Thermoelastic field of a transversely isotropic elastic medium containing a penny-shaped crack: exact fundamental solution

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.

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

Journal of Applied Mechanics ◽

10.1115/1.3167729 ◽

1984 ◽

Vol 51 (4) ◽

pp. 811-815 ◽

Cited By ~ 9

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.

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

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2020.0998 ◽

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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