Showing papers on "Linear elasticity published in 1990"

Mechanics of Solid Materials

Abstract: 1. Elements of the physical mechanisms of deformation and fracture 2. Elements of continuum mechanics and thermodynamics 3. Identification and theological classification of real solids 4. Linear elasticity, thermoelasticity and viscoelasticity 5. Plasticity 6. Viscoplasticity 7. Damage mechanics 8. Crack mechanics.

Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods

Abstract: This paper discusses the homogenization method to determine the effective average elastic constants of linear elasticity of general composite materials by considering their microstructure. After giving a brief theory of the homogenization method, a finite element approximation is introduced with convergence study and corresponding error estimate. Applying these, computer programs PREMAT and POSTMAT are developed for preprocessing and postprocessing of material characterization of composite materials. Using these programs, the homogenized elastic constants for macroscopic stress analysis are obtained for typical composite materials to show their capability. Finally, the adaptive finite element method is introduced to improve the accuracy of the finite element approximation.

Nonlinear elasticity and pressure-dependent wave speeds in granular media

Abstract: Following is an analysis of the small-strain nonlinear elasticity of granular media near states of zero stress, as it relates to the pressure-dependent incremental linear elasticity and wave speeds. The main object is elucidation of the p ½ dependence of incremental elastic moduli on pressure p , a dependence observed in numerous experiments but found to be at odds with the p ½ scaling predicted by various micromechanical models based on hertzian contact. After presenting a power-law continuum model for small-strain nonlinear elasticity, the present work develops micromechanical models based on two alternative mechanisms for the anomalous pressure scaling, namely: (1) departures at the single-contact level from the hertzian contact, due to point-like or conical asphericity; (2) variation in the number density of hertzian contacts, due to buckling of particle chains. Both mechanisms result in p ½ pressure scaling at low pressure and both exhibit a high-pressure transition to p ½ scaling at a characteristic transition pressure p *. For assemblages of nearly equal spheres, a non-hertzian contact model for mechanism (1) and percolation-type model for (2) yield estimates of p * of the form p * = c μ ˆ ∝ 3 . Here c is a non-dimensional coefficient depending only on granular-contact geometry, while α ≪ 1 is a small parameter representing spherical imperfections and μ ˆ is an appropriate elastic modulus of the particles. Then, with R representing particle radius and h a characteristic spherical tolerance or asperity height, it is found that α = ( h / R ) ½ for mechanism (1) as opposed to α = h / R for (2). Limited data from the classic experiments of Duffy & Mindlin on sphere assemblages tend to support mechanism (1), but more exhaustive experiments are called for. In addition to the above analysis of reversible elastic effects, a percolation model of inelastic ‘shake-down’ or consolidation is given. It serves to describe how prolonged mechanical vibration, leading to the replacement of point-like or inactive contacts by stiffer Hertz contacts may change the pressure-scaling behaviour of particulate media. The present analysis suggests that pressure-dependence of elasticity may provide a useful means of characterizing the state of consolidation and stability of dense particulate media.

Hypersingular integrals in boundary element fracture analysis

Abstract: A new general purpose boundary element method for domains with cracks has been recently developed. This technique avoids the use of a multi-domain decomposition by including an additional integral equation expressing the boundary condition on the crack. The principal requirement of this technique is the analytic determination of certain hypersingular integrals of the Green's function which arise from this equation. In order to establish the applicability of this method for fracture, these integrals are evaluated herein for the Kelvin solution of the three-dimensional Navier equations of linear elasticity. Numerical results for fracture problems using the single-domain boundary element analysis are also presented.

Nonlinear analysis in soil mechanics : theory and implementation

Abstract: Part I. FUNDAMENTALS. 1. Introduction. Characteristics of soil behavior.Idealizations and material modeling. Historical review of plasticity in soil mechanics. Nonlinear stress analyses in geotechnical engineering. Need, objectives and scope. References. 2. Basic Concept of Continuum Mechanics. Introduction. Notations. Stresses in three dimensions. Definitions and notations. Cauchy's formulas, index notation, and summation convention. Principal axes of stresses. Deviatoric stress. Geometrical representation of stresses. Strains in three dimensions. Definitions and notations. Deviatoric strain. Octahedral strains and principal shear strains. Equations of solid mechanics. Equations of equilibrium (or motion). Geometric (compatibility) conditions. Constitutive relations. Summary. References. Part II. MATERIAL MODELING-BASIC CONCEPTS. 3. Elasticity and Modeling . Introduction. Elastic models in geotechnical engineering. Linear elastic model (generalized Hooke's law). Cauchy elastic model. Hyperelastic model. Hypoelastic model. Uniqueness, stability, normality, and convexity for elastic materials. Uniqueness. Drucker's stability postulate. Existence of W and v. Restrictions - normality and convexity. Linear elastic stress-strain relations. Generalized Hooke's law. A plane of symmetry. Two planes of symmetry (orthotropic symmetry). Transverse and cubic isotropies. Full isotropy. Isotropic linear elastic stress-strain relations. Tensor forms. Three-dimensional matrix forms. Plane stress case. Plane strain case. Axisymmetric case. Isotropic nonlinear elastic stress-strain relations based on total formulation. Nonlinear elastic model with secant moduli. Cauchy elastic model. Hyperelastic (green) model. Isotropic nonlinear elastic stress-strain relations based on incremental formulation. Nonlinear elastic model with secant muduli. Cauchy elastic model. Hyerelastic model. Hypoelastic model. Summary. References. 4. Perfect Plasticity and Modeling. Introduction. Deformation theory. An illustrative example. Variable moduli models. Flow theory. Yield criteria. Flow rule. Basic requirements. Perfect plasticity models. Tresca and von Mises models. Coulomb model. Drucker-Prager model. Prandtl-Reuss stress-strain relations. Generalized stress-strain relations. Stiffness formulation. General description. Stiffness coefficients. Summary. References. 5. Hardening Plasticity and Modeling. Introduction. Flow theory. Loading function. Hardening rule. Flow rule. Drucker's postulate. Hardening plasticity models. Lade-Duncan model. Lade model. Nested yield surface models. Generalized multi-surface models. Bounding surface models. Prandtl-Reuss stress-strain relations. Prandtl-Reuss equations. Matrix form of Prandtl-Reuss equations. Generalized stress-strain relations. Incremental stress-strain relations. Isotropic hardening. Kinematic hardening. Mixed hardening. Stiffness formulation. General description. Stiffness coefficients. Summary. References. PART III.

Fracture Mechanics Criteria and Applications

Abstract: 1. Introductory chapter.- 1.1. Conventional failure criteria.- 1.2. Characteristic brittle failures.- 1.3. Griffith's work.- 1.4. Fracture mechanics.- References.- 2. Linear elastic stress field in cracked bodies.- 2.1. Introduction.- 2.2. Crack deformation modes and basic concepts.- 2.3. Eigenfunction expansion method for a semi-infinite crack.- 2.4. Westergaard method.- 2.5. Singular stress and displacement fields.- 2.6. Method of complex potentials.- 2.7. Numerical methods.- 2.8. Experimental methods.- 2.9. Three-dimensional crack problems.- 2.10. Cracks in bending plates and shells.- References.- 3. Elastic-plastic stress field in cracked bodies.- 3.1. Introduction.- 3.2. Approximate determination of the crack-tip plastic zone.- 3.3. Small-scale yielding solution for antiplane mode.- 3.4. Complete solution for antiplane mode.- 3.5. Irwin's model.- 3.6. Dugdale's model.- 3.7. Singular solution for a work-hardening material.- 3.8. Numerical solutions.- References.- 4. Crack growth based on energy balance.- 4.1. Introduction.- 4.2. Energy balance during crack growth.- 4.3. Griffith theory.- 4.4. Graphical representation of the energy balance equation.- 4.5. Equivalence between strain energy release rate and stress intensity factor.- 4.6. Compliance.- 4.7. Critical stress intensity factor fracture criterion.- 4.8. Experimental determination of KIc.- 4.9. Crack stability.- 4.10. Crack growth resistance curve (R-curve) method.- 4.11. Mixed-mode crack propagation.- References.- 5. J-Integral and crack opening displacement fracture criteria.- 5.1. Introduction.- 5.2. Path-independent integrals.- 5.3. J-integral.- 5.4. Relationship between the J-integral and potential energy.- 5.5. J-integral fracture criterion.- 5.6. Experimental determination of the J-integral.- 5.7. Stable crack growth studied by the J-integral.- 5.8. Mixed-mode crack growth.- 5.9. Crack opening displacement (COD) fracture criterion.- References.- 6. Strain energy density failure criterion.- 6.1. Introduction.- 6.2. Volume strain energy density.- 6.3. Basic hypotheses.- 6.4. Two-dimensional linear elastic crack problems.- 6.5. Uniaxial extension of an inclined crack.- 6.6. Three-dimensional linear elastic crack problems.- 6.7. Bending of cracked plates.- 6.8. Ductile fracture.- 6.9. Failure initiation in bodies without pre-existing cracks.- 6.10. Other criteria based on energy density.- References.- 7. Dynamic fracture.- 7.1. Introduction.- 7.2. Mott's model.- 7.3. Stress field around a rapidly propagating crack.- 7.4. Strain energy release rate.- 7.5. Transient response of cracks to impact loads.- 7.6. Standing plane waves interacting with a crack.- 7.7. Crack branching.- 7.8. Crack arrest.- 7.9. Experimental determination of crack velocity and dynamic stress intensity factor.- References.- 8. Fatigue and environment-assisted fracture.- 8.1. Introduction.- 8.2. Fatigue crack propagation laws.- 8.3. Fatigue life calculations.- 8.4. Variable amplitude loading.- 8.5. Mixed-mode fatigue crack propagation.- 8.6. Nonlinear fatigue analysis based on the strain energy density theory.- 8.7. Environment-assisted fracture.- References.- 9. Engineering applications.- 9.1. Introduction.- 9.2. Fracture mechanics design philosophy.- 9.3. Design example problems.- 9.4. Fiber-reinforced composites.- 9.5. Concrete.- 9.6. Crack detection methods.- References.- Author Index.

Stretching of Grafted Polymer Layers

Abstract: We calculate the osmotic pressure in a semi-dilute solution of stretched polymers in a good solvent. The results are applied to polymer brushes where polymers are stretched either directly or by application of shear. It is shown that when a brush is sheared against the solvent, the stretching proceeds always within the nonlinear regime of polymer elasticity. When the brush is subjected to a combination of compression and shear forces and the shear is gradually increased, there is a crossover between low-shear linear elastic behaviour to a nonlinear regime in which the brush swells and eventually reaches its full uncompressed thickness at high shear stress. The relevance of our results to present and future experiments is discussed.

Concentration dependence of the linear elastic behaviour of model microgel dispersions

Abstract: A model study of the linear viscoelastic behaviour of concentrated aqueous dispersions of deformable particles is described. Measurements on several grades of Sephadex beadlets with varying degrees of cross-linking were made in the linear range and demonstrated predominantly elastic behaviour. After allowing for differences in intrinsic swelling a universal concentration dependence was apparent. The detailed concentration dependence of the elastic modulus of the dispersions is consistent with a cell model with interparticle potentials given by Hertzian contact theory for elastic spheres. At very high concentrations, when swelling becomes restricted, the observed behaviour approaches that expected for deswelling of covalent gels with a scaling law exponent ≈0.6. The intrinsic elastic modulus of the granules inferred from the above cell model is in good agreement with independent estimates obtained from osmotic deswelling measurements on individual granules analysed according to Flory theory for cross-linked networks.

Micromechanics characterization of sublaminate damage

Abstract: A micromechanics analytical model is developed for characterizing the fracture behaviour of a fibre reinforced composite laminate containing a transverse matrix crack and longitudinal debonding along 0/90 interface. Both the matrix and the fibres are considered as linear elastic. A consistent shear lag theory is used to represent the stress-displacement relations. The governing equations, a set of differential-difference equations, are solved satisfying the boundary conditions appropriate to the damage configuration by making use of an eigenvalue technique. The properties of the constituents appear in the model explicitly. Displacements and stresses in the fibres and the matrix are obtained, and the growth of damage is investigated by using the point stress criterion. The investigation includes fibre stress distribution in zero degree plies, transverse crack and debonding intitiation as functions of laminate geometry, and the effect of fibre breaks in the zero degree ply on damage growth. The predicted damage growth patterns and the corresponding critical strains agree with the finite element and experimental results.

Negative Poisson’s ratio in a transversely isotropic foam structure

Abstract: Recent experimental work has shown that foams having a negative Poisson’s ratio may be fabricated by inverting the usual energetically preferred tetrahedral structure to form a reentrant structure. It is shown here that a linear elastic analysis of a two‐dimensional reentrant honeycomb provides a theoretical prediction of the negative Poisson’s ratio in the plane for a transversely isotropic material. The value of Poisson’s ratio depends upon the reticulation angle, which is related to the permanent volumetric compression ratio required to form the reentrant structure.

Prediction of soil stresses using the finite element method

Abstract: A finite element program has been developed that used a compaction model for agricultural soils developed at the National Soil Dynamics Laboratory (NSDL) and Auburn University to predict linear elastic parameters for each element in the model. Incremental loading was used by the finite element model to gradually load the soil so that these linear parameters could be varied many times over the loading period. The finite element model was compared with data obtained from soil bin research. Results showed that a flat disc load was modeled well but a spherical disc load was not.

Modeling of a folded plate

Abstract: It is shown that the solution of a three-dimensional linear elasticity problem in a thin folded plate converges strongly inH1 to a solution of a two-dimensional model as the thickness goes to 0. This model consists of two plate equations coupled through their common edge.

Residual Stress State of Brazed Ceramic/Metal Compounds, Determined by Analytical Methods and X-ray Residual Stress Measurements

Abstract: The residual stress state of brazed ceramic/metal compounds is described by means of X-ray residual stress determinations and analytical calculations using a model of three elastic infinite plates. It is shown that the residual stress state of the soldered compound depends on the materials combination and on the geometrical conditions. The combination of X-ray residual stress measurements and analytical calculations allows decisions on whether the assumption of a linear elastic model, based on elementary bending theory, is valid for the particular compounds.

Equivalence of finite element methods for problems in elasticity

Abstract: Modifications of the Morley method for the approximation of the biharmonic equation are obtained from various finite element methods applied to the equations of linear isotropic elasticity and the stationary Stokes equations, by elimination procedures analogous to those used in the continuous case. Problems with Korn’s first inequality for nonconforming $P_1 $ elements and its implications for the approximation of the elasticity equations are also discussed.

Mechanical conditions for stability and optimal convergence of mixed finite elements for linear plane elasticity

Abstract: In order to develop an efficient and manageable tool for checking stability and optimal convergence of mixed finite elements (LBB- and ‘equilibrium’ condition), three mechanical conditions are stated. The first requires the continuity of the normal component of the stress tensor across interelement boundaries, the second forbids spurious modes on a two element patch, and the third is to avoid zero-energy-stresses on an element. The mathematical proof shows that the conditions are necessary and sufficient. Finally, the hybrid implementation of two plane mixed elements is carried out, and comparisons are made with two standard displacement elements. In particular, the mixed element with constant displacement shape functions (MMC) surpasses the linear displacement element by far and also the quadratic displacement element if the computational effort is compared.

Linear elasticity of planar delaunay networks. Part II: Voigt and Reuss bounds, and modification for centroids

Abstract: The linear elastic Delaunay network model developed in a previous paper is used to obtain further results on mechanical properties of graph-representable materials. First, we investigate the error involved in the uniform strain approximation — a computationally inexpensive approach widely employed in the determination of effective moduli of granular and fibrous media. Although this approximation gives an upper bound on the macroscopic moduli, it results in very good estimates of their second order statistics. In order to derive a lower bound another window definition has to be introduced. Also, an energy-based derivation of both bounds is given. The final result relates to a modification of a Delaunay network so that its vertices correspond to the centroids of cells of the corresponding Voronoi tessellation; an increase of effective moduli and a decrease of their scatter are observed.

Analysis of elastic and elastic-plastic adhesive joints using a mathematical programming approach

Abstract: This paper is concerned with the stress analysis of adhesive joints. Aiming at a geometrically two-dimensional treatment of the adhesive, a linear displacement field through its thickness is assumed. Using the principle of virtual work and assuming linear elastic behaviour, equilibrium and constitutive equations are derived. In the case of a thin adhesive, these equations are simplified and the description so obtained is used also in the case of an elastic-perfectly plastic adhesive. In the case of elastic adhesive two different variational formulations are given: one in displacements and one mixed in displacements and adhesive tractions. In the elastic-plastic case a mixed formulation is given. Using these variational formulations finite element discretizations are performed. The matrix equations obtained in the elastic-plastic case are shown to be equivalent to a problem of mathematical programming: a parametric linear complementarity problem (LCP) involving derivatives. A solution algorithm previously used for contact problems with friction is proposed. As an application of the theory presented, an elastic-plastic analysis of a shear loaded adhesive joint test specimen is carried out.

Strain field measurement in fracture process zone

Abstract: Center‐notched mortar plate specimens are loaded in tension A multiple sensitivity vector holographic setup is developed to record several deformation stages during the stable crack propagation range The three‐sensitivity vector setup enables the calculation of both crack opening displacements and strain fields around the crack trajectories An image analysis system is used to isolate the interferometric effect from the sandwich holograms, resulting in fringe patterns with perfect contrast Image analysis is also used as a faster, more accurate, and more consistent method for fringe count After evaluation of the holograms, the existence of tensile forces transmitted through the crack faces is associated with the presence of tensile strain behind the crack tip A definition of the fracture process zone (FPZ) is proposed based on the difference between experimentally observed and linear elastic fracture mechanics (LEFM) strain fields Deviations from the linear elastic solution show a relatively small zo

Strain-dependent moduli and pressuremeter tests.

Abstract: High precision measurements of the deformation of stiff clay in the laboratory have shown that the response of these soils is non-linear from very small strains. This is a reflection of the occurrence of plastic irrecoverable strain in the soil and is an indication that the soil cannot realistically be described as linear elastic material. The object of this note is to show that when non-linear effects are to be studied care is necessary in order to ensure that the parameters that are being compared from pressuremeter and triaxial tests are truly comparable. The theory behind undrained triaxial compression tests and pressuremeter tests is outlined. The variation of secant shear stiffness with strain and its effect on the tangent shear modulus and pressuremeter shear modulus is examined. Problems associated with the pressuremeter test are noted and include the assumption of an undrained process and exclusion of the viscous element of response. It is also noted that non-linearity of the type considered here leads to different distribution of stress or strain around a test device or geotechnical structure from those which emerge from constant modulus elastic analysis. (TRRL)

Modulus 4.0: expansion and validation of the modulus backcalculation system. final report

Abstract: This report describes the Texas Transportation Institute's continuing efforts to upgrade the MODULUS backcalculation system. Enhancements have been made in several areas, including: (1) Inclusion of a procedure to estimate the depth to a stiff layer; (2) A method of assessing the non-linearity of the subgrade and computation of the optimum number of sensors to use in the backcalculation routine; and (3) The replacement of the BISAR linear elastic procedure with the WES5 procedure recently developed by the U.S. Corps of Engineers. The new MODULUS 4.0 is evaluated with monthly deflection data collected on 10 experimental sites for which all the layer materials have been tested in the laboratory. Validation of the system is attempted by using pavement sections instrumented with Multidepth Deflectometers. By simultaneously monitoring surface and depth deflections it is possible to quantify the effectiveness of the backcalculation system. Results show that the linear elastic model used in MODULUS produces reasonable layer moduli for pavements with thick asphalt surfacing. However, errors may result in using the linear elastic approach on thin pavements. The use of a stress dependent model which includes dilation substantially improves the match of measured and computed depth deflections on thin pavements. Preliminary results from a finite element backcalculation system have also been included.

Experimental Evaluation of Global Composite Laminate Stiffnesses by Structural Wave Propagation

Abstract: An experimental method based on a consistent second order theory for flex ural waves in an orthotropic fiber reinforced plate is described. The propagation of struc tural waves allows the accurate evaluation of four elastic constants E, E, G, Gin a single experiment. The flexural waves are induced centrally with a piezoceramic trans ducer. Phase velocities as a function of frequency are determined from the phase spectra of flexural waves measured with a heterodyn laser-Doppler-interferometer along straight lines through the loading point. The theory is developed by means of asymptotic expan sions of the three dimensional equations of linear elasticity and includes shear deforma tion. The measured behaviour and the accuracy of the moduli is shown to be very satisfac tory and in accordance with theoretical expectations.

Micromechanical investigation of an optical fiber embedded in a laminated composite

Abstract: A linear elastic study is performed as a first order approximation to investigate the geometry of theresin-rich region observed around optical fibers embedded in laminated composites. The Rayleigh-Ritzmethod is employed with beam bending functions as assumed trial functions. The total potential energyis formulated in terms of unknown force distributions and the length of the resin pocket. The resultingsystem of coupled nonlinear equations is solved by the Levenberg-Marquardt algorithm to compute theshape and size of the resin pocket. Results of this analysis show the effect of laminate stacking sequence onthe geometry of the resin pocket and are found to agree well with experimental observations.The computedgeometry is automatically discretized for FEM analysis in order to obtain stress intensity information atthe lateral ends of the resin pocket. 1. INTRODUCTION Optical fiber sensors have been developed during the past ten years to measure a wide rangeof physical observables in "smart structures". Despite all the advantages of optical sensors, it isessential to investigate their mechanical interactions with the host material before the concept ofembedding optical fibers in composite structures can be successfully inpiemented. It is necessaryto verify that the presence of optical fibers in the host will not affect its mechanical performance,and will not significantly influence the stress and strain field in the host composite.It is well known that optical fibers embedded in laminated composites are usually surrounded

Damage Development in Composite Laminates Under Monotonic Loading

Abstract: Damage development under monotonic loading was studied in five cross-ply graphite/epoxy laminates. The laminates exhibited three characteristic ranges of varying stiffness. The first range is linear elastic and is characterized by the absence of any measurable damage. The second range is one of decreasing stiffness and corresponds to transverse matrix cracking in the 90° layer up to the characteristic damage state (CDS). The third range is a nearly linear one of stabilized or even slightly increasing stiffness. Ultimate failure was governed by the ultimate tensile strain in the 0° plies with small variations attributed to residual stresses, statistical scatter and local strain concentration.

Effect of variable linear elastic parameters on finite element prediction of soil compaction

Abstract: An axisymmetric linear elastic finite element program was developed to investigate the effect that the two linear elastic parameters, Poisson*s ratio and Young's modulus, had on soil compaction. This program was verified against Boussinesq's linear elastic theory. It was found that increased values of Young's modulus had no effect on the stress state in the soil mass but that strain levels were decreased. Increased values of Poisson's ratio increased the stress state and decreased the strain levels. The interaction of these two parameters point to the need to be able to vary both over the entire stress range.

Elastic distortion field in single layer heterostructures in the presence of misfit dislocations

Abstract: The elastic distortion field in (001)‐grown lattice mismatched heterostructures of diamond or zinc‐blende structure having misfit dislocations at the interface is calculated within the linear elasticity theory. Not only the misfit dislocation densities in the two 〈110〉 directions but also the distribution of the possible Burgers vectors are taken into consideration. A transition layer where the elastic field is appreciably laterally nonuniform extends from the interface up to a distance of the same order of the mean dislocation spacing. It is shown that this transition layer affects x‐ray diffraction measurements. Beyond this region, the elastic distortion field is uniform and is found to depend only on the mean values of the Burgers vectors associated with the two dislocation distributions. In particular it is shown that in general the strain field, i.e., the symmetric part of the elastic distortion field, is not biaxial. The three independent parameters describing the lattice deformations in the epilaye...

MODULUS 4.0: expansion and validation of the MODULUS backcalculation system

Abstract: This report describes the Texas Transportation Institute's continuing efforts to upgrade the MODULUS backcalculation system. Enhancements have been made in several areas, including: 1. Inclusion of a procedure to estimate the depth to a stiff layer. 2. A method of assessing the non-linearity of the subgrade and computation of the optimum number of sensors to use in the backcalculation routine. 3. The replacement of the BISAR linear elastic procedure with the WES5 procedure recently developed by the US Corps of Engineers. The new MODULUS 4.0 is evaluated with monthly deflection data collected on 10 experimental sites for which all the layer materials have been tested in the laboratory. Validation of the system is attempted by using pavement sections instrumented with Multidepth Deflectometers. By simultaneously monitoring surface and depth deflections it is possible to quantify the effectiveness of the backcalculation system. Results show that the linear elastic model used in MODULUS produces reasonable layer moduli for pavements with thick asphalt surfacing. However, errors may result in using the linear elastic approach on thin pavements. The use of a stress dependent model which includes dilation substantially improves the match of measured and computed depth deflections on thin pavements. Preliminary results from a finite element backcalculation system have also been included.