Displacement discontinuity formulation for modeling cracks in orthotropic shear deformable plates

A displacement discontinuity formulation is presented for modeling cracks in orthotropic Reisnner plates. Fundamental solutions for displacement discontinuity are derived for the first time using a Fourier transform method. Boundary integral equations are presented in terms of discontinuity rotations on the crack surfaces for opening mode problems. As the fundamental solutions have singularity of O (1/r ²), Chebyshev polynomials of the second kind are used to evaluate the integral equations. By solving for coefficients of the Chebyshev polynomials, the stress intensity factors at the crack tips are obtained directly. Comparisons are made with solutions using the finite element method to demonstrate that the displacement discontinuity method is an efficient and accurate method for solving crack problems in orthotropic Reissner plates.     

Extension of generalized plane strain plates with reinforced cracks

Applying the transversally isotropic generalized plane strain plate of Mindlin and Kane, as refined by Kotousov and Wang, the authors use Fourier transforms to develop a hyper-singular line-spring model for a reinforced crack under extensional loads, enabling the analysis of cracks in plates of arbitrary thickness. The new model is used to develop fracture mechanics geometry correction factors to account for the effect of the transverse modulus, Poisson’s ratio, and plate thickness on the stress intensity and crack opening displacement. The model is then used to examine the reinforcing effect of springs bridging the crack faces and extend the Rose model to include plate thickness effects, allowing the inclusion of plate thickness effects in the fracture mechanics analysis of reinforced cracks.     

Stress intensity factors for cracks in thin plates

This paper presents stress intensity factor solutions for several crack configurations in plates. The loadings considered include internal pressure, and also combined bending and tension. The dual boundary element method is used to model the plate and mixed mode stress intensity factors are evaluated by a crack surface displacement extrapolation technique and the J-integral technique. Several cases including centre crack, edge crack and cracks emanating from a hole in finite width plates are presented.     

A boundary element method applied to the elastostatic bending problem of beam-stiffened plates

The present paper proposes a basic formulation for the static bending problem of beam-stiffened elastic plates. This problem has been so far analyzed using the Timoshenko theory in which the equivalent shear force and bending moments are assumed to act on the beam stiffener. Since fourth-order derivatives of unknown displacements are included in the formulation, in its numerical implementation fourth-order polynomials must be used as the interpolation functions.

In this paper, the interactive forces and moments between the plate and the stiffener are treated as line distributed unknown loads. In the numerical implementation of the formulation, these forces can be approximated using a suitable family of interpolation functions. The formulation is presented in detail and a computer code is developed. The numerical results obtained by the computer code are discussed, whereby the usefulness of the proposed solution procedure is demonstrated.     

A BEM formulation based on Reissner’s theory to perform simple bending analysis of plates reinforced by rectangular beams

In this work, the plate bending formulation of the boundary element method (BEM), based on the Reissner’s hypothesis, is extended to the analysis of plates reinforced by rectangular beams. This composed structure is modelled by a zoned plate, being the beams represented by narrow sub-regions with larger thickness. The integral equations are derived by applying the weighted residual method to each sub-region, and summing them to get the equation for the whole plate. Equilibrium and compatibility conditions are automatically imposed by the integral equations, which treat this composed structure as a single body. In order to decrease the number of degrees of freedom, some approximations are considered for both displacements and tractions along the beam width. The accuracy of the proposed model is illustrated by simple examples whose exact solution are known as well as by more complex examples whose numerical results are compared with a well-known finite element code.     

Formation mechanism and three-point bending behaviour of directly fabricated aluminium foam plates

New-style smooth-faced aluminium foam plates (AFPs) covered by dense skins have been fabricated directly. AFPs with porosity ranging from 79.3 to 85.2% and with an average diameter of 2.3 to 2.7?mm are obtained via the melt foaming method. The formation mechanism of external dense skins is studied. The dense skin is composed of a cell wall of non-ruptured bubbles, a coarse region formed from ruptured bubbles and plateau borders. The fracture mechanism of AFPs is studied based on static three-point bending tests. They show that the cracked region is only observed at the centre of the experimental samples and that brittle fracture is the main fracture mechanism of these AFPs.     

Nonlinear bending analysis of annular FGM plates using higher-order shear deformation plate theories

This paper addresses the axisymmetric nonlinear bending analysis of an annular functionally graded plate under mechanical loading based on FSDT and TSDT. Using nonlinear von-Karman theory, the discretized equations are solved using the dynamic relaxation (DR) method combined with the finite difference technique. The effects of the material constant n, boundary conditions, thickness-to-radius ratio and shear deformation are studied. The results show that although, the difference between TSDT and FSDT becomes greater with an increasing thickness-to-external radius ratio, the effects of different types of boundary conditions is also of great importance.     

Modified four-point bending specimen for determining the interface fracture energy for thin, brittle layers

A well-known four-point bending test has been modified such that the critical energy release rate for delaminating cracks propagating at the interface of a thin, brittle layer bonded to a substrate can be measured. The energy release rate required for crack delaminating at those interfaces is obtained by attaching a stiffening layer to the layer system. Another advantage of this modification is that segmentation of the layer and plastic deformation of the substrate during bending are avoided. The interface fracture energy of a plasma sprayed ZrO₂-ceramic layer on a flame sprayed high alloyed steel substrate has been measured. This revised version was published online in August 2006 with corrections to the Cover Date.     

A 2.5D finite element model for bending and straightening in continuous casting of steel slabs

This paper presents a bi‐dimensional slice model of the continuous casting process developed to focus on the risk of transverse cracking during bending and straightening of steel slabs. The model is based on the finite element method and it integrates both thermal and mechanical aspects: temperature evolution, solidification, stress and strain developments. Generalized plain strain conditions are applied in the casting direction, allowing taking account of the extraction force applied to the slab as well as strains in this direction. The model also includes an original solution to counteract the generally wrong modelling of slab bulging with such slice models. The model has been applied to an industrial case of slab casting. Some numerical results illustrate the accuracy of the model compared to results of other models, measurements and observations on the caster. Transverse cracks are predicted to be the most likely to occur at the edge on the upper face, at the end of straightening of the slab. This is due to the combination of low ductility of the material with tensile stress and elongation in the casting direction in the straightening zone. This conclusion has been confirmed by the examination of slabs that present transverse cracks. Copyright © 2006 John Wiley & Sons, Ltd.     

Three-dimensional linear elastic distributions of stress and strain energy density ahead of V-shaped notches in plates of arbitrary thickness

This paper presents an analytical solution, substantiated by extensive finite element calculations, for the stress field at a notch root in a plate of arbitrary thickness. The present approach builds on two recently developed analysis methods for the in-plane stresses at notch root under plane-stress or plane strain conditions, and the out-of-plane stresses at a three-dimensional notch root. The former solution (Filippi et al., 2002) considered the plane problem and gave the in-plane stress distributions in the vicinity of a V-shaped notch with a circular tip. The latter solution by Kotousov and Wang (2002a), which extended the generalized plane-strain theory by Kane and Mindlin to notches, provided an expression for the out-of-plane constraint factor based on some modified Bessel functions. By combining these two solutions, both valid under linear elastic conditions, closed form expressions are obtained for stresses and strain energy density in the neighborhood of the V-notch tip. To demonstrate the accuracy of the newly developed solutions, a significant number of fully three-dimensional finite element analyses have been performed to determine the influences of plate thickness, notch tip radius, and opening angle on the variability of stress distributions, out-of-plane stress constraint factor and strain energy density. The results of the comprehensive finite element calculations confirmed that the in-plane stress concentration factor has only a very weak variability with plate thickness, and that the present analytical solutions provide very satisfactory correlation for the out-of-plane stress concentration factor and the strain constraint factor.     

Local stiffening effect of 'double deck1 bending strain gauges: part 2

In stress measurement where it is difficult or impossible to access the inside surface of the structure, "double deck" (DD) or "sandwich" bending strain gauges can be used to measure both the membrane and bending strains. A DD gauge consists of a pair of constituent foil gauges which are bonded on the upper and lower surfaces of a plastic base or filling. It is mounted on the external surface of the test specimen to measure the external surface strain, and by extrapolation, the internal surface strain of the. specimen. The. sensitivity of the DD bending gauge increases when the separation between the pair of constituent foil gauges increases. However, this also increases the stiffness of the whole gauge. If the stiffness of the DD gauge is significant compared with that of the plate to which it is attached, the presence of the gauge may cause significant changes in local stiffness and result in errors of strain interpretation. This study is presented in two papers. Part I, appeared in the Feburary 1996 issue of Strain, presented an approximate analytical solution for the local stiffening effect of a DD gauge, and quantifies the errors in the strain measurement. Part 2 (this paper) presents the verification of the. solution using a three-dimensional finite element calculation, a parametric study and a correction method that can be applied easily. The extrapolation errors due to the stiffening effect and random reading noises are also explored. The figure and equation numbers in this part are continuedfrom Part 1, but the reference numbering is specific to Part 2.     

Antisymmetric bending analysis of symmetric laminated plates including transverse shear and normal effects

Antisymmetric bending analysis of symmetric laminated plates is presented here. The transverse shear and normal strain and stress effects on bending of such laminates are considered. The displacement fields and the transverse shear and normal stress fields are assumed to preserve the displacement and traction continuity conditions at the interface between layers. A set of twelfth-order governing equations and consistent boundary conditions are given from a mixed variational theorem. Solutions for simply-supported cross-ply plates and a strip are discussed. The numerical results are compared with elasticity solutions and results given from other theories. The present theory is found to agree closely with three-dimensional elasticity solutions.     

Evaluation of the fracture properties of polymer mortars reinforced with nanoparticles

Polymer-based nanocomposites have shown superior mechanical, thermal, as well as multifunctional properties when compared to plain polymer matrices. This fact has lead to a great research activity focused on incorporating polymeric materials with nanoparticles over the last years. The present research investigates the influence of nanoparticles reinforcement on fracture properties of polymer mortars, with particular regards to fracture and toughening mechanisms. This investigation was carried-out for a series of polymer mortars, containing varying amounts of aluminum oxide (Al₂O₃) and iron oxide (Fe₂O₃) nanoparticles dispersed in epoxy matrices.     

Quasi-isotropic bending responses of metallic sandwich plates with bi-directionally corrugated cores

In order to reduce anisotropic behaviors of sandwich plates with open channel cores under the bending load, bi-directionally corrugated cores were introduced. Bi-directionally corrugated core has two additional design parameters related with a corrugation pass than uni-directionally corrugated core, so that its properties with respect to core orientations can be controlled. Sandwich plate with bi-directionally corrugated core is designed optimally so that beam buckling of face sheets is reduced drastically and anisotropic buckling behavior in the face sheets is minimized. The cores fabricated by a sectional forming process were bonded with face sheets by adhesive bonding. Three-point bending experiments were carried out with respect to core orientations. It has been shown from the experiments that sandwich plates with bi-directionally corrugated cores exhibit quasi-isotopic bending behaviors and structural performances in sandwich plates.     

Crack Growth analysis of plates Loaded by bending and tension using dual boundary element method

This paper presents an extension of the dual boundary element method to analysis of crack growth in plates loaded in combine bending and tension. Five stress intensity factors, two for membrane behaviour and three for shear deformable plate bending are computed using the J-Integral technique. Crack growth processes are simulated with an incremental crack extension analysis based on the maximum principal stress criterion. The method is considered effective since no remeshing is required and the crack extension is modelled by adding new boundary elements to the previous crack boundaries. Several incremental crack growth analysis for different configurations and loadings are presented.     

Plastic strains and the macroscopic critical plane orientations under combined bending and torsion with constant and variable amplitudes

Round cross-section specimens made of 18G2A steel were subjected to different combinations of constant- and variable-amplitude bending and torsion. The fatigue tests were performed under bending and torsion with moment control in the high cycle fatigue regime. Two approaches were used to calculate stress courses from moment histories. In one approach, stresses and strains were computed using simple elastic beam theory (nominal stresses). In the other approach, time courses of moments were used to calculate stress and strain histories taking into account plastic strains and non-linear stress distribution along the specimen cross-section on the basis of the algorithm described in the paper. The loading histories computed according the two methods were used to calculate the critical plane orientations. It was assumed that the orientation of the critical plane is controlled only by shear or tensile fatigue mechanism. Moreover, the theoretical critical plane positions were compared to the experimental macroscopic fatigue fracture plane orientations.     

Electromechanical model of periodic cracks in piezoelectric materials

The electro-elastic problem for a periodic array of cracks in a piezoelectric medium subjected to coupled electro-mechanical loads is investigated. The mixed boundary value problem, which is formulated directly in terms of the crack surface displacements and electrical potentials, results in a system of hyper-singular integral equations in which the unknown functions are the crack surface displacement and electric potential. Numerical results include the crack surface displacement and the stress and electric intensity factors for the entire range of possible periodic crack spacing and medium size. The central contribution of this paper is the development of an analytical model that predicts crack-spacing effect. The resulting model is validated by a 2D finite element analysis.     

On the estimation of the first interpenetration point in the open model of interface cracks

A general expression for estimating the location of the first interpenetration point in the open model of interface cracks that, for a fixed reference system, can be directly applied to all material combinations is presented.     

Crack-tip fields for matrix cracks between dissimilar elastic and creeping materials

The stress fields near the tip of a matrix crack terminating at and perpendicular to a planar interface under symmetric in-plane loading in plane strain are investigated. The bimaterial interface is formed by a linearly elastic material and an elastic power-law creeping material in which the crack is located. Using generalized expansions at the crack tip in each region and matching the stresses and displacements across the interface in an asymptotic sense, a series asymptotic solution is constructed for the stresses and strain rates near the crack tip. It is found that the stress singularities, to the leading order, are the same in each material; the stress exponent is real. The oscillatory higher-order terms exist in both regions and stress higher-order term with the order of O(r°) appears in the elastic material. The stress exponents and the angular distributions for singular terms and higher order terms are obtained for different creep exponents and material properties in each region. A full agreement between asymptotic solutions and the full-field finite element results for a set of test examples with different times has been obtained.