Quarterly of Applied Mathematics

The analogy between potential theory and classical elasticity suggests an extension of the powerful method of integral equations to the boundary value problems of elasticity. A vector boundary formula relating the boundary values of displacement and traction for the general equilibrated stress state is derived. The vector formula itself is shown to generate integral equations for the solution of the traction, displacement, and mixed boundary value problems of plane elasticity. However, an outstanding conceptual advantage of the formulation is that it is not restricted to two dimensions. This distinguishes it from the methods of Muskhelishvili and most other familiar integral equation methods. The presented approach is a real variable one and is applicable, without inherent restriction, to multiply connected domains. More precisely, no difficulty of the order of determining a mapping function is present and unwanted Volterra type dislocation solutions are eliminated a priori. An indication of techniques necessary to effect numerical solution of the resulting integral equations is presented with numerical data from a set of test problems.

Earlier results [1,2] 2 ^{2} on safe loads for a Prandtl-Reuss material subject to surface tractions or displacements which increase in ratio are here extended to any perfectly plastic material and any history of loading.

Strain-displacement relations for thin shells valid for large displacements are derived. With these as a starting point approximate strain-displacement relations and equilibrium equations are derived by making certain simplifying assumptions. In particular the middle surface strains are assumed small and the rotations are assumed moderately small. The resulting equations are suitable as a basis for stability investigations or other problems in which the effects of deformation on equilibrium cannot be ignored, but in which the rotations are not too large.

This paper studies mathematical methods in the emerging new discipline ofComputational Anatomy. Herein we formalize the Brown/Washington University model of anatomy following the global pattern theory introduced in [1, 2], in which anatomies are represented as deformable templates, collections of 0, 1, 2, 3-dimensional manifolds. Typical structure is carried by the template with the variabilities accommodated via the application of random transformations to the background manifolds. The anatomical model is a quadruple(Ω,H,I,P)\left ( \Omega , H, I, P \right ), the background spaceΩ=˙UαMα\Omega \dot = {U_\alpha }{M_\alpha }of 0, 1, 2, 3-dimensional manifolds, the set of diffeomorphic transformations on the background spaceH:Ω↔Ω{H} : \Omega \leftrightarrow \Omega, the space of idealized medical imageryII, andPPthe family of probability measures onHH. The group of diffeomorphic transformationsHHis chosen to be rich enough so that a large family of shapes may be generated with the topologies of the template maintained. Fornormal anatomyone deformable template is studied, with(Ω,H,I)\left ( \Omega , H, I \right )corresponding to ahomogeneous space[3], in that it can be completely generated from one of its elements,I=HItemp,Itemp∈II = {HI_{temp}}, {I_{temp}} \in I. Fordisease, a family of templatesUαItempα{U_\alpha }I_{temp}^\alphaare introduced of perhaps varying dimensional transformation classes. The complete anatomy is a collection of homogeneous spacesItotal=Uα(Iα,Hα){I_{total}} = {U_\alpha }\left ( {I^\alpha }, {H^\alpha } \right ).

The two-dimensional convective motion generated by buoyancy forces on the fluid in a long rectangle, of which the two long sides are vertical boundaries held at different temperatures, is considered with a view to the determination of the rate of transfer of heat between the two vertical boundaries. The governing equations are set up; they reveal that the flow is determined uniquely by the Prandtl number σ \sigma , the Rayleigh number A = g ( T 1 − T 0 ) d 3 / ( T 0 κ ν ) A = g\left ( {{T_1} - {T_0}} \right ){d^3}/\left ( {{T_0}\kappa \nu } \right ) , and the ratio of the sides of the rectangle l / d l/d . In the case of cavities used for thermal insulation of buildings, which is kept specially in mind throughout the paper, A A is usually about 1000 d 3 ^{3} (where d d is in centimeters), and l / d l/d takes values between about 5 and 200.