Refined Variational Solutions of the Interfacial Thermal Stresses in a Laminated Beam

1992 ◽  
Vol 114 (2) ◽  
pp. 193-198 ◽  
Author(s):  
W.-L. Yin
Keyword(s):  
Thermal Stresses ◽  
Virtual Work ◽  
Free Edge ◽  
Interlaminar Stresses ◽  
Laminated Beam ◽  
Variational Solutions ◽  
Stress Functions ◽  
Polynomial Expansions ◽  
Resultant Forces ◽  
Derivatives Of

Efficient and accurate solutions of the interlaminar stresses in a layered beam under a temperature loading are obtained by a variational method using stress functions and the principle of complementary virtual work. Polynomial expansions of the fifth or lower degrees are used to approximate the variation of the stress functions in the thickness direction of each layer. Comparison of the solutions of the various orders with the existing numerical and analytical solutions indicates that the variational solutions converge rapidly as the degree of the polynomial expansion increases and that even the lowest-order variational solutions yield satisfactory results for the interlaminar stresses. Over short segments of the interface adjacent to the free edge, the resultant forces of the interlaminar normal and shearing stresses are given by the first-order derivatives of the stress functions. These global measures of the severity of interlaminar peeling and shearing action are predicted accurately by the lowest-order variational solution.

Thermal Stresses and Free-Edge Effects in Laminated Beams: A Variational Approach Using Stress Functions

1991 ◽  
Vol 113 (1) ◽  
pp. 68-75 ◽  
Author(s):  
Wan-Lee Yin
Keyword(s):  
Edge Effects ◽  
Degrees Of Freedom ◽  
Thermal Stresses ◽  
Virtual Work ◽  
Free Edge ◽  
Interlaminar Stresses ◽  
Equilibrium Equations ◽  
Laminated Beam ◽  
Stress Functions ◽  
Cubic Polynomial

A variational method involving stress functions is used to determine the interlaminar stresses and the free-edge effects in a laminated beam under a temperature loading. The stress function in each layer is approximated by a cubic polynomial function of the thickness coordinate. The equilibrium equations, the traction boundary conditions, and the continuity conditions of the interlaminar stresses are exactly satisfied in this analysis, while the compatibility equations and interfacial continuity of the tangential strains are enforced in an averaged sense by applying the principle of complementary virtual work. The method is highly efficient and accurate. A thermal stress analysis for a three-layer beam using only eight eigenfunctions yield results that are comparable in accuracy to finite-element solutions involving thousands of degrees of freedom.

Free-Edge Effects in Anisotropic Laminates Under Extension, Bending, and Twisting, Part II: Eigenfunction Analysis and the Results for Symmetric Laminates

1994 ◽  
Vol 61 (2) ◽  
pp. 416-421 ◽  
Author(s):  
W.-L. Yin
Keyword(s):  
Eigenvalue Problem ◽  
Edge Effects ◽  
Free Edge ◽  
Interlaminar Stresses ◽  
Particular Solutions ◽  
Variational Solutions ◽  
Stress Functions ◽  
Polynomial Expansions ◽  
Eigenfunction Analysis

The eigenvalue problem associated with the determination of the interlaminar stresses in a laminated strip is examined and physical interpretations are given to the (constant) particular solutions and the complementary solutions of the problem. The case of symmetric laminates is considered in detail, and variational solutions are computed for four-layer, symmetric, cross-ply, and angle-ply laminates subjected to the three fundamental types of strain loads. Solutions based on two sets of stress functions with polynomial expansions of different degrees are compared with each other and with the existing solutions to assess the accuracy. The interfacial values of the stress functions and their derivatives are identified as the resultant peeling and shearing forces over end intervals of the interface.

The Effects of Inclined Free Edges on the Thermal Stresses in a Layered Beam

1993 ◽  
Vol 115 (2) ◽  
pp. 208-213 ◽  
Author(s):  
Wan-Lee Yin
Keyword(s):  
Thermal Stresses ◽  
Stress Singularity ◽  
Approximate Solutions ◽  
Free Edge ◽  
Local Concentration ◽  
Interlaminar Stresses ◽  
Beneficial Effects ◽  
Stress Functions ◽  
Layered Beam ◽  
Free Edges

A stress-function-based variational method is used to determine the thermal stresses in a layered beam with inclined free edges at the two ends. The stress functions are expressed in terms of oblique cartesian coordinates, and polynomial expansions of the stress functions with respect to the thickness coordinate are used to obtain approximate solutions. Severe interlaminar stresses act across end segments of the layer interfaces. Local concentration of such stresses may be significantly affected by the inclination angle of the end planes. Variational solutions for a two-layer beam show generally beneficial effects of free-edge inclination in dispersing the concentration of interlaminar stresses. The significance of these effects is generally not indicated by the power of the stress singularity as computed from an elasticity analysis of a bimaterial wedge.

Free-Edge Effects in Anisotropic Laminates Under Extension, Bending and Twisting, Part I: A Stress-Function-Based Variational Approach

1994 ◽  
Vol 61 (2) ◽  
pp. 410-415 ◽  
Author(s):  
W.-L. Yin
Keyword(s):  
Free Boundary ◽  
Virtual Work ◽  
Solution Process ◽  
Free Edge ◽  
Interlaminar Stresses ◽  
Equilibrium Equations ◽  
Free Boundary Conditions ◽  
Single Solution ◽  
Loading Case ◽  
Stress Functions

A variational method involving Lekhnitskii’s stress functions is used to determine the interlaminar stresses in a multilayered strip of laminate subjected to arbitrary combinations of axial extension, bending, and twisting loads. The stress functions in each layer are approximated by polynomial functions of the thickness coordinate. The equilibrium equations, the traction-free boundary conditions, and the continuity conditions of the interlaminar stresses are exactly satisfied in the present analysis, while the compatibility equations and the continuity of the displacements across the interfaces are enforced in an averaged sense by applying the principle of complementary virtual work. This yields an eigenvalue problem for the interfacial values of the stress functions and their normal derivatives. Interlaminar stresses for all three distinct loading cases may be obtained, in a single solution process, by combining the eigenfunctions with appropriate particular solutions (peculiar to each loading case) so as to ensure satisfaction of the traction-free boundary condition at the free edge.

Interfacial Thermal Stresses in Layered Structures: The Stepped Edge Problem

1995 ◽  
Vol 117 (2) ◽  
pp. 153-158 ◽  
Author(s):  
Wan-Lee Yin
Keyword(s):  
Stress Field ◽  
Elastic Layer ◽  
Thermal Stresses ◽  
Free Edge ◽  
Multilayered Structure ◽  
Local Geometry ◽  
Temperature Load ◽  
Edge Problem ◽  
Stress Functions ◽  
Multilayer Substrate

The intense, localized stress field produced by a temperature load in a multilayered structure may be significantly affected by the local geometry of the free edge. We examine here the stepped edge problem associated with bonding an elastic layer (silicon chip) to a single or multilayer substrate with a slightly larger length. Stress functions are introduced in various rectangular regions and the continuity of tractions are enforced across all inter-region boundaries. Furthermore, continuity of displacements is enforced across the junction of the two segments of the base laminate. The analysis results indicate that even a minute protrusion of the edge of the base laminate relative to the attached chip may cause significant changes in the peeling and shearing stresses in the end region of the interface.

The Effects of Adhesive Nonlinearity on Thermal Stress in Layered Components

1994 ◽  
Vol 116 (2) ◽  
pp. 105-109 ◽  
Author(s):  
Wan-Lee Yin ◽  
James L. Dale
Keyword(s):  
Thermal Stress ◽  
Middle Layer ◽  
Free Edge ◽  
Boundary Region ◽  
Local Concentration ◽  
Interlaminar Stresses ◽  
Softening Behavior ◽  
Stress Strain Relation ◽  
Inelastic Material ◽  
Stress Functions

Interlaminar stresses near the free edge of a multi-layered structure under thermal and mechanical loads are significantly affected by nonlinear and inelastic material properties. Most previous studies of the subject ignored such effects and obtained singular or extremely severe and localized stress fields in boundary regions based strictly on the assumption of linearly elastic stress-strain relation. In the present paper, a variational method, using approximate stress functions and the principle of complementary energy, is developed to study the thermal stress in a three-layer beam including a thin, compliant, non-linearly elastic middle layer. It is found that the elastic softening behavior of the thin layer results in dispersion of the interlaminar stresses and widening of the boundary region. Hence the use of toughened, compliant bonding layers may produce a beneficial effect by alleviating local concentration of interlaminar stresses.

3D Thermal Stresses in Composite Laminates Under Steady-State Through-Thickness Thermal Conduction

2020 ◽  
Vol 12 (06) ◽  
pp. 2050065
Author(s):  
Yan Guo ◽  
Yanan Jiang ◽  
Ji Wang ◽  
Bin Huang
Keyword(s):  
Steady State ◽  
Plane Stress ◽  
Composite Laminates ◽  
Stress Function ◽  
Thermal Conduction ◽  
Thermal Stresses ◽  
Virtual Work ◽  
Ritz Method ◽  
Stress Fields ◽  
Stress Functions

In this study, 3D thermal stresses in composite laminates under steady-state through thickness thermal conduction are investigated by means of a stress function-based approach. One-dimensional thermal conduction is solved for composite laminate and the layerwise temperature distribution is calculated first. The principle of complementary virtual work is employed to develop the governing equations. Their solutions are obtained by using the stress function-based approach, where the stress functions are taken from the Lekhnitskii stress functions in terms of in-plane stress functions and out-of-plane stress functions. With the Rayleigh–Ritz method, the stress fields can be solved by first solving a standard eigenvalue problem. The proposed method is not merely computationally efficient and accurate. The stress fields also strictly satisfy the prescribed boundary conditions validated by the results of finite element method (FEM) results. Finally, some of the results will be given for discussion considering different layup stacking sequences, thermal conductivities and overall temperature differences. From the results, we find that the thermal conductivity greatly affects the stress distributions and peak values of stresses increase linearly for the present model. The proposed method can be used for predicting 3D thermal stresses in composite laminates when subjected to thermal loading.

Simple Solutions of the Free-Edge Stresses in Composite Laminates under Thermal and Mechanical Loads

1994 ◽  
Vol 28 (6) ◽  
pp. 573-586 ◽  
Author(s):  
Wan-Lee Yin
Keyword(s):  
Composite Laminates ◽  
Constitutive Relations ◽  
Free Edge ◽  
Normal Derivative ◽  
Boundary Region ◽  
Material Behavior ◽  
Interlaminar Stresses ◽  
Inelastic Material ◽  
Stress Functions ◽  
Temperature Loads

Intense and localized interlaminar stresses generally occur in a narrow boundary region near the free edge of a multilayered anisotropic laminate under mechanical and temperature loads. Quantitative measures of interlaminar action across interfaces may be readily obtained through purely algebraic operations, even if nonlinear and inelastic material behavior becomes significant in the boundary region due to severe strain concentration. These measures are the limiting values of the Lekhnitskii stress functions F and $$ (and of the normal derivative of F) along interfaces and toward the interior region of the laminate. In the present work, they are used as the basis of an exceedingly simple and efficient method of interlaminar stress analysis that is potentially applicable to free-edge problems involving nonlinear thermoelastic constitutive relations. Example solutions are obtained for symmetric, four-layer, cross-ply and angle-ply laminates under a temperature load and two different types of strain loads, and the results are found to be in reasonable agreement with the existing numerical and analytical solutions based on elaborate analysis methods.

Thermal Stresses in a Rectangular Plate Clamped Along an Edge

1949 ◽  
Vol 16 (2) ◽  
pp. 118-122 ◽  
Author(s):  
B. J. Aleck
Keyword(s):  
Boundary Conditions ◽  
Normal Stress ◽  
Rectangular Plate ◽  
Strain Energy ◽  
Thermal Stresses ◽  
Free Edge ◽  
Drop Out ◽  
Derivatives Of ◽  
Uniform Change ◽  
Clamped Edge

Abstract An approximate solution has been obtained for the stresses induced by a uniform change in temperature of a thin rectangular plate, clamped along an edge. The solution has been carried to completion for plates whose clamped edge is long, i.e., more than 5 times the length of the perpendicular free edge. The solution for smaller ratios of clamped to perpendicular lengths is expressed in terms of six determined functions whose coefficients are to be evaluated by satisfying two boundary conditions. The thermal-stress problem is first converted to one of specified boundary tractions. The normal stress, σx, parallel to the clamped edge is assumed of the form σx = f1 (x) + y f2(x) + y2f3(x), where fi(x) are as yet undetermined functions, and where y is the co-ordinate at right angles to the clamped edge. Using the equations of equilibrium and the boundary conditions, τxy and σy are expressed in terms of powers of y and the derivatives of fi(x). The integral representing the strain energy is then expressed in terms of the expressions for σx, σy, and τxy. In accordance with the principle of least work, the integral representing the strain energy is minimized, using the calculus of variations. The resulting simultaneous differential equations for fi(x) are solved as a linear combination of twelve functions (six of which drop out, by symmetry). Given f1(x), then f2(x) and f3(x) are determinate by virtue of the simultaneous equations. The six coefficients in the expression for f1 are evaluated by satisfying the boundary conditions along the free edges. The maximum normal stress concentration, over 10, occurs at the junction of the free and clamped edges.

scholarly journals Series Solution-Based Approach for the Interlaminar Stress Analysis of Smart Composites under Thermo-Electro-Mechanical Loading

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 268
Author(s):  
Salman Khalid ◽  
Jaehun Lee ◽  
Heung Soo Kim
Keyword(s):  
Mechanical Loading ◽  
Composite Laminates ◽  
Composite Structures ◽  
Series Solution ◽  
Eigenvalue Problems ◽  
Virtual Work ◽  
Coupling Effect ◽  
Free Edge ◽  
Smart Composite ◽  
Interlaminar Stresses

This paper introduces a new loading condition considering the combined thermo-electro-mechanical coupling effect in a series solution-based approach to analyze the free-edge interlaminar stresses in smart composite laminates. The governing equations are developed using the principle of complementary virtual work. The assumed stress fields satisfy the traction-free and free-edge boundary conditions. The accurate stress states of the composite structures are acquired through the procedure of generalized eigenvalue problems. The uniform temperature is employed throughout the laminate, and the electric field loading is applied to the symmetric piezo-bonded actuators to examine the combined effect of thermal and electrical stresses on the overall deformation of smart composite laminates. It was observed that the magnitude of the peeling stresses generated by mechanical loading was reduced by the combined thermal and electric excitation loading (up to 25.3%), which in turn resulted in expanding the service life of the smart composite structures. The proposed approach is implemented on three different layup configurations. The efficiency of the current methodology is confirmed by comparing the results with the 3D finite element (FEM) solution computed by ABAQUS.

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