On large deflection theories of plates.
Large elastic-plate deflection equations criticized for applicability and consistency
PLATE THEORY↗
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On refined Reissner-type theories of plates
Reissner-type equations for linear stress distribution determination through plate thickness
PLATE THEORY↗
A theory for the fracture of thin plates subjected to bending and twisting moments
Stress fields near the tip of a through crack in an elastic plate under bending and twisting moments are reviewed assuming both Kirchhoff and Reissner plate theories. The crack tip displacement and rotation fields based on the Reissner theory are calculated. These results are used to calculate the J-integral (energy release rate) for both Kirchhoff and Reissner plate theories. Invoking Simmonds and Duva's (1981) result that the value of the J-integral based on either theory is the same for thin plates, a universal relationship between the Kirchhoff theory stress intensity factors and the Reissner theory stress intensity factors is obtained for thin plates. Calculation of Kirchhoff theory stress intensity factors from finite elements based on energy release rate is illustrated. It is proposed that, for thin plates, fracture toughness and crack growth rates be correlated with the Kirchhoff theory stress intensity factors.
Hui, C. Y.↗
An assessment of four-noded plate finite elements based on a generalized third-order theory
Plate finite elements based on the generalized third-order theory of Reddy and the first-order shear deformation theory are analyzed and compared on the basis of thick and thin plate modeling behavior, distortion sensitivity, overall accuracy, reliability, and efficiency. In particular, several four-noded Reddy-type elements and the nine-noded Lagrangian and heterosis (Mindlin-type) plate elements are analyzed to assess their behavior in bending, vibration, and stability of isotropic and laminated composite plates. A four-noded Reddy-type element is identified which is free of all spurious stiffness and zero energy modes, computationally efficient, and suitable for use in any general-purpose finite element program.
Averill, R. C.↗
A refined shear deformation theory for the analysis of laminated plates
A refined, third-order plate theory that accounts for the transverse shear strains is presented, the Navier solutions are derived for certain simply supported cross-ply and antisymmetric angle-ply laminates, and finite-element models are developed for general laminates. The new theory does not require the shear correction factors of the first-order theory (i.e., the Reissner-Mindlin plate theory) because the transverse shear stresses are represented parabolically in the present theory. A mixed finite-element model that uses independent approximations of the generalized displacements and generalized moments, and a displacement model that uses only the generalized displacements as degrees of freedom are developed. The displacement model requires C sup 1-continuity of the transverse deflection across the inter-element boundaries, whereas the mixed model requires a C sup 0-element. Also, the mixed model does not require continuous approximations (between elements) of the bending moments. Numerical results are presented to show the accuracy of the present theory in predicting the transverse stresses. Numerical results are also presented for the nonlinear bending of plates, and the results compare well with the experimental results available in the literature.
Reddy, J. N.↗
A refined nonlinear theory of plates with transverse shear deformation
A higher-order shear deformation theory of plates accounting for the von Karman strains is presented. The theory contains the same dependent unknowns as in the Hencky-Mindlin type first-order shear deformation theory and accounts for parabolic distribution of the transverse shear strains through the thickness of the plate. Exact solutions of simply supported plates are obtained using the linear theory and the results are compared with the exact solutions of three-dimensional elasticity theory, the first order shear deformation theory, and the classical plate theory. The present theory predicts the deflections, stresses, and frequencies more accurately when compared to the first-order theory and the classical plate theory.
Reddy, J. N.↗
Theory of micropolar plates.
Linear theory of micropolar elasticity applied to micropolar plates stress and bending theory
Eringen, A. C.↗
Fracture Mechanics of Thin, Cracked Plates Under Tension, Bending and Out-of-Plane Shear Loading
Cracks in the skin of aircraft fuselages or other shell structures can be subjected to very complex stress states, resulting in mixed-mode fracture conditions. For example, a crack running along a stringer in a pressurized fuselage will be subject to the usual in-plane tension stresses (Mode-I) along with out-of-plane tearing stresses (Mode-III like). Crack growth and initiation in this case is correlated not only with the tensile or Mode-I stress intensity factor, K(sub I), but depends on a combination of parameters and on the history of crack growth. The stresses at the tip of a crack in a plate or shell are typically described in terms of either the small deflection Kirchhoff plate theory. However, real applications involve large deflections. We show, using the von-Karman theory, that the crack tip stress field derived on the basis of the small deflection theory is still valid for large deflections. We then give examples demonstrating the exact calculation of energy release rates and stress intensity factors for cracked plates loaded to large deflections. The crack tip fields calculated using the plate theories are an approximation to the actual three dimensional fields. Using three dimensional finite element analyses we have explored the relationship between the three dimensional elasticity theory and two dimensional plate theory results. The results show that for out-of-plane shear loading the three dimensional and Kirchhoff theory results coincide at distance greater than h/2 from the crack tip, where h/2 is the plate thickness. Inside this region, the distribution of stresses through the thickness can be very different from the plate theory predictions. We have also explored how the energy release rate varies as a function of crack length to plate thickness using the different theories. This is important in the implementation of fracture prediction methods using finite element analysis. Our experiments show that under certain conditions, during fatigue crack growth, the presence of out-of-plane shear loads induces a great deal of contact and friction on the crack surfaces, dramatically reducing crack growth rate. A series of experiments and a proposed computational approach for accounting for the friction is discussed.
Zehnder, Alan T.↗
A high order theory for uniform and laminated plates
A theory of plate deformation is derived which accounts for the effects of transverse shear deformation, transverse normal strain, and a nonlinear distribution of the in-plane displacements with respect to the thickness coordinate. The theory is compared with lower order plate theories through application to a particular problem involving a plate acted upon by a sinusoidal surface pressure. Comparison is also made with exact elasticity solution of this problem. It is found that when the ratio of the characteristic length of the load pattern to the plate thickness is of the order of unity, lower order theories are inadequate and the present high order theory is required to give meaningful results. Results are given for the bending of symmetric cross-ply and angle-ply laminates. Comparison with exact elasticity solutions indicates that the present plate theory is sufficiently accurate for predicting the behavior of thick laminates.
Lo, K. H.↗
Assessment of computational models for thermoelectroelastic multilayered plates
A study is made of the accuracy of the steady-state (static) thermoelectroelastic response of multilayered hybrid composite plates predicted by five modeling approaches, based on two-dimensional plate theories. The plates consist of a combination of fiber-reinforced and piezothermoelastic layers. The standard of comparison is taken to be the exact three-dimensional thermoelectroelastic solutions, and the quantities compared include gross response characteristics (e.g., strain energy components, and average through-the-thickness displacements); detailed, through-the-thickness distributions of displacements and stresses; and sensitivity coefficients of the response quantities (derivatives of the response quantities with respect to material parameters of the plate). The modeling approaches considered include first-order theory; third-order theory; discrete-layer theory (with piecewise linear variation of the in-plane displacements, temperature and electric potential, in the thickness direction); and two predictor-corrector procedures.
Tang, Yvette T.↗
Influence of Material Distribution on Impact Resistance of Hybrid Composites
Impact events occur in a wide variety of circumstances. A typical example is a bullet impacting a target made of composite material. These impact events produce time-varying loads on a structure that can result in damage. As a first step to understanding the damage resistance issue in composite laminates, an accurate prediction of the transient response during an impact event is necessary. The analysis of dynamic loadings on laminated composite plates has undergone considerable development recently. Rayleigh-Ritz energy method was used to determine the impact response of laminated plates. The impact response of composite plates using shear deformation plate theory was analyzed. In recent work a closed-form solution was obtained for a rectangular plate with four edges simply supported subjected to a center impact load using classical plate theory. The problem was further investigated and the analysis results compared of both classical plate theory and shear deformation theory, and found that classical plate theory predicts very accurate results for the range of small deformations considered. In this study, the influence of cross sectional material distribution on the comparative impact responses of hybrid metal laminates subjected to low and medium velocity impacts is investigated. A simple linear model to evaluate the magnitude of the impact load is proposed first, and it establishes a relation between the impact velocity and the impact force. Then a closed-form solution for impact problem is presented. The results were compared with the finite element analysis results. For an 11 layer-hybrid laminate, the impact response as a function of material distribution in cross-section is presented. With equal areal weight, the effect of the number of laminate layers on the impact resistance is also investigated. Finally, the significance of the presented results is discussed.
Abatan, Ayu↗
Small bending and stretching of sandwich-type shells
A theory has been developed for small bending and stretching of sandwich-type shells. This theory is an extension of the known theory of homogeneous thin elastic shells. It was found that two effects are important in the present problem, which are not normally of importance in the theory of curved shells: (1) the effect of transverse shear deformation and (2) the effect of transverse normal stress deformation. The first of these two effects has been known to be of importance in the theory of plates and beams. The second effect was found to occur in a manner which is typical for shells and has no counterpart in flat-plate theory. The general results of this report have been applied to the solution of problems concerning flat plates, circular rings, circular cylindrical shells, and spherical shells. In each case numerical examples have been given, illustrating the magnitude of the effects of transverse shear and normal stress deformation.
Reissner, Eric↗
Application of Normal Mode Expansion to AE Waves in Finite Plates
Breckenridge et al. (1975), Hsu (1985) and Pao (1978) adapted approaches from seismology to calculate the response at the surface of an infinite half-space and an infinite plate. These approaches have found use in calibrating acoustic emission (AE) transducers. However, it is difficult to extend this theoretical approach to AE testing of practical structures. Weaver and Pao (1982) considered a normal mode solution to the Lamb equations. Hutchinson (1983) pointed out the potential relevance of Mindlin's plate theory (1951) to AE. Pao (1982) reviewed Medick s (1961) classical plate theory for a point source, but rejected it as useful for AE and no one seems to have investigated its relevance to AE any further. Herein, a normal mode solution to the classical plate bending equation was investigated for its applicability to AE. The same source-time function chosen by Weaver and Pao is considered. However, arbitrary source and receiver positions are chosen relative to the boundaries of the plate. This is another advantage of the plate theory treatment in addition to its simplicity. The source does not have to be at the center of the plate as in the axisymmetric treatment. The plate is allowed to remain finite and reflections are predicted. The importance of this theory to AE is that it can handle finite plates, realistic boundary conditions, and can be extended to composite materials.
Gorman, M. R.↗
Iterative algorithms and numerical problems of the theory of plates and shells
Digital computer approach to elasticity theory problems using program based on iterative algorithms
ITERATIVE SOLUTION↗
Cubic Zig-Zag Enrichment of the Classical Kirchhoff Kinematics for Laminated and Sandwich Plates
A detailed anaylsis and examples are presented that show how to enrich the kinematics of classical Kirchhoff plate theory by appending them with a set of continuous piecewise-cubic functions. This analysis is used to obtain functions that contain the effects of laminate heterogeneity and asymmetry on the variations of the inplane displacements and transverse shearing stresses, for use with a {3, 0} plate theory in which these distributions are specified apriori. The functions used for the enrichment are based on the improved zig-zag plate theory presented recently by Tessler, Di Scuva, and Gherlone. With the approach presented herein, the inplane displacements are represented by a set of continuous piecewise-cubic functions, and the transverse shearing stresses and strains are represented by a set of piecewise-quadratic functions that are discontinuous at the ply interfaces.
Nemeth, Michael P.↗
An approximate three-dimensional theory of plates with application to crack problems
Elastic plate uniform extension with rectangular crack by three dimensional bending theory, using variational principle
Hartranft, R. J.↗
The surface and through crack problems in layered orthotropic plates
An analytical method is developed for a relatively accurate calculation of Stress Intensity Factors in a laminated orthotropic plate containing a through or part-through crack. The laminated plate is assumed to be under bending or membrane loading and the mode 1 problem is considered. First three transverse shear deformation plate theories (Mindlin's displacement based first-order theory, Reissner's stress-based first-order theory, and a simple-higher order theory due to Reddy) are reviewed and examined for homogeneous, laminated and heterogeneous orthotropic plates. Based on a general linear laminated plate theory, a method by which the stress intensity factors can be obtained in orthotropic laminated and heterogeneous plates with a through crack is developed. Examples are given for both symmetrically and unsymmetrically laminated plates and the effects of various material properties on the stress intensity factors are studied. In order to implement the line-spring model which is used later to study the surface crack problem, the corresponding plane elasticity problem of a two-bonded orthotropic plated containing a crack perpendicular to the interface is also considered. Three different crack profiles: an internal crack, an edge crack, and a crack terminating at the interface are considered. The effect of the different material combinations, geometries, and material orthotropy on the stress intensity factors and on the power of stress singularity for a crack terminating at the interface is fully examined. The Line Spring model of Rice and Levy is used for the part-through crack problem. The surface crack is assumed to lie in one of the two-layered laminated orthotropic plates due to the limitation of the available plane strain results. All problems considered are of the mixed boundary value type and are reduced to Cauchy type of singular integral equations which are then solved numerically.
Erdogan, Fazil↗
A three-dimensional nonlinear analysis of cross-ply rectangular composite plates
The results of a three-dimensional, geometrically nonlinear, finite-element analysis of the bending of cross-ply laminated anisotropic composite plates are presented. Individual laminae are assumed to be homogeneous, orthotropic and linearly elastic. A fully three-dimensional isoparametric finite element with eight nodes (i.e. linear element) and 24 degrees of freedom (three displacement components per node) is used to model the laminated plate. The finite element results of the linear analysis are found to agree very well with the exact solutions of cross-ply laminated rectangular plates under sinusoidal loading. The finite element results of the three-dimensional, geometrically nonlinear analysis are compared with those obtained by using a shear deformable, geometrically nonlinear, plate theory. It is found that the deflections predicted by the shear deformable plate theory are in fair agreement with those predicted by three-dimensional elasticity theory; however stresses were found to be not in good agreement.
Kuppusamy, T.↗