Reaz A. Chaudhuri

An equilibrium method for prediction of transverse shear stresses in a thick laminated plate

First two equations of equilibrium are utilized to compute the transverse shear stress variation through thickness of a thick laminated plate after in-plane stresses have been computed using an assumed quadratic displacement triangular element based on transverse inextensibility and layerwise constant shear angle theory (LCST). Centroid of the triangle is the point of exceptional accuracy for transverse shear stresses. Numerical results indicate close agreement with elasticity theory. An interesting comparison between the present theory and that based on assumed stress hybrid finite element approach suggests that the latter does not satisfy the condition of free normal traction at the edge. Comparison with numerical results obtained by using constant shear angle theory suggests that LCST is close to the elasticity solution while the CST is closer to classical (CLT) solution. It is also demonstrated that the reduced integration gives faster convergence when the present theory is applied to a thin plate.

On boundary-discontinuous double Fourier series solution to a system of completely coupled P.D.E.'s

Abstract A heretofore unavailable boundary-discontinuous double Fourier series based approach for solution to a system of completely coupled linear second-order partial differential equations with constant coefficients and subjected to general (completely coupled) boundary conditions is presented. This method facilitates the selection of the unknown Fourier coefficients, contributing to the complete solution and also takes into account the possible discontinuities of the assumed solution functions and/or their first derivatives at the boundaries. Dirichlet and Neumann types of boundary conditions can be treated as two special cases. The method is applied to obtain solutions of the hitherto unsolved class of problems, pertaining to arbitrarily laminated anisotropic (constant)-shear-flexible doubly-curved shells of rectangular planform, with arbitrary admissible boundary conditions and subjected to general transverse loading. Such specific cases of lamination as anti-symmetric angle-ply, symmetric angle-ply and general cross-ply, such particular case of loading as uniformly distributed transverse load and such specific case of geometry as a rectangular plate, can be obtained as special cases of the above. In addition, this method is shown to reproduce the available boundary-continuous solutions for unsymmetric cross-ply plates and doubly-curved shells with SS3 type simply-supported boundary conditions and anti-symmetric angle-ply plates with SS2 type simply-supported boundary conditions.

An approximate semi-analytical method for prediction of interlaminar shear stresses in an arbitrarily laminated thick plate

Abstract An approximate semi-analytical method for determination of interlaminar shear stress distribution through the thickness of an arbitrarily laminated thick plate has been presented. The method is based on the assumptions of transverse inextensibility and layerwise constant shear angle theory (LCST) and utilizes an assumed quadratic displacement potential energy based finite element method (FEM). Centroid of the triangular surface has been proved, from a rigorous methematical point of view (Aubin-Nitsche theory), to be the point of exceptional accuracy for the interlaminar shear stresses. Numerical results indicate close agreement with the available three-dimensional elasticity theory solutions. A comparison between the present theory and that due to an assumed stress hybrid FEM suggests that the (normal) traction-free-edge condition is not satisfied in the latter approach. Furthermore, the present paper is the first to present the results for interlaminar shear stresses in a two-layer thick square plate of balanced unsymmetric angle-ply construction. A comparison with the recently proposed Equilibrium Method (EM) indicates the superiority of the present method, because the latter assures faster convergence as well as simultaneous vanishing of the transverse shear stresses on both the exposed surfaces of the laminate. Superiority of the present method over the EM, in the case of a symmetric laminate, is limited to faster convergence alone. It has also been demonstrated that the combination of the present method and the reduced (quadratic order) numerical integration scheme yields convergence of the interlaminar shear stresses almost as rapidly as that of the nodal displacements, in the case of a thin plate.

Three-dimensional stress singularity at a bimaterial interface crack front

Abstract An eigenfunction expansion method is developed to obtain three-dimensional asymptotic stress fields in the vicinity of the front of a semi-infinite crack along a bimaterial interface of a plate subjected to far-field extension and bending. In comparison with the existing method, the present method is considerably easier to implement, and is also computationally more efficient in the sense that it does not need to resort to iterative schemes to solve the system of three coupled partial differential equations of three-dimensional elasticity theory. Explicit expressions for singular stress fields in the neighborhood of the front of a semi-infinite crack along a bimaterial interface are derived in the paper.

A novel eigenfunction expansion solution for three-dimensional crack problems

Abstract A novel eigenfunction expansion method is developed to obtain three-dimensional asymptotic stress fields in the vicinity of the front of a semi-infinite crack. The present method is relatively easy to implement, and is also computationally more efficient in the sense that it does not need to resort to either iterative schemes or more complicated analytical techniques such as the (Lure Al. Three dimension problems of the theory of elasticity. New York: Interscience, 1964.) symbolic method to solve the system of three coupled partial differential equations of three-dimensional elasticity theory. Explicit expressions for singular stress fields in the neighborhood of the front of a semi-infinite crack are derived in the paper. The order of stress singularity thus computed is compared with counterparts developed by earlier investigators. The primary goal of this study is to settle the controversy that has existed during the last quarter of a century as regards the order of stress singularity at the three-dimensional crack front. Additionally, the relationship between the strain–energy release rate and the stress–intensity factor is investigated from a three-dimensional standpoint Furthermore, numerical results pertaining to development of plastic yield zone ahead of the front of a semi-infinite crack are also obtained.

Prediction of the Compressive Strength of Thick-Section Advanced Composite Laminates:

An analytical study, with the twin objectives of (1) development of a simple, yet comprehensive approach for prediction of the compressive strength of thick-section advanced composite (graphitc/cpoxy) cross-ply laminates and (2) explanation of unacceptably low strength of the (90N°/0°)k cylindrical specimens tested at David Taylor Research Center (DTRC) under hydrostatic compression, has been undertaken. A simple closed-form expression for the kink mode compressive strength has been derived, under the simplifying assumptions of small deformation, material linearity, local flatness and piecewise linear distribution of the surface-parallel displacements through the thickness, using the minimum potential energy approach. Initial fiber misalignment (maximum tangent angle), ultimate fiber shear strain and the two transverse shear moduli are found to be the key parameters, which limit the compressive strength of the aforementioned thick-section composite laminates. An analysis pertaining to the elastic plane strain in extensional deformation of the compressed composite reveals that kink band formation is primarily due to rotational instability of the fibers, and regardless of the event or sequence of events that may act as precursor(s), kink band formation, once triggered, will, in general, be the dominant (lowest energy) failure mode, especially in the presence of such defect as fiber waviness or misalignment. Critical kink band angle has been found to be bounded well away from zero (a fact consistent with experimental observations), and has been shown to be dependent on the ultimate shear strain of the fiber, maximum initial fiber misalignment tangent angle, θ0 and transverse moduli, Er, Gir and Grr, of the composite laminate. Numerical results for graphite/epoxy and glass/epoxy laminates demonstrate the extreme sensitivity of the former to the initial fiber misalignment defects, responsible for the lowering of the compressive strength of the thick cylinders tested at DTRC and the observed scatter in the test data. These results clearly indicate the sensitivity of kink band angle, β, for graphite/epoxy composites, to the maximum initial fiber misalignment tangent angle, θ0, in the lower range of values (2–3° or less).

Free-edge stress singularity in a bimaterial laminate

A Williams type asymptotic solution pertaining to the free edge effect and singular stress field in a bonded rectangular plate of dissimilar isotropic materials is presented. Numerical results presented include the effects of material properties on the computed stress singularity. Additionally, the stress singularity computed using the present analysis is compared with the corresponding plane stress approximation. For example, the largest values for the stress singularities at the free edge of a model bimaterial laminate investigated here can be as high as 0.405 and 0.311, computed using (i) the present analysis, and (ii) the corresonding plane stress approximation, respectively.

Stress Singularity due to Kink Band Weakening a Unidirectional Composite under Compression

A Williams type of eigenfunction expansion approach is used to compute the local stress singularity representing a measure of the degree of inherent flaw sensitivity of unidirectional fiber reinforced composites subjected to compression. Previous experimental studies have qualitatively linked the formation of kink bands to the presence of fabrication defects, such as fiber misalignment. These local singular stress regions serve as the primary trigger mechanism for kink band propagation in 0°-plies. The present analysis also explains the previous test results relating to propagation of failure from a notch in a unidirectional composite under compression. Furthermore, the present investigation is the first to quantify, in the context of LEFM (linear elastic fracture mechanics), the sensitivity of these composites to inherent local flaws, such as fiber misalignments, and also shows the inadequacy of the conventional elastic micro-buckling type of analysis to fully explain the experimental results. Although because of the assumed isotropy of the fiber and matrix materials, the present study is primarily suited to glass fiber reinforced composites, the conclusions drawn here are general enough to apply to carbon fiber reinforced composites as well. Numerical results presented include the effects of fiber included wedge angle, and the ratios of fiber-matrix shear moduli and Poissons ratios on the strengths of the mode I and mode II singularities. Of special practical interest is the present LEFM type of analysis applied to quantitatively investigate the inherent flaw sensitivity of two E-glass/epoxy composites experimentally investigated earlier. Compression fracture type of failure of these composites can be fully explained and quantified by the present two-dimensional LEFM-based method, which is beyond the scope of one-dimensional micro-buckling approach.

Three-dimensional asymptotic stress field in the vicinity of the circumference of a penny shaped discontinuity

Abstract An eigenfunction expansion method is presented to obtain three-dimensional asymptotic stress fields in the vicinity of the front of a penny shaped discontinuity, e.g., crack, anticrack (infinitely rigid lamella), etc., subjected to the far-field torsion (mode III), extension/bending (mode I) and sliding shear/twisting (mode II) loadings. Five different discontinuity-surface boundary conditions are considered: (i) penny shaped crack, (ii) penny shaped anticrack or perfectly bonded thin rigid inclusion, (iii) penny shaped thin transversely rigid inclusion (frictionless planar slip permitted), (iv) penny shaped thin rigid inclusion in part perfectly bonded, the remainder with frictionless slip, and (v) penny shaped thin rigid inclusion alongside penny shaped crack. The computed stress singularity for a penny shaped anticrack is the same as that of the corresponding crack. The main difference is, however, that all the stress components at the circular tip of an anticrack depend on Poisson’s ratio under modes I and II.

Free vibration analysis of thin arbitrarily laminated anisotropic plates using boundary-continuous displacement Fourier approach

A boundary continuous displacement based Fourier series solution to the boundary-value problem of free vibration of an arbitrarily laminated thin rectangular plate is presented. This powerful approach is employed to solve a system of three highly coupled partial differential equations arising from the Kirchhoff hypothesis as applied to an anisotropic laminate, with the SS2-type simply supported boundary conditions prescribed at all four edges. The accuracy of the computed eigenvalues (natural frequencies) is ascertained by studying the convergence characteristics of the lowest seven natural frequencies, and also by comparison with the computed degenerate FEM (finite element methods) results. Other important numerical results presented include variation of the response quantities of interest with geometric and material parameters, such as fiber orientation angle and longitudinal-to-transverse modulus ratio.