Showing papers on "Linear elasticity published in 1992"

The dual boundary element method: Effective implementation for crack problems

Abstract: SUMMARY The present paper is concerned with the effective numerical implementation of the two-dimensional dual boundary element method, for linear elastic crack problems. The dual equations of the method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation on the other, general mixed-mode crack problems can be solved with a single-region formulation. Both crack surfaces are discretized with discontinuous quadratic boundary elements; this strategy not only automatically satisfies the necessary conditions for the existence of the finite-part integrals, which occur naturally, but also circumvents the problem of collocation at crack tips, crack kinks and crack-edge corners. Examples of geometries with edge, and embedded crack are analysed with the present method. Highly accurate results are obtained, when the stress intensity factor is evaluated with the J-integral technique. The accuracy and efficiency of the implementation described herein make this formulation ideal for the study of crack growth problems under mixed-mode conditions.

Introduction to the finite element method

Abstract: * Introduction. * Matrix Algebra. * Direct Approach. * Strong and Weak Formulations - One-dimensional Heat Flow. * Gradient - Gauss' Divergence Theorem - Green Theorem. * Strong and Weak Forms - Two-and Three-Dimensional Heat Flow. * Choice of Approximating Functions for the FE-method - Scalar Problems. * Choice of Weight Function - Weighted Residual Methods. * FE-formulation of One-Dimensional Heat Flow. * FE-formulation of Two-and-Three Dimensional Heat Flow. * Guidelines for Element Meshes and Global Nodal Numbering. * Stresses and Strains. * Linear Elasticity. * FE-formulation of Torsion and Non-circular Shafts. * Approximating Functions for the FE-method - Vector Problems. * FE-formulation of Three-and-Two Dimensional Elasticity. * FE-formulation of Beams. * FE-formulation of Plates. * Isoparametric Finite Elements. * Numerical Integration.

Dual boundary element method for three-dimensional fracture mechanics analysis

Abstract: In this paper, an effective numerical implementation of the three-dimensional dual boundary element method, for linear elastic crack problems, is presented. Displacement and traction integral equations which constitute the dual boundary formulation are used independently on crack surfaces to overcome the numerical difficulties associated with co-planar crack surfaces in boundary element analysis. The crack surfaces are modelled with discontinous quadrilateral quadratic elements. The use of discontinous elements allow for accurate integration of finite part integrals. The accuracy of the proposed method is demonstrated by solving a number of problems including edge and embedded cracks.

Linear finite element methods for planar linear elasticity

Abstract: A linear nonconforming (conforming) displacement finite element method for the pure displacement (pure traction) problem in two-dimensional linear elasticity for a homogeneous isotropic elastic material is considered. In the case of a convex polygonal configuration domain, error estimates in the energy (L[sup 2]) norm are obtained. The convergence rate does not deteriorate for nearly incompressible material. Furthermore, the convergence analysis does not rely on the theory of saddle point problems. 22 refs.

Composite and Reinforced Elements of Construction

Abstract: Part 1 Mechanics of inhomogeneous deformable solids: basic relations of continuum mechanics basic equations of thermoelasticity and electroelasticity mechanical models of composite materials. Part 2 Asymptotic homogenization of regular structures: homogenization techniques for periodic structures homogenization method for regions with a wave boundary local problems and effective coefficients. Part 3 Elasticity of regular composite structures: homogenization of the linear elasticity problem laminated composites - effective properties and fracture criteria effective characteristics of unidirectional fibre composites plane elasticity problem for a periodic composite with a crack homogenization of the geometrically nonlinear elasticity problem for a periodic composite elastic stability equatiions. Part 4 Thermoelasticity of regular composite structures: homogenization of thermoelasticity problem fibre composites - local stresses and effective properties laminated composite with prescribed thermoelastic properties composite material design. Part 5 General homogenization models for composite shells and plates with rapidly varying thickness: elasticity problem for a shell of a regularly nonhomogeneous material with wavy surfaces thermal conductivity of a curved thin shell of a regularly nonhomogeneous material with corrugated surfaces thermoelasticity of a curved shell of regularly nonhomogeneous material with corrugated surfaces geometrically nonlinear problem for a thin regularly nonhomogeneous shell with corrugated faces. Part 6 Structurally nonhomogeneous periodic shells and plates: local problem formulation for structurally nonhomogeneous shells and plates of orthotropic material an approximate method for determining the effective properties of ribbed and wafer shells effective properties of a three-layered shell with a honeycomb filler elastic moduli and local stresses in wafer type plates and shells, including the interaction between cell elements stretching of shells or plates reinforced by regular system of thin surface strips shells and plates with corrugated surfaces of regular structure. Part 7 Network and framework reinforced shells and plates with regular structure effective elastic moduli of a network reinforced shell effective thermoelastic properties of network reinforced shells heat conduction of network reinforced plates and shells constitutional equations for network reinforced plates and shells of rectangular, rhombic and triangular structure composite shells with high stiffness framework type reinforcement. Part 8 The fundamental solution of the periodic elasticity problem: the derivation of a doubly periodic fundamental solution of the three-dimensional elasticity problem transformation of the doubly periodic fundamental solution of the elasticity problem singly periodic fundamental solution of the plane elasticity. Appendices.

Strain relaxation by domain formation in epitaxial ferroelectric thin films.

Abstract: The origin of strain-induced, modulated domain structures observed in epitaxial ferroelectric lead titanate thin films is discussed using a phenomenological total-energy calculation Linear elasticity is used to account for the substrate contribution while a free-energy functional of the Landau-Ginzburg-Devonshire type is used to calculate the domain-wall and the polarization contributions from the film Good agreement between the predictions of this model and the experimental results is found for thickness-dependent properties such as the relative domain population and spontaneous strain

Low-Velocity Impact Response of Foam-Core Sandwich Composites

Abstract: The low-velocity impact response of foam-core sandwich composites with fiberglass/epoxy face sheets is treated by a combination of computational and experimental methods. Linear elastic constitutive models are used for the face sheets and epoxy bond layer in conjunction with a foam constitutive model that includes nonlinear hardening plasticity and coupling between volumetric and deviatoric deformation. A transient finite- element code, utilizing four-noded uniform strain quadrilaterals, is used to explicitly solve the equations for balance of mass and momentum. The resulting deformation histories are compared to the experimental results and show qualitative agreement. The computed transverse shear stresses are used to correlate ultrasonic measurement of damage in the core/epoxy interface. Comparison of the plate stiffness prior to and after impact illustrates the effect of damage on subsequent behavior.

Axisymmetric wave propagation in fluid-loaded cylindrical shells. I: Theory

Abstract: An analysis is carried out of axisymmetric waves propagating along fluid‐loaded cylindrical shells within the framework of linear elasticity and classical perfect‐slip boundary conditions at the solid–fluid interface. Numerical solutions are obtained for various axisymmetric eigenmodes for a cylindrical shell in vacuum; a cylindrical shell surrounded by a liquid of infinite radial extent; a hypothetical liquid column with both the stress‐free and displacement‐free boundary conditions; a cylindrical shell with a liquid core; and a cylindrical shell immersed in an infinite liquid. Numerical results are obtained for both the radiating (leaky) and nonradiating eigenmodes of the system by a careful search of the complex eigenfrequencies of the associated boundary value problem. In particular, attenuation of leaky modes due to radiation of energy into the surrounding medium is expressed in terms of the imaginary part of the eigenfrequency. Computational results are presented for the dispersion curves as well as the displacement and stress amplitude component distributions along the radial direction for various propagating modes of the system. Practical benefits from such analyses are discussed.

Use of the distinct element method to perform stress analysis in rock with non-persistent joints and to study the effect of joint geometry parameters on the strength and deformability of rock masses

Abstract: To use the distinct element method, it is necessary to discretize the problem domain into polygons in two dimensions (2 D) or into polyhedra in three dimensions (3 D). To perform distinct element stress analysis in a rock mass which contains non-persistent finite size joints, it is necessary to generate some type of fictitious joints so that when they are combined with the non-persistent joints, they discretize the problem domain into polygons in 2 D or into polyhedra in 3 D. The question arises as to which deformation and strength parameter values should be assigned to these fictitious joints so that they behave as intact rock. In this paper, linear elastic, perfectly-plastic constitutive models with the Mohr-Coulomb failure criterion, including a tension cut-off, were used to represent the mechanical behaviour of both intact rock and fictitious joints. It was found that, by choosing the parameter values for the constitutive models as given below, it is possible to make the fictitious joints behave as intact rock, in a global sense. Some examples are given in the paper to illustrate how to use the distinct element method to perform stress analysis of rock blocks which contain non-persistent joints and to study the effect of joint geometry parameters on strength and deformability of rock masses.

Resilient characterization of pavement materials

Abstract: The paper presents an approach for characterizing pavement materials using the modified linear elastic behaviour. The secant modulus of elasticity is expressed in terms of the stress invariants and an expression for the secant Poisson's ratio is derived using path independence of the total work along a closed loading cycle. Triaxial test results of granular base–subbase materials which exhibit strong non-linear behaviour and dilatancy are analysed and presented. The constitutive law is included in a finite element program and results of pavement analyses are discussed. It is found that the secant Poisson's ratio of granular base materials reaches values between 0·6 and 0·7, indicating a volume increase under high stress ratios. The pavement response predicted using the above material characterization is compatible with non-destructive test results.

Use of ‘simple solutions’ for boundary integral methods in elasticity and fracture analysis

Abstract: The 'simple solutions' or 'indirect' method of analysing Cauchy and hypersingular integrands in the gradient (flux or traction) boundary integral equation (BIE) is applied to linear elastic fracture analysis. Because of the geometric singularity of the crack surface, application of the simple solutions formulas on the crack face requires integration over a temporary 'closure surface' rather than the remainder of the body. Closure surface constructions are exhibited for crack surfaces, allowing the gradient BIE to be applied as a constraint equation on a crack surface where the primary BIE is degenerate. Computational results are given for two benchmark fracture problems.

Justification of the two-dimensional equations of a linearly elastic shallow shell

Abstract: We consider a problem in three-dimensional linearized elasticity, posed over a shell with a specific geometry, subjected to general loadings, and clamped on a portion of its lateral surface. We show that, as the thickness of the shell goes to zero, the solution of the three-dimensional problem converges to the solution of two-dimensional shallow shell equations. This approach, which provides in particular a mathematical definition of “shallowness”, clearly delineates conditions under which a three-dimensional problem may be deemed asymptotically equivalent to a two-dimensional shallow shell problem.

Bifurcations and instabilities in fracture of cohesive-softening structures: a boundary element analysis†

Abstract: The “cohesive-crack model” is adopted, together with the hypotheses of small deformations and linear elasticity outside the process zone or “craze”, for the simulation of fracture processes in structures of concrete-like materials. A “direct”, collocation, multidomain boundary element method is employed and shown to be computationally effective in the considered situations, which are characterized by non-linearity on interfaces only. Iterative algorithms for the direction search and interface adjustment in propagation analysis and for the determination of the response to a craze-tip advancement are developed and numerically tested. Softening as an instabilizing factor embodied in the cohesive-crack model may give rise to path bifurcations (“equilibrium branching”), instability under load control and intrinsic (“snapback”) instability. These phenomena are analysed by the proposed boundary element procedure and discussed.

Finite element calculations of the accommodation energy of a misfitting precipitate in an elastic-plastic matrix

Abstract: The elastic-plastic accommodation energy generated by the formation of a plate-shaped inclusion in an effectively infinite solid is calculated using two-dimensional (2-D) and three-dimensional (3-D) finite element techniques. A typical example of the occurrence of such an inclusion, modeled in detail in this article, is the formation of a zirconium hydride platelet in a zirconium alloy. To verify the finite element models, initial calculations were based on a linear elastic model of the inclusion and the surrounding matrix material, plus elastic-plastic solutions of an isotropically misfitting spherical inclusion expanding within an elastic/perfectly plastic, infinite solid. Good agreement with the corresponding exact analytical results was found. The finite element analysis was used to determine the accommodation energy of isotropically and anisotropically misfitting oblate spheroids contained in an elastic/perfectly plastic medium. Calculations were carried out for oblate spheroids with aspect ratios (semiminor to semimajor axes) of 0.75, 0.5, 0.25, and 0.1. In contrast to the elastic result, the elastic-plastic accommodation energy values increased with decreasing aspect ratio. This result is due to an increase in the hydrostatic component of the stress in the matrix and a consequent loss in ability to decrease the misfit stresses by plastic deformation. Three-dimensional analyses of cuboidal inclusions expanding into infinite elastic and elastic/plastic solids were also performed. The results depended on mesh density, but reasonable values could be obtained at moderate mesh densities.

Prediction of resin pocket geometry for stress analysis of optical fibers embedded in laminated composites

Abstract: A linear elastic study is performed, as a first-order approximation, to investigate the geometry of the resin-rich region observed around fiber-optic sensors embedded in laminated 'adaptive' composites. The Rayleigh-Ritz method is employed with beam-bending functions as assumed trial functions. The total potential energy is minimized with respect to unknown force distributions in each layer and the unknown length of the resin pocket. The resulting system of coupled nonlinear equations is solved by the Levenberg-Marquardt algorithm to compute the shape and size of the resin pocket. Results of this analysis show the effect of laminate stacking sequence, lamination pressure, and optical fiber diameter on the geometry of the resin pocket; and are found to agree well with experimental observations. The computed geometry is automatically discretized for finite-element analysis in order to obtain stress information in the vicinity of the resin pocket.

Finite element based mechanical models of the cornea for pressure and indenter loading

Abstract: Background In this article, we examine the finite element method and its use in mechanical modeling of the cornea. Methods Both linear elastic and geometrically nonlinear models are considered. The effects of corneal geometry, boundary conditions, thickness, and the number of layers in the model are determined. Results Results indicate that, for intraocular pressure loading, the thickness and the choice of boundary conditions at the limbus are significant. Conclusions A model of tonometry showed that simulating the layered nature of the cornea and including the increased flexibility of large deformation are critical.

Ill-posedness at the boundary for elastic solids sliding under Coulomb friction

Abstract: We consider an incompressible neo-Hookean elastic solid sliding on a rigid surface under the influence of Coulomb friction. It is shown that ill-posedness at the boundary due to failure of Agmon's condition can occur. If the friction coefficient is greater than one, this is the case even in the limit of linear elasticity. The effect of a dependence of the friction force on the sliding velocity is also considered.

A formulation of Stokes's problem and the linear elasticity equations suggested by the Oldroyd model for viscoelastic flow

Abstract: — We propose a three fields formulation of Stokes' s problem and the équations of linear elasticity, allowing conforming finite element approximation and using only the classical inf-sup condition relating velocity and pressure. No condition ofthis type is needed on the « non Newtonian » extra stress tensor. For the linear elasticity équations this method gives uniform results with respect to the compressibility. Résumé. — On propose une formulation à trois champs du problème de Stokes et des équations de V élasticité linéaire, permettant des approximations par éléments finis conformes et ne nécessitant que la classique condition inf-sup en vitesse pression à V exclusion de toute condition sur le tenseur « non Newtonien » des extra contraintes. Sur les équations de V élasticité linéaire la méthode est uniforme par rapport à la compressibilité.

Static and dynamic analysis of transversely isotropic, moderately thick sandwich beams by analogy

Abstract: A flexural theory of elastic sandwich beams is derived which renders quite precise results within a wide range of ratios of dimensions, mass densities, and elastic constants of the core and faces. The assumptions of the Timoshenko theory of shear-deformable beams are applied to each of the homogeneous, linear elastic, transversely isotropic layers individually. Core and faces are perfectly bonded. The principle of virtual work is applied to derive the equations of motion of a symmetrically designed three-layer beam and its boundary conditions. By definition of an effective cross-sectional rotation the complex problem is reduced to a problem of a homogeneous beam with effective stiffnesses and with corresponding boundary conditions. Thus, methods of classical mechanics become directly applicable to the higher-order problem. Excellent agreement of the results of illustrative examples is observed when compared to solutions of other higher-order laminate theories as well as to exact solutions of the theory of elasticity.

Micromechanical Analysis of Fracture of Concrete

Abstract: In this paper, a recently developed lattice model for simulating the fracture of concrete is presented. The material is modelled as a lattice of brittle-breaking beam ele- ments. The heterogeneity of the material is introduced in three manners: (1) by assigning random strength values to the beams in a regular lattice, (2) by generating a random parti- cle structure of the material and assigning different strength values to the beam elements appearing in the various composite materials, and (3) by assigning constant strength values to beam elements in a random lattice. The fracture law is extremely simple, and upon exceedence of the strength of a beam element, it is simply removed from the mesh. The analysis is completely linear elastic. With the model crack face bridging in tension, curvi- linear crack growth and the fracture mechanism of double-edge notched four-point-shear beams can be simulated realistically. The model seems very attractive because only a small number of single valued parameters is needed. These parameters can be tuned to experimental data in a relatively simple manner. It is important that the crack shapes found

Equilibrium position of misfit dislocations at planar interfaces

Abstract: The equilibrium position of misfit dislocations at interfaces is analysed within linear elasticity theory. Calculations are performed for an isolated dislocation as well as for an infinite array of dislocations in an infinite bicrystal. Analytic expressions are obtained for the image forces on the dislocation array due to the elasticity discontinuity. The position of the dislocations is determined by the balance between image forces and coherency forces on the dislocations. An equilibrium position is obtained in crystal I with smaller shear modulus G1. The equilibrium stand-off distance is proportional to the inverse of the lattice misfit and increases with the ratio of the shear moduli, G2/G1, and of the Poisson's ratios, ν2/ν1, respectively. The calculated stand-off distance of misfit dislocations in Nb adjacent to a Nb Al2O3 interface is smaller by a factor of 2 than the experimentally observed distance. This discrepancy can be explained qualitatively by the higher core energy of a dislocation in the vicinity to the elastically rigid Al2O3 compared to the core energy of a dislocation in bulk Nb.

Finite-element analysis of time-dependent large-deformation problems

Abstract: An updated Lagrangian finite-element formulation has been developed for time-dependent problems of soil consolidation involving finite deformations. Large plastic strains as well as rotations occur in such problems and nominal stress measures are introduced in the formulation to redefine stresses. This leads to corrective terms for equilibrium and yield violations in addition to geometric stiffening terms in the governing integral equations. The soil is considered to be either a linear elastic or an elastoplastic, critical-state material. Some simple numerical examples are studied to validate the formulation, followed by a detailed analysis of the problem of penetration of a pile into soil. The results of this problem are viewed with emphasis on the physical interpretation and practical significance.