Deformation, linear

Contour Length Relaxation. Doi and Edwards have proposed an additional, faster relaxation mechanism for which they use the term contour length relaxation (5). As shown in Figure 3, contour length relaxation is a process by which a deformed linear (or star shaped) chain should retract towards the center of mass of the chain. Since the overall contour length increases upon deformation, the proposal by Doi and Edwards is that the deformed chain would want to resume the same chain density along the overall contour of the chain as that... 

The terms are arranged into sections dealing with basic definitions of stress and strain, deformations used experimentally, stresses observed experimentally, quantities relating stress and deformation, linear viscoelastic behaviour, and oscillatory deformations and stresses used experimentally for solids. The terms which have been selected are those met in the conventional mechanical characterization of polymeric materials. 

Dumont and Bougeard (68, 69) reported MD calculations of the diffusion of n-alkanes up to propane as well as ethene and ethyne in silicalite. Thirteen independent sets of 4 molecules per unit cell were considered, to bolster the statistics of the results. The framework was held rigid, but the hydrocarbon molecules were flexible. The internal coordinates that were allowed to vary were as follows bond stretching, planar angular deformation, linear bending (ethyne), out-of-plane bending (ethene), and bond torsion. The potential parameters governing intermolecular interactions were optimized to reproduce infrared spectra (68). 

LVE fluids (3.3-8a) Constant Zero No Predicts small deformation linear response... 

If we now perform a creep experiment, applying a constant stress, a0 at time t = 0 and removing it after a time f, then the strain/ time plot shown at the top of Figure 13-89 is obtained. First, the elastic component of the model (spring) deforms instantaneously a certain amount, then the viscous component (dashpot) deforms linearly with time. When the stress is removed only the elastic part of the deformation is regained. Mathematically, we can take Maxwell s equation (Equation 13-85) and impose the creep experiment condition of constant stress da/dt = 0, which gives us Equation 13-84. In other words, the Maxwell model predicts that creep should be constant with time, which it isn t Creep is characterized by a retarded elastic response. 

On removal of the applied stress, the material experiences creep recovery. Figure 14.5 shows the creep and the creep recovery curves of the Maxwell element. It shows that the instantaneous application of a constant stress, Oo, is initially followed by an instantaneous deformation due to the response of the spring by an amount Oq/E. With the sustained application of this stress, the dashpot flows to relieve the stress. The dashpot deforms linearly with time as long as the stress is maintained. On the removal of the applied stress, the spring contracts instantaneously by an amount equal to its extension. However, the deformation due to the viscous flow of the dashpot is retained as permanent set. Thus the Maxwell element predicts that in a creep/creep recovery experiment, the response includes elastic strain and strain recovery, creep and permanent set. While the predicted response is indeed observed in real materials, the demarcations are nevertheless not as sharp. 

It will be clear that in the field of block copolymers, fallow aretis are present to a high degree. The difficulty of the measurement of viscoelastic properties is the sensitiven of the gels to (shear) deformations linear viscoelasticity is, in general, only measurable at very small deformations, so that very small stresses have to be measured. Special sensitive instruments are needed for these studies. 

Good fibrous reinforcement are generally britde in character they deform linearly to failure widiout yielding. This attribute creates a situation in which, in the presence of a notch or hole under static tension m conqiression test conditions, the fiber reinforced composite behaves more like a tnittle material than metal. This issue has been a source of concern in the materials development and selection activities, as well as in en eering design (27). 

Uniaxial Extension. A rubber strip of original length Lo is stretched uni-axially to a length L, as illustrated in Figure 1. The stretch and elongation are AL = L — Lq and k = LILo, respectively. The strain e (also known as the relative deformation, linear dilation, or extension) and the elongation or extension ratio X are related by... 

Dynamic mechanical methods (typically oscillatory parallel plate rheometry) are commonly used to measure the dynamic mechanical properties from the liquid state to the solid state. By using small-amplitude oscillatory deformations (linear viscoelastic regime), the dynamic storage and loss moduli can be obtained. From these quantities, the viscosity and modulus can be calculated (71) (see Dynamic Mechanical Analysis). 

FIGURE 4.31. Distribution of the director for (a) a deformation quadratic in the field (b) a deformation linear in the field and (c) diffraction patterns from phase lattices caused by the quadratic and linear flexoelectric effect [188]. 

Derivation of viscoelastic beam deflection equation in the time domain It is instructive to derive the deflection equation for a viscoelastic beam without resorting to Laplace transforms. Consider the undeformed and deformed beam shown in Fig. 8.5. Making the assumptions (the same as in elementary solid mechanics) of small deformations, linear behavior, and a non-warping cross-sections (plane sections remain plane) will give the relations. 

Incompressibility For small deformation linear elastic problems incompressibility is assured if Poisson s ratio is equal to 0.5, which also means that the bulk modulus is infinite (see Eq. 9.6). Under this assumption then, V = 0.5 and Ko = Under the same conditions Poisson s ratio for an incompressible viscoelastic material is also a constant 0.5 and. 

Elastic materials deform linearly according to Hooke s law, and as they are stretched their constituent molecules become displaced from their equilibrium positions. A simple mechanical model can be used to represent the behavior of the material as a lattice of balls connected by springs (Figure 1.18). This elastic behavior holds until the applied force is large enough that the material begins to deform irreversibly. Ductile materials exhibit a plastic deformation regime at stresses above the yield stress, whereas a brittle material will fracture at this point. 

H B) Nonlinear analysis with shear deformation ( - ) Nonlinear analysis without shear deformation ( ") Linear analysis with shear deformation ( ) Linear analysis with out shear deformation... 

To obtain an optimum it may be desirable to employ the variational theory for optimizing the total energy of the system with respect to any deformation under certain constrained conditions. Since highly crosslinked network systems are fairly "rigid," without losing the generality, they may be considered to deform linearly elastic. This is particularly true if deformation is small. 

---