By Morton E. Gurtin
For the decade, the writer has been operating to increase continuum mechanics to regard relocating limitations in fabrics focusing, specifically, on difficulties of metallurgy.
This monograph provides a rational remedy of the proposal of configurational forces; it truly is an attempt to advertise a brand new point of view. incorporated is a presentation of configurational forces inside of a classical context and a dialogue in their use in parts as assorted as part transitions and fracture.
The paintings could be of curiosity to fabrics scientists, mechanicians, and mathematicians.
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Additional info for Configurational Forces as Basic Concepts of Continuum Physics: v. 137
3 0, (3–6a) b. Working. Standard force and moment balances as consequences of invariance SF FS . 27 (3–6b) The assertion (3–5a) ⇔ (3–6a) is a direct consequence of the divergence theorem. To show that, granted (3–6a), (3–5b) ⇔ (3–6b), consider the tensor (y − o) ⊗ Sn da + (y − o) ⊗ b dv. M(P ) ∂P P Then (3–5b) is equivalent to the assertion that M(P ) be symmetric: M(P ) M(P ) . Since (y − o) ⊗ Sn da (y − o) ⊗ Div S dv + ∂P P F S dv, P (3–6a) yields the conclusion M(P ) F S dv, P and M(P ) M(P ) for all P if and only if (3–6b) is satisfied.
C. Pseudomomentum The external body force (7–2) may be written in the form e −p˙ + ∇(−k) + 1 2 y˙ ∇ρ, 2 (7–9) with p −F p −ρF y˙ (7–10) a field generally referred to as the pseudomomentum. Trivially, (7–3a) may be written as a momentum balance Div S p˙ . Similarly, (7–9) yields, as an alternative 1 Cf. Podio-Guidugli . 48 7. Inertia and Kinetic Energy. Alternative Versions of the Second Law to (7–3b), the configurational momentum balance2 Div(C − k1) + g + 1 2 y˙ ∇ρ 2 ˙ p. (7–11) Note that, by (6–9), the term C C − k1 1−F S (7–12) representing stress in (7–11) has the form of an Eshelby stress with the free energy replaced by the Lagrangian (cf.
The detailed discussion of Gurtin and Struthers [1990, §4]. 24 2. Kinematics Consistency requirement for vector fields: Those spatial vector fields that represent physical quantities should be invariant under changes in material observer; material vector fields that represent physical quantities should be invariant under changes in spatial observer. For example, the motion velocity y˙ represents the time derivative of the motion holding material points X fixed; because the transformation to X˜ does not affect this computation, y˙ is invariant under a change in material observer.