Solution of the plane stress problems of strain-hardening materials described by power-law using the complex pseudo-stress function

1991 ◽  
Vol 12 (5) ◽  
pp. 481-492 ◽  
Author(s):  
Wang Zi-kung ◽  
Wei Xue-xia ◽  
Gao Xin-lin
Keyword(s):  
Strain Hardening ◽  
Power Law ◽  
Plane Stress ◽  
Stress Function

Analysis of a Power-Law Material Containing a Single Hole Subjected to a Uniaxial Tensile Stress Using the Complex Pseudo-Stress Function

1988 ◽  
Vol 55 (2) ◽  
pp. 267-274 ◽  
Author(s):  
Y. S. Lee ◽  
L. C. Smith
Keyword(s):  
Tensile Stress ◽  
Power Law ◽  
Plane Stress ◽  
Stress Function ◽  
Uniaxial Tensile ◽  
Complex Conjugate ◽  
Second Derivative ◽  
Compatibility Equation ◽  
Single Hole ◽  
Uniaxial Tensile Stress

The two-dimensional compatibility equation for time-dependent materials described by the power law is expressed in terms of the second derivative of the stress function with respect to complex conjugate variables. The equation is solved by introducing the pseudo-stress function which satisfies the biharmonic equation resulting from the compatibility equation. The relationship between the second derivative of the stress function and the pseudo-stress function is established. The mixed derivative of the stress function associated with the dilatational stress is expressed by an integral of the complicated pseudo-stress function. The velocity and strain-rate components are expressed in terms of the pseudo-stress function. Therefore, responses of power-law creep materials subjected to various boundary conditions can be obtained. Using the pseudo-stress function, the stress distribution in power-law materials, containing a single hole under plane strain and subjected to a uniaxial tensile stress, is found. The Stress Concentration Factors (SCF) on the hole surface, obtained by using the pseudo-stress function, are compared with those under plane stress obtained by another investigator. The results show that the SCF under plane stress is approximately 8 percent higher than that obtained by using the analysis techniques described herein. The maximum tangential stress for m < 0.5 is obtained away from the hole whereas for 0.5 ≤ m ≤ 1 the maximum stress is found at the hole for θ = π/2.

Generalized plane stress in a semi-infinite strip under arbitrary end-load

1964 ◽  
Vol 281 (1385) ◽  
pp. 184-206 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Plane Stress ◽  
Stress Function ◽  
Infinite Strip ◽  
Order Ordinary Differential Equation ◽  
Orthogonal Functions

The problem involves the determination of a biharmonic generalized plane-stress function satisfying certain boundary conditions. We expand the stress function in a series of non-orthogonal eigenfunctions. Each of these is expanded in a series of orthogonal functions which satisfy a certain fourth-order ordinary differential equation and the boundary conditions implied by the fact that the sides are stress-free. By this method the coefficients involved in the biharmonic stress function corresponding to any arbitrary combination of stress on the end can be obtained directly from two numerical matrices published here The method is illustrated by four examples which cast light on the application of St Venant’s principle to the strip. In a further paper by one of the authors, the method will be applied to the problem of the finite rectangle.

Strain-Hardening Solutions to Plate Problems

1959 ◽  
Vol 26 (2) ◽  
pp. 276-284
Author(s):  
Nicholas Perrone ◽  
P. G. Hodge
Keyword(s):  
Strain Hardening ◽  
Plane Stress ◽  
Kinematic Model ◽  
Circular Plates ◽  
Flow Laws

Abstract Two sets of strain-hardening flow laws based on an analogy with a kinematic model are derived for circular plates made of an initially Tresca material. The two sets of flow laws, called complete and direct hardening, differ in the point at which the plane-stress assumption is introduced. The complete-hardening flow laws are more consistent since the plane-stress assumption is made at the latest possible stage in their derivation. Complete and direct-hardening solutions are obtained to each of three different plate problems. The results indicate that the direct-hardening solutions give a fair approximation to the complete-hardening ones.

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.

A Proposed Generalized Model for Forming Limit Diagram

2015 ◽  
Author(s):  
Aly El Domiaty ◽  
Abdel-Hamid I. Mourad ◽  
Abdel-Hakim Bouzid
Keyword(s):  
Constitutive Equation ◽  
Strain Rate ◽  
Strain Hardening ◽  
Power Law ◽  
Yield Criterion ◽  
Rate Of Change ◽  
Forming Limit ◽  
Strain Hardening Exponent ◽  
Sensitivity Factor ◽  
Generalized Model

One of the most significant approaches for predicting formability is the use of forming limit diagrams (FLDs). The development of the generalized model integrates other models. The first model is based on Von-Misses yield criterion (traditionally used for isotropic material) and power law constitutive equation considering the strain hardening exponent. The second model is also based on Von-Misses yield criterion but uses a power law constitutive equation that considers the effect of strain rate sensitivity factor. The third model is based on the modified Hill’s yield criterion (for anisotropic materials) and a power law constitutive equation that considers the strain hardening exponent. The current developed model is a generalized model which is formulated on the basis of the modified Hill yield criterion and a power law constitutive equation considering the effect of strain rate. A new controlling parameter (γ) for the limit strains was exploited. This parameter presents the rate of change of strain rate with respect to strain. As γ increases the level of the FLD raises indicating a better formability of the material.

Axisymmetric Plane Stress Problems in Anisotropic Plasticity

1969 ◽  
Vol 36 (1) ◽  
pp. 7-14 ◽  
Author(s):  
Wei Hsuin Yang
Keyword(s):  
Stress Concentration ◽  
Strain Hardening ◽  
Plane Stress ◽  
Analytic Solutions ◽  
Anisotropic Plasticity ◽  
Stress Plane ◽  
Method Of Solution ◽  
Degree Of Anisotropy ◽  
Concentration Factors ◽  
Stress Concentration Factors

Based on an established theory of anisotropic plasticity, a class of axisymmetric plane stress problems is solved for sheet metals which harden according to a power law and are isotropic in their plane. A new method of solution, the stress plane method, is used. The analytic solutions for the problems considered are obtained in the stress plane. The stress-concentration factors introduced by a hole or a rigid inclusion at the center of an infinite sheet are obtained for arbitrary degree of anisotropy and strain-hardening characteristics. The influence of anisotropy and strain-hardening on the deep-drawing problem is also studied. The results show that the type of anisotropy and strain-hardening assumed always influences the stress concentration and drawability in a favorable way.

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