Showing papers on "Linear elasticity published in 1998"

Prediction of Failure Load of R/C Beams Strengthened with FRP Plate Due to Stress Concentration at the Plate End

Abstract: Epoxy-bonding a composite plate to the tension face is an effective technique for repair and retrofit of reinforced concrete beams. Experiments have indicated local failure of the concrete layer between the plate and longitudinal reinforcement in retrofitted beams. This mode of failure is caused by local stress concentration at the plate end as well as at the flexural cracks. This paper presents a method for calculating shear and normal stress concentration at the cutoff point of the plate. This method has been developed based on linear elastic behavior of the materials. The effect of the large flexural cracks along the beam has also been investigated. The model has been used to find the shear stress concentration at these cracks. The predicted results have been compared with both the finite element method and experimental results. The analytical models provide closed form solutions for calculating stresses at the plate ends that can easily be incorporated into design equations.

A notch intensity factor approach to the stress analysis of welds

Abstract: In the context of linear elastic stress gradients that are present in welded joints, a stress field approach based on notch stress intensity factors is presented with the aim of describing stress distributions in the neighbourhood of weld toes, since fatigue strength is dependent on such distributions. This paper summarizes the analytical fundamentals and gives an appropriate definition of the parameters for stress components under opening and sliding modes. Then, by comparing the expected results with those obtained by numerical analysis, the contributions of the symmetric and skew-symmetric loading modes are quantified for different geometries, and summarized into concise expressions which also take into account the influence of the main geometrical parameters of the welded joint. The range of validity and the application limits of this field approach in the presence of weld toe radii are discussed. Finally, a synthesis of experimental fatigue strength data based on the new field parameters is reported.

Numerical simulation of crack growth in an isotropic solid with randomized internal cohesive bonds

Abstract: A virtual internal bond (VIB) model with randomized cohesive interactions between material particles is proposed as an integration of continuum models with cohesive surfaces and atomistic models with interatomic bonding. This approach differs from an atomistic model in that a phenomenological “cohesive force law” is assumed to act between “material particles” which are not necessarily atoms; it also differs from a cohesive surface model in that, rather than imposing a cohesive law along a prescribed set of discrete surfaces, a randomized network of cohesive bonds is statistically incorporated into the constitutive law of the material via the Cauchy-Born rule, i.e., by equating the strain energy function on the continuum level to the potential energy stored in the cohesive bonds due to an imposed deformation. This work is motivated by the notion that materials exhibit multiscale cohesive behaviors ranging from interatomic bonding to macroscopic ductile failure. It is shown that the linear elastic behavior of the VIB model is isotropic and obeys the Cauchy relation; the instantaneous elastic properties under equibiaxial stretching are transversely isotropic, with all the in-plane components of the material tangent moduli vanishing at the cohesive stress limit; the instantaneous properties under equitriaxial stretching are isotropic with a finite strain modulus. We demonstrate through two preliminary numerical examples that the VIB model can be applied in direct simulation of crack growth without a presumed fracture criterion. The prospect of this type of approach in numerical simulations of fracture seems to be highly promising.

Analysis of fault slip inversions: Do they constrain stress or strain rate?

Abstract: Fault slip data commonly are used to infer the orientations and relative magnitudes of either the principal stresses or the principal strain rates, which are not necessarily parallel or equal. At the local scale of an individual fault, the shear plane and slip direction define the orientations of the local principal strain rate axes but not, in general, the local principal stress axes. At a large scale, the orientations of P and T axes maxima for sets of fault slip data do not provide accurate inversion solutions for either strain rate or stress. The quantitative inversion of such fault slip data, however, provides direct constraints on the orientations and relative magnitudes of the global principal strain rates. To interpret the inversion solution as constraining the global principal stresses requires that (1) the fault slip pattern must have a characteristic symmetry no lower than orthorhombic; (2) the material must be mechanically isotropic; and (3) there must be a linear constitutive relationship between the global stress and the global strain rate. Isotropic linear elastic constitutive equations are appropriate to describe the local deformation surrounding an individual slip discontinuity. Fault slip inversions, however, constrain the characteristics of a large-scale cataclastic flow, which is described by constitutive equations that are probably, but to an unknown degree, anisotropic and nonlinear. Such material behavior would not strictly satisfy the requirements for the stress interpretation. Thus, at the present state of knowledge, fault slip inversion solutions are most reliably interpreted as constraining the principal strain rates.

A fast solution method for three‐dimensional many‐particle problems of linear elasticity

Abstract: A boundary element method for solving three-dimensional linear elasticity problems that involve a large number of particles embedded in a binder is introduced. The proposed method relies on an iterative solution strategy in which matrix—vector multiplication is performed with the fast multipole method. As a result the method is capable of solving problems with N unknowns using only O(N) memory and O(N) operations. Results are given for problems with hundreds of particles in which N"O(105). ( 1998 John Wiley & Sons, Ltd.

Shear stiffness of a solid–solid multicontact interface

Abstract: The macroscopic multicontact between two rough nominally flat surfaces is a common object whose physics is only partially understood. This paper is aimed at giving experimental evidence for the linear elastic response of a multicontact interface to a moderate shear force, i.e. below the threshold for incipient sliding. Non–intuitive properties of the interfacial shear stiffness are exhibited, in the spirit of macroscopic friction laws, which should be of practical interest when evaluating the performances of a built–up system. These are explained qualitively within the random surface framework prevailing in multicontact mechanics, and a numerical treatement of the three–dimensional profile of a real rough surface is proposed, which enables a direct quantitative simulation of the elastic stiffness. This is found to be compatible with experimental data on a polymer glass and an aluminium alloy. The sensitivity of interfacial stiffness measurements is discussed, and illustrated by the experimental evidence of the plastic deformation of aluminium alloy asperities under light nominal pressure. This emphasizes the need for an elastoplastic description of asperity deformation within a multicontact.

A posteriori error estimates for mixed FEM in elasticity

Abstract: A residue based reliable and efficient error estimator is established for finite element solutions of mixed boundary value problems in linear, planar elasticity. The proof of the reliability of the estimator is based on Helmholtz type decompositions of the error in the stress variable and a duality argument for the error in the displacements. The efficiency follows from inverse estimates. The constants in both estimates are independent of the Lame constant $\lambda$ , and so locking phenomena for $\lambda\to\infty$ are properly indicated. The analysis justifies a new adaptive algorithm for automatic mesh–refinement.

Elastic solutions for a transversely isotropic half-space subjected to a point load

Abstract: SUMMARY We rederive and present the complete closed-form solutions of the displacements and stresses subjected to a point load in a transversely isotropic elastic half-space. The half-space is bounded by a horizontal surface, and the plane of transverse isotropy of the medium is parallel to the horizontal surface. The solutions are obtained by superposing the solutions of two infinite spaces, one acting a point load in its interior and the other being free loading. The Fourier and Hankel transforms in a cylindrical co-ordinate system are employed for deriving the analytical solutions. These solutions are identical with the Mindlin and Boussinesq solutions if the half-space is homogeneous, linear elastic, and isotropic. Also, the Lekhnitskii solution for a transversely isotropic half-space subjected to a vertical point load on its horizontal surface is one of these solutions. Furthermore, an illustrative example is given to show the e⁄ect of degree of rock anisotropy on the vertical surface displacement and vertical stress that are induced by a single vertical concentrated force acting on the surface. The results indicate that the displacement and stress accounted for rock anisotropy are quite di⁄erent for the displacement and stress calculated from isotropic solutions. ( 1998 John Wiley & Sons, Ltd.

Non-linear model of small-strain behaviour of soils

Abstract: The paper describes a general approach to the development of a constitutive model for predict-ing the generalized small-strain behaviour of soils. The model incorporates non-linear stress— strain behaviour, stress-path dependence and a consistent definition of the small-strain region (SSR). Based on this general approach, a simple model is proposed which incorporates crossanisotropic elasticity within the linear elastic region (LER) and elliptical boundaries for the LER and SSR. The model leads to unique normalized stress—strain behaviour of undisturbed Bothkennar clay for the complete range of stress probe directions in triaxial stress space. The comparison confirms the uniqueness of the normalized volumetric and deviatoric stress— strain behaviour. The model can easily be incor-porated into numerical procedures for solving two- and three-dimensional boundary value problems. ĺarticle decrit une conception generale de la mise au point ´un modele constitutif pour predlre le comportement general des s...

Hartig's law and linear elasticity with initial stress

Abstract: The two constitutive equations which have hitherto been called upon in mathematical studies on static determination of residual stress by boundary measurements share the same deficiency, namely that they do not adequately describe the elastic response of any currently known real material. Here we substantiate this critical remark and present a physically more acceptable constitutive equation for further work on the subject.

First-Order System Least Squares (FOSLS) for Planar Linear Elasticity: Pure Traction Problem

Abstract: This paper develops two first-order system least-squares (FOSLS) approaches for the solution of the pure traction problem in planar linear elasticity. Both are two-stage algorithms that first solve for the gradients of displacement (which immediately yield deformation and stress), then for the displacement itself (if desired). One approach, which uses L 2 norms to define the FOSLS functional, is shown under certain H 2 regularity assumptions to admit optimal H 1 -like performance for standard finite element discretization and standard multigrid solution methods that is uniform in the Poisson ratio for all variables. The second approach, which is based on H -1 norms, is shown under general assumptions to admit optimal uniform performance for displacement flux in an L 2 norm and for displacement in an H 1 norm. These methods do not degrade as other methods generally do when the material properties approach the incompressible limit.

Effective Elastic Constants of Particulate Composites with Inhomogeneous Interphases

Abstract: We consider a composite material reinforced with spherical inclusions, which are arranged in a matrix in a statistically isotropic way and they are bonded to the matrix via an interphase. We represent the interphase as an inhomogeneous region (a functionally graded material) and consider an idealized model in which the interphase is linear elastic and isotropic and has the Young's modulus and Poisson's ratio varying in radial direction. We predict effective elastic constants of such a composite and study the influence of the spatial inhomogeneity of interphase on these constants. In the analysis we use the composite spheres assemblage method [1] to evaluate the effective bulk modulus and the generalized self-consistent method [21 to obtain the effective shear modulus. We show that the radial variation in the Young's modulus of interphase has an influence on the effective elastic moduli as compared with the corresponding uniform interphase case when particles are stiffer than the matrix and the effect is n...

Modelling of muscle behaviour by the finite element method using hill's three-element model

Abstract: We present a numerical algorithm for the determination of muscle response by the finite element method. Hill's three-element model is used as a basis for our analysis. The model consists of one linear elastic element, coupled in parallel with one non-linear elastic element, and one non-linear contractile element connected in series. An activation function is defined for the model in order to describe a time-dependent character of the contractile element with respect to stimulation. Complex mechanical response of muscle, accounting for non-linear force–displacement relation and change of geometrical shape, is possible by the finite element method. In an incremental-iterative scheme of calculation of equilibrium configurations of a muscle, the key step is determination of stresses corresponding to a strain increment. We present here the stress calculation for Hill's model which is reduced to the solution of one non-linear equation with respect to the stretch increment of the serial elastic element. The muscle fibers can be arbitrarily oriented in space and we give a corresponding computational procedure of calculation of nodal forces and stiffness of finite elements. The proposed computational scheme is built in our FE package PAK, so that real muscles of complex three-dimensional shapes can be modelled. In numerical examples we illustrate the main characteristic of the developed numerical model and the possibilities of solution of real problems in muscle functioning. © 1998 John Wiley & Sons, Ltd.

A 3-D Model for a Multilayered Body Loaded Normally and Tangentially Against a Rigid Body: Application to Specific Coatings

Abstract: Coatings are increasingly used to improve the mechanical and tribological behavior of surfaces. It is necessary to develop models to guide the initial choice of coating/ substrate combinations that can withstand the applied load, A three-dimensional model of an elastic multilayered body, loaded both normally and tangentially against an elliptical rigid body ( partial sliding, rolling/sliding conditions ), is presented here. This model is based on linear elasticity theory, integral transforms, Fast Fourier Transform, and unilateral contact analysis with friction. Normal and tangential contact conditions between the two bodies are first determined and then used to calculate the multilayered body stress field. One application is given here: The influence of the mechanical properties of coating and substrate, as well as coating thickness, is studied on contact conditions, internal stresses, and potential failure mechanisms.

Static and dynamic boundary element analysis in incompressible linear elasticity

Abstract: The boundary element formulation of incompressible, isotropic, linear elastostatics and frequency domain elastodynamics for the displacements and hydrostatic pressure is presented. Both two- and three-dimensional problems are considered. It is proven that the incompressible fundamental tensors can be obtained from the corresponding compressible ones by simply putting the Poisson ratio ν equal to 0.5 in elastostatics and the p -wavenumber k p equal to zero in elastodynamics. The various kernels employed in a complete static or dynamic boundary element analysis for the compressible case are written in a modified form that permits one to use already existing codes for compressible, incompressible or nearly incompressible cases without any problem. Numerical examples involving two- and three-dimensional problems under static and dynamic conditions are presented. These examples serve to illustrate the method and demonstrate its high accuracy and efficiency.

Discrete dislocation simulations and size dependent hardening in single slip

Abstract: Plastic deformation in two-dimensional monophase and composite materials is studied using a discrete dislocation dynamics method. In this method, dislocations are represented by line defects in a linear elastic medium, and their interactions with boundaries or second-phase elastic particles are incorporated through a complementary finite element solution. The formulation includes a set of simple constitutive rules to model the lattice resistance to dislocation glide, as well as the generation, annihilation and pinning of dislocations at point obstacles. The focus is on the predicted strain hardening of these materials when only a single slip system is active. When the particle morphology is such as to require geometrically necessary dislocations, hardening in the composite materials exhibits a distinct size effect. This size effect is weaker than that predicted by simple analytical estimates based on geometrically necessary dislocations.

Confined concrete subjected to uniaxial monotonic loading

Abstract: The behavior of concrete confined with carbon and glass fiber wraps is determined by uniaxial compression tests of filament wound composite cylinders filled with concrete. Experimental variables include concrete strength, type of fiber, and the ratio of fiber volume to concrete volume. Both axial and circumferential strains are measured in addition to axial stress. Two models are presented for the observed behavior. Both models assume that the total strain is comprised of linear elastic strain, strain due to void collapse, and strain due to slip along critical crack surfaces. The model for the crack strain is developed from simple mechanical principles, while the model for void strain is empirical. Some of the crack model parameters were determined by separate shear tests. A parameter study is presented that looks at the effects of varying some of the model parameters. A comparison of experimental and model results is also presented.

A thick hollow sphere compressed by equal and opposite concentrated axial loads: an asymptotic solution

Abstract: We consider an elastic hollow sphere with midsurface radius R and thickness 2h which is subjected to two equal and opposite concentrated loads acting at the ends of a diameter. The three-dimensional linear elasticity solution to this problem consists of (i) a narrow Saint Venant component extending a distance of order O(h) from each load point, (ii) a wider "edge bending" component extending a distance of order $O(\sqrt{Rh})$ from each load point, and (iii) a "membrane" component which permeates the whole sphere. Because of the stress singularities at the load points, the Saint Venant component is extremely complex and difficult to calculate. We determine the other two (outer) components of the solution without any explicit knowledge of the Saint Venant (or inner) component. This is achieved by a rigorously valid method which does not depend on any physical heuristic, such as Saint Venant's principle. Our method depends on the fact, established here for the first time, that the eigenfunctions for the sphe...

Utility of elastic models in predicting fault displacement fields

Abstract: Although elastic models have long been used to model earthquake deformation, their application to fault problems is questionable, as accumulated fault strain is higher and the relevant timescales are longer. We test the utility of using elastic models to predict fault displacement fields by independently measuring the three-dimensional slip (offset) distribution and displacement field of small normal faults. The displacement field is constrained from the topography of the deformed bedding planes; the slip distribution is constrained from stratal offsets in multiple sections of fault-normal saw-cuts. Using the observed slip distribution, we calculate both one- and three-dimensional elastic displacement fields. We find that the large strain associated with fault growth can be accommodated with linear elastic models. Much of the remaining misfit between the data and the model may result from elastic interaction with other nearby faults, the inelastic zone around surrounding fault tips, and prefaulting irregularities in the measured bedding plane surface.

A least‐squares approach for uniform strain triangular and tetrahedral finite elements

Abstract: A least-squares approach is presented for implementing uniform strain triangular and tetrahedral finite elements. The basis for the method is a weighted least-squares formulation in which a linear displacement field is fit to an element's nodal displacements. By including a greater number of nodes on the element boundary than is required to define the linear displacement field, it is possible to eliminate volumetric locking common to fully integrated lower-order elements. Such results can also be obtained using selective or reduced integration schemes, but the present approach is fundamentally different from those. The method is computationally efficient and can be used to distribute surface loads on an element edge or face in a continuously varying manner between vertex, mid-edge and mid-face nodes. Example problems in two- and three-dimensional linear elasticity are presented. Element types considered in the examples include a six-node triangle, eight-node tetrahedron, and ten-node tetrahedron. © 1998 John Wiley & Sons, Ltd.