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.

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.

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.

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.

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.

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.

scholarly journals Analysis of Free Edge Stresses in Composite Laminates Using Higher Order Theories

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Hamidreza Yazdani Sarvestani ◽  
Ali Naghashpour
Keyword(s):  
Composite Laminates ◽  
High Efficiency ◽  
Single Layer ◽  
Free Edge ◽  
Higher Order ◽  
Design Guidelines ◽  
Computational Time ◽  
Interlaminar Stresses ◽  
Layerwise Theory ◽  
Free Edges

This paper presents the determination of the interlaminar stresses close to the free edges of general cross-ply composite laminates based on higher order equivalent single-layer theory (HESL). The laminates with finite dimensions were subjected to a bending moment, an axial force, and/or a torque for investigation. Full three-dimensional stresses in the interior and the boundary-layer regions were determined. The computed results were compared with those obtained from Reddy’s layerwise theory. It was found that HESL theory predicts precisely the interlaminar stresses near the free edges of laminates. Besides, high efficiency in terms of computational time is obtainable when HESL theory is used as compared with layerwise theory. Finally, various numerical results were presented for the cross-ply laminates. Also design guidelines were proposed to minimize the edge-effect problems in composite laminates.

The Thermoelastic Analysis of Chip-Substrate System

2004 ◽  
Vol 126 (3) ◽  
pp. 325-332 ◽  
Author(s):  
Linzhi Wu
Keyword(s):  
Temperature Field ◽  
Stress Concentration ◽  
Integrated Circuits ◽  
Thermal Stresses ◽  
Free Edge ◽  
Thermal Gradients ◽  
Substrate Structure ◽  
Thermal Stress Field ◽  
Electronic Assemblies ◽  
Free Edges

The presence of dissimilar material systems and thermal gradients introduces thermal stresses in multi-layered electronic assemblies and packages during fabrication and operation. The thermal stresses of the chip-substrate structure near free edges play an important role in determining the reliability of electronic packaging structures. Therefore, it is important to provide designers a good estimate of free edge stresses. According to the heat conduction mechanism of integrated circuits, the temperature field distribution in the chip and substrate is derived and solved when the chip works in a steady state. Taking the temperature field in the chip and substrate as the heat source, we solve the thermal stress field in the chip and substrate by using the technique of Fourier’s series expansion. The effects of geometric parameters of the chip and substrate on thermal stresses are analyzed. From the analysis of thermal stresses in the chip-substrate structure, it can be found that the stress concentration near free edges is more prominent. In the design of electronic packagings, the stress concentration near free edges which may cause cracking and delamination leading to the failure or malfunction of electronic assemblies and packages should be taken into account in details.

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.

Nonlinear Theory for Flexural Motions of Thin Elastic Plate, Part 3: Numerical Evaluation of Boundary Layer Solutions

1982 ◽  
Vol 49 (2) ◽  
pp. 409-416
Author(s):  
N. Sugimoto
Keyword(s):  
Boundary Layer ◽  
Stress Singularity ◽  
Free Edge ◽  
Thin Elastic Plate ◽  
Normal Stresses ◽  
Interior Region ◽  
First Order ◽  
Plausible Values ◽  
Bending Stresses ◽  
Order Stress

The boundary layer solutions previoulsy obtained in Part 2 of this series for the cases of the built-in edge and the free edge are evaluated numerically. For the built-in edge, a characteristic penetration depth of the boundary layer toward the interior region is given by 0.13 εh, εh being the normalized thickness of the plate, while for the free edge, it is given by 0.32 εh. Thus the boundary layer for the free edge penetrates more deeply toward the interior region than that for the built-in edge. The first-order stress distribution in each boundary layer is displayed. For the built-in edge, the stress singularity appears on the edge. It is shown that, in the boundary layer, the shearing and normal stresses become comparable with the bending stresses. Similarly for the free edge, the shearing stress also becomes comparable with the twisting stress. It should be remarked that, in the boundary layer, the shearing or the normal stress plays a primarily important role as the bending or the twisting stress. But the former decays toward the interior region and remains higher order than the latter. Finally owing to these numerical results, the coefficients involved in the “reduced” boundary conditions for the built-in edge are evaluated for the various plausible values of Poisson’s ratio.

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