By Simon R. Eugster
This learn monograph discusses novel techniques to geometric continuum mechanics and introduces beams as constraint non-stop our bodies. within the coordinate unfastened and metric autonomous geometric formula of continuum mechanics in addition to for beam theories, the primary of digital paintings serves because the basic precept of mechanics. in accordance with the conception of analytical mechanics that forces of a mechanical process are outlined as twin amounts to the kinematical description, the digital paintings process is a scientific strategy to deal with arbitrary mechanical platforms. while this system is especially handy to formulate prompted beam theories, it truly is crucial in geometric continuum mechanics whilst the assumptions at the actual house are comfy and the gap is modeled as a tender manifold. The e-book addresses researcher and graduate scholars in engineering and arithmetic drawn to fresh advancements of a geometrical formula of continuum mechanics and a hierarchical improvement of brought about beam theories.
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Extra info for Geometric Continuum Mechanics and Induced Beam Theories (Lecture Notes in Applied and Computational Mechanics)
Berkeley, 1974) 17. J. Eells Jr, A setting for global analysis. Bull. Am. Math. Soc. 72(5), 751–807 (1966) 18. I. Eliasson, Geometry of manifolds of maps. J. Differ. Geom. 1(1–2), 169–194 (1967) 19. E. R. Fischer, The manifold of embeddings of a closed manifold. Differential Geometric Methods in Mathematical Physics (Springer, Berlin, 1981), pp. 310–325 20. E. Binz, J. R. Fischer, Geometry of Classical Fields. North-Holland Mathematics Studies, vol. 154 (Elsevier Science, Amsterdam, 1988) 21. C.
7) imply that ϕ(0, ˜ ·) = γ . Let f ∈ C ∞ (M) and P ∈ N . Then the composition function ϕ˜ induces the section ˜ ∈ C k (γ ∗ T M) defined by γ ∗ δϕ ˜ f ) = (P, (ϕ(0, ˜ P), δ ϕ(P)( ˜ f ))) = P, γ(P), ∂1 ( f ◦ ϕ)| ˜ (0,P) γ ∗ δ ϕ(P)( . Let (U, x) be a chart on M and γ(P) ∈ U . 9), the section through the pullback tangent bundle can locally be represented as ˜ = P, (x ◦ γ)(P), (δϕi ◦ γ)(P)(∂x i ◦ γ)| P γ ∗ δ ϕ(P) . A tangent vector can alternatively be defined, cf. , by an equivalence class of curves which pass with the same velocity through the same point on the manifold.
Hence, the covariant derivative is a C 0 section through the tensor bundle κ∗ T S ⊗ T B ∗ over B. We introduce the function ∇ : C 1 (κ∗ T S) → C 0 (κ∗ T S ⊕ (κ∗ T S ⊗ T ∗ B)) δκ → (δκ, (κ∗ ∇)(δκ)), where ⊕ denotes the direct sum. 2) holds. 2) and the representation theorem of Riesz-Markov, a force of a continuous body f ∈ C 1 (κ∗ T S) has a representation by a collection of tensor measures (f0 , f1 ) ∈ C 0 (κ∗ T S)∗ ⊕ C 0 (κ∗ T S ⊗ T ∗ B)∗ . Consequently, the virtual work of a continuous body can be represented as1 δW = f(δκ) = B δκdf0 + B (κ∗ ∇)(δκ)df1 .