Showing papers on "Linear elasticity published in 1969"

Wave front analysis in composite materials

Abstract: : The propagation of an initially sharp plane pressure pulse through a linear elastic composite medium is analysed. Wave front and ray theory analogous to geometrical optics is shown to determine the change in shape of the leading wave front and also the stresses immediately behind it. For certain circumstances the stress amplitudes on this front, or the corresponding tensile stresses on its reflection at the free back surface of a slab, may be critical in design. Examples are presented of an initially sharp plane pressure pulse transmitted through an elastic circular cylinder and an elastic spherical inclusion. The method can be applied to more general composite configurations, and can be extended to determine the stress gradient behind the front. For the latter, general formulae are derived by which the reflection and transmission coefficients can be determined for the stress gradient and the higher order derivatives at an arbitrary interface.

A modification of the Rayleigh-Ritz method for stress concentration problems in elastostatics☆

Abstract: Details are given of a modification to the Rayleigh-Ritz method which, by extending the field of definition of the coordinate functions, improves the convergence in predicting the values of stress concentrations in elastic continua. The extension to the field of definition requires that the coordinate functions are not only complete in energy but also allows early derivatives of the displacement to be continuous. The procedure is described for three-dimensional linear elasticity and, by way of illustration, for the problem of plate bending.

Effective Dynamic Properties of a Fiber‐Reinforced Material and the Propagation of Sinusoidal Waves

Abstract: A theoretical study is presented on the propagation of a plane sinusoidal wave through a material that is reinforced with parallel fibers in one direction. The wave propagates in a direction normal to the fibers, and both fiber and matrix are made of linear elastic materials. An integral formulation for the scattered‐wave solution of an isolated fiber is used to study the multiple scattering in an infinite slab of the composite material. The transmitted and reflected waves from the composite and from a homogeneous slab are shown to be similar. By matching the two sets of results, formulas expressed in terms of the isolated‐fiber solution are derived for the wave speed, the effective density, and the modulus of the composite. In general, the effective density and modulus so defined are complex numbers and depend on the wave frequency. This fact indicates the possible existence of dissipation and dispersion in the composite under dynamic loadings. A series solution is presented for a composite containing ci...

A matrix theory of elastic-locking structures

Abstract: The theory developed exhibits the following peculiar features: structures are discretizised in finite elements, the constitutive laws piecewise linearized, the problem is split in a preliminary linear elastic solution and a “corrective” nonlinear subproblem; concepts and techniques of quadratic and linear programming theory are utilized. The main results are: for the analysis under given loads and dislocations, a pair of extremum theorems for locking stresses, corresponding to dual quadratic programs; for the limit analysis with respect to locking situations two already known theorems, which are here deduced from the solvability conditions of the above quadratic programs and formulated as dual linear programs. The extension of the results to imperfectly locking behavior is carried out. Some examples illustrate the solution techniques based on the theory expounded.

First‐Branch Dispersion of Torsional Waves in Bimaterial Rods

Abstract: Torsional wave propagation is investigated for a rod consisting of two concentric, circular cylinders of dissimilar materials, perfectly bonded at the interface. Dispersion curves for the first branch of propagation are obtained from the frequency equation, which is derived through a solution satisfying the equations of linear elasticity. These dispersion curves are presented in dimensionless form with the ratio of constituent cylinder radii and ratios of like material properties as parameters. Only parameter values for which ρ2/ρ1>μ2/μ1 are considered. An expression for the first branch torsional wave velocity limit, as wavelength becomes infinite, is obtained from the exact frequency equation.

Dispersion of Flexural Waves in Circular, Bimaterial Cylinders—Theoretical Treatment

Abstract: The frequency equation for flexural waves travelling in infinitely long, circular, bimaterial cylinders is obtained, based oil the equations of linear elasticity. Several first branches of the dispersion curves are presented and discussed for various ratios of the constituent cylinder radii. Dispersion characteristics significantly different from those predicted by the theory for homogeneous cylinders are realized. Some implications in regard to equivalent properties are discussed.

Determination of stress in rock with linear or non-linear elastic characteristics

Abstract: A non-linear law of elasticity is proposed in which each principal stress is expressed as the summation of two series, one a function of the dilatory (hydrostatic) or octahedral normal strain and the other of the deviatory or octahedral shear strain. The constants in the series can be obtained from a simple uniaxial compression or tension test on the material. These expressions can be used to determine the stress from strain readings in rock having non-linear stress-strain characteristics and the application of these expressions when using the CSIR triaxial strain cell in such rock is demonstrated.

Dynamic Tear Test Definition of the Temperature Transition From Linear Elastic to Gross Strain Fracture Conditions

Abstract: : Transition temperature concepts for fracture-safe design have been based on the relatively narrow temperature range evidenced by the fracture-mode transition from plane strain to plane stress. Fracture mechanics theory has suggested that large increases of section size should provide sufficient mechanical constraint for retention of plane strain conditions through the transition temperature range. Recent investigations based on dynamic tear (DT) tests of thick-section reactor-grade steels (A-533 B) have provided clear evidence that the plane strain to plane stress transition is not eliminated. The DT test, as conducted using a small specimen, defines the temperature range of transition from linear elastic to gross strain mechanical conditions of fracture and, therefore, the required analytical treatment for flaw size-stress calculations. Fracture mechanics concepts are brought into consonance with transition temperature concepts of long standing, and the importance of considering fracture initiation in terms of limiting dynamic toughness values is emphasized. (Author)

Reference Stress for Redistribution Time in Creep of Structures

Abstract: When creep occurs in a structure subject to a step load the stresses redistribute with time. It is shown that if the initial stress distribution is the linear elastic one, and the material obeys an n-power creep law, the time for a particular stress to reach its steady state value may be estimated from the results of a single creep test at a selected value of stress with-out the parameters of the creep law being known. This reference stress is identified from the results of an appropriate elastic-creep analysis.

Waves on a Spherical Shell

Abstract: Within the framework of classical linear elasticity theory, the dynamic equations governing the motion of a spherical shell are given. These include the effects of transverse shear and rotatory inertia, and can be derived on the basis of three assumptions additional to those of linear elasticity. The response of an incomplete spherical shell subject to a suddenly applied, constant moment Mθ is studied. The wavefront behavior of the solution is found by taking the Laplace transform of the equations, and then making an asymptotic expansion for large values of p, the Laplace variable. The solution is given in the form of a traveling wave, which is followed into the focus point and on its reflection from there. The applied discontinuity at the wavefront grows as θ → π; it gives a square‐root singularity at θ = π, and a logarithmic singularity on the reflected wavefront. These results are synthesized by an elliptic function.

Shakedown Loads for Pressure Vessels of Strain Hardening Materials

Abstract: A power law, well known in creep analysis, embodies a family of curves which express the stress-strain relations for a family of materials ranging from linear elastic to rigid perfectly plastic. A linearization of the relationship between stress concentration factor and the reciprocal of strain hardening exponent for geometrically similar pressure vessels made of materials within the family has enabled a view of shakedown in vessels of strain hardening materials to be formulated. The absence of discontinuities in the power law, except at the rigid plastic end point, results in shakedown loads dependent on strain hardening exponent and previous loading history.

Development and evaluation of a triangular thin shell element based upon the hybrid assumed stress model.

Abstract: : A finite-element formulation for a general triangular thin doubly-curved Kirchhoff shell element has been carried out, based upon the hybrid assumed stress variational model. This formulation is then applied specifically to conical (and cylindrical) shells, and the associated finite-element properties are evaluated. Several example static problems involving isotropic cylindrical shells and an isotropic conical shell subjected to mechanical loading are solved by employing the present triangular element to evaluate this formulation; good agreement with independent reliable solutions for these problems has been found. The formulation presented, however, also includes skew orthotropic plane stress shells, variable thickness shells, and thermal loads, all for linear elastic behavior. (Author)

Lateral vibration of a nonlinear viscoelastic beam under initial axial tension.

Abstract: : The free lateral vibration of a simply-supported metallic beam under axial creep deformation is considered The constitutive law is formulated with stress power functions in the primary and secondary creep terms; the instantaneous linear elastic deformation is also included For better physical visualization, a nonlinear Maxwell-Kelvin model is used to represent the constitutive law A reduction to various special models is also obtained It is assumed that the initial stress is much greater than the increments of stress caused by the oscillation, and a perturbation technique is employed In the present study, the perturbation essentially results in replacing a nonlinear viscoelastic problem by an equivalent linear viscoelastic problem Analytical solutions for the Maxwell-Kelvin model and for the special models are obtained Numerical results for a stainless steel and an aluminum alloy are also presented and discussed (Author)

On complex potentials in two-dimensional linearized elasticity with couple stresses

Abstract: A semi-complex-variable technique has been developed for solving planestrain problems in linear elasticity with couple stresses when the body moments are zero. In conjunction with theMindlin couple-stress functionψ, two complex potentials are introduced which reduce to those of the non-couple-stress case. With this technique it is only necessary to solve the equationψ−l2∇2ψ=0 forψ.