1 Continuum mechanics

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Aerospace Structures – Exercises – 1 – Exercises #1 Continuum Mechanics Aerospace Structures – Exercises – 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       Where 95.0 18.1   R r L l R L   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 original undeformed radius of 6.5 mm. Evaluate the components of Cauchy, First Piola-Kirchoff and Second Piola-Kirchoff stress tensor. ----------------------- a) The evaluation of the Green-Lagrange strain tensor 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     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          IFFE T L R l r X Y Z x y z Aerospace Structures – Exercises – 3 – The components of the small strain tensor can be evaluated once the components of displacement are known. Such components can be derived by the description of mot ion, which has been provided in the exercise. Hence:     1 1 1    RR RR LL ZZZZzw YYYYyv XXXXxu    The small strain tensors results:         …