Bending of simply-supported,antisymmetrically laminated rectangular plate under transverse loading

The deformation of antisymmetric angle-ply laminated plate under transverse loading is analysed using the bending theory of laminated plate presented by Ren.^1,2 By expanding the load in a double Fourier series, a closed-form solution is obtained for special types of simply-supported boundary condition. The results are compared with similar results from the classical laminated plate theory and Mindlin theory. This shows that the theory presented by Ren is suitable not only for cylindrical bending but also for general bending problems.     

Diffusive relaxation of stress concentrations at grain boundary cavities in elevated temperature creep

This work deals with certain aspects of elevated temperatures creep cavitation of grain boundaries under cyclic and rapidly applied loading. The response of partially damaged materials (where damage is represented as crack-like cavities on the grain boundaries) following load alterations at temperatures in the vicinity of 0.5 T_m or higher is analyzed. The interaction between grain boundary diffusion and elastic deformation is important in alleviating local stress concentrations under these conditions. The stress levels considered are assumed to be low enough that plastic deformation is significant. Diffusive processes contribute to high temperature creep rupture by material redistribution from the void surfaces to the grain boundaries, and any non-uniform matter accommodation along the grain boundaries is accomplished by elastic deformation under the conditions assumed. The same material redistribution mechanism dominates in the stress relaxation process. The analysis of the stress and displacement fields is based on consideration of the coupled elasticity-diffusion boundary value problem, which leads to an integral equation. On the basis of the solution obtained, the detailed analysis of the process under cyclic, step and ramp loadings is given. For suddenly applied loading the results demonstrate that the elastic stress concentration is effectively relaxed by diffusion after t = 0.05τ where $\text{τ ≈}\text{L}{}^3\text{DE}$, L is half of the distance between adjacent tips of neighboring cracks along the grain boundary, E is Young's modulus, and D is diffusion parameter relating volumetric flux along the grain boundary to the stress gradient. By t = 0.25τ the stress distribution becomes essentially identical to that calculated for rigid grains, i.e. diffusion has become the dominant process and elastic deformability of the adjoining grains is then irrelevant.     

Stress intensity factors for cracks initiated from a center-holed plate with unsymmetrical lengths under tension

The values of stress intensity factors of unsymmetrical cracks initiated from hole edges under tension were calculated in a center-holed plate, that is a plate having a hole in the center of a front face. The correction factor for stress intensity in the case of the holed plate with cracks was compared with that of a center-cracked plate whose crack length was equal to the total of the hole diameter and the lengths of the cracks in the case of the holed plate. In the present study, to understand the variation of the stress intensity factor, counterplots of the stress intensity factor K_I and the correction factor F_I are employed. Also, the variation of ratio R_F of correction factors between the holed plate with crack and the center-cracked plate were investigated. Where non-dimensional hole diameter a/W is smaller than 0.2, the ratio R_F is less than 1.1 after crack growth length c/W reaches 0.1. Where, 2a, 2c and 2W are hole diameter, the growth length of the crack from hole edge, and plate width, respectively. Therefore, the stress intensity factor for cracks initiated from a holed plate can be approximated within 10% error to that of a center-cracked plate where a/W is smaller than 0.2, though the unsymmetrical cracks are initiated from the edges. Where a/W is larger than 0.3, R_F becomes larger than 1.1 in the wide range of c/W.     

Diffusion-controlled growth of ferrite plates in plain-carbon steels

Abstract

The implementation of the theory of diffusion-controlled growth of ferrite plates in plain-carbon steels is critically assessed. It is found that the use of empirically extrapolated diffusion coefficients, phase boundaries, and thermodynamic functions leads to errors in calculations of growth rate. The errors become most important for low transformation temperatures, leading to exaggerated growth rates. Ways of avoiding these difficulties are suggested, and a new analysis of experimental data indicates that the lengthening of Widmanstatten ferrite plates in Fe–C alloys occurs at a rate which is influenced by the diffusion of carbon in the austenite ahead of the interface, assuming that the plates adopt a tip radius consistent with the maximum growth velocity. However, there is a systematic discrepancy between theory and experiment: plate-growth theory seems to underestimate the lengthening rate by some 5 μm s^?1. This may have something to do with the lath shape of Widmanstatten ferrite, but an analysis using needle–growth theory does not resolve the problem for data obtained at low lengthening rates. In general, plate–growth theory gives a better explanation of experimental data. The growth of bainite sheaves occurs at a rate much faster than expected from carbon diffusion-controlled growth. If the maximum-velocity hypothesis is incorrect (as it is for dendritic solidification), the above-mentioned discrepancies would be larger.

MST/192     

Finite element analysis for surface diffusion-controlled shape instabilities of plate-like grains

Based on classical theory of surface diffusion and evaporation–condensation, a finite element program is developed to simulate the unstable shape evolution of plate-like grains. The program is used to analyze thermal grooving on a polycrystalline surface and compared with a non-linear solution and finite difference analysis. It shows that the finite element method used is robust, accurate and efficient. Then, the shape evolution kinetics of the plate-like grains are simulated as a function of the thermal grooving angle θ at the grain boundary–surface junctions and the initial aspect ratio of the plate β (plate width to thickness). When θ=0 (without internal boundary), the plate-like grain will evolve into cylinders directly. When an internal boundary exists, there is a critical thermal grooving angle θ_min for given β. If θ<θ_min, the plate cannot split, otherwise, the plate will split along the internal boundary of the plate-like grain. An approximate formulation of θ_min as a function of β is given based on a number of finite element analyses. The effect of initial termination shape of the plate on θ_min is also examined, and a weak effect was found. When β>10, its effect can be neglected.     

Path-independent integrals around two circular holes in an infinite plate under biaxial loading conditions

An analytic solution is presented for stresses induced in an infinite plate with two unequal circular holes by remote uniform loadings and arbitrary internal pressures in the holes. The solution is obtained by using the general expression for a biharmonic function in bipolar coordinates. The Airy stress function is decomposed in the sum of a fundamental stress function for an infinite plate remotely loaded, which gives non vanishing tractions on the circular boundaries, and an auxiliary stress function required to satisfy the boundary conditions on the pressures at the edges of the holes, which produces vanishing stresses at infinity. Correspondingly, the variations of the stress concentration factor are determined in terms of the holes geometry and loading conditions. The path independent J_k- (k = 1, 2), M- and L-integrals are analytically calculated on a closed contour encircling the two holes, under remote loading, in order to evaluate the energy release rates accompanying unit translation, self similar expansion and rotation of the holes, respectively. Results are then presented for varying loading orientation angle, biaxial loading ratio and holes geometry.     

Stress analysis of multilayered plates around circular holes

In this paper, a technique to study the 3-dimensional stress state around a circular hole in laminated plates is developed. First, the 3-dimensional elasticity problem for a thick plate with a circular hole is formulated in a systematic fashion by using the z-component of the Galerkin vector and that of Muki's harmonic vector function. This problem was originally solved by Alblas[1]. The reasons for reconsidering it are to introduce a technique which may be used in solving the elasticity problem for a multilayered plate and to verify and extend the results given by Alblas. Among the additional results of particular interest, one may mention the significant effect of the Poisson's ratio on the behavior and the magnitude of the stresses. Secondly, the elasticity problem for a laminated thick plate, which consists of two bonded dissimilar layers and which contains a circular hole, is considered. The problem is formulated for arbitrary axisymmetric tractions on the hole surface. Through the expansion of the boundary conditions into Fourier series, the problem is reduced to an infinite system of algebraic equations which is solved by the method of reduction. Of particular interest in the problem are the stresses along the interface as they relate to the question of delamination failure of the composite plate. These stresses are calculated and are observed to become unbounded at the hole boundary. An approximate treatment of the singular behavior of the stress state is presented, and the stress intensity factors are calculated. It is also observed that, the results compare rather well with those obtained from the finite element method.     

Plate bending analysis with hr-adaptive boundary elements

An adaptive scheme for the boundary element method (BEM), based on an error estimate using a sample point error analysis, is applied to a bending problem for a thin elastic plate. Both the boundary element refinement and relocation, i.e. the h- and r-version adaptive schemes, are implemented by employing the non-dimensional error indicator proposed in the present paper. Effectiveness of the proposed adaptive strategy for the boundary element plate bending analysis is demonstrated through some numerical applications of plate bending.     

Analysis of cracked plates using an iterative hybrid technique of boundary element method and distributed dislocation method

An iterative hybrid technique of boundary element method (BEM) and distributed dislocation method (DDM) is introduced for solving two dimensional crack problems. The technique decomposes the problem into (n + 1) subsidiary problems where n is the number of crack branches. The required solution will be the sum of these (n + 1) solutions. The first subsidiary problem is to find the stress distribution induced in the plate in the absence of the crack using BEM. All of the remaining subsidiary problems, are stress disturbance ones that will be solved using DDM. The results will be added and compared with the boundary conditions of the original problem. Iteration will be performed between the plate boundaries and crack faces until all of the boundary conditions are satisfied.     

Effect of displacement current in magneto–electro-elastic plates subjected to dynamic loading

Dynamic response of moderately thick magneto?Celectro-elastic plate using magnetic vector potential in finite element formulation is presented in this paper. Dynamic loading generate time varying electric and magnetic fields in magneto?Celectro-elastic continuum. Displacement current is associated with the generation of magnetic field due to time varying electric field. The non-conservative electric field is represented using electric scalar potential and magnetic vector potentials. Studies are carried out for CCCC, CCFC, CFFC and FCFC boundary conditions of the plate excited with time-harmonic mechanical excitation, the frequency range being chosen based on the critical frequency of the plate analyzed. The magnetic flux density in longitudinal x-direction is not affected by the electric displacement current for all the boundary conditions. The longitudinal y-direction and transverse direction components of magnetic flux density are showing variations for FCFC boundary condition when displacement current is accounted. The effect of displacement current is significant when two opposite edges of the plate are clamped.     

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.     

A research on the creep properties of titanium matrix composites rolled with different deformation degrees

(TiB + La₂O₃)/Ti composites were in situ synthesized and deformed with different deformation degrees. The influence of TiB whisker orientation and grain refinement on the creep properties of titanium matrix composites (TMCs) are discussed. The creep test reveals that the steady state creep rate of TMCs first decreases and then increases with the increase of deformation degree, which can be attributed to competing effects: TiB whisker rotating to the rolling direction, α plate grain boundary hindering and pinning dislocations can all decrease the creep rate, however, dislocation movement on the α plate grain boundary and dislocation emitting from the α plate grain boundary can both increase the creep rate.     

An efficient BEM formulation for analysis of bond-line cracks in thin walled aircraft structures

In this paper bond-line cracks in assembled plate structures are investigated. A multi-region boundary element formulation is used to model the assembly and to perform the analysis. Once the solution at the boundary has been obtained, energy methods are used to extract relevant fracture parameters from the boundary values. The energy release rate (G) method and a new J-integral formulation are implemented for interlaminar cracks in shear deformable plates. The energy release rate and the J-integral are then decomposed into stress intensity factors and their value along the bond-line is computed by means of a displacement discontinuity method.     

Two-parameter characterization of elastic-plastic crack front fields: Surface cracked plates under tensile loading

In this paper the J-Q two-parameter characterization of elastic-plastic crack front fields is examined for surface cracked plates under uniaxial and biaxial tensile loadings. Extensive three-dimensional elastic-plastic finite element analyses were performed for semi-elliptical surface cracks in a finite thickness plate, under remote uniaxial and biaxial tension loading conditions. Surface cracks with aspect ratios a/c = 0.2, 1.0 and relative depths a/t = 0.2, 0.6 were investigated. The loading levels cover from small-scale to large-scale yielding. In topological planes perpendicular to the crack fronts, the crack stress fields were obtained. In order to facilitate the determination of Q-factors, modified boundary layer analyses were also conducted. The J-Q two-parameter approach was then used in characterizing the elastic-plastic crack front stress fields along these 3D crack fronts. Complete distributions of the J-integral and Q-factors for a wide range of loading conditions were obtained. It is found that the J-Q characterization provides good estimate for the constraint loss for crack front stress fields. It is also shown that for medium load levels, reasonable agreements are achieved between the T-stress based Q-factors and the Q-factors obtained from finite element analysis. These results are suitable for elastic-plastic fracture mechanics analysis of surface cracked plates.     

Magneto stress analysis of a strip with a semi elliptical notch under uniform magnetic field

Two dimensional solutions of the magnetic field and magneto elastic stress are presented for a magnetic material of a thin strip with a semi-elliptical notch subjected to uniform magnetic field. The strip is a finite plate of a simply connected region. A linear constitutive equation is used for the stress analysis. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate. Therefore, the magneto elastic stress is analyzed using Maxwell stress. In the present problem, as a result, the plane stress state does not arise, and the σ_z in the direction of the plate thickness and the shear deflection (anti-plane shear stress) arise for soft ferromagnetic material. The stress σ_z in the plate is strong compressive stress for a soft ferromagnetic material. A rational mapping function is used for the stress analysis, and the each solution is obtained as a closed form. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition is completely satisfied without any linear assumptions on the boundary. The anti-plane shear stress causes Mode III stress intensity factor when the notch is a crack. Stress concentration values are investigated for a notch problem, of which expression is given. Figures of the anti-plane shear stress distribution, Mode III stress intensity factor, and stress concentration values are shown.     

A nth-order meshless generalization of Reddy’s third-order shear deformation theory for the free vibration on laminated composite plates

In this paper, a nth-order shear deformation theory is proposed to analyze the free vibration of laminated composite plates. The present nth-order shear deformation theory satisfies the zero transverse shear stress boundary conditions on the top and bottom surface of the plate. Reddy’s third-order theory can be considered as a special case of present nth-order theory (n = 3). Natural frequencies of the laminated composite plates with various boundary conditions, side-to-thickness ratios, material properties are computed by present nth-order theory and a meshless radial point collocation method based on the thin plate spline radial basis function. The results are compared with available published results which demonstrate the accuracy and efficiency of present nth-order theory.     

Deposition of solid particles in laminar boundary layer on a flat plate

Results are given of experimental and theoretical investigation of deposition of small solid particles on the surface of a flat plate under conditions of vertical laminar boundary layer. The present investigation is aimed at qualitatively and quantitatively estimating the effect made by the parameters of two-phase flow of the “gas—solid particles” type and by the adhesive properties of particles and surface on the deposition of particles on the plate surface. The flow velocity is 1.5 and 3 m/s. In so doing, the value of Reynolds number along the plate does not exceed 10⁵. Synthetic corundum powders with average sizes of 12, 23, and 32 μm are used as the dispersed phase of two-phase flow. The mass concentration of particles in the flow is 0.01 kg/m³. A flat plate of stainless steel is used as the object of investigation. The distributions of gas velocity and concentration of particles within the boundary layer are measured using laser optical diagnostics. The number of particles deposited along the plate surface is measured by the gravimetric method. The adhesive properties of the “particle-surface” pair are studied using the centrifugal method of detachment of particles from the surface. Logarithmic-normal dependences of the number of adhesion of particles on the force of detachment are obtained. The hydrodynamic parameters of two-phase flow in the vicinity of the plate surface are calculated using the model of two-phase laminar boundary layer. The mathematical expression is suggested for the calculation of the magnitude of deposition of solid particles along the surface of a flat plate, which includes the special features of hydrodynamics of flow, the adhesive properties of the particles and surface, and the probabilistic pattern of the process of entrapment of particles by the surface.     

Experimental study on the particle accumulation phenomenon in seeded flows around a near-bottom-wall horizontal cylinder

In experiments of flows around a cylinder in a water channel, an interesting phenomenon is that a particle accumulation line obviously forms on the bottom of the channel. The present paper focuses on this phenomenon, and the formation mechanism of the particle accumulation line is in detail investigated experimentally with particle image velocimetry (PIV). The circular cylinder was set in a fully developed turbulent boundary layer with 12 gap ratios S/D ranging from 0 to 1.5 under two Reynolds numbers (1371 and 902) based on the momentum loss thickness. The possible mechanism of this phenomenon has been demonstrated with the experimental results: the separation takes place due to the interaction between the wake of the cylinder and the boundary layer of the plane wall, the gap flow separates from the wall downstream of the cylinder and causes an attachment vortex of low velocity area at about 1 to 2 cylinder diameters from the cylinder, where the particle accumulation line forms steadily.