Unloading path

Assume pressure, needed to take the elastic—plastic boundary to radius r corresponds to point B (see Fig. 3). Then provided the cylinder unloads elasticady when the internal pressure is removed, ie, unloading path BE is paradel to OA, the residual shear stress distribution is as fodows. 

The effect of subjecting a thick-waded cylinder to a pressure greater than the yield pressure and then releasing the pressure is to put the material adjacent to the bore of the cylinder in compression while the outer layers remain in tension. On subsequent repressurization the cylinder wid, to a first approximation, retrace the unloading path BE (see Eig. 3) so that the cylinder withstands elasticady a pressure equal to that appHed originally. 

Figure 1.9. (a) Pressure release in a multianvil apparatus as function of decompression time. The shortest unloading path to ambient pressure takes about 1 s. (b) Fast unloading in a multianvil experiment with starting conditions of2400 0 and 25 GPa... 

Figure 2. Argillites mechanical behaviour under classic tr iaxial compression tests, 2a) Triaxial test with unloading paths for determining damage evolution, 2b) Triaxial test with increasing step by step the confining pressure on the post-failure phase.

Figure 8.1 shows stress-strain curves of atactic polystyrene (PS) in compression at 295 K for two structures with different initial states well annealed, i.e., furnace cooled from Tg + 20 K to room temperature, and rapidly quenched into ice water (Hasan and Boyce 1993). In both cases there is a gradual transition to fully developed plasticity that is reached at the peak of a yield phenomenon which is more prominent in the annealed material. Both curves show several unloading histories, starting with one close to the upper yield peak. All unloading paths show prominent Bauschinger effects of plastic strain recovery that is independent of the pre-strain. These indicate the presence of strain-induced back stresses and some recoverable stored elastic strain energy. In both cases the flow stress moves toward a unique flow state attained at a strain of around 0.3. 

Notable features were the pronounced hysteresis, unrecovered strain and Mullins effect (whereby re-loading follows a stress-strain path closer to the unloading path than the original loading path). From curves such as these we calculated several quantifiers of the inelasticity. Consider the first cycle for material PU1, shown above. It defines three zones A, B and C in the stress-strain diagram. 

The hypoplastic material law describes the stress rate as a function of stress, strain rate and void ratio and is well-suited for cohensionless, granular materials. It reliably predicts the non-linear and inelastic behaviour of soil due to its rate-type formulation which ensures a realistic modelling of loading and unloading paths. Also, by virtue of the rate-type formulation, numerical implementation of the hypoplastic material law is significantly simplified. The hypoplastic constitutive equation by von Wolffersdorff (1996) is given in equation 16. 

The Mullins effect [79] is a strain induced softening phenomenon, which is associated mainly with a significant reduction in the stress at a given level of strain during the unloading path as compared with the stress on initial loading in stress-strain cyclic tests [80] (Fig. 18). 

Seismic Analysis of Steel-Concrete Composite Buildings Numerical Modeling, Fig. 9 Loading and unloading paths of bilinear kinematic model... 

When a fine-grained, poly crystalline wire of a ductile material is deformed under an axial load the stress-strain curve obtained has the form shown by the full line in Fig. 3.2(a), where the initial area of cross-section is used to calculate the stress from the applied forces. Permanent deformation occurs when the stress exceeds the yield stress Oy so that, on reducing the stress to zero, a permanent deformation results. For example, if the specimen is unloaded when the point B is reached, the unloading path is not BAO but BC, which has approximately the same slope as OA. Of the total strain corresponding to the point B, the elastic strain... 

Material nonlinearity may be hyperelastic or elasto-plastic. The difference between the behavior of an elastic and elasto-plastic material is seen on unloading as in the former case the unloading path coincides with the loading path whereas in the latter case a different unloading path results in permanent deformation when the load has been removed. Elasto-plastic behavior is characterized by a linear region up to the yield point, after which soflening behavior is seen. Hyperelastic materials such as elastomers exhibit nonlinear elastic response for even large strains. 

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