A crack with surface effects in a piezoelectric material

2016 ◽  
Vol 22 (1) ◽  
pp. 3-19 ◽  
Author(s):  
Xu Wang ◽  
Kun Zhou
Keyword(s):  
Differential Equations ◽  
Piezoelectric Material ◽  
Logarithmic Singularity ◽  
Coupled System ◽  
First Order ◽  
Size Dependent ◽  
Crack Tips ◽  
Piezoelectric Solid ◽  
Interaction Approximation ◽  
The Continuum

We study the contribution of surface piezoelectricity to the anti-plane deformations of a hexagonal piezoelectric material weakened by a crack. The surface piezoelectricity is incorporated by using an extended version of the continuum-based surface/interface model of Gurtin and Murdoch. The original boundary value problem is finally reduced to a system of two coupled first-order Cauchy singular integro-differential equations by considering a distribution of line dislocations and electric-potential-dislocations on the crack. Through a diagonalization strategy, the coupled system can be transformed into two independent singular integro-differential equations, each of which contains only one single unknown function and can be numerically solved by the collocation method. Our solution demonstrates that the stresses, strains, electric displacements and electric displacements exhibit the logarithmic singularity at the crack tips. The obtained solution is further used to predict the size-dependent effective electroelastic properties of a piezoelectric solid containing multiple nanocracks with surface piezoelectricity within the framework of non-interaction approximation.

Interaction between a nanocrack with surface elasticity and a screw dislocation

2016 ◽  
Vol 22 (2) ◽  
pp. 131-143 ◽  
Author(s):  
Xu Wang ◽  
Hui Fan
Keyword(s):  
Screw Dislocation ◽  
Real Axis ◽  
Analytical Study ◽  
Logarithmic Singularity ◽  
Surface Elasticity ◽  
Spontaneous Generation ◽  
The Real ◽  
Crack Tips ◽  
Interaction Problem ◽  
The Continuum

In the present analytical study, we consider the problem of a nanocrack with surface elasticity interacting with a screw dislocation. The surface elasticity is incorporated by using the continuum-based surface/interface model of Gurtin and Murdoch. By considering both distributed screw dislocations and line forces on the crack, we reduce the interaction problem to two decoupled first-order Cauchy singular integro-differential equations which can be numerically solved by the collocation method. The analysis indicates that if the dislocation is on the real axis where the crack is located, the stresses at the crack tips only exhibit the weak logarithmic singularity; if the dislocation is not on the real axis, however, the stresses exhibit both the weak logarithmic and the strong square-root singularities. Our result suggests that the surface effects of the crack will make the fracture more ductile. The criterion for the spontaneous generation of dislocations at the crack tip is proposed.

Interaction Between an Edge Dislocation and a Crack With Surface Elasticity

2015 ◽  
Vol 82 (2) ◽  
Author(s):  
Xu Wang ◽  
Peter Schiavone
Keyword(s):  
Edge Dislocation ◽  
Analytical Study ◽  
Surface Elasticity ◽  
Image Force ◽  
Size Dependent ◽  
Interface Theory ◽  
Singular Integrodifferential Equations ◽  
Finite Crack ◽  
Crack Tips ◽  
The Continuum

We undertake an analytical study of the interaction of an edge dislocation with a finite crack whose faces are assumed to have separate surface elasticity. The surface elasticity on the faces of the crack is described by a version of the continuum-based surface/interface theory of Gurtin and Murdoch. By using the Green's function method, we obtain a complete exact solution by reducing the problem to three Cauchy singular integrodifferential equations of the first-order, which are solved by means of Chebyshev polynomials and a collocation method. The correctness of the solution is rigorously verified by comparison with existing analytical solutions. Our analysis shows that the stresses and the image force acting on the edge dislocation are size-dependent and that the stresses exhibit both the logarithmic and square root singularities at the crack tips when the surface tension is neglected.

Size-Dependent Nonlinear Vibrations of First-Order Shear Deformable Magneto-Electro-Thermo Elastic Nanoplates Based on the Nonlocal Elasticity Theory

2016 ◽  
Vol 08 (04) ◽  
pp. 1650053 ◽  
Author(s):  
Reza Ansari ◽  
Raheb Gholami
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Elasticity Theory ◽  
Shear Deformation ◽  
Nonlocal Elasticity ◽  
Nonlocal Parameter ◽  
Nonlocal Elasticity Theory ◽  
Nonlinear Free Vibration ◽  
First Order ◽  
Size Dependent

This paper deals with the size-dependent geometrically nonlinear free vibration of magneto-electro-thermo elastic (METE) nanoplates using the nonlocal elasticity theory. The mathematical formulation is developed based on the first-order shear deformation plate theory, von Kármán-type of kinematic nonlinearity and nonlocal elasticity theory. The influences of geometric nonlinearity, rotary inertia, transverse shear deformation, magneto-electro-thermal loading and nonlocal parameter are considered. First, the generalized differential quadrature (GDQ) method is utilized to reduce the nonlinear partial differential equations to a system of time-dependent nonlinear ordinary differential equations. Afterwards, the numerical Galerkin method, periodic time differential operators and pseudo-arc length continuation algorithm are employed to compute the nonlinear frequency versus the amplitude for the METE nanoplates. The presented methodology enables one to describe the large-amplitude vibration characteristics of METE nanoplates with various sets of boundary conditions. A detailed parametric study is carried out to analyze the important parameters such as the nondimensional nonlocal parameter, external electric potential, external magnetic potential, temperature change, length-to-thickness ratio, aspect ratio and various edge conditions on the nonlinear free vibration characteristics of METE nanoplates. The results demonstrate that considering the size effect on the vibration response of METE nanoplate results in decreasing the natural frequency, a remarkable increasing effect on the hardening behavior and subsequently increasing the nonlinear-to-linear frequency ratio.

Dynamics of Temperatures in Thermally-Coupled, Heated Rooms With PI Control

2007 ◽  
Author(s):  
Frederick W. Thwaites ◽  
Mihir Sen
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Coupled System ◽  
Time Dependent ◽  
Pi Control ◽  
Thermal Coupling ◽  
First Order ◽  
Dependent Behavior ◽  
The Stability ◽  
First Order Differential Equations

The purpose of this study is to analyze the behavior of a set of thermally-controlled rooms arranged in the form of a ring. Each room is heated and can exchange heat with its neighbors as well as with the environment. The heater in each room is PI controlled. A lumped capacitance approximation is used for the rooms leading to a system of first-order differential equations. Numerical methods are used to determine the time-dependent behavior of the coupled system. The linear stability of the system is analyzed for various parameters. The stability is found to be independent of the strength of the thermal coupling between rooms.

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