Strain energy density theory

The parameters governing crack propagation have been proposed by Sih and Madenci [8] with the strain energy density theory. The strain energy density function, d Wld V, in front of the crack tip has been expressed in the form... 

From Figure 27.12, it is interesting that the intercept of G, at Ta = 0 seems to be the same in these plots. From the strain energy density theory discussed above, Equation (17) can be rewritten as... 

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. 

Statistical network theory leads to the expression of the strain energy density (energy stored in unit volume of the rubber) in terms of the extension ratios ... 

If a crystal is subjected to small strain elastic deformation it is convenient to imagine the energetics of the strained solid in terms of the linear theory of elasticity. As we noted in chap. 2, the stored strain energy may be captured via the elastic strain energy density which in this context is a strictly local quantity of the form... 

Li, X.B. Zuo, YJ. Ma, C D. 2005. Failure Criterion of Strain Energy Density and Catastrophe Theory Analysis of Rock Subjected to Static-dynamic Coupling Loading. Chinese Journal of Rock Mechanics and Engineering 24(16) 2814-2824. 

There is an extensive body of literature describing the stress-strain response of rubberlike materials that is based upon the concepts of Finite Elasticity Theory which was originally developed by Rivlin and others [58,59]. The reader is referred to this literature for further details of the relevant developments. For the purposes of this paper, we will discuss the developments of the so-called Valanis-Landel strain energy density function, [60] because it is of the form that most commonly results from the statistical mechanical models of rubber networks and has been very successful in describing the mechanical response of cross-linked rubber. It is resultingly very useful in understanding the behavior of swollen networks. 

Here we begin with a sample of rabber having initial dimensions l, I2, I3. We deform it by an amount A/, A/2, A/3 and define the stretch (ratio) in each direction as A, = (/, -I- A/,)//, = ///,. The purpose of Finite Elasticity Theory has been to relate the deformations of the material to the stresses needed to obtain the deformation. This is done through the strain energy density function, which we will describe using the Valanis-Landel formalism as IT(A, A2, A3). Importantly, as we will see later, this is the mechanical contribution to the Helmholtz free energy. Vala-nis and Landel assumed [60] that the strain energy density function is a separable function of the stretches A, ... 

Von MiSOS Yiold Critorion. The Von Mises yield criterion (also known as the maximum distortional energy criterion or the octahedral stress theory) (25) states that yield will occur when the elastic shear-strain energy density reaches a critical value. There are a number of ways of expressing this in terms of the principal stresses, a common one being... 

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... 

Similar to the idea of finite elasticity theory, the K-BKZ model postulates the existence of a strain potential function U(Ji, I2, i). 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... 

In 1979, an elegant proof of the existence was provided by Levy [10]. He demonstrated that the universal variational functional for the electron-electron repulsion energy of an A -representable trial 1-RDM can be obtained by searching all antisymmetric wavefunctions that yield a fixed D. It was shown that the functional does not require that a trial function for a variational calculation be associated with a ground state of some external potential. Thus the v-representability is not required, only Al-representability. As a result, the 1-RDM functional theories of preceding works were unified. A year later, Valone [19] extended Levy s pure-state constrained search to include all ensemble representable 1-RDMs. He demonstrated that no new constraints are needed in the occupation-number variation of the energy functional. Diverse con-strained-search density functionals by Lieb [20, 21] also afforded insight into this issue. He proved independently that the constrained minimizations exist. 

Nonequilibrium conditions may occur with respect to disturbances in the interior of a system, or between a system and its surroundings. As a result, the local stress, strain, temperature, concentration, and energy density may vary at each instance in time. This may lead to instability in space and time. Constantly changing properties cannot be described properly by referring to the system as a whole. Some averaging of the properties in space and time is necessary. Such averages need to be clearly stated in the utilization and correlation of experimental data, especially when their interpretations are associated with theories that are valid at equilibrium. Components of the generalized flows and the thermodynamic forces can be used to define the trajectories of the behavior of systems in time. A trajectory specifies the curve represented by the flow and force components as functions of time in the flow-force space. 

Here, is again the surface work, S is the surface energy as previously defined and is the loss function dependent on crack speed, temperature and the strain, eo, applied to the specimen. The theory gives explicitly in terms of the energy density distribution in the specimen and the plastic or visco-elastic hysteresis of the material. 

The rigidity of bispidine ligands has been analyzed on the basis of cavity size calculations with molecular mechanics [and density functional theory (DFT) — in order to check the accuracy and rehability of the force-field calculations] and a comparison of the computed and corresponding experimental structures and their analysis, based on the computed strain energy curves (42, 69, 189, 201). 

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