Stress-strain behavior plastic deformation

This stress-strain behavior is consistent with the optic metallographic data which evidenced partial redistribution of hydrogen over the powder particles when the compacting temperature was increased to 400°C and uniform hydrogen distribution on additional annealing or during plastic deformation at T > 500°C. 

Fig. 10.60 Compressive stress-strain behavior of PS and LLDPE at 25°C and crosshead speed of 25.4 mm/min. At a compressive stress level of 20 MPa the deformation of the soft LLDPE is large, in the dissipative region and nearly twenty times the PS deformation, which is of the order of 0.04, in the elastic nondissipative range. [Reprinted by permission from B. Qian, D. B. Todd, and C. G. Gogos, Plastic Energy Dissipation (PED) and its Role in Heating/Melting of Single Component Polymers and Multi-component Polymer Blends, Adv. Polym. Techn., 22, 85-95 (2003).]...

A plastic material is defined as one that does not undergo a permanent deformation until a certain yield stress has been exceeded. A perfectly plastic body showing no elasticity would have the stress-strain behavior depicted in Figure 8-15. Under influence of a small stress, no deformation occurs when the stress is increased, the material will suddenly start to flow at applied stress a(t (the yield stress). The material will then continue to flow at the same stress until this is removed the material retains its total deformation. In reality, few bodies are perfectly plastic rather, they are plasto-elastic or plasto-viscoelastic. The mechanical model used to represent a plastic body, also called a St. Venant body, is a friction element. The... 

Figure 10.2. Stress-strain behavior. With elastic (reversible) deformation, stress and strain are linearly proportional in most materials (exceptions include polymers and concrete). With plastic (permanent) deformation, the stress-strain relationship is nonlinear.

Before concluding this discussion of cell walls, we note that the case of elasticity or reversible deformability is only one extreme of stress-strain behavior. At the opposite extreme is plastic (irreversible) extension. If the amount of strain is directly proportional to the time that a certain stress is applied, and if the strain persists when the stress is removed, we have viscous flow. The cell wall exhibits intermediate properties and is said to be viscoelastic. When a stress is applied to a viscoelastic material, the resulting strain is approximately proportional to the logarithm of time. Such extension is partly elastic (reversible) and partly plastic (irreversible). Underlying the viscoelastic behavior of the cell wall are the crosslinks between the various polymers. For example, if a bond from one cellulose microfibril to another is broken while the cell wall is under tension, a new bond may form in a less strained configuration, leading to an irreversible or plastic extension of the cell wall. The quantity responsible for the tension in the cell wall — which in turn leads to such viscoelastic extension — is the hydrostatic pressure within the cell. 

An effort to investigate the kinematics of plastic deformation in glassy atactic polypropylene was presented by Mott, Argon, and Suter.Using an atomistic simulation for strains up to 20%, the authors observed that the plastic rearrangement of the structure was revealed in the microstructural stress—strain behavior (i.e., smooth reversible portions bounded by irreversible sharp drops in the stress values). 

The load-displacement curves for C(T) tests of the neat EpoxyH were almost linear until the final unstable fracture. The fracture toughness value in 77K-LNj was 210 J/m and that in RT-air was 120 J/m. Thus the toughness increased by 1.8 times by changing the test environment from RT-air to 77K-LN. Brown and co-workers have found that amorphous polymers crazed in 77K-LNj, but not in a helium or vacuum at about 78K [20-22]. They have also reported that the stress-strain behavior of all polymers, amorphous and crystalline, is affected by at low temperatures [22]. Kneifel has reported that the fracture toughness of epoxy in 77K-LNj is higher than that in RT-air and 5K, and that the reason for this is the reduced notch effect by plastic deformation [23]. Then, the increase of the fracture toughness of the neat EpoxyH in this study is probably caused by the similar effect. 

The size and shape of the plastic zones are illustrated in Fig. 4.1 the plane strain zone is estimated for a Poisson ratio v of 0.3. Note that the actual zone sizes would be larger to reflect redistribution of stresses associated with plastic deformation near the crack tip. The shape of the plastic zone would also change to reflect the workhardening behavior of the material. The actual sizes and shapes would need to be determined experimentally for each class of materials. 

The consistency of a grease is a complex of related properties, easily demonstrable empirically but difficult to define precisely. We can single out yield stress as a truly definable, pertinent property and then have a quantitative parameter in terms of which we can treat consistency. Criddle and Dreher [1] observed typical solid-body stress-strain behavior in greases, with an elastic region, a region of plastic deformation and an ultimate yield or rupture point. At rest grease behaves like a solid body provided the specimen is not too big, it will not flow under the force of gravity. ... 

The stress-strain behavior of ceramic polycrystals is substantially different from single crystals. The same dislocation processes proceed within the individual grains but these must be constrained by the deformation of the adjacent grains. This constraint increases the difficulty of plastic deformation in polycrystals compared to the respective single crystals. As seen in Chapter 2, a general strain must involve six components, but only five will be independent at constant volume (e,=constant). This implies that a material must have at least five independent slip systems before it can undergo an arbitrary strain. A slip system is independent if the same strain cannot be obtained from a combination of slip on other systems. The lack of a sufficient number of independent slip systems is the reason why ceramics that are ductile when stressed in certain orientations as single crystals are often brittle as polycrystals. This scarcity of slip systems also leads to the formation of stress concentrations and subsequent crack formation. Various mechanisms have been postulated for crack nucleation by the pile-up of dislocations, as shown in Fig. 6.24. In these examples, the dislocation pile-up at a boundary or slip-band intersection leads to a stress concentration that is sufficient to nucleate a crack. 

Figure 6.25 Stress-strain behavior (curve 1) for a polycrystalline material that demonstrates significant plastic deformation and necking. In addition, the true stress-true strain (curve 2) is shown schematically.

O. H. Varga, Stress-Strain Behavior of Elastic Materials, Wiley-Interscience, New York, 1966. A. Peterlin, Plastic Deformation of Polymers, Marcel Dekker, New York, 1971. 

Unfortunately, Hooke s Law does not accurately enough reflect the stress-strain behavior of plastics parts and is a poor guide to good successful design. Assuming that plastics obey Hookean based deformation relationships is a practical guarantee of failure of the part. What will be developed in this chapter is a similar type of basic relationship that describes the behavior of plastics when subjected to load that can be used to modify the deformation equations and predict the performance of a plastics part. UnUke the materials that have been used which exhibit essentially elastic behavior, plastics require that even the simplest analysis take into account the effects of... 

If we perform undrained triaxial tests (namely, void ratio e=constant) on isotropically and normally consolidated clay at several confining pressures p, the stress-strain behavior is schematically shown in Fig. 6.2. It is observed that at the final stage of loading (namely, the failure state) the stress ratio rj = q/p becomes constant qf = q/p )f = M), which is referred to as the critical state, which means that the deformation is developed under a constant volumetric plastic strain and a constant shear stress at the critical state q = Mp. ... 

Figure 5.6. Typical stress-strain behavior for a ductile material showing initial elastic behavior (linear) followed by plastic deformation beyond the yield point and fractnre...

In all materials (plastics, metals, wood, etc.) elementary mechanical theory demonstrates that some shapes resist deformation from external loads. This phenomenon stems from the basic physical fact that deformation in beam or sheet sections depends upon the mathematical product of the modulus of elasticity (E) and the moment of inertia (I), commonly expressed as El (Chapter 3, Stress-strain behavior). It is applied to all types of constructions such as solids, foams, and sandwich structures. In many applications plastics can lend themselves in the form of a sophisticated lightweight stiff structure and the requirements are such that the structure must be of plastics. In other instances, the economics of fabrication and erection of a plastics lightweight structure and the intrinsic appearance and other desirable properties make it preferable to other materials. 

These cracks are much shallower and are formed on unloading at the boundary between elastic, nonpermanently deformed material and the plastically deformed material close to the indenter the interface between these two regions is the source of a stress field because material that has been plastically strained has a different stress-strain behavior than the normal material. 

---