Cracked Elastic Layer Under a Compressive Mechanical Load

2009 ◽  
Author(s):  
Vahagn Makaryan ◽  
Michael Sutton ◽  
Tatevik Yeghiazaryan ◽  
Davresh Hasanyan ◽  
Xiaomin Deng
Keyword(s):  
Elastic Layer ◽  
Integral Transformation ◽  
Mechanical Load ◽  
Physical Parameters ◽  
Algebraic Equations ◽  
Closed Form Solutions ◽  
Linear Algebraic Equations ◽  
Penny Shaped Crack ◽  
Shaped Crack ◽  
Parameter Values

In the present work, the problem of an elastic layer weakened by a finite penny shaped crack parallel to a layer’s surface that is loaded in compression is considered. Assuming that the surfaces of the crack have frictional slipping contact, Henkel and Legendre integral transformation techniques are employed to formulate solutions in the form of an infinite system of linear algebraic equations. The regularity of the equations is established and closed-form solutions are obtained for stresses and strains. Assuming shear stress on the crack surfaces is linearly distributed, numerical results show both geometric and physical parameters have an essential influence on the stress distribution around the crack, with specific parameter values indicating the normal stress along the crack surface can change its sign from negative to positive. The implications of the work will be discussed.

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.

The Stress Response of Radially Polarized Rotating Piezoelectric Cylinders

2003 ◽  
Vol 70 (3) ◽  
pp. 426-435 ◽  
Author(s):  
D. Galic ◽  
C. O. Horgan
Keyword(s):  
Analytic Solution ◽  
Coupled System ◽  
Special Problem ◽  
Algebraic Equations ◽  
Equilibrium Equations ◽  
Closed Form Solutions ◽  
Linear Algebraic Equations ◽  
Constant Potential ◽  
Radially Polarized ◽  
Fundamental Boundary

Recent advances in smart structures technology have lead to a resurgence of interest in piezoelectricity, and in particular, in the solution of fundamental boundary value problems. In this paper, we develop an analytic solution to the axisymmetric problem of an infinitely long, radially polarized, radially orthotropic piezoelectric hollow circular cylinder rotating about its axis at constant angular velocity. The cylinder is subjected to uniform internal pressure, or a constant potential difference between its inner and outer surfaces, or both. An analytic solution to the governing equilibrium equations (a coupled system of second-order ordinary differential equations) is obtained. On application of the boundary conditions, the problem is reduced to solving a system of linear algebraic equations. The stress distribution in the tube is obtained numerically for a specific piezoceramic of technological interest, namely PZT-4. For the special problem of a uniformly rotating solid cylinder with traction-free surface and zero applied electric charge, explicit closed-form solutions are obtained. It is shown that for certain piezoelectric solids, stress singularities at the origin can occur analogous to those occurring in the purely mechanical problem for radially orthotropic elastic materials.

Numerical simulation of magnetohydrodynamic micropolar fluid flow and heat transfer in a channel with shrinking walls

2014 ◽  
Vol 92 (9) ◽  
pp. 987-996 ◽  
Author(s):  
Kashif Ali ◽  
Muhammad Ashraf ◽  
Nimra Jameel
Keyword(s):  
Heat Transfer ◽  
Micropolar Fluid ◽  
Thermal Control ◽  
Computational Procedure ◽  
Physical Parameters ◽  
Algebraic Equations ◽  
Iteration Parameter ◽  
Electrically Conducting ◽  
Flow And Heat Transfer ◽  
Linear Algebraic Equations

We numerically study the steady hydromagnetic (magnetohydrodynamic) flow and heat transfer characteristics of a viscous incompressible electrically conducting micropolar fluid in a channel with shrinking walls. Unlike the classical shooting methodology, two distinct numerical techniques are employed to solve the transformed self-similar nonlinear ordinary differential equations (ODEs). One is the combination of a direct and an iterative method (successive over-relaxation with optimal relaxation parameter) for solving the sparse system of linear algebraic equations arising from the finite difference discretization of the linearised ODEs. For the second one, a pseudotransient method is used where time plays the role of an iteration parameter until the steady state is reached. The two approaches may be easily extended to other geometries (for example, sheets, disks, and cylinders) with possible wall conditions like slip, stretching, rotation, suction, and injection. Effects of some physical parameters on the flow and heat transfer are discussed and presented through tables and graphs. Detailed description of the computational procedure and the results of the study may be beneficial for the researchers in the flow and thermal control of polymeric processing.

scholarly journals Axisymmetric Torsion of an Elastic Layer Sandwiched between Two Elastic Half-Spaces with Two Interfaced Cracks

2019 ◽  
Vol 41 (2) ◽  
pp. 57-66
Author(s):  
Fateh Madani ◽  
Belkacem Kebli
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Elastic Layer ◽  
Integral Transformation ◽  
Mixed Boundary ◽  
Symmetry Plane ◽  
Algebraic Equations ◽  
Dual Integral Equations ◽  
The Moment

AbstractThe present article examines the problem related to the axisymmetric torsion of an elastic layer by a circular rigid disc at the symmetry plane. The layer is sandwiched between two similar elastic half-spaces with two penny-shaped cracks symmetrically located at the interfaces between the two bonded dissimilar media. The mixed boundary-value problem is transformed, by means of the Hankel integral transformation, to dual integral equations, that are reduced, to a Fredholm integral equation of the second kind. The numerical methods are used to convert the resulting system to a system of infinite algebraic equations. Some physical quantities such as the stress intensity factor and the moment are calculated and presented numerically according to some relevant parameters. The numerical results show that the discontinuities around the crack and the inclusion cause a large increase in the stresses that decay with distance from the disc-loaded. Furthermore, the dependence of the stress intensity factor on the disc size, the distance between the crack and the disc, and the shear parameter is also observerd.

Stress Distribution in Bonded Dissimilar Materials Containing Circular or Ring-Shaped Cavities

1965 ◽  
Vol 32 (4) ◽  
pp. 829-836 ◽  
Author(s):  
F. Erdogan
Keyword(s):  
Integral Equations ◽  
Complete Solution ◽  
Closed Form Solution ◽  
Form Solution ◽  
Fredholm Integral Equations ◽  
Infinite Plane ◽  
Algebraic Equations ◽  
Axially Symmetric ◽  
Linear Algebraic Equations ◽  
Penny Shaped Crack

The general problem of two semi-infinite elastic media with different properties bonded to each other along a plane and containing a series of concentric ring-shaped flat cavities is considered. Using the Green’s functions for the semi-infinite plane, the problem is formulated as a system of simultaneous singular integral equations. Closed-form solution of the corresponding dominant system with Cauchy kernels is given. To obtain the complete solution, a technique reducing the problem to solving a system of linear algebraic equations rather than a pair of Fredholm integral equations is outlined. The examples for which the contact stresses are plotted include the bonded media with an axially symmetric external notch subject to axial load or homogeneous temperature changes and the case of penny-shaped crack.

New Analytical Solutions of Buckling Problem of Rotationally-Restrained Rectangular Thin Plates

2019 ◽  
Vol 11 (10) ◽  
pp. 1950101 ◽  
Author(s):  
Salamat Ullah ◽  
Jinghui Zhang ◽  
Yang Zhong
Keyword(s):  
Boundary Conditions ◽  
Integral Transform ◽  
Integral Transformation ◽  
Solution Procedure ◽  
Thin Plates ◽  
Algebraic Equations ◽  
Transform Method ◽  
Linear Algebraic Equations ◽  
Title Problem ◽  
Finite Integral

A double finite sine integral transform method is employed to analyze the buckling problem of rectangular thin plate with rotationally-restrained boundary condition. The method provides more reasonable and theoretical procedure than conventional inverse/semi-inverse methods through eliminating the need to preselect the deflection function. Unlike the methods based on Fourier series, the finite integral transform directly solves the governing equation, which automatically involves the boundary conditions. In the solution procedure, after performing integral transformation the title problem is converted to solve a fully regular infinite system of linear algebraic equations with the unknowns determined by satisfying associated boundary conditions. Then, through some mathematical manipulation the analytical buckling solution is elegantly achieved in a straightforward procedure. Various edge flexibilities are investigated through selecting the rotational fixity factor, including simply supported and clamped edges as limiting situations. Finally, comprehensive analytical results obtained in this paper illuminate the validity of the proposed method by comparing with the existing literature as well as the finite element method using (ABAQUS) software.

scholarly journals An analytical solution for the axisymmetric problem of a penny-shaped crack in an elastic layer sandwiched between dissimilar materials

2018 ◽  
Vol 5 (3) ◽  
pp. 18-00125-18-00125 ◽  
Author(s):  
Kotaro MIURA ◽  
Makoto SAKAMOTO ◽  
Koichi KOBAYASHI ◽  
Jonas A. PRAMUDITA ◽  
Yuji TANABE
Keyword(s):  
Analytical Solution ◽  
Elastic Layer ◽  
Axisymmetric Problem ◽  
Dissimilar Materials ◽  
Penny Shaped Crack ◽  
Shaped Crack
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