Exercises of the course

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Space Structures – Exercises - A.Airoldi, M.Anghileri, C.Bisagni, L.Castelletti, L.Lanzi, A. Milanese – 1 – Exercises #1 Continuum Mechanics Space Structures – Exercises - A.Airoldi, M.Anghileri, C.Bisagni, L.Castelletti, L.Lanzi, A. Milanese – 2 – Continuum Mechanics – exercise #1 The figure sketches the uniaxial deformation of a cylinder. The process is described by the following relations: Zz Yy Xx R R L λ λ λ = = = Where 95 . 0 18 . 1 == == R r L l R L λ λ a) Determine the components of Green-Lagrange strain tensor and of the small strain tensor b) Assume that, in the deformed configuration, a uniformly distributed load P = 25 kN is applied to the end sections of the cylinders, which have an origin al undeformed radius of 6.5 mm. Evaluate the components of Cauchy, First Piola-Kirchoff and Second Piola-Kirchoff stress tensor. ------------ /rhombus6/rhombus6/rhombus6/rhombus6----------- a) The evaluation of the Green-Lagrange strain tens or requires the expression of the deformation gradient, which can be directly calculated from the definition: ( )           =                 ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =∂ ∂= R R L Z z Y z X z Z y Y y X y Z x Y x X x λ λ λ χ 00 00 00 X XF By applying the definition of Green-Lagrange strain tensor, we obtain: ( )           − −=           − − − =                     −                     = =−= 04875 . 000 004875 . 00 001962 . 0 100 010 001 2 1 100 010 001 00 00 00 00 00 00 2 1 2 1 2 2 2 R R L R R L R R L λ λ λ λ λ λ λ λ λ IFFE T The components of the small strain tensor can be ev aluated once the components of displacement are known. Such components can be derived by the descri ption of motion, which has been provided in the exercise. Hence:…