Analysis of the M-Integral in Plane Elasticity

2004 ◽  
Vol 71 (4) ◽  
pp. 572-574 ◽  
Author(s):  
Y. Z. Chen ◽  
Kang Yong Lee
Keyword(s):  
Stress Field ◽  
Infinite Plate ◽  
Plane Elasticity ◽  
Physical Stress ◽  
Paper Analysis ◽  
Elasticity Solution ◽  
Large Circle ◽  
Applied Forces ◽  
The Difference ◽  
M Integral

In this paper, analysis of the M-integral in plane elasticity is carried out. An infinite plate with any number of inclusions and cracks and with any applied forces and remote tractions is considered. To study the problem, the mutual work difference integral (abbreviated as MWDI) is introduced, which is defined by the difference of works done by each other stress field on a large circle. The concept of the derivative stress field is also introduced, which is a real elasticity solution and is derived from the physical stress field. It is found that the M-integral on a large circle is equal to a MWDI from the physical stress field and a derivative stress field. Finally, the expression for M-integral on a large circle is obtained. The variation for the M-integral with respect to the coordinate transformation is addressed. An illustrative example for the use of M-integral is presented.

Some Properties of J-Integral in Plane Elasticity

2001 ◽  
Vol 69 (2) ◽  
pp. 195-198 ◽  
Author(s):  
Y. Z. Chen ◽  
K. Y. Lee
Keyword(s):  
Function Method ◽  
Infinite Plate ◽  
Scalar Function ◽  
Plane Elasticity ◽  
Reciprocal Theorem ◽  
J Integral ◽  
Large Circle ◽  
Complex Variable Function ◽  
Variable Function ◽  
Complex Variable Function Method

Some properties of the J-integral in plane elasticity are analyzed. An infinite plate with any number of inclusions, cracks, and any loading conditions is considered. In addition to the physical field, a derivative field is defined and introduced. Using the Betti’s reciprocal theorem for the physical and derivative fields, two new path-independent D1 and D2 are obtained. It is found that the values of Jkk=1,2 on a large circle are equal to the values of Dkk=1,2 on the same circle. Using this property and the complex variable function method, the values of Jkk=1,2 on a large circle is obtained. It is proved that the vector Jkk=1,2 is a gradient of a scalar function Px,y.

Stress Field for a Coated Triangle-Like Hole Problem in Plane Elasticity

2019 ◽  
Vol 36 (1) ◽  
pp. 55-72 ◽  
Author(s):  
S. C. Tseng ◽  
C. K. Chao ◽  
F. M. Chen
Keyword(s):  
Stress Field ◽  
Edge Dislocation ◽  
Coating Layer ◽  
Infinite Plate ◽  
Material Constant ◽  
Plane Elasticity ◽  
Analytical Continuation ◽  
Shape Factors ◽  
Trapping Mechanism ◽  
The Matrix

ABSTRACTThe stress field induced by an edge dislocation or a point force located near a coated triangle-like hole in an infinite plate is provided in this paper. Based on the method of analytical continuation and the technique of conformal mapping in conjunction with the alternation technique, a series solution for the displacement and stresses in the coating layer and the matrix is obtained analytically. Examples for the interaction between an edge dislocation and a coated triangle-like hole for various material constant combinations, coating thicknesses and shape factors are discussed. The analysis discovers that the so-called trapping mechanism of dislocations is more likely to exist near a coated triangle-like hole. The result shows that the dislocation will first be repelled by the coating layer and then attracted by a hole when the coating layer is slightly stiffer than the matrix. However, when the coating layer is sufficiently thin, the dislocation will always be attracted by a hole even the coating layer is stiffer than the matrix.

Analytical-Numerical Determination of Stress Distribution around a Tip of Polygon-Like Inclusion

2016 ◽  
Vol 713 ◽  
pp. 94-98
Author(s):  
Ondřej Krepl ◽  
Jan Klusák ◽  
Tomáš Profant
Keyword(s):  
Stress Distribution ◽  
Stress Field ◽  
Numerical Procedure ◽  
Plane Elasticity ◽  
Necessary Condition ◽  
Singular Stress ◽  
Reliable Assessment ◽  
Generalized Stress Intensity Factors ◽  
Stress Behaviour

A stress distribution in vicinity of a tip of polygon-like inclusion exhibits a singular stress behaviour. Singular stresses at the tip can be a reason for a crack initiation in composite materials. Knowledge of stress field is necessary condition for reliable assessment of such composites. A stress field near the general singular stress concentrator can be analytically described by means of Muskhelishvili plane elasticity based on complex variable functions. Parameters necessary for the description are the exponents of singularity and Generalized Stress Intensity Factors (GSIFs). The stress field in the closest vicinity of the SMI tip is thus characterized by 1 or 2 singular exponents (1 - λ) where, 0<Re (λ)<1, and corresponding GSIFs that follow from numerical solution. In order to describe stress filed further away from the SMI tip, the non-singular exponents for which 1<Re (λ), and factors corresponding to these non-singular exponents have to be taken into account. Analytical-numerical procedure of determination of stress distribution around a tip of sharp material inclusion is presented. Parameters entering to the procedure are varied and tuned. Thus recommendations are stated in order to gain reliable values of stresses and displacements.

Multiterm Extended Kantorovich Method for Three-Dimensional Elasticity Solution of Laminated Plates

2012 ◽  
Vol 79 (6) ◽  
Author(s):  
Santosh Kapuria ◽  
Poonam Kumari
Keyword(s):  
Stress Field ◽  
Composite Laminates ◽  
Three Dimensional ◽  
Laminated Plates ◽  
Elasticity Solution ◽  
Kantorovich Method ◽  
Extended Kantorovich Method ◽  
Single Term ◽  
3D Elasticity ◽  
Laminated Structures

In an article recently published in this journal, the powerful single-term extended Kantorovich method (EKM) originally proposed by Kerr in 1968 for two-dimensional (2D) elasticity problems was further extended by the authors to the three-dimensional (3D) elasticity solution for laminated plates. The single-term solution, however, failed to predict accurately the stress field near the boundaries; thus limiting its applicability. In this work, the method is generalized to the multiterm solution. The solution is developed using the Reissner-type mixed variational principle that ensures the same order of accuracy for displacements and stresses. An n-term solution generates a set of 8n algebraic-ordinary differential equations in the in-plane direction and a similar set in the thickness direction for each lamina, which are solved in close form. The problem of large eigenvalues associated with higher order terms is addressed. In addition to the composite laminates considered in the previous article, results are also presented for sandwich laminates, for which the inaccuracy in the single-term solution is even more prominent. It is shown that considering just one or two additional terms in the solution (n = 2 or 3) leads to a very accurate prediction and drastic improvement over the single-term solution (n = 1) for all entities including the stress field near the boundaries. This work will facilitate development of near-exact solutions of many important unresolved problems involving 3D elasticity, such as the free edge stresses in laminated structures under bending, tension and torsion.

Cavities vis-a`-vis Rigid Inclusions and Some Related General Results in Plane Elasticity

1989 ◽  
Vol 56 (4) ◽  
pp. 786-790 ◽  
Author(s):  
John Dundurs
Keyword(s):  
Stress Field ◽  
Plane Strain ◽  
Elastic Constants ◽  
Plane Stress ◽  
Poisson Ratio ◽  
Plane Elasticity ◽  
By Products

There is a strange feature of plane elasticity that seems to have gone unnoticed: The stresses in a body that contains rigid inclusions and is loaded by specified surface tractions depend on the Poisson ratio of the material. If the Poisson ratio in this stress field is set equal to +1 for plane strain, or +∞ for plane stress, the rigid inclusions become cavities for elastic constants within the physical range. The paper pursues this circumstance, and in doing so also produces several useful by-products that are connected with the stretching and curvature change of a boundary.

On the Inversion of Residual Stresses From Surface Displacements

1989 ◽  
Vol 56 (3) ◽  
pp. 508-513 ◽  
Author(s):  
Zhanjun Gao ◽  
T. Mura
Keyword(s):  
Residual Stress ◽  
Stress Field ◽  
Integral Equations ◽  
Surface Displacement ◽  
Plastic Strains ◽  
Residual Stress Field ◽  
Residual Displacements ◽  
Plastic Damage ◽  
Surface Displacements ◽  
The Difference

When plastic damage regions are accumulated in a material, there exist residual displacements on the surface of the material after all the loadings are removed. The residual displacements are defined as the difference between before and after loading, and can be measured experimentally without destruction of the material. This paper addresses the problem of evaluating the residual stress field caused by the accumulation of the plastic damage regions in a subdomain of the material. The problem is formulated as a system of integral equations relating the surface displacements to the unknown plastic strains. The damage domain, which appears as the domain of integration of the integral equations, is also unknown. Determination of the shape of the damage domain, together with the plastic strains, is a very complicated nonlinear problem. In addition to the residual surface displacement data, it requires more information about the loading history or other restrictive assumptions. However, the residual stress field in the vicinity of the damage domain is obtained after the equivalent damage domain and the equivalent plastic strains are introduced. The problem is an inverse problem, which is substantially different from the conventional forward analysis of structural mechanics. Special attention is given to the uniqueness and stability of the solution.

scholarly journals Effect of Aquarobic and Weight Training on Cortisol Levels in Obese Women

2016 ◽  
Vol 12 (1) ◽  
Author(s):  
Siti Baitul Mukarromah ◽  
Hardhono Susanto ◽  
Tandiyo Rahayu
Keyword(s):  
Weight Training ◽  
Physical Stress ◽  
Control Group ◽  
Obese Women ◽  
Control Group Design ◽  
Handling Stress ◽  
Before And After ◽  
Cortisol Levels ◽  
The Difference ◽  
And Control

Exercise is physical stress which potentially causes disruption of homeostasis, especially in sports that is excessively done. Weight Training (LB) and Aquarobic Exercise (LA) can be modulators of handling stress. This research aims at investigating the effect of the difference between LB and LA to physical stress in obese women. The study was conducted in 2014. The method used in this study was randomized experimental pretest-posttest control group design in 36 obese women, aged 45-50 years who were divided into 3 groups, group LB 50% RM, 3 sets, 12 repetition, treatment two times a day for 8 weeks (n = 12), LA 75% HRmax, treatment 2 days for 8 weeks (n = 12) and control group (n = 12). Body Mass Index (BMI) and cortisol levels were measured before and after the treatment. Hypothesis testing was conducted using test (One-Way ANOVA and Kruskal-Wallis) and the mean difference test (Tukey HSD and Mann Whitneys). The results of BMI is increased in the WT group and is decreased in LA group as compared to control group (p <0.05). The decrease of cortisol level is higher than in LA and LB group and controls (p <0.05). LB and LA affect the physical stress that is characterized by the increase in cortisol levels in obese women. Conclusion: LB is more dominant than LA in increasing physical stress.

A non-circular Eshelby inclusion near a non-parabolic inhomogeneity admitting internal uniform stresses

2021 ◽  
pp. 108128652110454
Author(s):  
Xu Wang ◽  
Peter Schiavone
Keyword(s):  
Stress Field ◽  
Circular Inclusion ◽  
Plane Elasticity ◽  
The Other ◽  
Uniform Stress ◽  
Eshelby Inclusion ◽  
Parabolic Shape ◽  
Uniform Stress Field ◽  
Elastic Inhomogeneity ◽  
Open Shape

With the aid of conformal mapping and analytic continuation, we prove that within the framework of anti-plane elasticity, a non-parabolic open elastic inhomogeneity can still admit an internal uniform stress field despite the presence of a nearby non-circular Eshelby inclusion undergoing uniform anti-plane eigenstrains when the surrounding elastic matrix is subjected to uniform remote stresses. The non-circular inclusion can take the form of a Booth’s lemniscate inclusion, a generalized Booth’s lemniscate inclusion or a cardioid inclusion. Our analysis indicates that the uniform stress field within the non-parabolic inhomogeneity is independent of the specific open shape of the inhomogeneity and is also unaffected by the existence of the nearby non-circular inclusion. On the other hand, the non-parabolic shape of the inhomogeneity is caused solely by the presence of the non-circular inclusion.

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