Download Elasticity with Mathematica: An introduction to continuum by Andrei Constantinescu PDF

By Andrei Constantinescu

This booklet is meant for researchers, engineers and scholars in strong mechanics, fabrics technological know-how and physics who're drawn to utilizing the ability of recent computing to unravel a wide selection of difficulties of either functional and basic value in elasticity. vast use of Mathematica within the booklet makes to be had to the reader more than a few recipes that may be effectively adjusted to compare specific tastes or specifications, to imagine recommendations, and to hold out symbolic and numerical research and optimization.

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IntegrateStrain[strain_] := Module[ {theta, omega}, ( theta = IntegrateGrad[-Curl[strain]]; omega = Table[Sum[Signature[{i, j, k}] theta[[k]], {k,3}],{i,3},{j,3}]; IntegrateGrad[strain + omega] ) /; ( Inc[strain] === {{0,0,0},{0,0,0},{0,0,0}})] Strain integration and compatibility operators IntegrateGrad[gf] Integrates gradient of scalar or vector field gf to obtain scalar or vector potential f IntegrateStrain[eps] Integrates small strain field eps to obtain the displacement vector field u Inc[f] Incompatibility operator for tensor field f Equivalent compatibility conditions for small strains The compatibility equation for the strain tensor ε inc ε = −(curl (curl ε )T )T = 0 is enforced if and only if the following equation is satisfied: ε + grad grad (tr ε ) − (grad + grad T )εε = 0 .

The principle of virtual power becomes (div σ + f ) · v dv + (tt − σ · n ) · v ds = ρ u¨ · v dv. 9) ∂ Because the above equality holds for an arbitrary virtual velocity field v , it implies that the following local balance equations are satisfied: • for each interior point x ∈ : div σ + f = ρ u¨ • for each boundary point x ∈ ∂ : t = σ · n. 2 Virtual power and the concept of stress 45 The boundary condition is no longer restricted to a particular type of surface traction: a solid body can carry surface tractions in the form of both pressures and shear forces.

Compatibility of small strains generated by a temperature field (Polytechnique Collective 1990–2005) Let us assume that a body is heated to a temperature field θ(x, y, z). Assuming that thermal dilatation is isotropic, that is, material stretch in all directions is the same due to temperature change, the small strain tensor due to the thermal dilatation is ε (x, y, z) = αθ(x, y, z)II , where α is the coefficient of linear thermal expansion. (a) Find the general expression of the temperature field such that the dilatation strains are compatible.

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