Measures of Finite Strain

When a material translates or rotates, it moves as a rigid body. In addition, it can deform (i.e., the distances between neighboring points can change). In general, we can relate the distanee vector Jx at time t between two neighboring points in a body to the distanee veetor rix at time / between the same two points after motion and deformation through an equation of the type 

Subtracting Eq. (10.6.2) from Eq. (10.6.3) and using a Taylor series expansion 

In general, the deformation gradient depends on position. However, if it is independent of position, the displacement is said to be homogeneous. Note that a nonzero value of the deformation gradient does not, ipso facto, imply that deformation has taken place for this to happen, distances between neighboring points must change. Let us pursue this point further. 

Additional discussion of strain measures may be found in the Hterature [9]. 

Solution From the problem statement, it is elear that 

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We now create the theory of finite strain by replacing the engineering strains on the left-hand side of Equation (3.39) by measures of finite strain. We choose as a measure of finite strain the Cauchy-Green measure, which (see Equations (3.16) and (3.17)) has principal values X, and X jj. Then,... 

We have shown here that the Cauchy-Green and Finger tensors are not equivalent measures of finite strain, which is a very important fact to remember in the formulation of constitutive equations, as is discussed in Chapter 3. 

This consists of experimental measurements of stress-strain relations and analysis of the data in terms of the mathematical theory of elastic continua. Rivlin7-10 was the first to pply the finite (or large) deformation theory to the phenomenologic analysis of rubber elasticity. He correctly pointed out the above-mentioned restrictions on W, and proposed an empirical form... 

Consider the relationship of the finite-gap flow in the region outside the boundary layer with the semi-infinite flow in the inviscid region. Evaluate and plot profiles of alternative measures of the strain rate as... 

Such total finite strains are known as natural or logarithmic strains, the use of which was first suggested by Ludwig (1909). The engineering strain is frequently used as a measure of finite linear strain (e.g., the percentage elongation in a simple tensile test is usually quoted). The relationship between these two measures can be derived as follows ... 

Given limits to the time resolution with which wave profiles can be detected and the existence of rate-dependent phenomena, finite sample thicknesses are required. To maintain a state of uniaxial strain, measurements must be completed before unloading waves arrive from lateral surfaces. Accordingly, larger loading diameters permit the use of thicker samples, and smaller loading diameters require the use of measurement devices with short time resolution. 

In particular it can be shown that the dynamic flocculation model of stress softening and hysteresis fulfils a plausibility criterion, important, e.g., for finite element (FE) apphcations. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. From the simulations of stress-strain cycles at medium and large strain it can be concluded that the model of cluster breakdown and reaggregation for prestrained samples represents a fundamental micromechanical basis for the description of nonlinear viscoelasticity of filler-reinforced rubbers. Thereby, the mechanisms of energy storage and dissipation are traced back to the elastic response of tender but fragile filler clusters [24]. 

Fig. 15. Measured shear strain during creep under a constant shear stress and viscoelastic recovery after cessation of shear for PDMS near the gel point [71] plotted against the time. The solid lines are predicted by the gel equation for finite strain...

The rheological properties of gum and carbon black compounds of an ethylene-propylene terpolymer elastomer have been investigated at very low shear stresses and shear rates, using a sandwich rheometer [50]. Emphasis was given to measurements of creep and strain recovery at low stresses, at carbon black flller contents ranging between 20 and 50% by volume. The EPDM-carbon black compounds did not exhibit a zero shear rate viscosity, which tended towards in-Anity at zero shear stress or at a finite shear stress (Fig. 13). This was explained... 

Given a sufficiently long time of creep, the velocity of creep will decelerate to zero and y t) attains an equilibrium limit if a viscoelastic solid is being measured. On the other hand, if the material is a viscoelastic liquid, the velocity of creep will decelerate to a finite constant value. Viscoelastic steady state is achieved, and y t) increases indefinitely. The creep experiment has a second part when the stress is set to zero after a period of creeping. A portion or all of the strain accumulated during creeping is then recovered as a function of time for a viscoelastic liquid or solid, respectively.For a viscoelastic liquid, the portion that is permanent deformation and irrecoverable reflects the contribution of viscous flow to the total deformation accumulated during creep. Since a viscoelastic solid does not flow, all of its creep deformation is recoverable. 

Thus, G t) = —a t)/Xo. This result is expected from the definition of G t) as used in Boltzmann s superposition principle. The way in which G t) is obtained from Eq. (4.27) also illustrates an experimental problem encountered in the measurement of G t). Experimentally the application of a strain involves the movement of a mechanical device, often a motor, which has a rate limit. Thus, e cannot be infinitely small experimentally. At Ao 0.1, the order of 0.05 s for e is basically the state of the art. How an experiment is affected by a finite e is a relative matter. If the relaxation times of G t), which are the interest of study, are sufficiently larger than e, errors caused by the finite e are negligible. 

With very small strains E and e become equal. However, e is not a realistic measure of large finite strain thus the use of s is favored under such conditions. For example, in simple compression of a billet from ho to h,... 

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