EN2210: Continuum Mechanics
Homework 5: Thermodynamics and Constitutive Equations Due Wednesday November 2, 2016
School of Engineering Brown University
1. Define the expended power of external forces acting on a deformable solid (which could be a sub- volume within a larger body) by
d 1 W =tv ⋅+ρρ bv ⋅− vv ⋅ exp ∫∫dt ∫2 SR R Show that the expended power is zero for any rigid velocity field of the form vy( ,)t= v00 () tt +ω () ×− ( yy ) where v0 (),ttω () are vector valued functions of time (but independent of position).
It is helpful to re-write the time derivative of kinetic energy in terms of accelerations dd11 ddvv ρvv⋅= ρρ vv ⋅=dV ⋅=vdV ρ ⋅ v dV dt ∫∫22dt 00 ∫dt ∫dt RR00 R R Hence dv Wexp =tv ⋅+ρ b − ⋅ vdV ∫∫dt SR It is also convenient to re-write the velocity field as vy( ,)tt= v00 () +− Wy ( y ) where W is a skew tensor that has ω()t as its dual vector.
2. It is helpful to have versions of the first and second laws of thermodynamics in terms of quantities defined on the reference configuration. To this end, define : • Reference mass density ρ0 • Nominal stress and deformation gradient SF, • Material stress and Lagrange strain rate ΣE, − • Referential heat flux Θ= J Fq1 With these definitions, follow the same procedure used in class to derive the spatial laws of thermodynamics (but now express the classical laws as integrals over the reference configuration) to show the following identities: ∂ε dFji ∂Qi ρη0 =SJij −+ ∂∂t x=const dt xi ∂ε dEij ∂Qi ρη0 =Σ−+ij J ∂∂t x=const dt xi dFji 1 ∂θ ∂∂ ψθ SQij −i −ρ0 +≥ s0 dtθ ∂ x ∂∂ t t i dEij 1 ∂θ ∂∂ ψθ Sij −Qsi −ρ0 +≥0 dtθ ∂ xi ∂∂ t t
3. Starting with the local form of the second law of thermodynamics and mass conservation ∂∂sq∂∂(qv /)qρ ρ ρ +ii −≥00 + = ∂txy=const ∂ yiiq ∂∂ ty
(the symbols have their usual meaning), derive the statement of the second law for a control volume ∂⋅qn q ρρsdV+ s()vn ⋅+ dA dA − dV ≥0 ∂t ∫∫ ∫qq ∫ RB B R
4. Determine how the following quantities transform under a change of observer (i) The spatial heat flux vector q (recall that qn⋅ dA gives the heat flux across a surface element with area dA and normal n, and note that n is an objective vectors, and that the all observers must see the same heat flux….) − (ii) The referential heat flux vector Θ= J Fq1 T (iv) The infinitesimal strain tensor εuu=∇ +∇( ) /2 xx
(iv) The spatial gradient of a scalar function of position in a deformed solid gy= ∇yφ() (v) The material gradient of a function of particle position in a solid Gx= ∇φ()