Creep anelastic

Figure 14-5 defines the creep below the critical shear stress. It is called anelastic creep and is to a large extent recoverable. The anelastic strain for a given stress below °crk approaches its equilibrium value u0 as... 

Rheology - The study of the flow of liquids and deformation of solids. Rheology addresses such phenomena as creep, stress relaxation, anelasticity, nonlinear stress deformation, and viscosity. 

In many materials, the mechanical response can show both elastic and viscous types of behavior the combination is known as viscoelasticity. In elastic solids, the strain and stress are considered to occur simultaneously, whereas viscosity leads to time-dependent strain effects. Viscoelastic effects are exhibited in many different forms and for a variety of structural reasons. For example, the thermoelastic effect was shown earlier to give rise to a delayed strain, though recovery of the strain was complete on unloading. This delayed elasticity is termed anelastic-ity and can result from various time-dependent mechanisms (internal friction). Figure 5.9 shows an example of the behavior that occurs for a material that has a combination of elastic and anelastic behavior. The material is subjected to a constant stress for a time, t. The elastic strain occurs instantaneously but, then, an additional time-dependent strain appears. On unloading, the elastic strain is recovered immediately but the anelastic strain takes some time before it disappears. Viscoelasticity is also important in creep but, in this case, the time-dependent strain becomes permanent (Fig. 5.10). In other cases, a strain can be applied to a material and a viscous flow process allows stress relaxation (Fig. 5.11). 

Time-dependent hysteresis effects can also occur in crystalline materials and these lead to mechanical damping. Models, such as the SLS and the generalized Voigt model, have been used extensively to describe anelastic behavior of ceramics. It is, thus, useful to describe the sources of internal friction in these materials that lead to anelasticity. The models discussed in the last section are also capable of describing permanent deformation processes produced by creep or densification in crystalline materials. For polycrystalline ceramics, creep is usually considered from a different perspective and this will be discussed further in Chapter 7. 

The experiment we introduced at the beginning of the previous subsection is also called the creep experiment. A small stress of Gq is imposed on a solid sample for a time period of to at a constant temperature after the stop of stress, the strain of changing with the time period of t monitors the relaxatirMi curve. There are four typical responses separately corresponding to viscous, elastic, anelastic and viscoelastic responses, as illustrated in Fig. 6.8. The creep curve of polymer viscoelasticity exhibits both instant and retarded elastic responses upon imposing and removal of the stress, and eventually reaches the permanent deformation. 

When a material is loaded under a fixed stress, after a certain time the strain continues to increase at a rate depending on the type of material. This slow continuing deformation of a material when subjected to a constant stress is called the creep mechanism, which is a typical anelastic behavior. The rate at which the strain change occurs is called the strain rate and is denoted ds/dt and expressed in s . For each material loaded under a constant stress it is... 

These last numerical results show that the plastic strain is obviously greatly influenced by the presence of the creep/relaxation phenomenon. In fact, the level of the plastic strain was considerably reduced from 0.004 to 0.0018, i.e a reduction ratio of 2.2 based on the reference case (Figure 6), which does not take into accormt creep behaviour of the ramming paste. Also, the anelastic strain level at the end of the simulation (i = 40 hours) is almost negligible compared to the other strains (e.g., plastic, thermal, etc.). This result directly ensues from the assumption that the baked ( = 1) ramming p>aste creep/relaxation behaviour is similar to that of the carbon cathode block (Richard et al., 2006). This case study thus shows the importance of taking all the relevant phenomena including creep behaviour into account in similar problems. A similar analysis could be done for all other deformations (chemical, thermal, plastic, etc.). 

The external manifestation of reorientation relaxation under an applied stress is the anelastic strain that accompanies a net change of orientational order. In contrast to the elastic strain, the anelastic strain develops in a time-dependent manner governed by the rate of the reorientation jump. Under a static stress, the relaxation may therefore be observed as a limited (and recoverable) creep process. Frequently, however, it is more desirable for reasons of sensitivity or convenience to observe the relaxation dynamically as a loss-peak, via internal friction measurements made as a function of temperature and/or vibration frequency. Figure 2 shows the oxygen Snoek peak in polycrystalline thin film niobium, tested in the same vibrating-reed apparatus" - used for our studies of... 

Figure 11.12 Creep rates as a function of total creep strain for polymethyl methacrylate at indicated temperatures for a stress level of 56 MPa. (Reproduced from Sherby, O.D. and Dorn, ].B. (1958) Anelastic creep of polymethyl methacrylate. ]. Mech. Phys. Solids, 6, 145. Copyright (1958) Elsevier Ltd.)...

Sherby, O.D. and Dorn, J.B. (1958) Anelastic creep of polymethyl methacrylate. 

Fig. 5.1. Creep curve of a polymer sample under tension (schematic). The elongation ALz induced by a constant force applied at zero time is set up by a superposition of an instantaneous elastic response dashed /me), a retarded anelastic part dash-dot line) and viscous flow dotted line). An irreversible elongation is retained after an unloading and the completion of the recovery process...

J(t) has a characteristic shape composed of several parts. Subsequent to the glassy range with a solid-like compliance in the order of 10 N m, an additional anelastic deformation emerges and eventually leads to a shear compliance in the order of 10 N m. The latter value is typical for a rubber. For a certain time a plateau is maintained but then there finally follows a steady linear increase of J, as is indicative for viscous flow. The displayed creep curve of polystyrene is really not a peculiar one and may be regarded as representative for all amorphous, i.e. noncrystalline polymers. One always finds these four parts... 

For a Newtonian low molar mass liquid, knowledge of the viscosity is fully sufficient for the calculation of flow patterns. Is this also true for polymeric liquids The answer is no under all possible circumstances. Simple situations are encountered for example in dynamical tests within the limit of low frequencies or for slow steady state shears and even in these cases, one has to include one more material parameter in the description. This is the recoverable shear compliance , usually denoted and it specifies the amount of recoil observed in a creep recovery experiment subsequent to the unloading. Jg relates to the elastic and anelastic parts in the deformation and has to be accounted for in all calculations. Experiments show that, at first, for M < Me, Jg increases linearly with the molecular weight and then reaches a constant value which essentially agrees with the plateau value of the shear compliance. 

The creep compliance /(f) = e(f)/time-dependent strain and a is the constant stress is, at very small strains, less than 1%, approximately independent of stress. The material is said to be linear anelastic or linear visco-elastic. Materials of this category follow the so-called Boltzmann superposition principle which can be expressed as follows ... 

Figure 1.11 Temporal behavior of a step-like stress, (Tq. applied between t and ti (a), and of the corresponding y i) (b) for a Newtonian liquid (red line), a Hookean solid (blue line) and a viscoelastic material (continuous black curve) showing a creep-recovery dynamics. The anelastic component to the viscoelastic response is also shown (green line).

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