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.

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.

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.

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.

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 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 Reduction of Free Edge Peeling Stress of Laminated Composites Using Active Piezoelectric Layers

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Bin Huang ◽  
Heung Soo Kim
Keyword(s):  
Boundary Conditions ◽  
Virtual Work ◽  
Laminated Composites ◽  
Free Edge ◽  
Governing Equations ◽  
Kantorovich Method ◽  
Free Boundary Conditions ◽  
Extended Kantorovich Method ◽  
Piezoelectric Layers ◽  
Iteration Processes

An analytical approach is proposed in the reduction of free edge peeling stresses of laminated composites using active piezoelectric layers. The approach is the extended Kantorovich method which is an iterative method. Multiterms of trial function are employed and governing equations are derived by taking the principle of complementary virtual work. The solutions are obtained by solving a generalized eigenvalue problem. By this approach, the stresses automatically satisfy not only the traction-free boundary conditions, but also the free edge boundary conditions. Through the iteration processes, the free edge stresses converge very quickly. It is found that the peeling stresses generated by mechanical loadings are significantly reduced by applying a proper electric field to the piezoelectric actuators.

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.

scholarly journals Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 461
Author(s):  
Kenta Oishi ◽  
Yoshihiro Shibata
Keyword(s):  
Free Surface ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Mhd Flow ◽  
Transmission Conditions ◽  
Well Posedness ◽  
Anisotropic Space ◽  
Free Boundary Conditions ◽  
Incompressible Magnetohydrodynamics

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space Hp1((0,T),Hq1)∩Lp((0,T),Hq3) for the velocity field and in an anisotropic space Hp1((0,T),Lq)∩Lp((0,T),Hq2) for the magnetic fields with 2<p<∞, N<q<∞ and 2/p+N/q<1. To prove our main result, we used the Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author.

Sign in / Sign up

Export Citation Format

Share Document