Showing papers on "Linear elasticity published in 1994"
TL;DR: In this paper, the authors present analytical tip region solutions for fracture width and pressure when a power law fluid drives a plane strain fracture in an impermeable linear elastic solid, where the tip region stress is dominated by a singularity which is particular to the hydraulic fracturing problem.
Abstract: We present analytical tip region solutions for fracture width and pressure when a power law fluid drives a plane strain fracture in an impermeable linear elastic solid. Our main result is an intermediate asymptotic solution in which the tip region stress is dominated by a singularity which is particular to the hydraulic fracturing problem. Moreover this singularity is weaker than the inverse square root singularity of linear elastic fracture mechanics. We also show how the solution for a semi-infinite crack may be exploited to obtain a useful approximation for the finite case.
288 citations
TL;DR: In this paper, a micromechanical framework is proposed to investigate effective mechanical properties of elastic multiphase composites containing many randomly dispersed ellipsoidal inhomogeneities.
Abstract: A micromechanical framework is proposed to investigate effective mechanical properties of elastic multiphase composites containing many randomly dispersed ellipsoidal inhomogeneities. Within the context of the representative volume element (RVE), four governing micromechanical ensemble-volume averaged field equations are presented to relate ensemble-volume averaged stresses, strains, volume fractions, eigenstrains, particle shapes and orientations, and elastic properties of constituent phases of a linear elastic particulate composite. A renormalization procedure is employed to render absolutely convergent integrals. Therefore, the micromechanical equations and effective elastic properties of a statistically homogeneous composite are independent of the shape of the RVE. Various micromechanical models can be developed based on the proposed ensemble-volume averaged constitutive equations. As a special class of models, inter-particle interactions are completely ignored. It is shown that the classical Hashin-Shtrikman bounds, Walpole's bounds, and Willi's bounds for isotropic or anisotropic elastic multiphase composites are related to the “noninteracting” solutions. Further, it is demonstrated that the Mori-Tanaka methodcoincides with the Hashin-Shtrikman bounds and the “noninteracting” micromechanical model in some cases. Specialization to unidirectionally aligned penny-shaped microcracks is also presented. An accurate, higher order (in particle concentration), probabilistic pairwise particle interaction formulation coupled with the proposed ensemble-volume averaged equations will be presented in a companion paper.
237 citations
TL;DR: In this article, a formulation of isotropic large strain elasticity and computational aspects of its finite element implementation are discussed. But the authors focus on the finite element implementations of the model.
Abstract: The paper presents a formulation of isotropic large strain elasticity and addresses some computational aspects of its finite element implementation. On the theoretical side, an Eulerian setting of isotropic elasticity is discussed exclusively in terms of the Finger tensor as a strain measure. Noval aspects are a direct representation of the Eulerian elastic moduli in terms of the Finger tensor and their rigorous decomposition into decoupled volumetric and isochoric contributions based on a multiplicative split of the Finger tensor into spherical and unimodular parts. The isochoric stress response is formulated in terms of the eigenvalues of the unimodular part of the Finger tensor. A constitutive algorithm for the computation of the stresses and tangent moduli for plane problems is developed and applied to a model problem of rubber elasticity. On the computational side, the implementation of the constitutive model in three possible finite element formulations is discussed. After pointing out algorithmic techniques for the treatment of incompressible elasticity, several numerical simulations are presented which show the performance of the proposed constitutive algorithm and the convergence behaviour of the different finite element fomulations for compressible and incompressible elasticity.
215 citations
TL;DR: In this paper, the results of measured internal strain and displacement fields from gel-based phantoms are compared with theoretical predictions of a linear elastic model, and some aspects of elasticity reconstruction based on measured displacement and strain fields are discussed.
Abstract: Evaluation of internal displacement and strain distributions in tissue under externally applied forces is a necessary step in elasticity imaging To obtain a quantitative image of the elastic modulus, strain and displacement fields must be measured with reasonable accuracy and inverted based on an accurate theoretical model of soft tissue mechanics In this paper, results of measured internal strain and displacement fields from gel-based phantoms are compared with theoretical predictions of a linear elastic model In addition, some aspects of elasticity reconstruction based on measured displacement and strain fields are discussed >
169 citations
166 citations
TL;DR: In this article, a micromechanistic deformation model is proposed where loading direction and weave parameters are allowed to vary and the model provides improved estimates for the linear elastic stiffness.
Abstract: A micromechanistic deformation model, that could realistically be incorporated into structural finite element codes, is proposed where loading direction and weave parameters are allowed to vary. Comparisons are made to previous models and experimental results for woven materials, indicating that the proposed model provides improved estimates for the linear elastic stiffness. The model further provides predictions for internal stresses in the longitudinal, transverse, and interlace regions of the woven laminate which qualitatively correspond to the experimentally observed failure mechanisms
108 citations
TL;DR: In this paper, a geometrically exact nonlinear beam model for composite rotor blades undergoing large vibrations in 3D space was developed, based on an energy approach, but the derivation is fully correlated with the Newtonian approach.
91 citations
TL;DR: In this paper, a simple, accurate technique for recovery of displcements and derivatives, such as strains, is presented, based on local interpolation of nodal displacements using a moving least square method.
86 citations
TL;DR: In this article, the authors review the most usual homogeneization approaches for linear elastic behavior in order to obtain their own range of validity in comparison with experimental data, and determine experimentally and numerically the effect of a third phase, named interphase, on the linear elastic behaviour of fiber composites.
Abstract: The study and design of composite structures raises numerous difficulties due to the high level of heterogeneity and to the small size of the fillers. One of the most usual solutions consists in replacing the composite material by a homogeneous medium which is considered as "macroscopically" equivalent. This kind of approach, named homogeneization, aims to provide every mechanical characteristic of the composite according to the mechanical and geometrical features of its constituents. The aim of the present paper is twofold: (1) to review the most usual homogeneization approaches for linear elastic behavior in order to obtain their own range of validity in comparison with experimental data, and (2) to determine experimentally and numerically the effect of a third phase, named interphase, on the linear elastic behavior of fiber composites.
82 citations
TL;DR: In this paper, two preconditioning techniques for the numerical solution of linear elasticity problems are described and studied based on spectral equivalence approach, and the efficiency and robustness of the described techniques are illustrated by numerical experiments.
Abstract: Two preconditioning techniques for the numerical solution of linear elasticity problems are described and studied. Both techniques are based on spectral equivalence approach. The first technique consists in an incomplete factorization of the separate displacement component part of the stiffness matrix. The second technique uses an incomplete factorization of the isotropic approximation to the stiffness matrix. Results concerning existence, stability and efficiency of these preconditioning techniques are presented. The efficiency and robustness of the described techniques are illustrated by numerical experiments.
67 citations
TL;DR: In this paper, an analysis of laminated composite plates forced into cylindrical bending by the application of voltages to piezoelectric actuators attached to the top and bottom surfaces is performed using the equations of linear elasticity.
Abstract: An analysis of laminated composite plates forced into cylindrical bending by the application of voltages to piezoelectric actuators attached to the top and bottom surfaces is performed using the equations of linear elasticity. In the treatment the actuator is modeled as a thin surface film and mixed edge conditions are employed to simulate simple supports. The results obtained from the linear elastic solution are compared with those obtained using the elementary equations of the extension and flexure of thin composite plates, which are the ones usually employed in practice. When the properties of adjacent laminates are very different the comparison reveals that for large span-length-to-thickness ratios the agreement is quite good but for small ones it is not good at all. The existence of a shearing stress singularity under the edge of the fully electroded actuator is exhibited.
TL;DR: In this paper, the authors applied the dual boundary element method and the path independent integral for the evaluation of dynamic stress intensity factors of stationary cracks in a linear elastic material and compared the results with solutions obtained by using other methods.
Abstract: The application of the dual boundary element method and the path independentĴ-integral for the evaluation of dynamic stress intensity factors of stationary cracks in a linear elastic material is presented. The distinct set of boundary equations of elastodynamics is obtained by using the dual boundary element method and the dual reciprocity approach. Numerical implementation of the path-independentĴ-integral and the decomposition technique is presented. The method is applied for several cracked structures and the results are compared with solutions obtained by using other methods.
TL;DR: In this paper, a plane-strain elasticity problem is studied by means of an adaptation of the Rayleigh multipole method for a domain with a set of circular elastic inclusions.
Abstract: A plane-strain elasticity problem is studied by means of an adaptation of the Rayleigh multipole method for a domain with a set of circular elastic inclusions. The complex potentials of Kolosov-Muskhelishvili are obtained in the form of Laurent series outside the inclusions. The results of the calculation of the multipole coefficients have been compared with those obtained by means of an integral approximation for two cases: a pair of identical inclusions and a square array of inclusions.
TL;DR: In this article, a computational micromechanics study of stress-assisted martensitic transformation using finite elements is carried out within a thermomechanical framework including the aspect of plastic deformation.
Abstract: A computational micromechanics study of stress-assisted martensitic transformation using finite elements is carried out within a thermomechanical framework including the aspect of plastic deformation. The phase transformation is treated by a stress-free transformation tensor corresponding to a certain habit plane variant involving a shape change resulting from shear and dilatational deformations which are eigenstrains within the emerging martensitic phase. General plane strain and axisymmetric analyses are carried out introducing different micromechanical models with appropriate boundary conditions required for the microfield approach. The introduced models are compared with a micromechanical model developed by Tvergaard in the viewpoint of thermodynamic aspects concerning the formation of martensite. The representative volume elements are specified with respect to the Schmid factor for the transformation shear corresponding to an average crystallographic orientation of a grain. We examine elastic-plastic deformations and distribution of transformation-induced microstress fields in both the parent and the emerging product phase due to accommodation of the transformation shape change. Analytical results for the linear elastic case based upon the Eshelby approach will be compared with numerical results. The plastic behavior of austenitic and martensitic microconstituents is described in the context of J2-flow theory with isotropic hardening. The thermomechanical framework is based on an explicit form of a potential suggesting the form of Gibbs free energy, from which transformation conditions on the mesodomain-, interface- and nucleation-site level are derived. The arrangement of martensitic microregions transforming in a stress field is studied considering the thermomechanical coupling of orientation and accommodation effects.
TL;DR: In this paper, the effect of nonlinear elasticity on seismic wave propagation has been investigated and it has been shown that the magnitude of the harmonics created by nonlinear interactions grows linearly with propagation distance in one-dimensional systems.
Abstract: Experiments in rock show a large nonlinear elastic wave response, far greater than that of gases, liquids and most other solids. The large response is attributed to structural discontinuities in rock such as microcracks and grain boundaries. The magnitude of the harmonics created by nonlinear interactions grows linearly with propagation distance in one-dimensional systems. In the earth, a large nonlinear response may be responsible for significant spectral alteration of a seismic wave at amplitudes and distances currently considered to be within the linear elastic regime. The authors argue, based on observations at ultrasonic frequencies, that the effect of nonlinear elasticity on seismic wave propagation may be large, and should be considered in modeling. 19 refs., 3 figs.
TL;DR: In this paper, the forced cylindrical bending vibrations of a laminated elastic plate with actuators at the top and bottom surfaces and forced into vibrations by applying time harmonic voltages to the actuators were analyzed.
Abstract: We analyze the forced cylindrical bending vibrations of a laminated elastic plate with actuators at the top and bottom surfaces and forced into vibrations by applying time harmonic voltages to the actuators. The actuators are modeled as thin films and mixed edge conditions are employed to simulate simple supports. The analysis is performed by using the method of Fourier series and the solution is exact within the assumptions of linear elasticity and plane strain deformations. Numerical results are computed for an aluminum plate with actuators affixed to its two major surfaces.
TL;DR: In this paper, the initial shear modulus distributions obtained from [0] 20, [90] 20 and [0/90] 5s Iosipescu and [45] 20 off-axis T300/914 specimens are compared.
TL;DR: In this article, the numerical behavior of elliptic problems with a small parameter is studied and an example concerning the computation of elastic arches is analyzed using this mathematical framework, and conditions ensuring that a numerical scheme will converge uniformly with respect to the small parameter are given.
Abstract: In this paper we study the numerical behaviour of elliptic problems in which a small parameter is involved and an example concerning the computation of elastic arches is analyzed using this mathematical framework. At first, the statements of the problem and its Galerkin approximations are defined and an asymptotic analysis is performed. Then we give general conditions ensuring that a numerical scheme will converge uniformly with respect to the small parameter. Finally we study an example in computation of arches working in linear elasticity conditions. We build one finite element scheme giving a locking behaviour, and another one which does not.
TL;DR: In this article, a simple finite element formulation for geometrical nonlinear shell structures was derived based on a direct introduction of the isoparametric finite element (FME) formulation into the shell equations.
Abstract: In this paper we derive a simple finite element formulation for geometrical nonlinear shell structures. The formulation bases on a direct introduction of the isoparametric finite element formulation into the shell equations. The element allows the occurrence of finite rotations which are described by two independent angles. A layerwise linear elastic material model for composites has been chosen. A consistent linearization of all equations has been derived for the application of a pure Newton method in the nonlinear solution process. Thus a quadratic convergence behaviour can be achieved in the vicinity of the solution point. Examples show the applicability and effectivity of the developed element.
TL;DR: In this article, the authors analyzed the stress field arising in the single-fiber pull-out test within the framework of the linear theory of elasticity, and showed that the stress distribution in the vicinity of the fiber is strongly inhomogeneous and exhibits severe concentrations of shear and radia stresses at fiber entry and at the fiber end.
TL;DR: In this article, the authors investigated the three-dimensional nature of the state of deformation in a single-lap test specimen in a linear elastic finite element analysis in which the boundary conditions account for the geometrically nonlinear effects.
Abstract: The three-dimensional nature of the state of deformation in a single-lap test specimen is investigated in a linear elastic finite element analysis in which the boundary conditions account for the geometrically non-linear effects. The validity of the model is demonstrated by comparing the resulting displacement fields with those obtained from a moire inteferometry experiment. The three-dimensional adherend and adhesive stress distributions are calculated and compared with those from a two-dimensional non-linear numerical analysis, Goland and Reissner's solution, and experimental measurements. The nature of the three-dimensional mechanics is described and discussed in detail. It is shown that three-dimensional regions exists in the specimen, where the adherend and adhesive stress distributions in the overlap near (and especially on) the free surface are quite different from those occurring in the interior. It is also shown that the adhesive peel stress is extremely sensitive to this three-dimensiona...
TL;DR: In this paper, a micromechanical model is presented which predicts the behaviour of a unidirectional composite subjected to axial shear load using standard finite elements, and the results show good agreement for low and intermediate fibre volume fractions.
TL;DR: Numerical examples for linear elasticity problems in two dimensions together with a comparison to the classical adaptive h-refinement strategy based on the equidistribution of errors are presented to validate the present formulation.
Abstract: An adaptive global–local refinement strategy based on the interior error estimates of the h-method is proposed. Adaptive global–local refinement strategy is aimed at constructing nearly optimal finite element meshes, where the force transfer to the local region of interest is sufficiently accurate so that the local phenomena of interest is resolved with a user-specified accuracy. Numerical examples for linear elasticity problems in two dimensions together with a comparison to the classical adaptive h-refinement strategy based on the equidistribution of errors are presented to validate the present formulation.
TL;DR: In this article, an incremental linear elastic constitutive equation was used to calculate the residual stresses generated in polymer systems, with emphasis on thermosetting systems which are curing under dimensionally constrained conditions.
TL;DR: In this paper, it was shown that in a class of less regular regions Korn's second inequality fails whereas the desired compact embedding still holds true for regions with the cone property.
Abstract: For the theory of boundary value problems in linear elasticity, it is of crucial importance that the space of vector-valued L2-functions whose symmetrized Jacobians are square-integrable should be compactly embedded in L2. For regions with the cone property this is usually achieved by combining Korn's inequalities and Rellich's selection theorem. We shall show that in a class of less regular regions Korn's second inequality fails whereas the desired compact embedding still holds true.
DOI•
01 Jan 1994
TL;DR: In this article, a constitutive law for trabecular (or spongious) bone is developed in the framework of continuum mechanics, implemented in a mechanical analysis computer program, validated by a number of in vitro experiments and illustrated by simulation of a femoral total hip component.
Abstract: Motivated by applications in orthopaedic surgery, new constitutive laws for trabecular (or spongious) bone are developed in the framework of continuum mechanics, implemented in a mechanical analysis computer program, validated by a number of in vitro experiments and illustrated by the simulation of a femoral total hip component. Current knowledge about the morphological and mechanical properties of trabecular bone is reviewed for setting the background and clarifying the contributions of the thesis. Comprehensive 1D and 3D theoretical models based on the approach of standard generalized materials are developed with a specific attention towards irreversible phenomena. The 1D model includes linear elasticity and rate-independent as well as rate-dependent plastic strain flow with damage. Based on a second order fabric tensor, the 3D model includes inhomogeneous, orthotropic linear elasticity and rate-independent plasticity with damage. In order to solve boundary value problems involving complex bone or bone-implant structures, implicit projection algorithms are developed for integrating the plastic flow rules with damage and implemented in the computer program TACT combining the finite element method, the linear iteration method and the finite difference method. The resulting numerical models are illustrated by the means of traction, bending and torsion benchmark tests. A number of pilot in vitro experiments are undertaken on human and bovine trabecular bone specimens in order to validate the theoretical models and identify the material constants. Quasistatic uniaxial and torsion experiments are performed with a method avoiding artefacts due to the inhomogeneous boundary conditions associated with porosity. Anisotropic elasticity, plasticity and damage of trabecular bone prove to be successfully described by the models in terms of structural density and morphology. Finally, the 3D constitutive law is applied to the biomechanical problem of primary stability of a cementless femoral total hip component in order to illustrate its potential.
TL;DR: Various preconditionsing methods are developed for use in preconditioned conjugate gradient iterative solution techniques and incomplete factorizations based on levels of fill, drop tolerance, and a two-level hierarchical basis are developed.
Abstract: Finite element modelling of three-dimensional elasticity problems give rise to large sparse matrices. Various preconditioning methods are developed for use in preconditioned conjugate gradient iterative solution techniques. Incomplete factorizations based on levels of fill, drop tolerance, and a two-level hierarchical basis are developed. Various techniques for ensuring that the incomplete factors have positive pivots are presented. Computational tests are carried out for problems generated using unstructured tetrahedral meshes. Quadratic basis functions are used. The performance of the iterative methods is compared to a standard direct sparse matrix solver. Problems with up to 70 000 degrees of freedom and small (<<1) element aspect ratio are considered