By Hathaway A.S.

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**Additional resources for Analytical dynamics: being a synopsis of leading topics**

**Example text**

41) we may obtain the local form of the second law, which reads η˙ ≥ −div (θ−1 q) + θ−1 r. 44) are known as the Clausius–Duhem form of the second law of thermodynamics. It is customary in elasticity and elastoplasticity to work with the Helmholtz free energy ψ, deﬁned by ψ = e − ηθ, rather than with the internal energy. 42), the local form of the second law becomes ψ˙ + η θ˙ − σ : ˙ + θ−1 q · ∇θ ≤ 0. 45) is known as the local dissipation inequality. Now we specialize to the situation in which subsequent developments will take place, namely, that of isothermal processes.

That is, we have x = x(y, t) after carrying out the inversion, and so, for example, the velocity u˙ has the alternative representation ∂ ˙ t), t) ≡ v(y, t). y(x, t) = u(x(y, ∂t Similar transformations can be carried out with respect to all variables, and the principles of balance of linear and angular momentum are then expressed in terms of integrals over the current conﬁguration Ωt . 2 Balance of Momentum; Stress 27 the stress goes, an argument identical to that which leads to the existence of the ﬁrst Piola–Kirchhoﬀ stress tensor gives the existence of a tensor σ, called the Cauchy stress, that has the property that the force per unit current area tν on an elemental surface having unit normal ν is given by σν = tν .

4. In this situation there is a single internal variable ξ, so that the generalized strain is P = (p, ξ), while the corresponding generalized stress is taken to be 54 3. 6: (a) Stress–strain curves for kinematic hardening; (b) the situation in generalized stress space Σ = (σ, χ), where χ, known as the back-stress, represents the translation of the initial elastic region in stress space. For this simple case of linear kinematic hardening the back-stress is related to the plastic strain through χ = −k1 p, as we saw in the example earlier.

### Analytical dynamics: being a synopsis of leading topics by Hathaway A.S.

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