ill-conditioned systems of linear equations.

147 citations


Journal ArticleDOI
TL;DR: In this paper, large foam unit cells are created using Voronoi techniques and a smooth transition from regular to random geometries is made, showing the strong sensitivity of the mechanical properties from the geometry of the microstructure.
Abstract: Foams can be created from coagulation of gas bubbles in liquid. After removal of cell faces, an open-cell foam remains consisting of a strut framework. In the past, mechanical properties were estimated by a small unit cell consisting of only a few struts. However, the random geometry of the foam can be of importance for the linear elastic properties. Here, large foam unit cells are created using Voronoi techniques. A smooth transition from regular to random geometries is made, showing the strong sensitivity of the mechanical properties from the geometry of the microstructure. Uniaxial global loads are transmitted through chains of highly loaded struts. The deformation of the struts in the foam is a mixture of bending and normal deformation, the ratio of which shown here to be dependent on the magnitude of the density.

Journal ArticleDOI
TL;DR: In this paper, the authors determined the mode I and II stress intensities for notched PMMA tensile specimens and notched mode II flexure specimens using a combination of the Williams (1952) asymptotic method, dimensional considerations, and detailed finite element analysis.
Abstract: In the context of linear elasticity, a stress singularity of the type Knrδ(δ<0) may exist at sharp re-entrant corners, with an intensity Kn In general the order of the stress singularity δ and the stress intensity differ for symmetric (mode I) and antisymmetric (mode II) loading Under general mixed-mode loadings, the magnitudes of the mode I and II intensities fully characterize the stress state in the region of the corner A failure criterion based on critical values of these intensities may be appropriate in situations where the region around the corner dominated by the singular fields is large compared to intrinsic flaw sizes, inelastic zones, and fracture process zone sizes We determined the mode I and II stress intensities for notched mode I tensile specimens and notched mode II flexure specimens using a combination of the Williams (1952) asymptotic method, dimensional considerations, and detailed finite element analysis We carried out a companion experimental study to extract critical values of the mode I and II stress intensities for a series of notched polymethyl methacrylate (PMMA) tensile and flexure specimens with notch angles of 90- The data show that excellent failure correlation is obtained, in both mode I and II loading, through the use of a single parameter, the critical stress intensity We then analyzed and tested a series of T-shaped structures containing 90- corners The applied tensile loading results in mixed-mode loading of the 90- corners Failure of the specimens is brittle and can be well-correlated with a critical mode I stress intensity criterion using the results of the notched mode I tensile tests This is attributed to large difference in the strength of the stress singularities in modes I and II: δ= -04555 and -00915 for modes I and II for a 90- notch As a result, the mode I loading dominates the failure process for the 90- corner in the T-structure

Journal ArticleDOI
TL;DR: A feed-back approach is developed here for primal as well as mixed FE discretisations of the fundamental problem in linear elasticity, yielding almost optimal meshes for various kinds of error measures.
Abstract: Recently a refined approach to error control in finite element (FE) discretisations has been proposed, Becker and Rannacher (1995b), (1996), which uses weighted a posteriori error estimates derived via duality arguments. The conventional strategies for mesh refinement in FE models of problems from elasticity theory are mostly based on a posteriori error estimates in the energy norm. Such estimates reflect the approximation properties of the finite element ansatz by local interpolation constants while the stability properties of the continuous model enter through a global coercivity constant. However, meshes generated on the basis of such global error estimates are not appropriate in cases where the domain consists of very heterogeneous materials and for the computation of local quantities, e.g., point values or contour integrals. This deficiency is cured by using certain local norms of the dual solution directly as weights multiplying the local residuals of the computed solution. In general, these weights have to be evaluated numerically in the course of the refinement process, yielding almost optimal meshes for various kinds of error measures. This feed-back approach is developed here for primal as well as mixed FE discretisations of the fundamental problem in linear elasticity.

Journal ArticleDOI
TL;DR: In this article, a finite element analysis of squeeze flow has been implemented for a material that exhibits elasto-viscoplasticity, based upon the assumption that linear elastic deformation occurs prior to yield and that the yield surface is strain rate hardening as defined by an associated viscoplastic flow rule.
Abstract: A finite element analysis of squeeze flow has been implemented for a material that exhibits elasto-viscoplasticity. The formulation is based upon the assumption that linear elastic deformation occurs prior to yield and that the yield surface is strain rate hardening as defined by an associated viscoplastic flow rule. Both no-slip and lubricated wall boundary conditions are considered. The numerical simulation results are compared with experimental measurements involving a model elasto-viscoplastic material for which the material parameters were derived from tensile and ram extrusion measurements. Satisfactory agreement was obtained for the compressive forces as a function of displacement, the radial displacement fields and the wall normal and shear stress distributions.

Journal ArticleDOI
TL;DR: In this paper, the authors study the linear stability of the full, time-independent, equations by introducing a new are length preserving perturbation scheme, which gives a direct proof of the existence of dynamical instabilities and provides the selection mechanism for the shape of unstable filaments.

Journal ArticleDOI
TL;DR: In this article, a linear elastic solution of an axisymmetric boundary value problem is used as a basis to generate its inelastic solution, where the material parameters are treated as field variables in an iterative manner.
Abstract: Linear elastic solution of an axisymmetric boundary value problem is used as a basis to generate its inelastic solution. This method treats the material parameters as field variables. Their distribution is obtained as a part of solution in an iterative manner. Two schemes of updating material parameters are discussed and compared. A procedure for calculation of residual stress field is presented. Application of the method to autofrettage is presented. Residual stress calculation based on actual material curve, isotropic and kinematic hardening models, and variable Bauschinger effect factor (BEF) is carried out. It is concluded that consideration of dependency of BEF on plastic strain makes significant changes to residual hoop stress near the bore for low-level autofrettage. However, this dependency is insignificant for high-level autofrettage. Results obtained here are shown to be in good agreement with experimental and finite element results.

Journal ArticleDOI
TL;DR: In this paper, the steady-state vibrations of a simply-supported rectangular linear elastic laminated plate with embedded PZT layers are analyzed using the three-dimensional linear theory of elasticity.

Journal ArticleDOI
TL;DR: In this article, an iterative method of determining the limit state of a perfectly plastic body for the Von Mises yield condition is described, where a sequence of incompressible linear elastic solutions are defined with a spatially varying shear modulus.

Journal ArticleDOI
TL;DR: In this paper, a cross-anisotropic model is proposed to predict the performance of granular bases in flexible pavements, which can account for the dilative behavior observed under the wheel load and the effects of compaction-induced residual stresses.
Abstract: A new cross-anisotropic model is proposed to predict the performance of granular bases in flexible pavements. A cross-anisotropic representation has different material properties (i.e., elastic modulus and Poisson's ratio) assigned in the horizontal and vertical directions. Repeated-load triaxial tests with vertical and lateral deformation measurements can be used to establish these anisotropic properties. Simple stress-dependent granular material models, obtained from analysis of the laboratory test data, are used in a nonlinear finite element program, named GT-PAVE, to predict pavement responses. The horizontal and shear stiffnesses are typically found to be less than the vertical. The nonlinear anisotropic approach is shown to account effectively for the dilative behavior observed under the wheel load and the effects of compaction-induced residual stresses. The main advantage of using a cross-anisotropic model in the base is the drastic reduction or elimination of significant tensile stresses generally predicted by isotropic linear elastic layered programs.

Journal ArticleDOI
TL;DR: A unified approach to construct finite elements based on a dual-hybrid formulation of the linear elasticity problem is given in this article, where the stress tensor is considered but its symmetry is relaxed by a Lagrange multiplier which is nothing else than the rotation.
Abstract: A unified approach to construct finite elements based on a dual-hybrid formulation of the linear elasticity problem is given. In this formulation the stress tensor is considered but its symmetry is relaxed by a Lagrange multiplier which is nothing else than the rotation. This construction is linked to the approximations of the Stokes problem in the primitive variables and it leads to a new interpretation of known elements and to new finite elements. Moreover all estimates are valid uniformly with respect to compressibility and apply in the incompressible case which is close to the Stokes problem.

Journal ArticleDOI
TL;DR: In this paper, the existence of SH surface waves in a half-space homogeneous material (i.e., antiplane shear wave motions which decay exponentially with the distance from the free surface) is shown to be possible within the framework of the generalized linear continuum theory of gradient elasticity with surface energy.
Abstract: The existence of SH surface waves in a half-space homogeneous material (i.e. anti-plane shear wave motions which decay exponentially with the distance from the free surface) is shown to be possible within the framework of the generalized linear continuum theory of gradient elasticity with surface energy. As is well-known such waves cannot be predicted by the classical theory of linear elasticity for a homogeneous half-space, although there is experimental evidence supporting their existence. Indeed, this is a drawback of the classical theory which is only circumvented by modelling the half-space as a layered structure (Love waves) or as having non-homogeneous material properties. On the contrary, the present study reveals that SH surface waves may exist in a homogeneous half-space if the problem is analyzed by a continuum theory with appropriate microstructure. This theory, which was recently introduced by Vardoulakis and co-workers, assumes a strain-energy density expression containing, besides the classical terms, volume strain-gradient and surface-energy gradient terms.

Journal ArticleDOI
TL;DR: In this article, the elastic response of aqueous foam to oscillating shear strain is probed by diffusing-wave spectroscopy, and the local bubble displacements are shown to be compatible with linear elastic behavior.
Abstract: The elastic response of aqueous foam to oscillating shear strain is probed by diffusing-wave spectroscopy. At low strains, the local bubble displacements are shown to be compatible with linear elastic behavior. At higher strains, a crossover to nonlinear but still fully periodic displacements is observed. To explain these results, we present a phenomenological model based on unstable bubble configurations.

Journal ArticleDOI
TL;DR: In this paper, an absolute nodal co-ordinate dynamic formulation is developed for the large deformations and rotations of three-dimensional plate elements, which does not require the use of coordinate transformation to define the global inertia properties of the plates.
Abstract: In this investigation, an absolute nodal co-ordinate dynamic formulation is developed for the large deformations and rotations of three-dimensional plate elements. In this formulation, no infinitesimal or finite rotations are used as nodal co-ordinates, instead global displacements and slopes are used as the plate coordinates. Using this interpretation of the plate coordinates the new method does not require the use of co-ordinate transformation to define the global inertia properties of the plates. The resulting mass matrix is the same constant matrix that appears in linear structural dynamics. The stiffness matrix, on the other hand, is a non-linear function of the nodal co-ordinates of the plate even in the case of a linear elastic problem. It is demonstrated in this paper that, unlike the incremental finite element formulations, the proposed method leads to an exact modelling of the rigid body inertia when the plate element moves as a rigid body. It is also demonstrated that by using the proposed method the conventional plate element shape function has a complete set of rigid body modes that can describe an exact arbitrary rigid body displacement. Using this fact, plate elements in the proposed new formulation can be considered as isoparametric elements. As a consequence, an arbitrary rigid body motion of the element results in zero strain. © 1997 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: In this paper, the Simple Asymptotic Body (SAB) model was used to simulate the elasticity of geomaterials at small strain levels, and the model showed that the peak-to-peak secant modulus has maximum and minimum values at very high and low loading frequencies where the damping value is zero, while damping has the maximum value at an intermediate loading frequency.

Journal ArticleDOI
TL;DR: In this article, a semi-infinite mode III crack dynamically propagates in a two-dimensional linear elastic infinite body is considered, where the crack tip is assumed to be a cohesive zone whose size is determined so as to cancel the classical crack tip stress singularity caused by the applied loads.

Journal ArticleDOI
TL;DR: In this article, the linear elastic solution of a boundary value problem is used as a basis to generate its inelastic solution, which is obtained in an iterative manner using the projection method, the arc length method, and Nueber's rule.

Journal ArticleDOI
TL;DR: In this paper, the relation between frequency and complex wave number of axisymmetric wave modes in an isotropic, thin-walled, cylindrical shell containing a linear viscoelastic medium is derived.
Abstract: In this study the relation between frequency and complex wave number of axisymmetric wave modes in an isotropic, thin-walled, cylindrical shell containing a linear viscoelastic medium is derived. Shell wall bending and longitudinal motion are coupled in an empty cylindrical shell. When a viscoelastic medium is enclosed, the shell motion is affected by the complex bulk and shear modulus, as well as by the density of the medium enclosed. A Maxwell model is used for both complex Lame constants λ and μ to describe the constitutive equations of the medium. By varying the complex moduli, the medium can be modeled as an inviscid fluid, an elastic material, or anything between these two extremes. The interaction of the thin-walled linear elastic shell and the viscoelastic medium is discussed numerically by calculating the complex dispersion relation. Numerical results are presented for an empty shell and a shell filled with three types of core material: an inviscid fluid, a shear dissipative fluid, and a shear elastic fluid. In a companion paper [J. Vollmann et al., J. Acoust. Soc. Am. 102, 909–920 (1997)], the experimental setup and the signal processing used to perform the high-resolution measurement of the dispersion relation are described in detail. Theoretical and experimental results are compared.

Journal ArticleDOI
TL;DR: In this paper, the complex frequency spectrum of axisymmetric wave modes in a circular cylindrical shell containing various viscoelastic media is measured by combining a high-resolution laser interferometer with modern spectrum estimation methods.
Abstract: The complex frequency spectrum of axisymmetric wave modes in a circular cylindrical shell containing various viscoelastic media is measured. A new measurement technique has been developed for this purpose by combining a high-resolution laser interferometer with modern spectrum estimation methods. To decompose the complex wave-number dependence, a complex spectrum estimation method has been implemented. Up to 40 dispersion curves of traveling, axisymmetric modes are decomposed simultaneously in a frequency range between 1 kHz and 2 MHz. The guided structural waves are excited by piezoelectric transducers. Linear elasticity can be considered as an extreme case of viscoelasticity (long relaxation times compared with the deformation periods). To ascertain the validity of the theory, dispersion curves are calculated for a shell containing a viscoelastic material behaving like the elastic shell and are compared with the measured curves of an isotropic aluminium rod. The phenomenon of “backward wave propagation,...

Journal ArticleDOI
TL;DR: In this paper, the Cosserat brothers' theory of porous media (TPM) was extended by micropolar degrees of freedom in the sense of the COSSERAT brothers, and the authors derived the non-symmetric effective skeleton stress and the couple stress tensor.
Abstract: Elastoplastic deformations of cohesive-frictional liquid-saturated granular solid materials can be described by use of a macroscopic continuum mechanical approach within the well-founded framework of the theory of porous media (TPM). In the present contribution, the TPM formulation of the skeleton material is extended by micropolar degrees of freedom in the sense of the Cosserat brothers. Proceeding from two basic assumptions, material incompressibility of both constituents (skeleton material and pore liquid) and geometrically linear solid deformations, the non-symmetric effective skeleton stress and the couple stress tensor are determined by linear elasticity laws. In the framework of the ideal plasticity concept, the plastic yield limit is governed by a smooth and closed single-surface yield function together with non-associated flow rules for both the plastic strain rate and the plastic rate of curvature tensor. Fluid viscosity is taken into account by the drag force. The inclusion of micropolar degrees of freedom, in contrast to the usual continuum mechanical approach to the TPM, allows, on the one hand, for the determination of the local average grain rotations and, on the other hand, additionally yields a regularization effect on the solution of the strongly coupled system of governing equations when shear banding occurs. However, in the framework of the original TPM formulation of fluid-saturated porous materials, the inclusion of the fluid viscosity alone also yields a certain regularization on shear band computations. The numerical examples are solved by use of finite element discretization techniques, where, in particular, the computation of shear band localization phenomena is carried out by the example of the well-known base failure problem of geotechnical engineering.

Journal ArticleDOI
TL;DR: In this article, a semi-inverse method was used to study deformations of a straight, prismatic, homogeneous body made of a porous, linear elastic, and isotropic material and loaded only at its end faces by self-equilibrated forces.
Abstract: We use a semi-inverse method to study deformations of a straight, prismatic, homogeneous body made of a porous, linear elastic, and isotropic material and loaded only at its end faces by self equilibrated forces As in the classical theory, the problem is reduced to solving plane elliptical problems It is shown that the Clebsch/Saint-Venant and Voigt hypotheses are not valid for this problem