Cauchy-Green strain measure

Any second-order tensor has a number of invariants associated with it. One such is the trace of the tensor, equal to the sum of its diagonal terms, applicable to any strain tensor. We define the first invariant h as the trace of the Cauchy-Green strain measure tr(C) ... 

Note 2.4 (Generalized strain measure (Hill 1978)9t). Since the right Cauchy-Green tensor C = is symmetric and the components are real numbers, there are three real eigenvalues that are set as A ( = 1,2,3) and the corresponding eigenvectors are given by Ni then we have... 

We now create the theory of finite strain by replacing the engineering strains on the left-hand side of Equation (3.39) by measures of finite strain. We choose as a measure of finite strain the Cauchy-Green measure, which (see Equations (3.16) and (3.17)) has principal values X, and X jj. Then,... 

The strain energy function 77 is a function of the components of some measure of strain, such as the stretch V or the Cauchy-Green measure C. 77 is a physical quantity with a numerical value upon which all observers will agree - it is independent of the axis set. On the other hand, the eomponents of V and C are entirely dependent on the axis set. Unless a function of these eomponents is chosen with care, it will itself be dependent on the axis set and so will be inadmissible as a strain energy function. This places restrictions on the form of 77, which can be approached in two ways ... 

In the nonlinear analysis of solids, there are two kinds of nonlinearities - the material nonlinearity and the geometric nonlinearity. The material nonlinearity is basically due to the existence of a nonlinear relation toween the stresses and the strains. The geometric nonlinearity implies that the strains involved are very large so that all the stress measures (Cauchy stress, Kirchhoff stress, first and second order Piola-Kirchhoff stresses, etc.) and the strain measures (engineering strain, natural strain, Green-Lagrange strain, etc.) are very much different in meaning and in numerical values. 

We have shown here that the Cauchy-Green and Finger tensors are not equivalent measures of finite strain, which is a very important fact to remember in the formulation of constitutive equations, as is discussed in Chapter 3. 

Before proceeding, some definitions are useful. Stress is the ratio of the force on a body to the cross-sectional area of the body. The true stress refers to the infinitesimal force per (instantaneous) area, while the engineering stress is the force per initial area. Strain is a measure of the extent of the deformation. Normal strains change the dimensions, whereas shear strains change the angle between two initially perpendicular lines. In correspondence with the true stress, the Cauchy (or Euler) strain is measured with respect to the deformed state, while the Green s (or Lagrange) strain is with respect to the undeformed state. 

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