Boundary integral formula of elastic problems in circle plane

A necessary optimality condition for the problem of the minimum-resistance shape for a rigid three-dimensional inclusion displaced in an unbounded isotropic elastic medium subject to a constraint on the volume of the inclusion is obtained through Betti's reciprocal work theorem. It generalizes Pironneau's optimality condition for the minimum-drag shape for a rigid body immersed into a uniform Stokes flow and is specialized for axisymmetric inclusions in axisymmetric and transversal translations. In both cases of translation, the three-dimensional displacement field is represented in terms of generalized analytic functions, and the three-dimensional elastostatics problem is reduced to boundary-integral equations (BIEs) via the generalized Cauchy integral formula. Minimum-resistance shapes are found in the semi-analytical form of functional series from an iterative procedure coupling the optimality condition and the BIEs. They are compared with the minimum-resistance spheroids and with the minimum-resistance spindle-shaped and lens-shaped bodies. Remarkably, in the axisymmetric translation, the minimum-resistance shapes transition from spindle-like shapes to almost prolate spheroidal shapes as the Poisson ratio changes from 1/2 to 0, whereas in the transversal translation, they are close to oblate spheroidal shapes for any Poisson ratio.

Download Full-text

A New Boundary Integral Equation Method for Cracked Piezoelectric Bodies

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.306-308.465 ◽

2006 ◽

Vol 306-308 ◽

pp. 465-470 ◽

Cited By ~ 2

Author(s):

Kuang-Chong Wu

Keyword(s):

Integral Equation ◽

Integral Equations ◽

Present Method ◽

Boundary Integral Equations ◽

Integral Formula ◽

Electric Displacement ◽

Integral Equation Method ◽

Boundary Integral ◽

Equation Method ◽

Intensity Factors

A novel integral equation method is developed in this paper for the analysis of two-dimensional general piezoelectric cracked bodies. In contrast to the conventional boundary integral methods based on reciprocal work theorem, the present method is derived from Stroh’s formalism for anisotropic elasticity in conjunction with Cauchy’s integral formula. The proposed boundary integral equations contain generalized boundary displacement (displacements and electric potential) gradients and generalized tractions (tractions and electric displacement) on the non-crack boundary, and the generalized dislocations on the crack lines. The boundary integral equations can be solved using Gaussian-type integration formulas without dividing the boundary into discrete elements. The crack-tip singularity is explicitly incorporated and the generalized intensity factors can be computed directly. Numerical examples of generalized stress intensity factors are given to illustrate the effectiveness and accuracy of the present method.

Download Full-text

Minimum-drag shapes in magnetohydrodynamics

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

10.1098/rspa.2011.0224 ◽

2011 ◽

Vol 467 (2136) ◽

pp. 3371-3392 ◽

Cited By ~ 2

Author(s):

Michael Zabarankin

Keyword(s):

Magnetic Field ◽

Reynolds Number ◽

Optimality Condition ◽

Boundary Integral Equations ◽

Integral Formula ◽

Magnetic Reynolds Number ◽

Boundary Integral ◽

The Body ◽

Mhd Equations ◽

Minimum Drag

A necessary optimality condition for the minimum-drag shape for a non-magnetic solid body immersed in the uniform flow of an electrically conducting viscous incompressible fluid under the presence of a magnetic field is obtained. It is assumed that the flow and magnetic field are uniform and parallel at infinity, and that the body and fluid have the same magnetic permeability. The condition is derived based on the linearized magnetohydrodynamic (MHD) equations subject to a constraint on the body’s volume, and generalizes the existing optimality conditions for the minimum-drag shapes for the body in the Stokes and Oseen flows of a non-conducting fluid. It is shown that for any Hartmann number M , Reynolds number Re and magnetic Reynolds number Re m , the minimum-drag shapes are fore-and-aft symmetric and have conic vertices with an angle of 2 π /3. The minimum-drag shapes are represented in a function-series form, and the series coefficients are found iteratively with the derived optimality condition. At each iteration, the MHD problem is solved via the boundary integral equations obtained based on the Cauchy integral formula for generalized analytic functions. With respect to the equal-volume sphere, drag reduction as a function of the Cowling number S= M 2 /( Re m Re ) is smallest at S=1. Also, in the considered examples, the drag values for the minimum-drag shapes and equal-volume minimum-drag spheroids are sufficiently close.

Download Full-text