Kinematics_of_CM_03_Deformation_Strain_Further

# Kinematics_of_CM_03_Deformation_Strain_Further - Section...

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Section 2.3 Solid Mechanics Part III Kelly 233 2.3 Deformation and Strain: Further Topics 2.3.1 Volumetric and Isochoric Deformations When analysing materials which are only slightly incompressible, it is useful to decompose the deformation gradient multiplicatively, according to ( ) F F I F 3 / 1 3 / 1 J J = = (2.3.1) From this definition { Problem 1}, 1 det = F (2.3.2) and so F characterises a volume preserving ( distortional or isochoric ) deformation. The tensor I 3 / 1 J characterises the volume-changing ( dilational or volumetric ) component of the deformation, with ( ) J J = = F I det det 3 / 1 . This concept can be carried on to other kinematic tensors. For example, with F F C T = , C F F C 3 / 2 T 3 / 2 J J = . (2.3.3) F and C are called the modified deformation gradient and the modified right Cauchy-Green tensor , respectively. The square of the stretch is given by { } X C X X C X ˆ ˆ ˆ ˆ 3 / 2 2 d d J d d = = λ ( 2 . 3 . 4 ) so that 3 / 1 J = , where is the modified stretch , due to the action of C . Similarly, the modified principal stretches are i i J 3 / 1 = , 3 , 2 , 1 = i (2.3.5) with 1 det 3 2 1 = = F (2.3.6) The case of simple shear discussed earlier is an example of an isochoric deformation, in which the deformation gradient and the modified deformation gradient coincide, I I = 3 / 1 J . 2.3.2 Relative Deformation It is usual to use the configuration at ) 0 , ( = t X as the reference configuration, and define quantities such as the deformation gradient relative to this reference configuration. As mentioned, any configuration can be taken to be the reference configuration, and a new

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Section 2.3 Solid Mechanics Part III Kelly 234 deformation gradient can be constructed with respect to this new reference configuration. Further, the reference configuration does not have to be fixed, but could be moving also. In many cases, it is useful to choose the current configuration ) , ( t x to be the reference configuration, for example when evaluating rates of change of kinematic quantities (see later). To this end, introduce a third configuration: this is the configuration at some time τ = t and the position of a material particle X here is denoted by ) , ( ˆ X
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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_03_Deformation_Strain_Further - Section...

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