Configurational Forces as Basic Concepts of Continuum by Morton E. Gurtin

By Morton E. Gurtin

For the decade, the writer has been operating to increase continuum mechanics to regard relocating obstacles in fabrics focusing, specifically, on difficulties of metallurgy. This monograph offers a rational therapy of the suggestion of configurational forces; it truly is an attempt to advertise a brand new perspective. incorporated is a presentation of configurational forces inside a classical context and a dialogue in their use in parts as diversified as section transitions and fracture. The paintings might be of curiosity to fabrics scientists, mechanicians, and mathematicians.

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5–10) To account for non-Galilean material observers as defined in (2–11), the working (5–6) should have a replaced by a + γ × (X − o) and should be augmented by the term {m · γ + (g + e) · [γ × (X − o)]} dv, P with m a material vector field that represents configurational body moments. ) The configurational moment balance (X − o) × Cn da + (X − o) × (g + e) dv + ∂P P m dv 0 (5–11) P then follows from invariance of the working under changes in material observer. The local form of (5–11), namely, C−C −m×, (5–12) establishes the need for the body moment m, as C need not be symmetric (cf.

Finally, for G and T tensors and Λ a 3-tensor, G(T:Λ) T:(ΛG ). Tij (∂Fij /∂Xk ), (1–27) j4. Functions of tensors The derivative of a scalar function (T) of a tensor T is written ∂T (T) and is defined by the chain rule: For any tensor function T(α) of a scalar variable α, d dα (T(α)) ˙ [∂T (T(α))] · T(α), j. General notation. Tensor analysis 17 or, more succintly, (T). In components, (∂T )ij T T(X), ˙ ∂T (T) · T. (1–28) ∂ /∂Tij . A consequence of this definition is that, for ∇ (T) ∂T (T):∇T (1–29) (where the gradient on the left is the gradient of (T(X)) with respect to X).

6–18) d. Fluids. Current configuration as reference Most of the previous discussion was linked to solids, but the mathematical theory itself is independent of the specific constitutive theory. Further, while a fixed reference configuration may be used to describe a fluid—and often is in the study of shock waves—constitutive equations for a fluid are independent of the specific choice of reference. That is why fluids are generally described using the current (deformed) configuration as reference.

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