Isotropic Elastic Plate Research Articles

Three-Dimensional Stress Field Near the Crack Front of a Thin Elastic Plate

Based on very detailed full-field finite element analysis of the near tip region of a thin isotropic elastic plate, the three-dimensional stress state in the vicinity of a through-crack front is characterized. The computed stress field reveals strong three-dimensional effects within a radial distance of about one-half thickness from the crack-tip. Further away from the tip, through-thickess variation of field quantities decreases and, at the radial distance of approximately 1.5 times the thickness, in-plane stresses merge with the dominant two-dimensional plane stress solutions. These “two-dimensional-three-dimensional” transition distances are essentially independent of the material Poisson’s ratio, yet the amplitude of variation is greatly affected by its value. The influence of Poisson’s ratio is clearly illustrated by local J along the crack front, which shows much higher variation through the thickness for nearly incompressible solids. At points very close to the crack front, relative magnitudes of out-of-plane strain components become very small, and asymptotic plane strain conditions prevail locally. On the mid-plane of the plate, the crack tip field converges to that given by the local plane strain stress intensity factor solution within a radial distance from the tip of less than 0.5 percent of thickness. In addition, it is found that the field near the intersection of crack front and free surface may be characterized by the corner singularity of a quarter infinite crack in a half space. The size of this domain is inferred from the gradient of local stress intensity factor with respect to distance from the free surface, and it appears that the corner singularity region extends up to a spherical radius of about 3 percent of plate thickness away from the intersection. Also the amplitude of the corner singularity field is described by a corner stress intensity factor, and its magnitude is determined for thin plates of various Poisson’s ratios.

Nonlinear bending analysis of thin circular plates by differential quadrature

The behavior of thin, circular, isotropic elastic plates with immovable edges and undergoing large deflections is investigated by the numerical technique of differential quadrature. Approximate results are determined with the aid of a symbolic manipulation computer program and a Newton-Raphson technique to solve the nonlinear systems of equations. Bending stresses, membrane stresses, and deflections are calculated for clamped and simply supported flexural edge conditions and for a uniform pressure load and a concentrated load at the center.

Non-linear standing waves of an elastic plate floating on the surface of a heavy liquid of infinite depth

The problem of non-linear standing waves in an ideal incompressible liquid on whose surface an elastic isotropic plate floats is solved by a perturbation method. The case of bifurcation of the solution is investigated. The results of the research are discussed briefly in /1/.

Analysis of stress intensity factors for surface cracks in pre/post penetration.

This paper presents stress intensity factors for surface cracks in pre/post penetration for finite elastic isotropic plates subjected to tension or bending loads. These have been analyzed by a three-dimensional boundary element analysis code, BEM3D, with a supercomputer and will be useful for Leak Before Break Design. The ratio of crack depth to plate thickness of the cracked plates, a/t, ranges from 0.80 to 1.20, and the ratio of crack depth to crack length, a/c, ranges from 0.20 to 1.20. The results have a good coincidence with those of Raju and Newman at a/t = 0.8. And using these results, the validity of the stress intensity factor equation proposed by Newman and Raju for a/t > 0.8 has been investigated. Furthermore, crack opening area in back surface has also been analyzed.

Rayleigh-Lamb Waves in an Elastic Plate With Voids

Rayleigh-Lamb waves in a homogeneous and isotropic linear elastic plate containing a distribution of vacuous pores (voids) are studied. Assuming that the plate is of uniform thickness and that its faces are stress-free, it is found that the waves move, in general, in two uncoupled families, of which one is symmetrical with respect to the midplane of the plate and the other antisymmetrical; each of these families is affected by the presence of voids. If the plate is thin and the frequency is small, the voids influence only the symmetric waves and, because of this influence, the waves propagate slower than their classical counterparts. If the plate thickness and the frequency are large, each of the two families degenerates into two uncoupled waves; one of these is a classical Rayleigh wave and the other is a new wave not encountered in the classical theory.

Natural frequencies of a linearly tapered non-homogeneous isotropic elastic circular plate resting on an elastic foundation

An analysis is presented here for the design of engineering systems where composite materials and non-uniform structures are used. A simple model for free vibrations of a non-homogeneous isotropic elastic circular plate of linearly varying thickness resting on an elastic foundation is considered. The governing differential equation of motion is solved by the Frobenius method. The frequencies, deflections and moments corresponding to the first five modes of vibration are computed for clamped and simply supported edge conditions for various values of taper constant, non-homogeneity parameter and foundation modulus.

A Transversely Isotropic Elastic Plate Pressed between Two Rigid Cylindrical Surfaces

This paper considers a fundamental contact problem of a transversely isotropic elastic plate pressed between two identical rigid cylindrical surfaces under concentrated loads. Two harmonic type stress functions are introduced under plane strain conditions and the method of Fourier transforms leads to a pair of dual integral equations which are reduced to Fredholm integral equations of the 2nd kind through the method of Abel transforms. The effects of anisotropy and plate thickness on the stresses and deformations as well as the contact pressure and the relations between load and contact area are examined.

On wave propagation and energy flux in thin transversely isotropic plates

A thin, unidirectionally reinforced, composite plate is considered and modelled as a transversely isotropic linear elastic plate. The structure is subjected to a oscillating load perpendicular to the plane of the plate at a central point. The resulting transverse (bending) waves are studied. Theoretical calculations using asymptotic expansions of the three-dimensional basic equations are presented. With the help of analytical solutions based on Fourier-transform techniques, calculated constant-phase lines, amplitude decay and energy flux are shown to be in good agreement with experiments performed on a thin unidirectionally reinforced plate made of carbon-fibre prepregs.

Certain variational problems with a small parameter in the theory of elasticity

For the mixed problem of elasticity theory on the deformation of a transversely isotropic cylinder, it is proved than in the selection of a specific small parameter the zeroth approximation equations agree with the bending equations for an elastic isotropic plate based on the Kirchhoff-Love hypotheses /1/. Certain singularly perturbed contact problems of the Signorini type are considered.

Natural frequencies of a non homogeneous isotropic elastic infinite plate of variable thickness resting on elastic foundation

The dynamic free response of a nonhomogeneous isotropic elastic infinite plate of parabolically varying thickness resting on an elastic foundation has been studied. The frequencies, deflections and moments corresponding to the first five modes of vibration have been computed for the two combinations of boundary conditions, clamped-clamped (C-C) and clamped-simply supported (C-SS) and various values of taper constant, nonhomogeneity parameter and foundation modulus by applying the method of Frobenius for the solution of the governing differential equation of motion.

On the Nonlocal Theory of Wave Propagation in Elastic Plates

Propagation of longitudinal waves in isotropic homogeneous elastic plates is studied in the context of the linear theory of nonlocal continuum mechanics. To determine the nonlocal moduli, the dispersion equation obtained for the plane longitudinal waves in an infinite medium is matched with the parallel equation derived in the theory of atomic lattice dynamics. Using the integroalgebraic representation of the stress tensor and the Fourier transform, the system of two coupled differential field equations is solved in the standard manner giving the frequency equations for the symmetric and antisymmetric wave modes. It is found that the short wave speed in the Poisson medium differs by about 13 percent from the speed established in the classical theory. A numerical example is given.

The influence of inplane deformation on the buckling loads of isotropic elastic plates

The paper deals with buckling of isotropic, homogeneous elastic plates of constant thickness and shows that the effect of inplane deformation is of the same order of magnitude as the shear effect. The two effects should therefore be considered together.

Axisymmetric vibrations of an isotropic elastic non-homogeneous circular plate of linearly varying thickness

Free axisymmetric vibrations of an isotropic, elastic, non-homogeneous circular plate of linearly varying thickness have been studied on the classical theory of plates. The non-homogeneity of the material of the plate is assumed to arise due to the variation of Young's modulus and density with the radius vector whereas Poisson's ratio is assumed to remain constant. The governing differential equation of motion is solved by the method of Frobenius. The transverse displacement of the plate has been expressed as a power series in terms of the radial co-ordinate. The frequency parameters corresponding to the first two modes of vibration have been computed for different values of the non-homogeneity parameter and taper constant and for clamped and simply supported edge conditions of the plate. A comparison between the numerical results for homogeneous and non-homogeneous material of the plate is also made.

An Application of Finite-Element Method To The Deformation Analysis of Coated Plain-Weave Fabrics

Methods for analysing coated plain-weave fabric which has properties of nonlinear elasticity have not yet been satisfactorily developed. In this paper, a method which is promis ing for use in engineering applications like the strength analysis of membrane structures is presented. The finite element method using a rectangular element consisting of plain-weave fabric and coating material which is assumed to be an isotropic elastic plate of plane stress is applied to the method. Verification of the me thod is made by using uniaxial stress-strain responses. A square piece of coated plain-weave fabric with a square hole in it is analyzed as an example of application of the present method. Key Words: coated plain-weave fabrics; finite element method; nonlinearly elastic biaxial response; geometrically nonlinear prob lem ; incremental approach.

On the interaction between two inclined cracks under biaxial load

An analysis of the in-plane biaxial loading of an infinite homogeneous and isotropic elastic plate with two inclined collinear cracks of equal length is performed. The boundary value problem is solved by using the complex potentials method. The influence of the lateral load on the local stress distributions as well as on the crack tips behaviour is analyzed and the features of the interaction are described.

Solid mechanics applications of boundary fitted coordinate systems

A numerical method which utilizes the boundary fitted coordinate method is presented and applied to the solution of three solid mechanics problems. The three problems considered are the elastic torsion of uniform shafts of arbitrary cross section, the elastic torsion of non-uniform shafts of arbitrarily varying circular cross section, and the bending of thin isotropic elastic plates of arbitrary shape with simple or clamped boundaries. The boundary fitted coordinate method is utilized to transform the arbitrary simply connected or multiply connected region under study onto a fixed rectangular domain where computations are easily done. The governing equations and boundary conditions are transformed and solved on the rectangular domain by SOR iteration. Numerical results for all three problems show close agreement with analytical solutions, although there is local error introduced when the coordinate system is severely skewed. Numerical results check closely with experimental results obtained for problems which have no analytical solution.

Refined theory for flexural motions of isotropic elastic plates

A technical theory for the flexural motions of isotropic elastic plates has been developed, taking into account the influence of transverse normal strain and transverse normal stress, together with rotatory inertia and transverse shear. The theory is tested by studying the classical wave propagation problem and results indicate the influence of the transverse normal strain on the wave speed at large values of h λ . In addition, a constant magnitude for the shear coefficient κ 2 = 5 6 is obtained, which is in contrast to an undetermined coefficient form in previous flexural motion formulations but consistent with the value obtained in the Reissner static technical theory of plate bending.

Analysis of unloading waves from suddenly punched hole in an axially loaded elastic plate

The propagation of plane-stress unloading waves emanating from a suddenly punched circular hole in an axially loaded isotropic elastic plate is studied by analytical and numerical methods.

On the Analysis of First and Second-Order Shear Deformation Effects for Isotropic Elastic Plates

Technical Briefs On the Analysis of First and Second-Order Shear Deformation Effects for Isotropic Elastic Plates E. Reissner E. Reissner Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla, Calif. 92093 Search for other works by this author on: This Site PubMed Google Scholar Author and Article Information E. Reissner Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla, Calif. 92093 J. Appl. Mech. Dec 1980, 47(4): 959-961 (3 pages) https://doi.org/10.1115/1.3153824 Published Online: December 1, 1980 Article history Received: March 1, 1980 Online: July 21, 2009

Influence of a longitudinally compressed elastic plate on the nonstationary wave motion of a homogeneous liquid

In a linear formulation, a study is made of the influence of a longitudinally compressed elastic isotropic plate on the nonstationary wave motion of a stream of homogeneous liquid of finite depth on which the plate floats. The waves are generated by periodic (in time) normal stresses applied to a restricted region of the plate surface and beginning at a certain initial time.