Abstract
This chapter contains the fundamental definitions and theorems relative to finite and infinitesimal deformations of a continuous system. Starting from the deformation gradient, the rotation tensor, right and left stretching tensors are defined making clear their use in defining the different aspects of deformation. Then, the left and right Cauchy–Green tensors are introduced with a complete analysis of their invariants. Starting from the displacement gradient, the infinitesimal deformations are defined. Finally, the compatibility conditions of a deformation are analyzed. After a section of exercises, the chapter ends with the introduction of the program Deformation, written with MathematicaⓇ [69].
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Notes
- 1.
From now on, capital indices are used to identify quantities defined in C ∗, while lowercase indices identify quantities in C.
- 2.
This hypothesis will be motivated in the next chapter.
- 3.
Here and in the following discussion the equality is intended to be satisfied by neglecting second-order terms in H.
- 4.
Note that I + W is orthogonal if we neglect second-order terms in H. In fact, \(\mathbf{(I + W)(I + W)}^{T} \simeq \mathbf{I + W + W}^{T}\mathbf{= I}\), since W is skew-symmetric.
- 5.
The Riemann–Christoffel tensor R is also referred as the curvature tensor and the condition (3.62) as Riemann’s theorem.
- 6.
If both the choices option=symbolic and simplifyoption=true are input, then the program cannot give the requested outputs. In this situation, the computation can be stopped from the pop-up menu of MathematicaⓇ [69]. To interrupt the computation relative to the last input line, choose Kernel → Interrupt Evaluation or Kernel → Abort Evaluation; to interrupt all the running processes, choose Kernel → Quit → Local → Quit. In this last case, the package Mechanics.m must be launched again before any other application.
References
J. Marsden, T. Hughes, Mathematical Foundations of Elasticity (Dover Publications, Inc. New York, 1983)
S. Wolfram, Mathematica ®;. A System for Doing Mathematics by Computer (Addison-Wesley Redwood City, California, 1991)
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© 2014 Springer Science+Business Media New York
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Romano, A., Marasco, A. (2014). Finite and Infinitesimal Deformations. In: Continuum Mechanics using Mathematica®. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-1604-7_3
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DOI: https://doi.org/10.1007/978-1-4939-1604-7_3
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