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Elements of Continuum Mechanics and Conservation Laws by Sergei K. Godunov, Evgenii I. Romenskii (auth.)

By Sergei K. Godunov, Evgenii I. Romenskii (auth.)

Elements of Continuum Mechanics and Conservation Laws offers a systematization of other types in mathematical physics, a examine of the constitution of conservation legislation, thermodynamical identities, and reference to standards for well-posedness of the corresponding mathematical difficulties.
The concept awarded during this ebook stems from learn performed by way of the authors about the formulations of differential equations describing explosive deformations of metals. In such tactics, elasticity equations are utilized in a few zones, while hydrodynamics equations are said in different zones. Plastic deformations seem in transition zones, which results in residual stresses. The advised version includes a few leisure phrases which simulate those plastic deformations. convinced legislation of thermodynamics are utilized in order to explain and examine differential equations simulating the actual techniques. This ends up in the specific formula of differential equations utilizing generalized thermodynamical potentials.

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Consider a cube 0 ~ XlO ~ h, 0 ~ X20 ~ h, 0 ~ X30 ~ h containing the mass poh3. We compute the work under deformation of this cube. During the time-interval (to, t), the energy E changes: Poh 3E[k1(t), k2(t), k3(t), S] - POh 3E[k 1(to), k 2(to), k3(tO), S] = POh3E[k1(t), k2(t), k3(t), Sj- POh3E(1, 1, 1, S). 19) On the other hand, this work can be found from the stresses and displacements relative to the faces of the cube. , at each time t the matrix is independent of Xb X2, X3 and the components Uij ofthe stress tensor are the same at all points of the cube (Uij = Uij(t)).

7) in the form dE = -Pd( ~) + TdS. 8) Let E = E(p, S). 9) p from which we can find the dependence of the pressure p and temperature T on p and S: p =p(p, S) = p2 Ep(p, S), T =T{p, S) = Es(p, S). 10) used. Then + T(V, S)dS, which implies (cf. 10)) p{V, S) = -Ev(V, S), T(V, S) = Es(V, S). 11) = A gas with the equation of state E E(p, S) is the simplest example of an elastic medium. In this case, E depends only on the entropy S and the density p of the medium (pi Po is the seal of the medium under deformation and Po is the density before deformation).

Hence (J'12 = (J'21 = 0 for k2 =1= k 1 · To prove (J'12 = 0 for k2 = k1, one can assume that the stresses are continuous functions of C or use the isotropy property of the medium. We do not consider this in detail but simply assume in the further considerations that (J'12 0 for kl =1= k 2, as well as for kl k 2. Similarly, (J'23 (J'32 0 and = (J'13 = = (J'31 = o. 7. 26) I. Elementary Properties of Deformations and Stresses 40 where 81 82 83 = k~~3 Ek,(k 1,k2,k3,S) = k1~:k3 k1Ek\ = pk1Ek,(k1,k2,k3,S), = pk2Ek2(k1' k2' k3, S), = pk3Ek3(k1' k2' k3, S).

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