Plate Of Infinite Width Research Articles

Semi-Analytical Finite Element Method for the Analysis of Guided Wave Dispersion in the Pre-stressed Composite Plates

Ultrasonic guided wave techniques offer an accurate and efficient procedure for damage monitoring in the structures. To develop reliable damage monitoring systems, it is essential to have a thorough understanding of the quantitative nature of ultrasonic guided waves that can be transmitted in composite laminates. In the present paper, the Semi-Analytical Finite Element (SAFE) method is employed due to its efficiency in the treatment of wave propagation problems involving complex materials and geometry. SAFE method is considered for the analysis of dispersion behaviour of guided waves in composite laminates by accounting the effect of in-plane load. The present study considers an infinite width plate such that the cross-section of the waveguide is modelled using 3 noded isoparametric 1-D elements representing the thickness of the plate. Equation of motion is formulated by using Hamilton’s equation. Finally, various parametric studies are carried out. The study includes analysing the effect of wave velocity in the plate subjected to applied in-plane load, and understanding the effect of dispersion characteristics on the direction of propagation (slowness curve). The study shows that at the lower frequency thickness product, the group velocity of the wave increases with the increase in the applied tensile load and vice-versa.

Optimizing the Thickness/Stiffness Distribution of Infinitely Wide Porous FGM Plates Subjected To Supersonic Flutter Constraints

The analytical solutions of supersonic flutter of porous functionally graded plates are investigated. The material properties of the plates (stiffness and density) vary according to a power law across the thickness and axial coordinates depending on the volume fraction of constituent materials. Their equilibrium and stability equations are derived based on the classical plate theory. The solution of the problem is derived using the Rayleigh–Ritz method and the Mathematica package. An optimization problem is formulated and solved for plates of infinite width.

A three-step model for optimizing coil spacings inside cuboid-shaped magnetic shields

A three-step model for calculating the magnetic field generated by coils inside cuboid-shaped shields like magnetically shielded rooms (MSRs) is presented. The shield is modeled as two parallel plates of infinite width and one tube of infinite height. We propose an improved mirror method that considers the effect of the parallel plates of finite thickness. A reaction factor is introduced to describe the influence of the vertical tube, which is obtained from finite element method (FEM) simulations. By applying the improved mirror method and then multiplying the result with the reaction factor, the magnetic flux density within the shielded volume can be determined in a fast computation. The three-step model is verified with both FEM and measurements of the field of a Helmholtz coil inside an MSR with a superconducting quantum interference device. The model allows a fast optimization of shield-coupled coil spacings compared to repetitive, time-consuming FEM calculations. As an example, we optimize the distance between two parallel square coils attached to the MSR walls. Measurements of a coil prototype of 2.75 m side length show a magnetic field change of 18 pT over the central 5 cm at the field strength of 2.7 µT. This obtained relative field change of 6 ppm is a factor of 5.4 smaller than our previously used Helmholtz coil.

裂纹面局部均布荷载下Ⅰ型裂纹有限宽板应力强度因子

With the crack line analysis method for stress intensity factors, the stress intensity factor of a finite-width plate with a mode-Ⅰ center crack subjected to uniform stress on the crack surface near the crack tip was studied analytically. Through analytical solution of the stress field and correction of the stress components on the crack line of an infinite-width plate with a mode-Ⅰ crack under uniform stress on the crack surface near the crack tip, the corrected stress components on the crack line of the corresponding finite-width plate were further deduced. Reasonable requirements on the corrected stress field at the crack line were proposed, the stress intensity factor of the finite-width plate was derived, and the correction coefficient of the stress intensity factor of the finite-width plate relative to the corresponding infinite-width plate was obtained. When the width of the finite-width plate approaches infinity, the stress intensity factor of the finite-width plate will be consistent with that of the corresponding infinite-width plate.

Longitudinal wave speed in auxetic plates with elastic constraint in width direction

This paper evaluates the longitudinal wave speed through a plate in which its two opposing sides are elastically restrained in the width direction, taking into consideration the material auxeticity and strain as well as changes to the density and cross-sectional area. Apart from the known role of Young’s modulus and density, the present results reveal that the wave speed can be enhanced by increasing the width elastic restraint. In the case of high elastic restraint, the speed of both tensile and compressive waves can be minimized by selecting plate materials with Poisson’s ratio of low magnitude. In the case of low elastic restraint, the speed of tensile and compressive waves can be greatly reduced by selecting plate materials with large positive and large negative Poisson’s ratio, respectively. For the special case of negligible strain, the longitudinal wave speed reduces to the elementary wave speed in prismatic rods and in plates of infinite width when the width elastic restraint stiffness approaches zero and infinity, respectively. The obtained results not only avail more parameters for adjusting the longitudinal waves in plates, but also identify the differing methods of effectively controlling the wave speed between tensile and compressive waves when the strain magnitude is non-negligible.

The analysis of stress intensity factors in two interacting collinear asymmetric cracks in a finite plate

This paper uses a Fourier transform technique to solve the problem of two interacting collinear unequal cracks in a finite width plate. This approach reduces the problem to the solution of two coupled integral equations each with a singular kernel which is the solved using Cauchy–Chebyshev polynomials. The solution is first validated by comparing the solutions for the case of two equal cracks in a finite width plate and for the case of two unequal cracks in an infinite width plate with published solutions. The problem of (interacting) cracks that grow from collocated corrosion pits that lie close to the edge of a structural member is then studied. Here it is shown that the closer that the two cracks are to a boundary accelerates crack growth towards the boundary and decelerates linking of the cracks.

T-stress solutions for cracks in rectangular plates with multiple holes

The elastic T-stress is increasingly being recognized as an important second parameter to the stress intensity factor for fracture and fatigue assessments. In this paper, the mutual or M-contour integral approach is employed in conjunction with the Boundary Element Method (BEM) to determine the numerical T-stress solutions for cracks in plates with multiple holes. The problems investigated include plates of infinite width with multiple holes at which single or double, symmetric cracks have grown from. Comparisons of these results are also made with the corresponding solutions of finite plates with a single hole. For completeness, stress intensity factor solutions for the cracked geometries analyzed are presented as well. These results will be useful for failure assessments using the two-parameter linear elastic fracture mechanics approach.

Geometry effects in squeeze flow of Bingham plastics

Squeeze flow of Bingham plastics shows small unyielded regions confined only near the center of the disks. Previous simulation results on axisymmetric coaxial disks are extended to aspect ratios ranging from 0.01 to 1, and to planar parallel plates of infinite width. New results are given for a wide range of Bingham numbers using the continuous modification of Papanastasiou for the Bingham model. Axisymmetric geometries give smaller unyielded regions than planar ones, while big aspect ratios give larger unyielded regions than small ones for the same Bingham number. Calculations of the squeeze force along disks or plates for different aspect ratios are also given and fitted to easy-to-use equations.

Elastic guided waves in a layered plate with rectangular cross section.

Guided waves in a layered elastic plate of rectangular cross section (finite width and thickness) has been studied in this paper. A semianalytical finite element method in which the deformation of the cross section is modeled by two-dimensional finite elements and analytical representation of propagating waves along the length of the plate has been used. The method is applicable to arbitrary number of layers and general anisotropic material properties of each layer, and is similar to the stiffness method used earlier to study guided waves in a laminated composite plate of infinite width. Numerical results showing the effect of varying the width of the plate on the dispersion of guided waves are presented and are compared with those for an infinite plate. In addition, effect of thin anisotropic coating or interface layers on the guided waves is investigated.

Elastic Guided Waves in a Layered Plate With a Rectangular Cross Section

Ultrasonic guided waves in a layered elastic plate of rectangular cross section (finite width and thickness) are studied in this paper. A semi-analytical finite element method in which the deformation of the cross section is modeled by two-dimensional finite elements and analytical representation of propagating waves along the long dimension of the plate is used. The method is applicable to an arbitrary number of layers of anisotropic properties and is similar to that used earlier to study guided waves in layered anisotropic plates of infinite width. Numerical results are presented for acoustic phonon modes of quasi-one-dimensional (QID) wires. For homogeneous wires, these agree well with recently reported results for dispersion of these modes.

Couple stress based strain gradient theory for elasticity

The deformation behavior of materials in the micron scale has been experimentally shown to be size dependent. In the absence of stretch and dilatation gradients, the size dependence can be explained using classical couple stress theory in which the full curvature tensor is used as deformation measures in addition to the conventional strain measures. In the couple stress theory formulation, only conventional equilibrium relations of forces and moments of forces are used. The couple's association with position is arbitrary. In this paper, an additional equilibrium relation is developed to govern the behavior of the couples. The relation constrained the couple stress tensor to be symmetric, and the symmetric curvature tensor became the only properly conjugated high order strain measures in the theory to have a real contribution to the total strain energy of the system. On the basis of this modification, a linear elastic model for isotropic materials is developed. The torsion of a cylindrical bar and the pure bending of a flat plate of infinite width are analyzed to illustrate the effect of the modification.

Improved Inherent Flaw Model for Tensile Fracture of Cracked Composite Laminates

Modifications are made in the inherent flaw model to predict the tensile fracture strength of cracked composite laminates. A simple relation is used for the inherent flaw length to improve the accuracy while evaluating the fracture strength of laminates. A quartic (fourth-order polynomial) equation is developed for the fracture strength of an infinite width plate containing a crack. This model is successfully applied to centre crack tensile specimens of graphite/epoxy and nylon/glass polyamide.

A New Approach in Utilising Inherent Flaw Model for Tensile Fracture of Composite Laminates with a Circular Hole

This paper improves the inherent flaw model proposed by Waddoups, Eisenman and Kaminski (known as the WEK model) to predict the strength of composite laminates with a circular hole. A simple relation is used for the inherent flaw length to improve the accuracy while evaluating the notched strength of laminates. A cubic equation is developed for the fracture strength of an infinite width plate having a circular hole. This model is applied to a host of experimental data from the open literature. Most of the experimental results are found to be in very good agreement with the analytically produced ones. The proposed model which is more simple could be more applicable for fracture strength evaluation of notched composite laminates.

Analytical expressions for crack opening displacements of edge cracked specimens under a segment of uniform crack face pressure

In this paper, Muskhelishvili's complex variable method was used to derive the primary equations of crack opening displacements (CODs) for edge cracked specimens when only the boundary conditions of the crack surface are satisfied. By means of fitting the COD weight function solutions based on the obtained primary COD equations, approximate equations of COD were obtained for edge cracked specimens under a segment uniform pressure acting on the crack surfaces. The equations are more accurate outside the near crack tip region. Because the crack closure level is sensitive to the variation of CODs in the near crack tip region, approximate COD-expressions with high accuracy in the near crack tip region were also developed for the edge cracked specimens under the corresponding loading condition based on the COD-expression for a central crack in an infinite width plate subjected to segment uniform pressure acting on crack surface. The equations with better accuracy outside and in the near crack tip region can be combined to rapidly calculate the COD on the whole crack surface, thus greatly facilitating fatigue crack growth analysis for edge cracked specimens.

Stress intensity factors and weight functions for a corner crack in a finite thickness plate

Weight functions for the two surface points of a quarter-elliptical corner crack in a finite thickness plate of infinite width are derived from a general weight function and two reference stress intensity factors. The weight functions have been validated for several linear and non-linear crack face stress fields against finite element data. The weight functions appear to be particularly suitable for fatigue and fracture analysis of quarter-elliptical cracks in complex stress fields.

Spectral investigation of spatial scales associated with substructures on a fluid-loaded plate

A wave-number-based formulation of the surface variational principle is used to investigate the effects of substructures on fluid-loaded systems. The paper examines issues of scale associated with the spatial distribution and representation of substructure attachments. The primary emphasis is suspended discrete spring-mass systems. A further extension considers an elastic beam attached at multiple locations. The main structure is a simply supported plate of infinite width contained in an infinite baffle and exposed to water on one side. Scales are introduced to the formulation by using a spectral Fourier series to represent point attachments. Previous work had considered the case where line masses are fastened to the plate. The concept that evolved entailed comparing the displacement and surface pressure obtained from analyses whose only difference is the number of terms in the spectral series used to represent the attached masses. The difference between such analyses is an indicator of the significance of the scales associated with the additional terms in the series. The work reported here increases the complexity of the system by allowing for the masses to be flexibly attached to the plate. This generalized problem is comparable to the types of systems considered to form fuzzy structures. However, the viewpoint here is deterministic. Results to be presented examine the influence of the scales associated with substructure attachments when the parameters of the suspended substructures have a range of values.

The rupture of a thin ligament associated with a part-through crack in a plate subject to tensile deformation

This paper focuses on the criterion for the plastic rupture of a thin ligament associated with a part-through crack in a tensile plate subjected to an applied tensile stress. With a very wide (infinite width) plate, the critical crack length for ligament rupture is shown to decrease with increasing applied stress. At low applied tensile stress levels, i.e. less than 50% of the yield stress, ligament rupture requires a very thin ligament and a very long crack. With a finite width plate, the required crack length for ligament rupture is less than the infinite width plate value, provided that the crack and plate dimensions are such that general yield occurs. The implications of these conclusions with regard to the development of a leak-before-break condition are discussed.

Effect of viscosity on the lift-drag ratio of a thin blunt wing at hypersonic rates of flow round it

The effect of viscosity on the carrying properties of hypersonic aircraft appears at great flight altitudes, where an important factor is the interaction of the laminar boundary layer with inviscid flow. In the present study the method of bands is used to make an approximate calculation of this effect for a regime of weak viscous interaction [1]. The results of [2] are used for conditions of inviscid flow round a body. The local coefficient of friction and coefficients of the additional pressure induced by the boundary layer are determined from the data for a plate of infinite width [3]. Simple relationships are obtained which make it possible to estimate the effect of viscosity on the magnitude of the maximum lift-drag ratio and the value of the angle of attack corresponding to it. The results are given of an experimental study of hypersonic flow round a plane triangular wing in a broad range of Reynolds numbers, and these confirm the relationships obtained and indicate the region in which they are applicable.

Heat transfer between two parallel plates with laminar pulsating flow

The problem of heat transfer from two parallel plates of infinite width is formulated for the case where the flow between these plates consists of a periodic motion imposed on a fully developed laminar steady flow. The results indicate an increase in the heat transfer rate with pulsation. This increase is proportional to the amplitude of pulsation and inversely proportional to the Prandtl number.

Analysis of tapered (in steps) laminated plates under uniform inplane load

A quasi three-dimensional isoparametric finite element has been developed for the linear elastic static stress analysis of variable thickness (stepped) laminated plates of infinite width subjected to a uniform inplane load. Numerical results are given for a single step plate with various arrangements for the ply fibre directions. Very high peak stresses are predicted in the corner region of the step, but these are reduced when a resin fillet is included in the theoretical model for the stepped region.