Showing papers on "Continuum mechanics published in 1992"

The discrete energy-momentum method: conserving algorithms for nonlinear elastodynamics

Abstract: In the absence of external loads or in the presence of symmetries (i.e., translational and rotational invariance) the nonlinear dynamics of continuum systems preserves the total linear and the total angular momentum. Furthermore, under assumption met by all classical models, the internal dissipation in the system is non-negative. The goal of this work is the systematic design of conserving algorithms that preserve exactly the conservation laws of momentum and inherit the property of positive dissipation forany step-size. In particular, within the specific context of elastodynamics, a second order accurate algorithm is presented that exhibits exact conservation of both total (linear and angular) momentum and total energy. This scheme is shown to be amenable to a completely straightforward (Galerkin) finite element implementation and ideally suited for long-term/large-scale simulations. The excellent performance of the method relative to conventional time-integrators is conclusively demonstrated in numerical simulations exhibiting large strains coupled with a large overall rigid motion.

Transformation field analysis of inelastic composite materials

Abstract: A new method is proposed for evaluation of local fields and overall properties of composite materials subjected to incremental thermomechanical loads and to transformation strains in the phases. The composite aggregate may consist of many perfectly bonded inelastic phases of arbitrary geometry and elastic material symmetry. In principle, any inviscid or time-dependent inelastic constitutive relation that complies with the additive decomposition of total strains can be admitted in the analysis. The governing system of equations is derived from the representation of local stress and strain fields by novel transformation influence functions and concentration factor tensors, as discussed in the preceding paper by G. J. Dvorak and Y. Benveniste. The concentration factors depend on local and overall thermoelastic moduli, and can be evaluated with a selected micromechanical model. Applications to elastic-plastic, viscoelastic, and viscoplastic systems are discussed. The new approach is contrasted with some presently accepted procedures based on the self-consistent and Mori-Tanaka approximations, which are shown to violate exact relations between local and overall inelastic strains.

Intense Dynamic Loading Of Condensed Matter

Abstract: MECHANICS OF CONTINUOUS MEDIA Stresses and Strains in a Solid Body Equations of One-Dimensional Motion of Compressible Media, Shock Waves Interpretation of Detection Data on Compression and Rarefaction Waves EXPERIMENTAL TECHNIQUES OF THE PHYSICS OF HIGH DYNAMIC PRESSURE Explosive Generation of Dynamic Pressure Ballistic Experiments with a Shock Wave Promising Sources of High Dynamic Pressure Discrete Measurement of Wave and Mass Velocities Pressure Profiles Recorded with Manganin Sensors Measurement of the Velocity of Matter ELASTOPLASTIC PROPERTIES OF SHOCK-LOADED SOLIDS Basic Relationships and Models Moduli of Elasticity and the Velocity of Sound in Shock Compressed Metals Dynamic Yield Point Structure of Plastic Compression Waves Compression and Rarefaction Waves in Shock-Compressed Metals Compaction of Porous Media in Shock Waves Catastrophic Thermoplastic Shear under Dynamic Deformation. Impact Compression of Brittle Materials Methods of Studying High Dynamic Deformations Microscopic Models of Strain Dynamics EVOLUTION OF LOAD PULSES IN MEDIA WITH POLYMORPHIC PHASE TRANSITIONS The Structure of Compression and Release Waves in Iron The Graphite to Diamond Transition under Shock Compression Properties of the Phase Transition Induced by Shock FRACTURE UNDER PULSED LOADING. SPALLING STRENGTH Dynamics of Wave Interactions During Spalling Spalling Strength of Metals Work of Spalling Fracture Determination if Tensile Stress Behind the Spall Plane Resistance of Polymers, Brittle Materials, and Liquids to Spalling Fracture Mechanism and Kinetics of Dynamic Fracture of Metals SHOCK AND DETONATION WAVES IN SOLID EXPLOSIVES Basic Concepts and Models Kinetics of Dissociation of Explosives Deduced from Analyses of Evolution of Shock Waves Semiempirical Macrokinetic Equations of Solid Explosives MODEL EQUATIONS OF STATE FOR A WIDE RANGE OF PRESSURE AND TEMPERATURE General Analysis of Phase Diagram Quasi-Harmonic Model of a Solid Equation of State for Condensed Phase at High Temperatures Evaporation Effects and Generalized Equations of State Tabulated and Approximate Equations of States GENERALIZED EQUATIONS OF STATE FOR METALS Cold Compression Curve Electron Component Thermal Excitation of the Crystal Lattice Liquid Phase Procedure for Constructing Semiempirical Equations of State for Metals THERMODYNAMIC PROPERTIES OF METALS Aluminum Copper Lead Lithium FURTHER BRIEF NOTES ON CONTINUUM MECHANICS Equation of Motion Shock Waves Characteristic Form of Gas-Dynamic Equations. Simple Waves The Structure of a Shock Wave Decay of a Random Discontinuity in Hydrodynamics Equation of Motion for Porous Condensed Media DYNAMICS OF CONDENSED MEDIA WITH ALLOWANCE FOR THEIR STRENGTH Equations of Motion. Divergence Form Characteristics Form on Equations of Motion Simple Waves Shock Waves in a Hyperelastic Medium Decay of an Arbitrary Discontinuity Equations of Motion for a Hyperelastic Medium in an Arbitrary Curvilinear Set of Coordinates Equations of Motion of a Maxwell Viscoelastic Medium Nonlinear Waves in a Viscoelastic Medium BRIEF REVIEW OF COMPUTATIONAL TECHNIQUES FOR THE DYNAMICS OF CONDENSED MEDIA Method of Particles in Cells Method of Large Particles Godunov's Method Lagrangian Methods NUMERICAL MODELING OF CONDENSED MEDIA UNDER INTENSIVE PULSED LOADING Tabulated Form of the Equation of State High-Speed Collisions Irregular Collisions of Strong Shock Waves in Metals Numerical Modeling of the Effect of Relativistic High-Current and High-Energy Ion Beams on Metal Targets Effect of an Explosion on an Iron Plate Impactor Penetration of an Obstacle of Finite Thickness Impact of a Micrometerorite on a Spacecraft Shield References Index

Analyses of Plastic Flow Localization in Metals

Abstract: The continuum mechanics framework for analyzing plastic flow localization is reviewed. The prediction of the localization of deformation into shear bands is sensitive to the constitutive description. The classical isotropic hardening elastic-plastic solid with a smooth yield surface and normality is very resistant to localization, but deviations from these idealizations have a strong effect. Thus, a material that forms a sharp vertex on the yield surface, as predicted by crystal plasticity, shows flow localization at quite realistic levels of strain, and even the formation of a rounded vertex on the yield surface has an important influence. Also softening induced by material damage or by the heating due to plastic dissipation have significant influence in promoting the onset of flow localization. In a practical situation one effect, such as thermal softening under high deformation rates, may be the dominant cause of localization, but often the interaction of different effects appears to be the more realistic explanation of observed flow localization. Some relevant constitutive models are reviewed and the effect of the different material models on localization predictions is illustrated. Important information on localization behavior in uniformly strained solids is obtained by a relatively simple material stability analysis, but often failure bymore » flow localization occurs in nonuniformly strained regions, where numerical solution procedures are necessary to obtain theoretical predictions. The numerical results reviewed cover localization under dynamic as well as quasi-static loading conditions. 81 refs., 16 figs.« less

A Visco-Plastic Constitutive Model for 60/40 Tin-Lead Solder Used in IC Package Joints

Abstract: A new unified visco-plastic constitutive model for the 60 Sn-40 Pb alloy used in solder joints of surface-mount IC packages and semiconductor devices is proposed. The model accounts for the measured stress-dependence of the activation energy and for the strong Bauschinger effect exhibited by the solder. The latter is represented by a back stress state variable which, in turn, evolves according to a hardening-recovery equation. Based on the observed hardening behavior, it is assumed that the isotropic resistance to plastic flow does not evolve within the deformation range covered in this study (e< 3 percent). The deformation phenomena associated with the solder’s monotonic and steady-state cyclic responses are accurately predicted for −55°C≦T≦150°C and 8 x 10−2 s−1 ≦ e ≦ 8 x 10−5 s−1 . The model also predicts well the overall trend of steady-state creep behavior. The constitutive model is formulated within a continuum mechanics framework and is therefore well suited for implementation into finite element or other structural codes.

An Anatomical Heart Model with Applications to Myocardial Activation and Ventricular Mechanics

Abstract: 4A three-dimensional finite element model of the mechanical and electrical behavior of the heart is being developed in a collaboration among Auckland University, New Zealand; the University of California at San Diego, U.S.; and McGill University, Canada. The equations of continuum mechanics from the theory of finite deformation elasticity are formulated in a prolate spheroidal coordinate system and solved using a combination of Galerkin and collocation techniques. The finite element basis functions used for the dependent and independent variables range from linear Lagrange to cubic Hermite, depending on the degree of spatial variation and continuity required for each variable. Orthotropic constitutive equations derived from biaxial testing of myocardial sheets are defined with respect to the microstructural axes of the tissue at the Gaussian quadrature points of the model. In particular, we define the muscle fiber orientation and the newly identified myocardial sheet axis orientation throughout the myocardium using finite element fields with nodal parameters fitted by least-squares to comprehensive measurements of these variables. Electrical activation of the model is achieved by solving the FitzHugh–Nagumo equations with collocation at fixed material points of the anatomical finite element model. Electrical propagation relies on an orthotropic conductivity tensor defined with respect to the local material axes. The mechanical constitutive laws for the Galerkin continuum mechanics model are (1) an orthotropic “pole–zero” law for the passive mechanical properties of myocardium and (2) a Wiener cascade model of the active mechanical properties of the muscle fibers. This chapter concentrates on two aspects of the model: first, grid generation, including both the generation of nodal coordinates for the finite element mesh and the generation of orthotropic material axes at each computational point, and, second, the formulation of constitutive laws suitable for numerically intensive finite element computations. Extensions to this model and applications to the mechanical and electrical function of the heart are described in Chapter 2 by McCulloch and co-workers.

Electronic Sputtering: From Atomic Physics to Continuum Mechanics

Abstract: The surprising fact that even very complex molecules can be ejected intact into the vapor phase when a material is electronically excited by incident particles provides a new probe of the behavior of condensed matter at high excitation densities. The physics of the conversion of the electronic excitation energy into mechanical and chemical energy links atomic physics in a solid at low excitation densities to nanometer‐scale continuum mechanics at high excitation densities

Elasto-plastic deformations in multibody dynamics

Abstract: The problem of formulating and numerically solving the equations of motion for a multibody system undergoing large motion and clasto-plastic deformations is considered here. Based on the principles of continuum mechanics and the finite element method, the equations of motion for a flexible body are derived. It is shown that the use of a lumped mass formulation and the description of the nodal accelerations relative to a nonmoving reference frame lead to a simple form of these equations. In order to reduce the number of coordinates that describe a deformable body, a Guyan condensation technique is used. The equations of motion of the complete multibody system are then formulated in terms of joint coordinates between the rigid bodies. The kinematic constraints that involve flexible bodies are introduced in the equations of motion through the use of Lagrange multipliers.

Incorporation of polymer diffusivity and migration into constitutive equations

Abstract: Given a general one-particle constitutive equation for the stress tensor, we discuss how to incorporate the additional effects of polymer diffusivity and migration into that constitutive equation within the framework of continuum mechanics. For the example of an upper-convected Maxwell model representing the polymer contribution to the stress tensor of a dilute polymer solution, we describe i) how to modify the constitutive equation for the stress tensor to include diffusion and migration effects, ii) how to formulate a balance equation for the polymer mass density in order to describe the nonhomogeneous composition of the polymer solution resulting from migration, and iii) how to close the extended set of coupled equations by means of further constitutive equations for the migration velocity and the diffusion tensor. In order to guarantee the material objectivity for all equations, we formulate them in the body tensor formulation of continuum mechanics (and then translate them into Cartesian space). The proposed equations are compared to results of a recent kinetic-theory approach.

An exact three-dimensional solution for normal loading of inhomogeneous and laminated anisotropic elastic plates of moderate thickness

Abstract: An exact three-dimensional solution is presented for the deformation and stress distribution in an elliptical plate under uniform normal loading of the lateral surfaces, and clamped along its edge. The plate is assumed to be of constant, moderate thickness and composed of anisotropic elastic material which is inhomogeneous in the through-thickness direction but symmetric about the mid-plane. The only material symmetry assumed is that of reflectional symmetry in planes parallel to the mid-plane. A transfer matrix method is used which, without making any further assumptions, gives the exact solution at each point in the plate in terms of the stress and displacement at the mid-plane. The two-dimensional differential equations governing these mid-plane values are found to be the same as those for an equivalent homogeneous plate whose constant elastic moduli are determined by appropriate through-thickness weighted averages of the inhomogeneous moduli. The solution of the two-dimensional problem is known for such a plate when subject to the specified surface and edge conditions, and yields a closed form analytical solution that satisfies all the governing equations and surface conditions of the full three-dimensional elasticity problem, with edge displacement conditions satisfied on the mid-plane. The important special case of an anisotropic laminated plate is given by assuming piecewise constant properties through the thickness.

Massively parallel computer simulation of plane‐strain elastic–plastic flow via nonequilibrium molecular dynamics and Lagrangian continuum mechanics

Abstract: Application of massively parallel low‐cost computers to the simulation of plane‐strain elastic‐plastic flow is discussed. Two different approaches, atomistic molecular dynamics and continuum mechanics, are applied to this problem. A hybrid scheme combining the two is also discussed.

The flutter instability in granular flow

Abstract: I n the flutter instability (so-named by Rice), the dynamic partial differential equations of continuum mechanics become ill-posed in the sense of Hadamard because the wave speed turns complex. Both real and imaginary parts of the wave speed are nonzero, meaning that waves propagate as they are amplified. For over a decade there was uncertainty whether any viable constitutive law led to the flutter instability. In this paper it is proved that the flutter instability does occur in widely accepted, two-dimensional models for granular flow.

Strain‐Based Constitutive Model with Mixed Evolution Rules for Concrete

Abstract: The theories of continuum damage mechanics and plasticity are combined in a strain‐based phenomenological approach to yield an effective constitutive model for plain concrete. The model reproduces the majority of the typical behavior exhibited by plain concrete: anisotropic stiffness evolution, pressure‐dependent ductility and strength, postpeak dilation, recovery of stiffness upon reverse loading, and permanent deformations. The proposed combination of continuum damage mechanics and plasticity theory is unique in that: (1) A strain‐based formulation is used; (2) a separate “inelastic” surface is postulated for the tensile regime and the compressive regime; (3) the inelastic surfaces are used for both damage evolution and permanent deformation; and (4) an isotropic evolution law is used for compression and a kinematic law for tension. The model requires a modest number of material constants (10). Strain‐softening considerations are discussed, relative to implementation of the model into numerical methods ...

Applications of an energy-momentum tensor in nonlinear elastodynamics : Pseudomomentum and eshelby stress in solitonic elastic systems

Abstract: In continuum mechanics the pseudomomentum is the covariant material momentum whose associated flux in the balance law is Eshelby's “energy-momentum” tensor. The unbalance of pseudomomentum was previously shown to play a basic role in the formulation of configurational forces and path-independent integrals in the theory of elastic inhomogeneities and brittle fracture. Here, it is further shown to provide a fundamental conservation law for dispersive nonlinear elastic systems which exhibit soliton solutions. This is illustrated by both classical and grade-two elasticity with applications to the sine-Gordon, Boussinesq. sine-Gordon -d'Alembert and generalized Zakharov systems encountered in various bulk or surface wave-propagation problems. In these systems the nonconservation of global pseudomomentum may be used for a perturbational approach to nearly integrable systems so as to study the influence of dissipation and external sources (e.g. defects).

A stabilized 9‐node non‐linear shell element

Abstract: A non-linear 9-node stress resultant shell finite element with six degrees of freedom per node is formulated. The material non-linearity is based on an implicit integration scheme using the von Mises yield criterion and linear isotropic bardening. The small strain geometric non-linearity is formulated using the polar decomposition theorem of continuum mechanics via a corotational updated Lagrangian method, which represents finite rotations with accuracy. Reduced integration is used to remove locking and calculate the stresses at their optimal stress accuracy points. A practical procedure is employed to stabilize the troublesome spurious zero energy modes. A number of tests covering the non-linear material and geometry ranges and buckling show the good performance of the new element.

Configuration design sensitivity analysis of built‐up structures part I: Theory

Abstract: A continuum based configuration design sensitivity analysis method is developed for built-up structures that include truss, beam, plane elastic solid and plate design components. The configuration design variation of a structural component can be characterized by changes in the domain shape and in the orientation of the component. A variational approach is then used to incorporate both shape and orientation effects in the same energy equation. A continuum shape design sensitivity analysis method that utilizes the material derivative idea of continuum mechanics is employed to account for effects of shape variation. An approach that is similar to the continuum shape design sensitivity analysis method is developed in this paper to account for the effect of orientation variation. Variations of energy bilinear and load linear forms, with respect to both shape and orientation design variables, are derived for each structural component. Using the adjoint variable or direct differentiation method, configuration design sensitivity results for built-up structures are obtained in terms of the design velocity fields. Configuration design sensitivity analyses of both static and eigenvalue responses are considered.

Design sensitivity analysis for nonlinear magnetostatic problems by continuum approach

Abstract: Using the material derivative concept of continuum mechanics and an adjoint variable method, in a 2-dimensional nonlinear magnetostatic system the sensitivity formula is derived in a line integral form along the shape modification interface. The sensitivity coefficients are numerically evaluated from the solutions of state and adjoint variables calculated by the existing standard finite element code. To verify this method, the pole shape design problem of a quadrupole is provided

Continuum theory of microstretch liquid crystals

Abstract: A continuum theory is proposed for liquid crystals whose molecular elements can expand and contract, in addition to undergoing translations and rotations. This is the microstretch continuum theory, generalizing microplar theory of liquid crystals. Constitutive equations are developed for arbitrarily shaped molecular elements. Thermodynamical restrictions are studied. The theory is then specialized for breathing rodlike liquid crystals. Equilibrium configuration for the microstretch motion is studied.

Deformations and stresses with and without microslip

Abstract: Kinematical definitions of deformations with and without microslip are presented. Transformation properties for such deformations are shown to follow directly from their definitions, and Burgers vector is related to the deformation without microslip. A limit procedure provides a concept of stress without microslip and leads to a natural concept of elastic response. Various decompositions of local deformation into elastic and plastic parts proposed in the literature are shown to be compatible with this kinematical setting. INTRODUCTION One of the principal methods for incorporating the inelastic behavior of metals into continuum models of materials is to employ classical notions of deformation and stress in describing the kinematics and dynamics of an inelastic body, but to introduce additional measures of deformation, usually called elastic and plastic deformation, into the constitutive equations for the material under consideration. Although generally successful, this approach has drawbacks that merit consideration: 1) The variety ol possible choices of elastic and plastic deformation and the difficulty in comparing models based on different choices has led to a lingering controversy in the literature; 2) the underlying classical kinematics does not directly incorporate the physical processes of slip at the microscopic level. My goal in this paper is to present a different method for incorporating the inelastic behavior of metals into a continuum model that is not subject to these drawbacks. This utilizes the research still in progress of Del Piero and Owen (forthcoming) on the kinematics of fractured continua in which classes of deformations broad enough to include explicitly slip at the microscopic level are defined and developed. In the second and third sections I describe a collection of deformations called invertible structured deformations that includes classical deformations, and I show how the results of Del Piero and Owen (forthcoming) lead to natural notions of deformation without microslip and deformation due to microslip. All of the considerations in the second and third sections are purely kinematical, so that these new notions of deformation are not variables that appear for . the first time in constitutive equations. Among the kinematical properties of invertible structured deformations summarized in the second section is the Approximation Theorem, which shows that each invertible structured deformation is a limit of "piecewise classical deformations", or "simple deformations". It is this property of invertible structured deformations that permits one to interpret their effect on a body in terms of complicated slip mechanisms occurring in the approximating simple deformations. The other results in the second section motivate and justify not only multiplicative, but also additive decompositions of local deformation into parts without microslip and parts due to microslip. All of the terms and factors in these decompositions have definite transformation properties under changes in observer and reference configuration that are direct consequences of the kinematical definitions. In the fourth section, new results are given that lead to a notion of stress without microslip for an invertible structured deformation, This stress takes into account differences between a given element of surface area in the deformed configuration and a corresponding element of surface generated by an approximation to the given invertible structured deformation. A passage to the limit yields a simple formula for the stress without microslip in terms of the Cauchy stress and the left microslip tensor introduced in the third section. The discussion in the fifth section shows how the availability of both deformation without microslip aiid stress without microslip provides a natural concept of elastic response for a body undergoing invertible structured deformations: the stress without microslip is a function of deformation without microslip. If the body undergoes a classical deformation and, in particular, undergoes no microslip, then the elastic response reduces to the classical relation: the Cauchy stress is a function of the deformation gradient. The relation between stress without microslip and deformation without microslip is equivalent to an equation that gives the Cauchy stress as a function of deformation without microslip and of deformation due to microslip in which the dependence on deformation due to microslip enters through a multiplicative factor (the transpose of the left microslip tensor). The final section of the paper is devoted to showing how various decompositions of deformation into elastic and plastic parts that have been proposed in the literature (Lee and Liu, 1967), (Clifton, 1972), (Nemat-Nasser, 1979), (Green and Naghdi, 1965) fit naturally into the present tramework of invertible structed deformations. If for each proposed decomposition the elastic deformation in that decomposition is identified as deformation without microslip, then the plastic deformation in that decomposition is easily related to one of the measures of deformation due to microslip introduced in the third section, and the transformation properties of both elastic and plastic deformation immediately follow. Thus, in the setting of invertible structured deformations, all the decompositions considered in the last section are consistent with one another: none has any special status, and each one identifies a particular measure of deformation due to microslip in the present theory. The present approach has some connections with theories of continuous distributions of dislocations and theories of defective crystals. The emergence in a natural way of Burgers vector, as shown in relation (10), shows that important physical ideas incorporated into theories of continuous distribution of dislocations (Kroner, 1958) have counterparts in the kinematics of fractured continua (Del Piero and Owen, forthcoming). Within the class of invertible structured deformations described here one can identify a smaller collection of deformations, the neutral deformations discussed by (Davini and Parry, 1991) (Fonseca and Parry, forthcoming): these correspond in the present theory to, the invertible structured deformations whose right microslip tensor is the gradient of a vector field. INVERTIBLE STRUCTURED DEFORMATIONS Let a region 0, and consider two invertible structured deformations: (s, vs), called a classical simple shear, and (s, I), called a simple shear due to microslip. Here, I is the identity tensor. Results presented below justify the following description: the simple shear due to microslip is accomplished as if the body were sliced into infinitely many, infinitely thin parallel slabs, each of which is rigidly displaced an infinitesimal amount parallel to itself, whereas the classical simple shear is accomplished by smooth deformation without slicing. The collection of invertible structured represents a broadening of the collection of classical deformations used in continuum mechanics. In fact, classical deformations are invertible structured deformations of the form (g, vg), i.e., deformations in which the local deformation vg is the same as the deformation without microslip G. Invertible structured deformations were introduced by Del Piero and Owen (forthcoming). We here need only consider a few of the principal results of that study. Approximation Theorem: Every invertible structured deformation is a limit of simple deformations. A simple deformation is determined by giving a "piecewise classical deformation" f which fractures the body into pieces along crack sites K in ji and then deforms each piece via a classical deformation. For example, the rectangular block 9t undergoes a simple deformation (a , s ) if it is sliced into m slabs by m—1 equally spaced parallel planes with the slabs then displaced by parallel translations giving relative displacement of magnitude c / m t o adjacent slabs. Here, the crack site c is the union of the slicing planes, and the mapping s displaces each slab by a translation. It is natural to think of s as a shearing displacement of a deck of cards, where each card represents one of the m slabs. The term "limit of simple deformations" in the statement of the Approximation Theorem means that there is a sequence m i—• (K , f ) of simple deformations such that the given invertible structured deformations (g, G) satisfies

Continuum damage mechanics for softening of brittle materials

Abstract: Continuum damage theory is used to model the failure behaviour of brittle materials. In the constitutive equations a damage parameter is incorporated. A damage criterion is postulated such that large differences between tension and compression strength can be described. A damage growth law is quantified based on experimental data for concrete. For the elaboration of the mathematical formulation the finite element method is applied. Numerical results obtained for a plane strain example show the merits of the procedure.

Analysis of Welded Tubular Connections Using Continuum Damage Mechanics

Abstract: A new modeling technique for the ultimate strength analysis of welded tubular connections subjected to general loading is presented. Based upon the finite element method, the model includes shell and solid elements, large deformations, and elastoplasticity. Unlike other approaches, the ability to simulate the initiation and propagation of fracture is included through the application of continuum damage mechanics (CDM). With this method, cracking is predicted on the basis of a damage variable, and its effect is applied as a reduction in the element stiffness. The development of the material constants is described and numerical results are obtained for two tubular joints that exhibit fracture failure. A comparison between the computed failure loads and those observed experimentally shows a close correlation. With this approach, welded connections that are subjected to arbitrary loading can be efficiently evaluated for strength.

The propagation of elastic waves in a certain class of inhomogeneous anisotropic materials. I. The refraction of a horizontally polarized shear wave source

Abstract: This paper is the first in a series of articles describing the refraction and propagation of infinitesimal disturbances in a \`coarse grained' inhomogeneous anisotropic material which is fused to an isotropic substrate. Here, the basic constitutive law for the material is motivated by applications to the non-destructive evaluation of austenitic steel welds, although it is clear that the phenomena described and the mathematical analysis used is also of interest in geophysics, the study of composite materials and several other areas of continuum mechanics. This work is concerned with the refraction of a horizontally polarized shear wave source at the fusion interface between a homogeneous isotropic material and transversely isotropic material. The latter is inhomogeneous by virtue of the fact that the zonal axis or axis of symmetry of the crystals varies in direction with the distance from the interface. The mathematical boundary-value problem is solved exactly, and, in the high-frequency limit, a uniform asymptotic expansion for the displacement vector is found. It is shown that in this limit, and for a wide range of material constants, the refracted energy which penetrates certain regions of the \`weld material' is totally internally reflected. This conclusion is highly significant in the design of inspection procedures for structurally important welds.

A continuum approach in shape design sensitivity analysis of magnetostatic problems using the boundary element method

Abstract: A novel shape design sensitivity formula for magnetostatic problems is derived analytically using the material derivative concept in continuum mechanics. In order to express the design sensitivity as a function of shape variation only, the adjoint variable is defined. Since the formula is expressed as boundary integration of state and adjoint variables over a deformed boundary, the boundary-element method (BEM) is used to evaluate the variables accurately on the boundary. The proposed algorithm is applied to the pole-shape optimization of a quadrupole magnet, through which the validity is proved. >

On the local measures of mean rotation in continuum mechanics

Abstract: We implement Cauchy's concept of a rotation-angle function on an oriented plane, and characterize situations when a rotation-angle function exists, and hence when measuring mean rotations in the manner of Cauchy or Novozhilov makes sense. We also discuss in passing the role of the skew part of the deformation gradient in measuring the mean deformation.