Potential elastic

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

The J value is defined as the elastic potential difference between the linear and nonlinear elastic bodies with the same geometric variables [52,53]. The elastic potential energy for a nonlinear elastic body is expressed 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... 

Nearest neighbors along a chain interact by means of a FENE (finitely extendible nonlinear elastic) potential... 

The bow stores elastic potential energy as the hunter pulls on it. The energy is transterred to the arrow as kinetic energy upon release. (Corbis Corporation)... 

Two repulsive contributions, osmotic and elastic contributions [31, 32], oppose the van der Waals attractive contribution where the osmotic potential depends on the free energy of the solvent-ligand interactions (due to the solvation of the ligand tails by the solvent) and the elastic potential results from the entropic loss due to the compression of ligand tails between two metal cores. These repulsive contributions depend largely on the ligand length, solvent parameters, nanopartide radius, and center-to-center distance ... 

There is an alternative and very direct way to generalize the Rouse-Zimm model for non-Gaussian chains. This approach takes advantage of the expression given by the original theory for the chain elastic potential energy in terms of normal coordinates ... 

FENE Finite extensible non-linear elastic potential... 

Elastic potential caused by an external stress Electric potential caused by an external electric field Constants... 

The next n flowing units (segments or molecules) will also change their elastic potential due to one single molecular displacement so that the total energy transfer is given by A ei = n A.i ei,. The shift r0 is related to n flowing units and with the law of Hooke we can write... 

Experimental Measuring the strain behavior (creep) at a given constant stress. At time t = 0 we subject our sample to an external stress a0, connected to an elastic potential A = o0rq/3 at every flow element. This elastic deformation is a disturbance of the thermodynamical equilibrium. At the same time this stress creates a purely elastic deformation y0 = o0/G0 of the whole body. 3... 

In contrast to the broad spectrum of activation energies AUt, for calculations we use only a single elastic potential A. The external shearing force causes an unequal internal stress leading to higher values for o0 at undangered spots, thus r0 in Eq. (23) may be lower (/ <> < 10-6 cm). We should not forget that every shear or tensile deformation y must be accomplished by a purely elastic deformation... 

We subject a sample of solid polymer material to a sudden deformation process y0, with an elastical stress o0 = Gy0 (G shear modules). The original strain gives rise to an increase of the molecular valency angles and the intermolecular distances. From these molecular deformations an elastic molecular potential A0 arises which in turn causes molecular displacements. These prevail in the direction of the original strain, decreasing the elastic potential A in that neighborhood. We can calculate... 

In order to derive the determining physical equation for the stress-deformation state of a two-component mixture, let us consider the expression for the change in the elastic potential energy during a continious transition from a liquid to a solid phase. Let this transition occur at time t = to and let the quantity of material that undergoes the transition be equal to the increase in the degree of crystallinity Act. The specific elastic potential characterizing the new state of the material up to the time of a new transition can be written as... 

The detailed expressions for both components of the total elastic potential are as follows ... 

Let us consider the n step of the transformation Aan at time tn. Now, it is possible to write down the equation for the elastic potential of the material up to time tn+i, when the next transformation Aan+l occurs ... 

Fig. 9.1. Left-hand side Representation of an elastic potential energy surface. It has the general form (6.35) with coupling strength parameter e = 0. In case (a), the equilibrium bond distance in the electronic ground state equals the equilibrium separation of the free BC fragment. The heavy arrows schematically indicate two representative trajectories starting at the respective FC points. Right-hand side The corresponding final state distributions.

The free energy of the solid matrix at time t is denoted by fl s(t). It involves at least two components the elastic potential iand the chemical potential x(rc of the crystals bound in the solid phase. At any time t, the elastic energy stored in the solid phase is denoted by I el(f) ... 

Figure 1.41. Potential energies for the bead-spring model LJ1—Lennard-Jones potential LJ2—van der Waals potential EXP1, EXP2—short-range polar potential FENE—finitely extensible nonlinear elastic potential.

---