Kinematics_of_CM_02_Deformation_Strain - Section 2.2 2.2...

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Section 2.2 Solid Mechanics Part III Kelly 207 2.2 Deformation and Strain A number of useful ways of describing the deformation of a material are discussed in this section. Attention is restricted to the reference and current configurations. No consideration is given to the particular sequence by which the current configuration is reached from the reference configuration and so the deformation can be considered to be independent of time. In what follows, particles in the reference configuration will often be termed “undeformed” and those in the current configuration “deformed”. In a change from Chapter 1, lower case letters will now be reserved for both vector- and tensor- functions of the spatial coordinates x , whereas upper-case letters will be reserved for functions of material coordinates X . There will be exceptions to this, but it should be clear from the context what is implied. 2.2.1 The Deformation Gradient The deformation gradient F is the fundamental measure of deformation in continuum mechanics. It is the second order tensor which maps line elements in the reference configuration into line elements (consisting of the same material particles) in the current configuration. Consider a line element X d emanating from position X in the reference configuration which becomes x d in the current configuration, Fig. 2.2.1. Then, using 2.1.3, ( ) ( ) ( ) X χ X χ X X χ x d d d Grad = + = (2.2.1) A capital G is used on “Grad” to emphasise that this is a gradient with respect to the material coordinates 1 , the material gradient , X χ / . Figure 2.2.1: the Deformation Gradient acting on a line element 1 one can have material gradients and spatial gradients of material or spatial fields – see later X x F X d x d
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