Infinite Orthotropic Plate Research Articles

Complex Potentials and Integral Equations for Thermoelasticity Problem of Orthotropic Cracked Plate

In the present article, the plane thermoelasticity problem of an infinite orthotropic cracked plate, is studied. The interaction between thermal and mechanical field, in a plate containing a crack and a heat source is investigated. The plate is submitted to normal stresses at infinity, where a heat flow is also activated. Applying the complex potentials method and the technique of Cauchy integrals, a system of singular integral equations is obtained. These equations, allow the determination of the temperature’s distribution, but also of the stress state at any point of the medium.

Analytical fracture parameters of two unequal collinear interface cracks in an orthotropic bimaterial

This paper presents an analytical solution for the fracture parameters of two unequal collinear interface cracks (UCIC) in an infinite orthotropic or isotropic bimaterial subject to in-plane loading. The analytical expressions for the stress intensity factors (SIF), Jk-integrals and crack opening displacements of two UCIC in an infinite orthotropic bimaterial plate are derived using the complex variable method and evaluated in conjunction with the Jacobian elliptic functions and elliptic integrals. The presented theory is verified using the boundary element method. The results show that the fracture parameters are greatly influenced by mismatch of the engineering constants and proximity of the cracks. The inner tip of the longer crack always experiences the maximum SIFs and Jk values. These fracture parameters will increase exponentially when the distance between the cracks inner tips decreases. This study provides a baseline for the verification of future numerical and experimental studies of two UCIC which is not currently available in the SIFs handbooks.

A Non-Singular Analytical Technique for Reinforced Non-Circular Holes in Orthotropic Laminate

The intricacy in Lekhnitskii’s available single power series solution for stress distribution around hole edge for both circular and noncircular holes represented by a hole shape parameter ε is decoupled by introducing a new technique. Unknown coefficients in the power series in ε are solved by an iterative technique. Full field stress distribution is obtained by following an available method on Fourier solution. The present analytical solution for reinforced square hole in an orthotropic infinite plate is derived by completely eliminating stress singularity that depends on the concept of stress ratio. The region of validity of the present analytical solution on reinforcement area is arrived at based on a comparison with the finite element analysis. The present study will also be useful for deriving analytical solution for orthotropic shell with reinforced noncircular holes.

Stress concentration factor near loaded hole contacting with the bolt in the monolayer of orthotropic composite material

Purpose. To suggest a method to predict the stress concentration factor near loaded hole contacting with the bolt in the monolayer of orthotropic composite material to be independent of difficult calculations. And to verify the factor with use results of numerical calculations using the finite element method in contact formulation. Methodology. The analysis of known outcomes in the literature is carried out. Several assumptions have been formulated about the changes of stress concentration factor depending on the ratio of the hole diameter to the plate width, as well as on orthotropy in the plate material. Assumptions are confirmed by calculated results, graphs. The finite element method was used. Results. A formula is proposed that predicts the stress concentration factor without carrying out difficult calculations of contact problems using known approximations for an isotropic material and the Lekhnitsky-Soldatov solution for a stretched orthotropic infinite plate with an empty hole. Using the results of numerical calculations of contact boundary problems with the finite element method the proposed formula was verified, its good accuracy was shown in the practically applicable range of the ratio of the hole diameter to the plate width. Originality. A four-step method is developed - simple formulas that allow rapid prediction of the stress concentration factor near loaded hole contacting with the bolt in a monolayer of orthotropic composite material using only of the elastic characteristics of the plate material, proven approximations and simple solutions.

STRESSES IN ORTHOTROPIC CIRCULAR CYLINDRICAL SHELLS WITH LARGE-SIZE REINFORCED HOLES - A FOURIER SERIES APPROACH

An analytical solution for tangential stress distribution around large-size reinforced circular holes up to a shell curvature parameter of β = 2 in an orthotropic circular cylindrical shell under uniaxial load is derived. For this purpose, a new parameter a linearly proportional to β for an unreinforced large-size hole is determined using a finite element analysis and introduced in the existing Fourier series solution for small holes. For various hole sizes, optimum uniform thickness over the shell surface around the hole is assessed based on the unreinforced shell-to-plate stress ratio. For each hole size corresponding to each stress ratio, the reinforced shell solution converges to the case of unreinforced circular hole in an orthotropic infinite plate.

Effects of Cut-Out Orientation and Fiber Angle on Stress Concentration in an Infinite Orthotropic Plate

The presence of cutout in structures may result in strength degradation due to alteration of load lines around the cutout. The effects of these cutouts on stress field should be studied carefully while designing of any material. This article presents the optimum design of an infinite orthotropic plate weakened by complex cutout using a genetic algorithm where cutout orientations and fiber angles are considered as design variables. The stress concentration factor is used as fitness function derived using the Muskhelishvili’s complex variable approach for various in-plane loadings. The effects of cutout orientation and fiber angles on stress concentration are discussed.

Thermal stress analysis of orthotropic plate containing a rectangular hole using complex variable method

One of the most important issues in the design of engineering structures is the prevention of structural failure due to the stress caused by geometric discontinuities. Hole geometry is one of the effective parameters in the distribution of stress and displacement around it. In this study, using the two-dimensional thermoelastic theory and based on the Likhnitskiiʼ complex variable technique, the stress analysis of orthotropic infinite plate with a circular hole under a uniform heat flux is developed to the plate containing a rectangular hole. To achieve this goal, a conformal mapping function is used to map an infinite plate containing a rectangular hole into the outside of a unit circle. Stress and displacement distributions around the rectangular hole in an orthotropic infinite plate are investigated in thermal steady state condition. The plate is under uniform heat flux at infinity and Neumann boundary conditions and thermal-insulated condition at the edge of the hole are considered. The rotation angle of the hole, fiber angle, the flux angle, bluntness and the aspect ratio of hole size are important parameters investigated in the present study. The obtained results show that these parameters have a significant effect on the stress and displacement distributions around the rectangular hole.

On the stress concentration around a polygonal cut-out of complex geometry in an infinite orthotropic plate

The best values of fiber volume fractions, fiber arrangements, and cut-out orientations in an orthotropic infinite plate weakened by a polygonal discontinuity of regular and complex geometry are investigated in the present work. Considering the stress concentration factor as a fitness minimization function, the genetic algorithm is employed and, elastic constants and stresses are computed utilizing the Mori-Tanaka theory and the Muskhelishvili’s complex variable method, respectively. The upshot of present work shows a substantial impact of fiber volume fraction, fiber arrangement and, corner radius and orientation of cut-out, on values of stress concentration factor for various in-plane loading conditions. Furthermore, the database of the complex constants used in the Schwarz–Christoffel mapping to develop distinct complex shapes is also reported.

Using dragonfly algorithm for optimization of orthotropic infinite plates with a quasi-triangular cut-out

Abstract This paper aims at optimizing the parameters involved in the stress analysis of perforated orthotropic plates, to achieve the lowest value of stress around the quasi-triangular cut-out located in an infinite orthotropic plate by using the Dragonfly Algorithm (DA) method. This method is used to calculate the stress distribution based on Lekhnitskii's analytical solution. The study design variables include fiber angle, load angle, bluntness, orientation angle of cut-out and the material properties. In addition, the performance of the DA algorithm is compared with the Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). The comparison of these methods indicates the appropriateness of the DA algorithm in optimizing the perforated plates. The finite element method has been used to verify the accuracy of the analytical results. The results indicate that by selecting the aforementioned parameters properly, we can increase the structural load-bearing capacity.

Bending vibration prediction of orthotropic plate with wave-based method

A novel numerical predictive approach for steady-state response of thin orthotropic plate is presented based on wave-based method (WBM) that is applied in bending vibration prediction of thin and thick plate in mid-frequency range. The wavenumber parameters for orthotropic material and the particular solution of an infinite orthotropic plate with Fourier transform are derived. The proposed method is validated by numerical examples with simply supported boundary and clamped boundary. The compared result shows that the computational accuracy and efficiency of WBM is significantly higher than element based method, which is the ability of WBM for mid-frequency problems. The predictive ability of WBM is extended to process the dynamic response of orthotropic plate.

Optimization of infinite orthotropic plates with hypotrochoid cutout under tensile loading using genetic algorithm

The effect of geometric discontinuities can be reduced by appropriate choice of parameters affecting the stress distribution around cutout. This is possible if the effective parameters are accurately calculated. In this study, using genetic algorithm, the optimum parameters are introduced in order to achieve the minimum value of stress around cutout. Here, design variables are fiber angle, load angle, aspect ratio of cutout, shape of cutout, rotation angle of cutout, and bluntness parameters. Using the Lekhnitskii’s method, stress distribution around various cutouts is determined. The effect of the aforementioned parameters on the stress values around various shapes of cutouts such as quasi-triangular, quasi-square, and hypotrochoid cutouts is examined. Also, the optimal parameters for each cutout are introduced. The results showed that these parameters have significant effects on stress distribution around the cutouts and the structural load-bearing capacity will increase without changing the type of material if the parameters are correctly chosen.

Local buckling of profiled skin sheets resting on tensionless elastic foundations under in-plane shear loading

Abstract In this paper, the local buckling behaviour of profiled skin sheets resting on tensionless elastic Winkler foundations under in-plane shear loadings is studied. The profiled sheets are modelled as thin orthotropic plates with the buckling behaviour being expressed through a group of nonlinear partial differential equations. For very long plates with both ends clamped, the buckling mode is composed of a series of periodically repeating buckling waves and hence an infinite plate model with only one buckle wave is effective to predict the buckling behaviour. The infinite orthotropic plate is further simplified to a one-dimensional mechanical model by assuming a lateral buckling mode function. After solving the governing equations of the one-dimensional model in both contact and non-contact regions, shear buckling coefficients of the system and the related buckling modes are obtained. Fitted formulae for the contact shear buckling coefficients in terms of relative foundation stiffness and skin profile parameters are developed. The analytical solutions are verified through examining extreme cases from previous studies and a series of finite element (FE) models in ABAQUS. Finally, a practical example is given to show the efficiency of the developed method.

Dynamic Stress Intensity Factors of Three Collinear Cracks in an Orthotropic Plate Subjected to Time-Harmonic Disturbance

AbstractAbstract-The dynamic stresses around three collinear cracks in an infinite orthotropic plate are solved. In this configuration, two equal cracks are situated symmetrically on either side of a central crack, and time-harmonic elastic waves impinge perpendicularly to the cracks. The problem is solved by superimposing two types of solutions. One solution is that for a crack in an infinite orthotropic plate, and the other is for two collinear cracks. The unknown coefficients in the superimposed solution are determined by applying the boundary conditions at the surfaces of the three cracks using the Schmidt method. The dynamic stress intensity factors are calculated numerically for an orthotropic plate that corresponds to the elastic properties of a boron-epoxy composite.

Antiplane Problem of Periodically Stacked Parallel Cracks in an Infinite Orthotropic Plate

The antiplane problem of the periodic parallel cracks in an infinite linear elastic orthotropic composite plate is studied in this paper. The antiplane problem is turned into the boundary value problem of partial differential equation. By constructing proper Westergaard stress function and using the periodicity of the hyperbolic function, the antiplane problem of the periodic parallel cracks degenerates into an algebra problem. Using the complex variable function method and the undetermined coefficients method, as well as with the help of boundary conditions, the boundary value problem of partial differential equation can be solved, and the analytic expressions for stress intensity factor, stress, and displacement near the periodical parallel cracks tip are obtained. When the cracks spacing tends to infinity, the antiplane problem of the periodic parallel cracks degenerates into the case of the antiplane problem of a single central crack.

Analysis of Mode I Periodic Parallel Cracks-Tip Stress Field in an Infinite Orthotropic Plate

The mechanical behavior near crack tip for periodic parallel cracks in an orthotropic composite plate subjected to the uniformly distributed load within the cracks surface is studied. The mechanical problem is turned into the boundary value problem of partial differential equation. By using the periodicity of the hyperbolic function in the complex domain and constructing proper Westergaard stress function, the periodicity of parallel cracks can be removed. Using the complex variable function method and the undetermined coefficients method, the boundary value problem of partial differential equation can be solved with the help of boundary conditions. The analytic expressions for stress intensity factor, stress, and displacement near the crack tip of periodical parallel cracks are obtained. When the vertical distance of cracks tends to infinity, the stress intensity factor degenerates into a single central crack situation. The stress intensity factor around the crack tip of periodic parallel cracks in an orthotropic composite plate depends on the shape factor. The interaction happens between the cracks. Finally, a numerical analysis of the stress and displacement changed with the polar angle is done.

Numerical determination of statistical energy analysis parameters of directly coupled composite plates using a modal-based approach

Statistical energy analysis (SEA) parameters such as average modal spacing, coupling loss factor and input power are numerically determined for point connected, directly coupled symmetrically laminated composite plates using a modal-based approach. The approach is an enhancement of classical wave transmission formula. Unlike most of the existing numerical or experimental techniques, the approach uses uncoupled plate modal information and treats substructure by means of averaged modal impedances. The procedure introduced here is verified using analytical definitions of infinite orthotropic plates which physically resemble to laminated plates for (under) specific conditions, and is tested by performing experimental power injection method (PIM) for an actual, right-angled composite structure. In the development process, force and moment transmissions are individually considered in order to be consistent with analytical formulations. Modal information of composite plates is statistically evaluated by the discrete singular convolution method with random boundary conditions. Proposed methodology not only provides an efficient use of SEA method in high frequency vibration analysis of composite structures, but also enhances SEA accuracy in mid frequency region in which conventional SEA fails. Furthermore, the effect of orientation angles of laminations on SEA parameters are also discussed in mid and high frequency regions.

Stress Concentration for In-Plane Stressed Orthotropic Plate with Elliptic Opening

The paper discusses the analytical calculation procedure of stress at points along the elliptic opening boundary in an in-plane stressed infinite orthotropic plate. The presented procedure is a first step in the strength analysis of such plates. The stress dependence on material elasticity characteristics and a multilayer plate stacking sequence as well as on a position of the ellipse semi-axes for prescribed stress conditions has been presented on numerical examples while the calculation results have been illustrated by diagrams and numeric data. The specific attention is paid on the case of equal or close values of complex parameters of the orthotropic plate. The derived conclusions, pointing out a danger of stress concentration and orthotropy effects, can be very useful in engineering practice.

Analysis of Mode III Collinear Periodic Cracks-Tip Stress Field of an Infinite Orthotropic Plate

The cracks-tip field on collinear periodic cracks of infinite orthotropic fiber reinforcement composite plate subjected to anti-plane shear force is studied in this paper. With the introduction of the Westergaard stress function and application of complex function theory and undetermined coefficients method, mechanical problem is changed into partial differential boundary value problem. The undetermined coefficients and the stress function are obtained with the help of boundary conditions. Due to the distribution of periodic cracks, stress intensity factor depends on the shape factor. The results show that interaction happens between the collinear periodic cracks. When the ratio of crack length and crack spacing is less than 1/3, the interaction between the cracks is very small. When the ratio gradually increases to 1, strong interaction between the cracks will be found. Cracks-tip field has scale effect. When the ratio of crack length and the given reference crack length decreases, the displacement field is significantly smaller.

Analysis of Mode III Collinear Periodic Cracks-Tip Stress Field of an Infinite Orthotropic Plate

The cracks-tip field on collinear periodic cracks of infinite orthotropic fiber reinforcement composite plate subjected to anti-plane shear force is studied in this paper. With the introduction of the Westergaard stress function and application of complex function theory and undetermined coefficients method, mechanical problem is changed into partial differential boundary value problem. The undetermined coefficients and the stress function are obtained with the help of boundary conditions. Due to the distribution of periodic cracks, stress intensity factor depends on the shape factor. The results show that interaction happens between the collinear periodic cracks. When the ratio of crack length and crack spacing is less than 1/3, the interaction between the cracks is very small. When the ratio gradually increases to 1, strong interaction between the cracks will be found. Cracks-tip field has scale effect. When the ratio of crack length and the given reference crack length decreases, the displacement field is significantly smaller.

An Analysis of Collinear Periodic Crack-Tip Stress Field of an Infinite Plate Subjected to Concentrated Force

The Stress field of collinear periodic cracks in an infinite orthotropic composite plate subjected to the concentrated force is studied. The mechanical problem is turned into partial differential boundary value problem. The solution of partial differential equations can be obtained by the complex function methods, which contains unknown coefficients. Using undetermined coefficients method and with the help of boundary conditions, the partial differential boundary value problem is solved. The expressions for stress and displacements near crack tip are derived. Since the distribution of periodic cracks, the stress intensity factor is related to crack shape. Finally, a numerical example shows that the stress changes with the polar angle.