A Circular Crack System in an Infinite Elastic Medium Under Arbitrary Normal Loads

1994 ◽  
Vol 61 (3) ◽  
pp. 582-588 ◽  
Author(s):  
Zhou Yong ◽  
M. T. Hanson
Keyword(s):  
Stress Component ◽  
Circular Crack ◽  
Transversely Isotropic ◽  
Isotropic Body ◽  
Coefficient Matrix ◽  
Algebraic Equations ◽  
Flat Punch ◽  
Normal Loading ◽  
Crack System ◽  
Penny Shaped Crack

This analysis considers a circular crack system containing a penny-shaped crack and a concentric, coplanar external circular crack under arbitrary normal loading in a transversely isotropic body. The solution is obtained by transforming the governing two-dimensional integral equation to a set of algebraic equations which are easily solved numerically due to the special coefficient matrix. The normal stress component coplanar with the crack system is determined in power series form. The equations are solved and solutions for the stress intensity factors around the crack fronts are given for several different loading conditions. Including the rigid-body displacements at infinity allows the contact problem of an annular flat punch on an elastic half-space to be solved simultaneously.

External Circular Crack Under Normal Load: A Complete Solution

1994 ◽  
Vol 61 (4) ◽  
pp. 809-814 ◽  
Author(s):  
V. I. Fabrikant ◽  
B. S. Rubin ◽  
E. N. Karapetian
Keyword(s):  
Normal Load ◽  
Complete Solution ◽  
Circular Crack ◽  
Transversely Isotropic ◽  
Elementary Functions ◽  
Normal Loading ◽  
Method Of Solution ◽  
External Circular Crack ◽  
Concentrated Loading ◽  
Crack Faces

For the first time, a complete solution in terms of elementary functions is given to the problem of a transversely isotropic elastic space weakened by an external circular crack and subjected to arbitrary normal loading applied symmetrically to both crack faces. A complete field of displacements and stresses due to a concentrated loading is given for both transversely isotropic and purely isotropic cases. The method of solution is based on the results published earlier by the first author.

A penny-shaped crack in a transversely isotropic piezoelectric layer

2001 ◽  
Vol 20 (6) ◽  
pp. 997-1005 ◽  
Author(s):  
Bao-Lin Wang ◽  
Naotake Noda ◽  
Jie-Cai Han ◽  
Shan-Yi Du
Keyword(s):  
Piezoelectric Layer ◽  
Transversely Isotropic ◽  
Penny Shaped Crack ◽  
Shaped Crack

Near-Tip Fields for Penny-Shaped Cracks in Magnetoelectroelastic Media

2006 ◽  
Vol 312 ◽  
pp. 41-46 ◽  
Author(s):  
Bao Lin Wang ◽  
Yiu Wing Mai
Keyword(s):  
Closed Form ◽  
Transversely Isotropic ◽  
Crack Plane ◽  
Axially Symmetric ◽  
Crack Configuration ◽  
Penny Shaped Crack ◽  
Electric And Magnetic Fields ◽  
Shaped Crack ◽  
Transversely Isotropic Solids ◽  
Isotropic Solids

This paper solves the penny-shaped crack configuration in transversely isotropic solids with coupled magneto-electro-elastic properties. The crack plane is coincident with the plane of symmetry such that the resulting elastic, electric and magnetic fields are axially symmetric. The mechanical, electrical and magnetical loads are considered separately. Closed-form expressions for the stresses, electric displacements, and magnetic inductions near the crack frontier are given.

A Kind of Preconditioned IGMRES(m) Algorithm

2012 ◽  
Vol 482-484 ◽  
pp. 413-416
Author(s):  
Chun Xiao Yu
Keyword(s):  
Krylov Subspace ◽  
Matrix Decomposition ◽  
Coefficient Matrix ◽  
Algebraic Equations ◽  
Residual Method ◽  
Minimal Residual Method ◽  
Algorithm Convergence ◽  
Minimal Residual

Fundamental theories are studied for an Incomplete Generalized Minimal Residual Method(IGMRES(m)) in Krylov subspace. An algebraic equations generated from the IGMRES(m) algorithm is presented. The relationships are deeply researched for the algorithm convergence and the coefficient matrix of the equations. A kind of preconditioned method is proposed to improve the convergence of the IGMRES(m) algorithm. It is proved that the best convergence can be obtained through appropriate matrix decomposition.

Indentation of a Penny-Shaped Crack by an Oblate Spheroidal Rigid Inclusion in a Transversely Isotropic Medium

1984 ◽  
Vol 51 (4) ◽  
pp. 811-815 ◽  
Author(s):  
Y. M. Tsai
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Contact Stress ◽  
Rigid Inclusion ◽  
Isotropic Medium ◽  
Transversely Isotropic ◽  
Transversely Isotropic Medium ◽  
Penny Shaped Crack ◽  
Shaped Crack

The stress distribution produced by the identation of a penny-shaped crack by an oblate smooth spheroidal rigid inclusion in a transversely isotropic medium is investigated using the method of Hankel transforms. This three-part mixed boundary value problem is solved using the techniques of triple integral equations. The normal contact stress between the crack surface and the indenter is written as the product of the associated half-space contact stress and a nondimensional crack-effect correction function. An exact expression for the stress-intensity is obtained as the product of a dimensional quantity and a nondimensional function. The curves for these nondimensional functions are presented and used to determine the values of the normalized stress-intensity factor and the normalized maximum contact stress. The stress-intensity factor is shown to be dependent on the material constants and increasing with increasing indentation. The stress-intensity factor also increases if the radius of curvature of the indenter surface increases.

Time-Differentiable Null Space Method for Constrained Mechanical Systems (Application to a Rheonomic System)

2013 ◽  
Author(s):  
Keisuke Kamiya
Keyword(s):  
Lagrange Multipliers ◽  
Null Space ◽  
Constraint Equation ◽  
Coefficient Matrix ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Accurate Analysis ◽  
Constrained Mechanical Systems ◽  
Space Matrix ◽  
Space Method

The governing equations of multibody systems are, in general, formulated in the form of differential algebraic equations (DAEs) involving the Lagrange multipliers. For efficient and accurate analysis, it is desirable to eliminate the Lagrange multipliers and dependent variables. Methods called null space method and Maggi’s method eliminate the Lagrange multipliers by using the null space matrix for the coefficient matrix which appears in the constraint equation in velocity level. In a previous report, the author presented a method to obtain a time differentiable null space matrix for scleronomic systems, whose constraint does not depend on time explicitly. In this report, the method is generalized to rheonomic systems, whose constraint depends on time explicitly. Finally, the presented method is applied to four-bar linkages.

scholarly journals An axisymmetric problem for a penny-shaped crack under the influence of the Steigmann–Ogden surface energy

2021 ◽  
Vol 477 (2248) ◽  
Author(s):  
Anna Y. Zemlyanova
Keyword(s):  
Surface Energy ◽  
Numerical Procedure ◽  
Material System ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Penny Shaped Crack ◽  
Normal Traction ◽  
Parametric Studies ◽  
Shaped Crack ◽  
Isotropic Elastic Medium

A problem for a nanosized penny-shaped fracture in an infinite homogeneous isotropic elastic medium is considered. The fracture is opened by applying an axisymmetric normal traction to its surface. The surface energy in the Steigmann–Ogden form is acting on the boundary of the fracture. The problem is solved by using the Boussinesq potentials represented by the Hankel transforms of certain unknown functions. With the help of these functions, the problem can be reduced to a system of two singular integro-differential equations. The numerical solution to this system can be obtained by expanding the unknown functions into the Fourier–Bessel series. Then the approximations of the unknown functions can be obtained by solving a system of linear algebraic equations. Accuracy of the numerical procedure is studied. Various numerical examples for different values of the surface energy parameters are considered. Parametric studies of the dependence of the solutions on the mechanical and the geometric parameters of the system are undertaken. It is shown that the surface parameters have a significant influence on the behaviour of the material system. In particular, the presence of surface energy leads to the size-dependency of the solutions and smoother behaviour of the solutions near the tip of the crack.

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