By Simon R. Eugster
This examine monograph discusses novel methods 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 main of digital paintings serves because the basic precept of mechanics. in response to the conception of analytical mechanics that forces of a mechanical procedure 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 technique is especially handy to formulate precipitated beam theories, it truly is crucial in geometric continuum mechanics while the assumptions at the actual house are comfortable and the distance is modeled as a gentle 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 triggered beam theories.
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Extra resources for Geometric Continuum Mechanics and Induced Beam Theories (Lecture Notes in Applied and Computational Mechanics)
23, the virtual displacement field of a continuous body δκ ∈ C 1 (κ∗ T S) can be identified with an element of the tangent space Tκ Q. This follows the tradition of analytical mechanics, where the virtual displacements are tangent vectors to the finite-dimensional configuration manifold, cf. . 25 (Affine Connection) Let u, v ∈ Γ (T M), f ∈ C ∞ (M). An (affine) connection on M is a mapping ∇ which assigns to every pair u, v another vector field ∇u v ∈ Γ (T M) with the following properties: (a) ∇u v is bilinear in u and v , (b) ∇ f u v = f ∇u v , (c) ∇u ( f v) = f ∇u v + u( f )v .
Segev, Forces and the existence of stresses in invariant continuum mechanics. J. Math. Phys. 27(1), 163–170 (1986) 2. W. G. Tait, Treatise on Natural Philosophy, vol. 1 (Clarendon Press, Oxford, 1867) 3. R. Kirchhoff, Vorlesungen über Mathematische Physik: Mechanik, vol. G. Teubner, Leipzig, 1876) 4. W. Noll. The foundations of classical mechanics in the light of recent advances in continuum mechanics, in: The Axiomatic Method, with Special Reference to Geometry and Physics, p. 266–281 (1959) 5.
A nowhere vanishing volume form on B. 11) where Pi = ρ−1 πi123 and P = Pi (dx i ◦ κ) ⊗ ∂θ j . Notice that P as an independent object, generally denoted as the stress tensor, is not a tensor field. By splitting-off of the volume element dV from the variational stress π, the tensor property is destroyed. At most, P can be considered as a tensor valued density. This explains the awkward transformation rules for the different stress tensors in classical nonlinear continuum mechanics, cf. . 12) where we introduced the commonly used notation of the double contraction.