Invariant energy density

The value of the dot product is a measure of the coalignment of two vectors and is independent of the coordinate system. The dot product therefore is a true scalar, the simplest invariant which can be formed from the two vectors. It provides a useful form for expressing many physical properties the work done in moving a body equals the dot product of the force and the displacement the electrical energy density in space is proportional to the dot product of electrical intensity and electrical displacement quantum mechanical observations are dot products of an operator and a state vector the invariants of special relativity are the dot products of four-vectors. The invariants involving a set of quantities may be used to establish if these quantities are the components of a vector. For instance, if AiBi forms an invariant and Bi are the components of a vector, then Az must be the components of another vector. 

From the viewpoint of the mechanics of continua, the stress-strain relationship of a perfectly elastic material is fully described in terms of the strain energy density function W. In fact, this relationship is expressed as a linear combination erf the partial derivatives of W with respect to the three invariants of deformation tensor, /j, /2, and /3. It is the fundamental task for a phenomenologic study of elastic material to determine W as a function of these three independent variables either from molecular theory or by experiment. The present paper has reviewed approaches to this task from biaxial extension experiment and the related data. The results obtained so far demonstrate that the kinetic theory of polymer network does not describe actual behavior of rubber vulcanizates. In particular, contrary to the kinetic theory, the observed derivative bW/bI2 does not vanish. 

No solution flow or ion-selective membrane is required and the volume of electrolyte required is low. The total amount of acid is invariant since only proton transfers are involved. The OCV of the cell is 0.6 V and the theoretical energy density is 67 Wh/kg. 

The free energy density of the network Fj V is written as a power series in the difference of these invariants from their values in the undeformed network (A c = A., = A = 1) ... 

Note that the first two terms on the right hand side tell us that there is a change in the strain energy density that is incurred because the defect has moved and it has dragged its elastic fields with it. However, these terms do not reflect the presence of broken translational invariance. By exploiting equality of mixed partials, and by rearranging terms via a simple application of the product rule for differentiation, this expression may be rewritten as... 

A natural extension of linear elasticity is h rperelasticity.l l H rperelasticity is a collective term for a family of models that all have a strain energy density that only depends on the applied deformation state. This class of material models is characterized by a nonlinear elastic response, and does not capture yielding, viscoplasticity, or time-dependence. The strain energy density is the energy that is stored in the material as it is deformed, and is typically represented either in terms of invariants... 

The definition of a invariant with respect to positioning of the dividing surface can be worked out, if one analyzes trends in the/z)-pc(z) function within the discontinuity surface. The specified quantity has the same value in the bulk of both phases, equal to the negative external pressure (Fig. 1-4). Within the discontinuity surface, pressure p has a tensor nature, making Pascal s law invalid. Meanwhile, the concentration and pressure dependence of the surface energy density,/ given by eq. (1.1), is valid only in the regions where Pascal s law holds, i.e., where pressure is a scalar quantity (direct summation of a scalar and a tensor within the same equation is not permitted). 

An alternative isotropic strain energy density function which can predict the appropriate type of stress stiffening for blood vessels is an exponential where the arguments is a polynomial of the strain invariants. 

The invariant probability density, which is associated to a surface of constant kinetic energy, can be written as a generalized function using the Dirac notation ... 

Isotropic hyperelastic materials For this model, the strain energy density function is written in terms of the principal stress invariants I, h, h). Equation 1 becomes... 

The K-BKZ Theory Model. The K-BKZ model was developed in the early 1960s by two independent groups. Bernstein, Kearsley, and Zapas (70) of the National Bureau of Standards (now the National Institute of Standards and Technology) first presented the model in 1962 and published it in 1963. Kaye (71), in Cranfield, U.K., published the model in 1962, without the extensive derivations and background thermodynamics associated with the BKZ papers (82,107). Regardless of this, only the final form of the constitutive equation is of concern here. Similar to the idea of finite elasticity theory, the K-BKZ model postulates the existence of a strain potential function U Ii, I2, t). This is similar to the strain energy density function, but it depends on time and, now, the invariants are those of the relative left Cauchy-Green deformation tensor The relevant constitutive equation is... 

The invariants are needed to express the bending energy of membranes. This is because the bending energy density of a fluid, laterally isotropic membrane can consist only of terms that do not depend on the orientation of the xy cross. Multiplying each of them with a coefficient leads to one of the standard formulas for the bending energy surface density. 

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