Cyclic stress-strain behaviour

We already mentioned in section 10.2.1 that the stress-strain behaviour of metals may change during cyclic loading. Depending on the initial state, different effects may occur. 

If we perform cyclic experiments at different strain amplitudes and plot the stabilised values of the stress amplitude, we arrive at the cyclic stress-strain diagram sketched in figure 10.29. Usually, it does not coincide with the result of a monotonous tensile or compressive test. If cyclic hardening occurs, 

The cyclic stress-strain curve is frequently approximated by the Ram-berg-Osgood law, equation (3.15), using modified parameters K and n  

To reduce the experimental efforts in measuring cyclic stress-strain curves, the incremental-step test can be used. In this test, the strain amphtude is varied block-wise between zero and a maximal value as sketched in figure 10.30. After the block has been repeated several times, the material behaviour does not change an5rmore and a stationary state is arrived at. If the stress is measured at each of the strain maxima, the 

The cyclic stress-strain curve can be used, for example, to perform finite element simulations of cyclic loadings. To simulate the complete experiment in the computer, it would be necessary to obtain information on the hardening of the material (isotropic and kinematic hardening) and to determine a material model that correctly describes it. This would be an extremely comphcated procedure. Furthermore, the entire number of cycles would have to be calculated, which would require an immense amount of computing time. Instead, the flow curve, taken by the finite element software to be monotonous, can be replaced by the cyclic stress-strain curve. A single, monotonous loading of the component is then simulated. Stresses and strains calculated in this way correspond well with those in the cyclically loaded component. 

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Dynamic properties are taken to mean the results from mechanical tests in which the plastic is subjected to a deformation pattern from which the cyclic stress-strain behaviour is calculated. These do not include cyclic tests in which the main objective is to fatigue the material. 

Fig. 10.28. Cyclic stress-strain behaviour at the beginning of strain-controlled fatigue experiments (after [130]). The controlled variable is shown on the left, the material answer in the centre...

Avanzini A. Mechanical characterization and finite element modelling of cyclic stress-strain behaviour of ultra high molecular weight polyethylene. Mater Des 2008 29(2) 330-43. 

The term dynamic test is used here to describe the type of mechanical test in which the rubber is subjected to a cyclic deformation pattern from which the stress strain behaviour is calculated. It does not include cyclic tests in which the main objective is to fatigue the rubber, as these are considered in Chapter 12. Dynamic properties are important in a large number of engineering applications of rubber including springs and dampers and are generally much more useful from a design point of view than the results of many of the simpler static tests considered in Chapter 8. Nevertheless, they are even today very much less used than the "static" tests, principally because of the increased complexity and apparatus cost. 

A series of studies was also made by us, of the PUs cyclic stress-strain response. The range of structures achieved by us was widened by inclusion of DBDI, as a diisocyanate with a very strong tendency to packing due to its constitutional mobility. A systematic investigation (as shown in Table 4.5), was made of the effects of varying HS and SS chemistry, crosslinking and preparation procedures, on the hysteresis behaviour and Mullins effect of melt-cast polyurethane elastomers. The... 

In Fig 3 the uniaxial fibre stress-strain relation is shown for a constant strain rate of 1.67 10" Hz. Equation (1) can also be used to determine cyclic stress strain response (an example is shown in Fig. 3) or even an analytical description of the relaxation behaviour of the fibres, however time dependent phenomena will not be discussed here. 

Figure 8.48 Loose sand behaviour showing (a) stress paths and (b) stress-strain relations for initiation of static (A, B) and cyclic (C) instability at small strains and subsequent liquefaction at large strains.

Hardin BO (1965) The natine of damping in sands. Proc Soil Mech Found Div, ASCE 91(SMl) 63-97 Hardin BO, Dmevich VP (1972) Shear modulus and damping in soils measurement and parameter effects. J Soil Mech Found Div. ASCE 98(SM6) 603-624 Ishihara K (1996) The representation of stress-strain relations in cyclic loading. Chapter 3. In Ishihara K -(ed) Soil behaviour in earthquake geotechnics, Oxford engineering science series, 46. Clarendon, Oxford, pp 16-39... 

The results of temperature and phase angle deviations discussed so far consider the case of time based compensation of the thermal strain (0 in Eq. 21.1. Another possibility is to define th(T) as a function of the temperatiue which is referred to as temperature based strain compensation. In this case, deviations of the peak temperatures and of the phase angle between temperature and mechanical strain affect the cyclic deformation and lifetime behaviour distincly less pronounced than for time-based compensation. However, a fit function has to be applied to feed (T) in the control circuit of the testing machine which may result in considerable strain and stress errors especially in the case that one fit function is used for the heating as weU as for the cooling part of the TMF cycle. 

A model of the post-peak cyclic behaviour of the concretes investigated for the tensile-tensile and the tensile-compressive loading is proposed. The average stress-total deformation curves are split into two parts the ascending parts in which a tmique relation exists between the stress and the strain that consists of an elastic ccanponent and an iirreversible one ... 

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