Density strain energy

Newton s law of motion F = md r/df applied to the volume element takes the form 

Now we can use the Taylor expansion for the strain energy density in terms of small 

This last equation demonstrates that the elastic constant tensor can be obtained by calculating the variations to second order in the potential energy density with respect to small strains. Using this expression, we can write the strain energy density as 

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Consider the vector percolation experiment shown in Fig. 11 applied to any 3D lattice in general. The stored strain energy density U in the lattice due to an... 

An alternative energy approach to the fracture of polymers has also been developed on the basis of non-linear elasticity. This assumes that a material without any cracks will have a uniform strain energy density (strain energy per unit volume). Let this be IIq. When there is a crack in the material this strain energy density will reduce to zero over an area as shown shaded in Fig. 2.65. This area will be given by ka where )k is a proportionality constant. Thus the loss of elastic energy due to the presence of the crack is given by... 

The stiffness matrix, Cy, has 36 constants in Equation (2.1). However, less than 36 of the constants can be shown to actually be independent for elastic materials when important characteristics of the strain energy are considered. Elastic materials for which an elastic potential or strain energy density function exists have incremental work per unit volume of... 

Kawabata, S. and Kawai, H. Strain Energy Density Functions of Rubber Vulcanizates from Biaxial Extension. Vol. 24, pp. 89 — 124. 

FIGURE 1.5 Energy release rate Gc versus length c of a bond line crack. U is the strain energy density. Block thickness H—2 mm, length L — 20 mm. (From Gent, A.N., Suh, J.B., and Kelly, III, S.G., Int. J. Non-Linear Mech., 42, 241, 2007. With permission.)... 

In order to apply the crack nucleation approach, the mechanical state of the material must be quantified at each point by a suitable parameter. Traditional parameters have included, for example, the maximum principal stress or strain, or the strain energy density. Maximum principal strain and stress reflect that cracks in rubber often initiate on a plane normal to the loading direction. Strain energy density has sometimes been applied as a parameter for crack nucleation due to its connection to fracture mechanics for the case of edge-cracked strips under simple tension loading. ... 

Due to the plane-specific namre of crack nucleation under multiaxial tests. Mars and Fatemi proposed the cracking energy density as an equivalence parameter that represents the portion of strain energy density available to be released as crack growth on a specific material plane. The form of the cracking energy density Wc is... 

Strain Energy Density Functions of Rubber Vulca-... 

II. The Strain Energy Density Function and the Phenomenolojpc Equation... 

III. Search for the Strain Energy Density Function of Vulcanized Rubbers. 95... 

In the present article, we summarize typical approaches to the evaluation of the strain energy density function from biaxial extension experiments and illustrate some intportant data. This article is not a review in the ordinary sense, as it deals to a large extent with a series of experiments carried out in our laboratory. By this we do not mean to bias or ignore any of the many important contributions by other authors. 

Our own experience, as well as that of other authors, has shown that very precise measurement for the stress-strain relationship under general biaxial deformation is required to investigate the behavior of the strain energy density function of rubber vulcanizates. Unfortunately, available biaxial extension data are still too meager to deduce the general form of the strain energy density function of rubber-like substances. We wish to take this opportunity to summarize the principal results from our recent efforts, in the hope that they may serve to illustrate the interesting and complex nature of the derivatives 31V/9/,- of such substances. 

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. 

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