By Constantine M. Dafermos (auth.)
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Extra info for Hyberbolic Conservation Laws in Continuum Physics
The map X ∗ that carries (x, t) to (χ (x, t), t) is a bilipschitz homeomorphism of X to some subset X ∗ of IR m+1 , with Jacobian matrix (cf. 2) J= Fv . 3). 4) ∗ = (det F)−1 , ∗ ∗ 29 1 (X ∗ ; IM n×m ) and ∈ L loc = (det F)−1 F , P ∗ = (det F)−1 P. 4) and are equivalent within the function class of ﬁelds considered here. 5) H= ∂ x¯ , ∂x det H = 1, . −1 o . Fig. 1. 6) ¯ = , ¯ = H . 4): ¯ ∗ = (det F) ¯ By the chain rule, deformation gradient relative to the new reference conﬁguration B. , as was to be expected, the spatial ﬁelds are not affected by changing the reference conﬁguration of the body.
2 Theorem. 2). 3). 5) [Q(U+ , X ) − Q(U− , X )]N ≥ 0 22 I Balance Laws holds for almost all (with respect to Hk−1 ) X ∈ J . Proof. 7) N = [DQ ◦ U , grad U ] + ∇ · Q − h. 8) DG ◦ U (X ) = DG(U˜ (X ), X ), DQ ◦ U (X ) = DQ(U˜ (X ), X ). 3), we deduce that N vanishes on C. 4, we infer (Q ◦ U )± = Q ◦ U± . 8. 9) N (F) = F [Q(U− , X ) − Q(U+ , X )]N dHk−1 . 5) holds. This completes the proof. 2) [G(W ) − G(V )]N = 0 holds for some states V, W in O and N ∈ S k−1 . 1) on IR k by the following procedure: Consider any ﬁnite family of parallel (k − 1)-dimensional hyperplanes, all of them orthogonal to N , and deﬁne a function U on IR k which is constant on each slab conﬁned between adjacent hyperplanes, taking the values V and W in alternating order.
11) q(F ˆ H −1 , s, g) = q(F, ˆ s, g). 11) hold forms a subgroup G of the special linear group SL(m), called the symmetry group of the medium. In certain media, G may contain only the identity matrix I in which case material symmetry is minimal. 11) conditions on the constitutive functions of the medium. Maximal material symmetry is attained when G ≡ SL(m). In that case the medium is a thermoelastic ﬂuid. 12) ε = ε˜ (ρ, s), q = q(ρ, ˜ s, g). 3)3 . 14) T = − p I, p = ρ 2 ∂ρ ε˜ (ρ, s), θ = ∂s ε˜ (ρ, s).
Hyberbolic Conservation Laws in Continuum Physics by Constantine M. Dafermos (auth.)