Elastic unloading

For, suppose that this inequality is not true, and that becomes negative at t = t. The smoothness assumptions that have been made imply that the integrand must continue to be negative for at least a short finite-time interval Ar. Now a second cycle may be chosen which is identical with tj) over the time interval (tf, tf + At), but in which elastic unloading commences at time t f + At. Since is zero during elastic portions of the cycle, the integral of is negative for this second cycle, which contradicts (5.50). 

Fig. 74 Schematic representation of the tensile curve of a fibre during (1) first loading, (2) hypothetical elastic unloading, (3) unloading and (4)second loading. Note that the second part of curve 4 practically coincides with curve 1 [1,6]... 

At the peak force (position A in Fig. 8.5a), there are two possibilities for the next strain state Elastic unloading along path AU, and further plastic straining along the path AN. A non-uniform strain state develops, as parts of the specimen elastically unload, and the plastic strain in one region increases to form a neck. The plastic deformation of the neck is partially driven by elastic energy release from the rest of the specimen. The condition that A is at the maximum in the force-extension or force-strain curve can be written... 

As indicated in Reference [31], in the case when elastic unloading to zero stress is not physically achievable due to the possible onset of reverse plastic deformation before the state of zero stress is reached, the intermediate configuration can be conceptually introduced by a virtual unloading to zero stress, locking all inelastic stmctural changes that would occur during the actual unloading. 

Figure 4.4 Multiplicative decomposition of an undeformed body to an intermediate body by elastically unloading tbe deformed body to zero stress...

Neglecting the fibre interaction through the matrix is a crude approximation, and the fibre bundle strength can only be considered a lower limit. An improved model is obtained by considering that when one microfibril breaks, the elastic unloading in the microfibril below and above the fracture leads to the development of a shear stress r at the fibre-matrix interface. This shear stress transmits the load from the matrix back to the fibre, and the stress in the fibre reaches its former value ct at a distance x below and above the fracture section, which is given by ... 

The experimental technique is shown in Fig. 7.27. The value of Kic is calculated from the highest load, where the elastic unloading comphance line deviates from linearity (Fig. 7.27a). The form of the cycle is also shown. 

Fig. 7.15. Schematic representatioii of the variation of substrate curvature k as a function of the temperature change T for an arbitrary bilayer for which as < f- It is assumed that the film material is an elastic-ideally plastic solid whose properties do not vary with tei er ure. The onset of plastic yielding at T = Ty and of complete yielding at T = Tpi of the film material and the corresponding substrate curvatures, Ky and Kpi, respectively, are marked in the figure. Note that reversing the temperature from T < T ei to T = 0 produces only elastic unloading with residual curvature Kres-...

Conditions governing the onset of reverse yielding during thermal cycling can also be identified for the present problem ( ). As the temperature is raised beyond Ty, there exists another critical temperature, say T = Tr ei) such that unloading at or prior to this temperature leads only to elastic deformation without reversed plastic flow in the film. If the tensile and compressive yield strengths of the elastic-ideally plastic film are equal, elastic unloading is ensured when the temperature is reversed from a value smaller than... 

Kd = Post-elastic stiffness (kip/inch) (KN/mm ) = Elastic (unloading) stiffness (kip/inch)... 

Figure 3. The distribution of effective remanent strain near a growing crack in a ferroelastic material. The active switching zone, elastically unloaded wake, and unloaded elastic sector behind the crack tip are each denoted on the illustration. The material law is given in Section 2, and the material parameters for this computation are s if/cro 5, ffo oq, m 0.01 and v 0.25.

There are some departnres or exceptions to the above desaiptions of stress (strain) behavior for viscoelastic polymers. For example, a stress-strain cnrve is shown in Figure 2.2a for an injection molded acetal copolymer. It shows strain softening with the yield and ultimate strengths approximately eqnal. It also shows an elastic unloading modulus much less than the initial elastic modnlns when loaded. This results in a permanent strain that is much smaller than the nltimate strain. For example, an ultimate... 

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