Continuum mechanics : elasticity, plasticity, by Ellis H. Dill

By Ellis H. Dill

"Most books on continuum mechanics specialize in elasticity and fluid mechanics. yet no matter if scholar or training specialist, glossy engineers want a extra thorough remedy to appreciate the habit of the complicated fabrics and structures in use at the present time. Continuum Mechanics: Elasticity, Plasticity, Viscoelasticity bargains an entire travel of the topic that comes with not just elasticity and fluid mechanics but additionally covers Read more...

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deals an entire travel of the topic that comes with not just elasticity and fluid mechanics but in addition covers plasticity, viscoelasticity, and the continuum version for fatigue and fracture mechanics. Read more...

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Sample text

We expect the stress to depend upon the strain, which is determined by the deformation gradient. In the absence of a general principle that prohibits it, we will initially allow the possibility that each of those quantities may depend on all those variables. However, we will find that such general relations are not possible because they are restricted by the requirement of objectivity and dissipation. 1) T = T(θ , h, F) , In this model, the stress depends only on the deformation from a particular reference configuration and not on the rate of deformation, nor on the history of deformation.

16) We next consider a special case. 17) where L(θ ) is a fourth-order tensor. 18) where the viscosities η and ζ depend in general on temperature. 16) apply and require that k ≥ 0, η ≥ 0, and 3ζ + 2η ≥ 0. 19) The complete stress–strain relation for an isotropic material in this case is Tˆ = 2 µ E + λ ( tr E)1 + 2η E + ζ (trE)1 . 20) Four material constants occur: two elastic moduli µ and λ, and two viscosities η and ζ, which may depend on temperature. 14 FLUID MECHANICS This section is devoted to the general theory of a material for which the stress depends on the rate of strain, but only on the strain through the volume change.

Substituting the constitutive relations and using h = g ⋅ F, we find 34 Continuum Mechanics − ρ( ∂ψˆ ∂ψˆ 1 ∂ψˆ ˆ ∂ψˆ + s )θ − ρ ⋅ h − (ρ − T) : E − ρ : E − qˆ ⋅ h ≥ 0. 11) Because this relation must hold for arbitrary θ , h, and E, we conclude that ψˆ must be independent of h and E, and ψ = ψˆ (θ , E), s=− ∂ψˆ . 12) Further study of the third term is required because Tˆ depends on E . 8), we find for h = 0 that −(ρ ∂ψˆ (θ , E) ˆ − T0 (θ , E)) : E + Tˆ e (θ , 0, E, E): E ≥ 0. 8) that the second term is of higher order in E than the first.

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