Exact Elasticity Theory Research Articles

An analysis of effective shear modulus for flexural and extensional waves and its application to reflection of sound by a plate

An analytical expression is presented for the correction factor that relates the effective shear modulus in Timoshenko–Mindlin plate theory to the actual shear modulus, for an unloaded plate. This expression is obtained by comparison of the approximate theory with exact elasticity theory. A thick-plate theory is developed for extensional waves, which also introduces an effective shear modulus, with a corresponding correction factor. It is shown analytically that both correction factors produce the proper high-frequency value for the phase speed, namely, the Rayleigh wave speed. Reflection of sound by a plate is described when both flexural and extensional waves are excited. Both types of waves are described by the expressions derived in thick-plate theory. A structural response function of the plate for reflection is given for each of the two wave types separately. The structural response function pertaining to the case where both wave types occur simultaneously is expressed in terms of the two individual response functions in the same way that the total impedance of a pair of impedances in parallel is expressed in terms of the values of the individual impedances. The distribution of kinetic energy over the two types of waves as a function of the angle of incidence is shown.

Analytic determination of effective shear modulus for flexural and extensional waves in an unloaded thick plate

Thin plate theory does not predict the proper phase speed of flexural and extensional waves in the limit of high frequency or large plate thickness. The Mindlin‐Timoshenko theory corrects this for the case of flexural waves by introduction of transverse shear; the “effective shear modulus” is represented by a correction factor that is fixed in such a way that the ensuing dispersion relation approaches the Rayleigh wave speed at high frequency. In this paper an analytic determination of the correction factor is presented, obtained by applying the results of exact elasticity theory (Lamb waves). The expression for the correction factor can be shown analytically to reproduce the Rayleigh wave speed at high frequency. An analogous treatment of symmetric waves leads to a thick plate theory for extensional waves, with a corresponding effective shear modulus. The expression for the pertinent correction factor leads also in this case to the Rayleigh wave speed at high frequency.

Piecewise continuous eigenfunction solutions for laminated slabs

A piecewise continuous eigenfunction procedure is developed to handle the two and three dimensional stress fields of rectangular anisotropic laminated slabs. In particular, each lamina of the slab is considered to be modelled by exact anisotropic elasticity theory such that perfect bonding between adjacent laminae is assumed. Since the eigenfunction basis arising from the development is complete but nonorthogonal, a piecewise weighted reorthogonalization procedure or alternatively a multiply connected version of the Gram Schmidt process is employed in solving the eigenfunction expansion. In order to obtain the completely general boundary value solution, complex series and adjoint differential forms are also employed to handle inhomogeneities in the governing DEQ. Several numerical experiments are included to reveal the capabilities of the solution procedure developed as well as the potential local importance of material anisotropy.

Comments on ’’Influence of rotary inertia and shear deformation on acoustic transmission through a plate’’ [J. Acoust. Soc. Am. 58, 741–745 (1975)

The transmission loss of an acoustic plane wave through a steel plate in water has been computed using Timoshenko–Mindlin plate theory and exact elasticity theory. Good agreement between the two models is found up to a frequency–thickness product of 20 kHz in. At 40 kHz in. and above, limitations of the plate theory approach become apparent. Subject Classification: [43]30.50; [43]20.60; [43]40.24; [43]20.30.

Comparison of classical plate theory, Timoshenko-Midlin plate theory, and exact elasticity theory for calculation of transmission loss through fluid-loaded plates

Two approximate plate theories, “classical” (CP) and Timoshenko-Midlin (TM), are commonly used in computing radiation and scattering from submerged structures. These models have previously been compared with an exact elasticity theory calculation of the phase velocity of the first antisymmetric mode of an unloaded plate. On this basis, CP is limited to frequencies below 0.7 times critical for steel in water. No indication of the limitations of TM is, however, found by this calculation. A more appropriate problem for underwater applications is transmission loss through the plate. We have calculated this quantity versus frequency-thickness product and angle of incidence using both the plate models and the exact theory. This approach shows that CP agrees well up to 1.5 times critical frequency. TM correctly predicts the first and second flexural mode transmission peaks but fails in other ways at frequencies greater than 4.0 times critical. In particular, no effects of symmetric modes are included, and an incorrect limit is reached at grazing incidence.

Modal Analysis for Impact of Layered Plates

An analysis is presented of the plane strain response of an arbitrarily layered elastic plate to surface impact loads of specified distribution over the plate surface and in time. Each layer has specified orthotropic material properties, with coincident material and plate axes. The method is based on the extended Ritz technique and employs a normal mode procedure to accurately predict the motion through the plate thickness of stress waves generated by the impact load. The case of impact on an isotropic plate is studied in detail, both with the present method and other methods based on the exact elasticity theory to indicate the accuracy and convergence properties of this method. The procedure is applied to the case of a two-layer plate, where each layer is assigned orthotropic elastic properties typical of a unidirectional boron-epoxy composite. The stress waves generated by a normal surface impact are presented to indicate both the quality and quantity of information that can be generated by the analysis. A tension wave that trails the initial compression wave is shown to exist.

A Refined Theory for Laminated Anisotropic, Cylindrical Shells

A set of governing equations and boundary conditions are derived which describe the static deformation of a laminated anisotropic cylindrical shell. The theory includes both transverse shear deformation and transverse normal strain, as well as expansional strains. The validity of the theory is assessed by comparing solutions obtained from the shell theory to results obtained from exact theory of elasticity. Reasonably good agreement is observed and both shear deformation and transverse normal strain are shown to be of importance for shells having a relatively small radius-to-thickness ratio.

Stress Analysis of Thick Laminated Composite and Sandwich Plates*

Because of the relatively soft interlaminar shear modulus in high per formance composites, laminated plate theory based on the Kirchhoff hypothesis becomes inaccurate for determining gross plate response and internal stresses of thick composites and sandwich type laminates. In this paper a procedure is developed for accurately calculating the mechanical behavior of a thick laminated composite or sandwich plate of arbitrary stacking sequence. The procedure is an extension of an existing laminated plate theory which includes the effect of transverse shear deformation, and is easily computerized for design application with a minimum of running time. Accuracy of the approach is ascertained by comparing solutions from the modified plate theory to exact elasticity theory.

Stress Analysis of Thick Laminated Composite and Sandwich Plates

Because of the relatively soft interlaminar shear modulus in high per formance composites, laminated plate theory based on the Kirchhoff hypothesis becomes inaccurate for determining gross plate response and internal stresses of thick composites and sandwich type laminates. In this paper a procedure is developed for accurately calculating the mechanical behavior of a thick laminated composite or sandwich plate of arbitrary stacking sequence. The procedure is an extension of an existing laminated plate theory which includes the effect of transverse shear deformation, and is easily computerized for design application with a minimum of running time. Accuracy of the approach is ascertained by comparing solutions from the modified plate theory to exact elasticity theory.

On the Use of Shell Theory for Determining Stresses in Composite Cylinders

Applying Vlasov-Ambartsumyan shell theory to anisotropic and laminated cylinders, equations are developed for calculating the stresses in a composite tube under combined axial load, torsion, and internal pressure. Comparison to results obtained from exact elasticity theory shows that the shell equations are capable of predicting, with a reasonable degree of accuracy, the large stress gradients found in highly anisotropic tubes. Thus the shell theory provides the experimentalist with a set of closed form expressions for readily defining the proper specimen dimensions for precise characterization of unidirectional and laminated composite tubes. A modification to the shell theory, in which the effects of transverse normal strain are included is also discussed. Numerical results show that such a modification is necessary for determining stresses induced by free thermal expansion. It is also shown that certain classical thin shell kinematic relations are incapable of predicting stresses in composite tubes.

On the propagation of flexural waves in anisotropic bars

Propagation of flexural waves in circularly cylindrical bars of anisotropic material on the basis of the exact three-dimensional theory of elasticity of small deformations is considered, the rigor of the analysis being the same as the one of the Pochhammer-Chree theory. A system of the governing equations of motion for the cylindrical orthotropy (its axis coinciding with the axis of the bar) is derived and expressed in terms of radial, tangential and axial displacement components. For a steady-state sinusoidal flexural wave the system reduces to coupled ordinary second order differential equations with a regular singular point. Solution for a transversely isotropic is obtained by means of Frobenius series method. Upon using two or three terms of the series approximate dispersion relations for isotropic material as well as for the pyrolytic graphite type material and two other types of anisotropic materials are obtained and illustrated by graphs. Higher order modes than those already known are found.

A shell theory compared with the exact three dimensional theory of elasticity

A theory for the equilibrium deformation of homogeneous isotropic shells is derived and compared with the exact theory of elasticity. The shell theory is derived axiomatically as follows. The potential energy is considered as a functional depending on displacements which are polynomials in the undeformed distance to the middle surface. The degree of the displacements tangent to the middle surface is a prescribed integer n and that of the displacement normal to the middle surface is n+ 1. The shell theory is then obtained by requiring that the potential energy be stationary with respect to all variations in this set of displacements. For each fixed n it is shown that a solution to the shell theory is an approximate solution to the exact theory at points not too near the edge of the shell provided that the displacement gradients, strains, loading, and thickness are sufficiently small. It is also shown that the agreement between the theories improves as n increases. Many of the proofs have been restricted to the case of zero body force, zero stress vector on the faces, and the strain energy density function of the linear theory, but the work has been carried out in more generality in a research report.

Double-Cantilever Cleavage Mode of Crack Propagation

The quantitative cleavage method for determining specific surface energies of solids is subject to several possible sources of error. Some of these are investigated analytically in this study, and a general expression for the surface energy is derived from beam theory for a linearly elastic material. The exact equation of motion for the propagating crack is derived but not solved. Errors associated with each term of the general expression are assessed. The ratio of shear to bending strain energy is small if the initial crack length is long compared with the specimen depth. The term arising from strains in the specimen past the tip of the crack is not developed, but is estimated; and a procedure is outlined for determining this term experimentally. Comparison of simple beam theory with more exact elasticity theory shows that in the main spans the difference is small because it involves only the shear strain energy. Also, restriction of anticlastic curvature near the crack tip is unimportant if the initial crack length is long compared with the specimen width. Elastic anisotropy is unimportant when shear can be neglected, provided the appropriate modulus is used. For nonlinear elastic material, as the crack propagates a small amount, the work increment is either zero or double the strain energy increment. A surface energy expression is developed in terms of Timoshenko's ``reduced modulus,'' and upper and lower bounds are fixed. For plastic materials the analysis is based on pseudoinstantaneous plastic response. Justification of this assumption is difficult to defend or deny. Therefore, an important conclusion is that whenever possible specimens should be designed to remain grossly within the elastic limit of the material. Superposition of large axial forces is a potential source of large errors in surface energy determinations, and possible eccentricities in the application of such forces makes even the sign of these errors difficult to determine. Hence, this method for stabilizing the crack plane is not recommended.

High-Frequency Extensional Vibrations of Sandwich Plates

In this paper there is derived a set of approximate equations governing the extensional motion of a sandwich plate, together with the associated initial and boundary conditions. Both the sandwich core and the facings are assumed homogeneous and isotropic, and the facings are assumed thin and identical to each other. The coupling effect of the thickness deformation of the core is considered in the derivation. Frequency-wavelength curves are then obtained from these equations for an infinite plate in plane strain. The corresponding curves for the same plate according to the three-dimensional exact theory of elasticity are also obtained. These two sets of curves are then used as a basis for discussion of the accuracy of the approximate equations. The curves indicate that for commonly used core materials and for wavelengths about three times the plate thickness and longer, the approximate equations are reasonably good up to frequencies somewhat higher than the fundamental thickness-stretch frequency of the plate.

Vibrations of Thick and Thin Cylindrical Shells Surrounded by Water

This report treats the free and forced vibrations of infinitely long, thick and thin cylindrical shells surrounded by water. Exact elasticity theory is used to treat unpressurized shells and an approximate shell theory is employed to treat the effects of static pressure, internal fluid, and structural damping. Comparisons are made between the results of the exact and approximate theories.

Flexural Vibrations of Elastic Sandwich Plates

The new flexural thecny of elastic sandwich plates developed in two recent papers2 includes the effects of transverse shear deformations, rotatory and translatory inertias, and flexural and extensional rigidities, with no limitations imposed upon the magnitudes of the ratios between the thicknesses, material densities, and elastic constants of the core and faces. On the basis of the new sandwich plate theory and the exact elasticity theory, flexural vibrations of sandwich plates are investigated in this paper. Numerical results yielded by the two theories show excellent agreement with each other, and the new sandwich plate theory is seen to b^ good for a very wide frequency range that is of practical interest. The importance of the various effects involved in the theory is assessed.

Flexural Vibrations of a Thick-Walled Circular Cylinder According to the Exact Theory of Elasticity

This paper gives results of the exact theory for flexural or nonaxially symmetric vibrations of a thick-walled cylindrical shell having freely supported ends. The results also apply to flexural wave propagation in infinitely long cylindrical shells. Curves of phase velocity and frequency as a function of wavelength ratio and wall thickness ratio are given. Included are some comparisons between the exact theory, the Timoshenko beam theory, and shell theory. ki = shear constant (such that Q = ki'AGy) Q = shear force A = cross-sectional area y = angle of shear at neutral axis