Green strain, defined

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) ... 

The inverse of the Cauchy-Green tensor, Cf, is called the Finger strain tensor. Physically the single-integral constitutive models define the viscoelastic extra stress Tv for a fluid particle as a time integral of the defonnation history, i.e. 

As in the green state, the strain amplification, due to the limited volume of the actually deformable phase, remains the first-order result of filler incorporation. For a given macroscopic deformation, the actual deformation of the polymeric matrix will always be much higher, obviously depending on the filler volume and its structure, which defines occluded rubber volume. 

The next strain measure is defined on the undeformed configuration, known as the Green— Lagrange strain E ... 

Although the deformation is described by U, other measures of strain can be useful. One example is provided by Green s strain tensor G, defined as... 

Green s strain tensor vanishes in an undeformed system G = 0. For small deformations, it converges to the strain tensor e, defined in section 2.2.2. 

For our studies, axenic cultures of Ochromonas danica were grown in a chemically defined, heterotrophic medium, as described by Aaronson (5). The green photosyn hetic cells were cultured under continuous illumination (about 5 W/m" ), at 24°C the dark-bleached strain was obtained keeping the cells in complete darkness at the same temperature. 

The intemal-vaiiables formulation is not the only way to define the internal energy. Following the seminal work of [122, 148] Boltzmann and Green and Rivlin and successively Coleman and Noll [124] proposed constitutive relations for which the stress a time t depends upon the entire history of deformation up to the current time instant. However, the definition of the internal energy was developed accordingly and in agreement with the fading memory properties, i.e., strains which occurred in the distant past have less influence on the present value of yr than those which occurred in the more recent past. 

The simplest (and weakest ) definition of an elastic material is one for which the stress depends only on the current strain these materials are termed Cauchy elastic. A subset of these materials is occupied by those for which the strain energy depends only on the current strain. These are termed Green elastic or hyperelastic and for these the strain energy is a function of the current strain only, and fully defines the material behaviour. For Cauchy... 

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