Nonlinear Continuum Mechanics, Problem Sheet 3 1. A Green-elastic material has stored energy function W per unit referece volume and is subject to two constraints: Ă˜1(F)=0 and Ă˜2(F)=0. Show that the relation between nominal stress and deformation gradient for this material is PIi = aw + q1 Fil + 92 Fil 001 , where q1 and q2 are undetermined. A body comprising such material occupies a domain $2 in its reference configuration and is in equilibrium in the presence of dead-load body force and some combination of dead-load traction and placement (x prescribed) conditions on its boundary Jn. Show that its equilibrium configuration is stable, and the solution branch is incrementally unique, if Z aXJ ? OXI OFiIOFjJ 02W(F) @201(F) OFiIOFjJ a2¢2(F) + 92 OFiIOFjJ # Ouj dx > 0 for all non-zero perturbations &x = Au (\ << 1) compatible with the constraints and any placement boundary conditions. Here, F, q1 and q2 are the actual equilibrium fields. 2. Show that the "upper convected Maxwell" constitutive relation for incompressible fluid, ? = 0ª - pI; St Sad +00 - 24 D, can be expressed in the integral form Copyright @ 2006 University of Cambridge. Not to be quoted or reproduced without permission. 09(1)-24 /* e-(t-t)/TF(t)F-1(t)D(t)F-T(')FT (t) d'. [Recall that the relation o = FTFT between Cauchy and second Piola-Kirchhoff stress for incom- pressible material equivalently interprets the components of T as contravariant components of o on convected coordinates.] Check directly that the integral form satisfies the principle of material indifference. Develop corresponding relations (differential and integral) for the "lower convected Maxwell" model. [Recall: 0IJ = Fi Foij defines the required covariant components of o.] 3. Obtain od(t) explicitly for the two models discussed in question 2, in the case of a time-dependent simple shear: 1 y(t) 0 0). 0 F(t) = 0 1 0 1) 4. Develop the solution for laminar flow between two plates in the presence of a pressure gradient, for "upper convected Maxwell" fluid. [Use the integral form of the constitutive relation.] Now make the problem non-trivial by considering a time-dependent pressure gradient, in the "slow flow" limit so that inertia is disregarded. If (x2, t) = Ov(x2, t)/dx2, show that y(x2, t) must satisfy 1 e-(+)Tj(x2,t)dt =- [G(t) + - + A(t) , ? 8 where G(t) is the pressure gradient and A(t) is an arbitrary function. Deduce v(x2, t). 5. Solve the problem of Couette flow, in the region a < r < b, -> < > < 0, with the boundary conditions ve(a) = vo, (constant) ve(b) = 0, for "upper convected Maxwell" fluid. [Try to avoid a lot of work by using the integral form of the constitutive relation and a suitable rotating frame of reference.]