Showing papers on "Linear elasticity published in 1976"
2,329 citations
TL;DR: In this paper, a finite element solution of structural mechanics problems with surface nonlinearities is presented, where only a few of the total number of degrees of freedom involved in the nonlinearity are eliminated by using the superelement technique.
162 citations
143 citations
Book•
01 Jan 1976
TL;DR: In this article, the authors present an analysis of variational inequalities with highly oscillating coefficients for nonlinear elasticity problems and their application to the optimisation of a temperature profile.
Abstract: Inequations quasi-variationnelles dans les problemes a frontiere libre en hydraulique.- The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics.- Asymptotic behaviour of solutions of variational inequalities with highly oscillating coefficients.- Application of convex analysis to the treatment of elastoplastic systems.- Theory of mixed and hybrid finite-element approximations in linear elasticity.- Perturbation of domains in elliptic boundary value problems.- Frost propagation in wet porous media.- Viscous fluid flow in chemically reacting and diffusing systems.- Local invertibility conditions for geometrically exact nonlinear rod and shell theories.- Some applications of functional analysis in the mathematical theory of structures.- Functional analysis applied to the optimisation of a temperature profile.- Global free boundary problems and the calculus of variations in the large.- Proof of existence and uniqueness of tidal waves with general vorticity distributions.- A critical appraisal of certain contemporary ship model testing techniques.- Solitary-wave solutions for some model equations for waves in nonlinear dispersive media.- Hilbertian unilateral problems in viscoelasticity.- On the norm-dependence of the concept of stability.- The hodograph method in fluid-dynamics in the light of variational inequalities.- A new formulation of diphasic incompressible flows in porous media.- Convergence of solutions in problems of elastic plastic torsion of cylindrical bars.- On an evolution problem in linear acoustics of viscous fluids.- On the mechanics of materials with fading memory.- Contraction semigroups and trend to equilibrium in continuum mechanics.- The buckling of a thin elastic plate subjected to unilateral conditions.- Problemes de contact entre corps solides deformables.- On the existence and uniqueness of a warpening function in the Elastic-plastic torsion of a cylindrical bar with multiply connected cross-section.- A method for computing the eigenfrequencies of an acoustic resonator.- Secondary bifurcation of a steady solution into an invariant torus for evolution problems of Navier-Stokes' type.- A basic open problem in the theory of elastic stability.- Some applications and methods of nonlinear functional analysis in finite displacement plate theory.- Criteres de validite de la theorie non-lineaire des coques elastiques.- Functional analysis approach for the derivation of hybrid variational functionals.- Stability of equilibrium in elastic-plastic solids.- Solutions in the large for certain nonlinear hyperbolic systems arising in shock-wave theory.- Cauchy problem in a scale of banach spaces and its application to the shallow water theory justification.- Perturbation results and their applications to problems in structural dynamics.- On the physical interpretation of certain inner products as a guide to the application of functional analysis.- Branching and stability for nonlinear shells.- On a free surface problem.- Theoretical constructions of selection of actual events from the virtual ones.- Steadily rotating chains.- Generating functionals and extremum principles in nonlinear elasticity with applications to nonlinear plate and shallow shell theory.- Determination de la configuration d'equilibre d'un plasma.- Elatic-plastic torsion of cylindrical pipes.
61 citations
TL;DR: The general theory of linear constraints in linear elasticity theory is outlined in this article, and a modified form of constraint theory is proposed for problems that are illposed in constraint theory although well-posed in the absence of any constraint.
Abstract: The general theory of linear constraints in linear elasticity theory is outlined. For problems that are ill-posed in constraint theory although well-posed in the absence of any constraint, a modified form of constraint theory is proposed.
41 citations
TL;DR: In this article, a rigorous procedure to determine the nonlinear impedance function of a rigid plate of arbitrary shape, only in partial contact with the elastic half-space, is developed, which can either be determined with the finite element method or based on solutions of displacements on the surface of an elastic half space at a certain distance from a rigid subdisk.
37 citations
TL;DR: In this article, the duality concepts for linear elastic analysis by the finite element method are extended to the plasticity problem using classical variational principles; in this way, both primal and dual quasi-direct approaches to the limit analysis problem are identified.
36 citations
TL;DR: In this paper, the linear elastic, dynamic transient analysis of some circular plate bending problems is considered by using axisymmetric, parabolic isoparametric, elements with an explicit time marching scheme.
30 citations
01 Jan 1976
TL;DR: In this paper, the authors present an analysis of stress and infinitesimal strain, friction, elasticity and strength of rock, and the effects of size and stress gradient.
Abstract: Chapters are included on: rock as a material; analysis of stress and infinitesimal strain; friction; elasticity and strength of rock; linear elasticity; laboratory testing; effects of size and stress gradient; fluid pressure and flow in rocks; behaviour in ductile materials; further problems in elasticity; time-dependent effects, crack phenomena and the mechanism of fracture; strain waves; the state of stress underground; underground measurements; granular materials; geological and geophysical applications; mining and other engineering applications. In the second edition, additions have been made in particular to the chapters on friction, elasticity and strength of rock, the state of stress underground, and geological and geophysical applications. The chapter on granular materials has undergone a major revision. /TRRL/
23 citations
TL;DR: In this paper, a matched asymptotic expansion is proposed for a cracked specimen which is subjected to longitudinal shear (mode III) loading, which gives the small-scale yielding estimate of linear fracture mechanics as a first approximation.
Abstract: A method of analysis based upon matched asymptotic expansions is proposed for a cracked specimen which is subjected to longitudinal shear (mode III) loading. This gives the small-scale yielding estimate of linear fracture mechanics as a first approximation, and provides systematic refinements which take account of the nonlinear interaction between the elastic and the plastic regions. Explicit solutions can be generated for any specimen which is amenable to a linear elastic analysis. Fracture parameters, such as crack opening displacement and the J integral, are expressed as power series in the ratio of applied stress to yield stress, and three terms are given explicitly. These are defined from linear elastic solutions alone. The edge-cracked strip and cracking from a semi-circular notch are studied as examples. Comparison with an exact solution for the former geometry suggests that the three-term expansions give useful results up to 75 % of limit load. The latter example is new and shows the effect of a notch on a crack at loads beyond the normal range of validity of linear elastic fracture mechanics.
22 citations
01 Jan 1976
TL;DR: Assuming a spherical geometry for the left ventricle, passive elastic stiffness-stress relations have been obtained on the basis of linear elasticity theory and large deformation theory and this implies that stiffness of muscleper se can be assessed from left ventricular pressure-volume data.
TL;DR: The equilibrium configurations of symmetric, 3-fold stacking fault nodes in fcc crystals were determined by a linear elastic self-stress method which accounts for the combined effects of dislocation interaction and elastic anisotropy as mentioned in this paper.
TL;DR: Surprisingly, inclusion of fairly large pretwist angles had little affect on the first three frequencies of transverse vibration in either the cranial or lateral directions, implying the presence of some inhomogeneities in material properties around the bone cross-section and/or along its length.
Book•
01 Oct 1976TL;DR: In this paper, the authors present a general theory of elasticity in the context of Cartesian tensors, and apply it to the problem of small deformation in the neighborhood of a point.
Abstract: I General Principles.- 1 Vectors and Cartesian Tensors.- 1.1. Scalars and Vectors.- 1.2. Coordinate Transformations.- 1.3. Orthogonality Relations.- 1.4. Addition of Vectors and Multiplication by a Scalar.- 1.5. Scalar and Vector Products of Two Vectors.- 1.6. Definition of Cartesian Tensors.- 1.7. Addition of Cartesian Tensors.- 1.8. Multiplication of Cartesian Tensors.- 1.9. Quotient Rule for Second-Order Tensors.- 1.10. Symmetric and Antisymmetric Tensors.- 1.11. Antisymmetric Tensor Components.- 1.12. Eigenvalues and Eigenvectors of Symmetric Tensors.- 1.13. Principal Axes of a Symmetric Tensor.- Selected Reading.- Exercises.- 2 Kinematics of Continuum Motion.- 2.1. Material and Spatial Variables.- 2.2. Definitions of Displacement, Velocity, and Acceleration.- 2.3. Deformation Gradients.- 2.4. Stretch and Angular Distortion of Line Elements.- 2.5. Condition for Rigid-Body Motion of Material about a Point.- 2.6. Decomposition of Deformation Gradients.- 2.7. General Motion of Material in the Neighborhood of a Point.- 2.8. Approximations Valid for Small Deformations.- 2.9. Motion in the Neighborhood of a Point for Small Deformations.- 2.10. Geometric Interpretation of Strain and Rotation Components of Small Deformation.- 2.11. Examples of Small Deformation.- 2.12. Unabridged Notation.- 2.13. Cylindrical Polar Coordinates.- Selected Reading.- Exercises.- 3 Governing Equations of Motion.- 3.1. Conservation of Mass.- 3.2. Balance of Linear Momentum.- 3.3. Balance of Angular Momentum.- 3.4. Evaluation of Time Derivative of Volume Integral.- 3.5. Green's Theorem.- 3.6. The Stress Vector.- 3.7. The Stress Tensor.- 3.8. Change of Stress Components with Rigid Rotations.- 3.9. Local Form of Mass Conservation.- 3.10. Local Form of Linear Momentum Balance.- 3.11. Local Form of Angular Momentum Balance.- 3.12. Some Simple Examples of Stress.- 3.13. Stress Boundary Conditions.- 3.14. Approximations Valid for Small Deformations.- 3.15. Unabridged Notation.- 3.16. Cylindrical Polar Coordinates.- Selected Reading.- Exercises.- II Classical Elasticity.- 4 Theory of Elasticity.- 4.1. Constitutive Relations for an Elastic Solid.- 4.2. Restrictions Placed on Constitutive Relations by Principle of Material Indifference.- 4.3. Material Symmetry Restrictions on the Constitutive Relations.- 4.4. Elastic Constitutive Relations Applicable to Small Deformations.- 4.5. Restriction on Elastic Constants Due to Existence of a Strain Energy Function.- 4.6. Restrictions on Elastic Constants Due to Material Symmetries.- 4.7. Constitutive Relations for Isotropic Elastic Materials.- 4.8. Alternate Form of Elastic Constitutive Relations.- 4.9. Governing Equations for Linear Elastic Deformation of an Isotropic Solid.- Selected Reading.- Exercises.- 5 Problems in Elasticity.- 5.1. Longitudinal and Transverse Elastic Waves.- 5.2. Static Twisting of Rods and Bars.- 5.3. Saint-Venant's Principle.- 5.4. Compatibility Equations.- 5.5. Plane Strain and Plane Stress.- 5.6. Bending of a Thin Beam by Uniform Loading.- 5.7. Equations for Plane Strain and Plane Stress in Polar Coordinates.- 5.8. Thick-Walled Cylinder under Internal Pressure.- 5.9. Circular Hole in a Strained Plate.- 5.10. Strength-of-Material Formulations.- 5.11. Bending and Extension of Beams.- 5.12. Bending and Extension of Thin Rectangular Plates.- 5.13. Axisymmetric Bending and Extension of Thin Cylindrical Shells.- Selected Reading.- Exercises.- III Thermal Elasticity.- 6 Theory of Thermal Elasticity.- 6.1. First Law of Thermodynamics.- 6.2. Second Law of Thermodynamics.- 6.3. Definition of a Thermoelastic Solid.- 6.4. Restrictions Placed on Constitutive Relations by the Second Law of Thermodynamics.- 6.5. Restrictions Placed on Constitutive Relations by Principle of Material Indifference.- 6.6. Restriction to Small Deformations and Small Temperature Changes.- 6.7. Restriction to Isotropic Materials.- 6.8. Governing Equations for Linear Thermoelastic Deformation of an Isotropic Solid.- Selected Reading.- Exercises.- 7 Problems in Thermal Elasticity.- 7.1. Thermoelastic Vibrations.- 7.2. Periodic Temperature Variation on the Boundary of a Thermoelastic Half-Space.- 7.3. Plane Strain and Plane Stress Thermoelastic Problems.- 7.4. Thermal Stresses in a Thin Elastic Strip.- 7.5. Plane Strain and Plane Stress Equations in Polar Coordinates.- 7.6. Hollow Circular Cylinder with Elevated Bore Temperature.- 7.7. Thermal Effects in Beam Deformations.- Selected Reading.- Exercises.- IV Viscous Elasticity.- 8 Theory of Viscous Elasticity.- 8.1. Definition of a Standard Viscoelastic Solid.- 8.2. Restrictions Placed by Principle of Material Indifference.- 8.3. Restriction to Small Deformations.- 8.4. Restriction to Isotropic Materials.- 8.5. Reduction of Constitutive Relations for Special Cases.- 8.6. Governing Equations for Linear Viscoelastic Deformation of an Isotropic Solid.- Selected Reading.- Exercises.- 9 Problems in Viscous Elasticity.- 9.1. Free Vibration of a Standard Viscoelastic Solid.- 9.2. Time-Dependent Uniaxial Response of a Standard Viscoelastic Solid.- 9.3. Hollow Circular Cylinder of Kelvin-Voigt Material Subjected to Periodic Bore Pressure.- 9.4. Viscous Effects in Beam Deformations.- 9.5. Viscoelastic Correspondence Principle.- 9.6. Laterally Constrained Bar.- Selected Reading.- Exercises.- V Plasticity.- 10 Theory of Plasticity.- 10.1. Definition of an Elastic-Plastic Solid.- 10.2. Restrictions Placed by Principle of Material Indifference.- 10.3. Restriction to Quasilinear Response Independent of Mean Stress.- 10.4. Plastic Constitutive Relations Applicable for Negligible Elastic Deformations.- 10.5. Governing Equations.- Selected Reading.- Exercises.- 11 Problems in Plasticity.- 11.1. Initial Yielding of a Thin-Walled Tube under Combined Tension-Torsion Loading.- 11.2. Initial Yielding of a Hollow Cylinder under Internal Pressure Loading.- 11.3. Twisting of a Circular Rod.- 11.4. Plastic Extension of a Cylindrical Bar under Simple Tension Loading.- 11.5. Plane Strain Compression.- 11.6. Plane Strain Deformation of Rigid-Perfectly Plastic Solids.- 11.7. Reduction of Plane Strain Equations.- 11.8. Slip-Line Theory.- 11.9. Numerical Solutions Using Slip-Line Theory.- 11.10. Wedge Penetration in a Rigid-Plastic Material.- Selected Reading.- Exercises.- Appendix A.- Similitude and Scale Modeling in Solid Mechanics.- Appendix B.- to Numerical Methods in Solid Mechanics.
TL;DR: In this paper, the authors extended the analysis given bymunds and W illis (1976) to deal with cracks in elastic work-hardening plastic specimens subjected to longitudinal shear loads and proved that a "plastic-zone correction" obtained by solving a linear elastic problem for a crack which is a length r y longer than the actual crack, provides a two-term asymptotic expansion for the J -integral, if r y is defined suitably in terms of the linear elastic stress concentration factor and the initial slope of the workhardening curve.
Abstract: Earlier analysis given by T.M. E dmunds and J.R. W illis (1976) is extended to deal with cracks in elastic work-hardening plastic specimens subjected to longitudinal shear loads. Solutions are expressed in terms of a set of parameters that are determined from linear elastic solutions alone. It is proved, for any specimen geometry and any loading symmetric about the plane of the crack, that a ‘plastic-zone correction’, obtained by solving a linear elastic problem for a crack which is a length r y longer than the actual crack, provides a two-term asymptotic expansion for the J -integral, if r y is defined suitably in terms of the linear elastic stress concentration factor and the initial slope of the work-hardening curve. The general method is applied in detail to a strip of finite width containing an edge crack, for which the effect of the work-hardening on the maximum extent of the plastic zone and on the J -integral is summarized graphically.
TL;DR: In this paper, the boundary between the X and Z sectors of growth in a specimen of synthetic quartz has been studied and a model of the lattice deformation in the region of the boundary is proposed to account satisfactorily for the direct image contrast seen in X-ray topographs.
Abstract: The boundary between the - X and Z sectors of growth in a specimen of synthetic quartz has been studied. Proportional differences in the unit cell dimensions a and c between material from the adjacent growth sectors were found to be 7-4±5-4 × 10−5 and -8-9±5-2 × 10−5, respectively. The interface between the two sectors was shown to be coherent. A model of the lattice deformation in the region of the boundary is proposed using linear elasticity theory and found to account satisfactorily for the direct image contrast seen in X-ray topographs.
TL;DR: An identity generalizing the Prager-Synge relationship in linear elasticity is deduced for a certain class of nonlinear elasticity laws in this article, and it is proved that the root-mean-square value (over the volume of a plate) of the error in the solution of the plate equations derived from the volume problem by means of the Kirchhoff hypothesis, does not exceed ch 1 2, where c is a constant and h is the relative thickness.
TL;DR: In this paper, a formal solution for traveling waves in isotropic linear elastic bars of rectangular cross section which have traction-free lateral surfaces is presented by crosswise superposition of two single series, each term of which gives zero shearing stresses on lateral surfaces.
Abstract: A formal solution for travelling waves in isotropic linear elastic bars of rectangular cross section which have traction-free lateral surfaces is presented. The solution is obtained by crosswise superposition of two single series, each term of which gives zero shearing stresses on the lateral surfaces. Frequency equations, which define the dispersion relations of elastic waves, are given by infinite determinants. Based on the approximate frequency equations, dispersion relations for a longitudinal mode, a torsional mode, and bending modes were calculated with the aid of a digital computer. Numerical results were compared with those given by other authers.
TL;DR: In this paper, an approximation to the stress distribution is taken as the sum of the periodic linear elastic solution and a residual stress constant in time, which is derived for two simple structures.
TL;DR: In this article, a theory based on linear elasticity was proposed for the shear stress distribution of wide flange or box tapered Hooken beams, where significant shears are carried by the flanges.
Abstract: The shear stress distribution of tapered beams is investigated using a theory based on linear elasticity. The theory is applicable to wide flange or box tapered Hooken beams. Conventionally the shear stress distribution is assumed to be uniform with the external shear carried solely by the web. By the proposed theory significant shears are carried by the flanges. Based on the proposed theory a simplified analysis procedure is described. Simply calculate the vertical components of the flange flexural load and subtract and add them from the total vertical shear. The resulting shear is assumed to be carried by the web as a relatively uniform stress distribution. In comparison with the finite element analysis and the classifical wedge theory close agreement was observed. /ASCE/
TL;DR: In this paper, it was shown that the solution to the equations of a linear micropolar elastic solid, in an exterior domain in R3, depends continuously on initial and boundary data, body forces and material coefficients.
TL;DR: In this article, the finite element method is applied to the small deflection bending analysis of nonuniform thin axisymmetric circular plates made of linear elastic material, which gives better results compared to other finite element methods besides offering savings in computer storage and time.
01 Jan 1976
TL;DR: In this article, a two-parameter fracture criterion was developed for thin-gaged or high-tough materials containing cracks and a close correlation was found between experimental and predicted failure stresses.
Abstract: Thin-gaged or high toughness materials containing cracks usually fail in a ductile manner with nominal failure stresses approaching the ultimate strength of the material. For such materials, a two-parameter fracture criterion was developed. An equation which related the linear elastic stress-intensity factor, elastic nominal stress, and two material parameters was previously derived and has been used as a fracture criterion for surface- and through-cracked specimens under tensile loading. This two-parameter fracture criterion was rederived in a more general form and was extended to compact and notch-bend fracture specimens. A close correlation was found between experimental and predicted failure stresses.
01 Sep 1976
TL;DR: In this article, a one-dimensional constitutive relation for metals under stress in a high temperature and high neutron flux field is deduced by using physical arguments, which contains modified superposition integrals in which the temperature and flux dependence of the material parameters is included via the use of two reduced time scales; linear elastic, thermal expansion and swelling terms also included.
Abstract: Employing an analogy between thermally induced and irradiation induced creep, physical arguments are used first to deduce a one-dimensional constitutive relation for metals under stress in a high temperature and high neutron flux field. This constitutive relation contains modified superposition integrals in which the temperature and flux dependence of the material parameters is included via the use of two reduced time scales; linear elastic, thermal expansion and swelling terms are also included. A systematic development based on thermodynamics, with the stress, temperature increment and defect density increment as independent variables in the Gibbs free energy, is then employed to obtain general three-dimensional memory integrals for strain; the entropy and coupled energy equation are also obtained. Modified superposition integrals similar to these previously obtained by physical argument are then obtained by substituting special functions into the results of the thermodynamic analysis, and the special case of an isotropic stress power law is examined in detail.
01 Dec 1976
TL;DR: In this paper, the inadequacy of a two noded beam-column element with a linear axial and a cubic transverse displacement field for inelastic analysis is demonstrated for complete equilibrium satisfaction in the linear elastic range.
Abstract: The inadequacy of a two noded beam-column element with a linear axial and a cubic transverse displacement field for inelastic analysis is demonstrated For complete equilibrium satisfaction in the linear elastic range a three noded beam-column element is shown to be consistent Next, the sensitivity of the inelastic response to numerical solutions of the inelastic response of a cantilever beam resulting from approximate integration of strain energy are brought out and finally, consequences of this on the nonlinear transient response of structures are considered