Kinematics_of_CM_02_Deformation_Strain

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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Section 2.2 Solid Mechanics Part III Kelly 208 It is customary to denote the motion vector-function χ in 2.1.3 simply by x , i.e. () t , X x x = , so that J i iJ X x F = = = , Grad x X x F Deformation Gradient (2.2.2) with J iJ i dX F dx d d = = , X F x action of F (2.2.3) Lower case indices are used in the index notation to denote quantities associated with the spatial basis {} i e whereas upper case indices are used for quantities associated with the material basis I E . Note that X X x x d d = is a differential quantity and this expression has some error associated with it; the error (due to terms of order 2 ) ( X d and higher, neglected from a Taylor series) tends to zero as the differential 0 X d . The deformation gradient (whose components are finite) thus characterises the deformation in the neighbourhood of a point X , mapping infinitesimal line elements X d emanating from X in the reference configuration to the infinitesimal line elements x d emanating from x in the current configuration, Fig. 2.2.2. Figure 2.2.2: deformation of a material particle Example Consider the cube of material with sides of unit length illustrated by dotted lines in Fig.
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 3 taught by Professor Staff during the Fall '11 term at Auckland.

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Kinematics_of_CM_02_Deformation_Strain - Section 2.2 2.2...

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