Elasticity ideal

Fig. 4. (a) Shear stress diagram for elastic ideal plastic material (b) partiady plastic thick-waded cylinder. 

If the sum of the mechanical allowances, c, is neglected, then it may be shown from equation 15 that the pressure given by equation 33 is half the coUapse pressure of a cylinder made of an elastic ideal plastic material which yields in accordance with the shear stress energy criterion at a constant value of shear yield stress = y -... 

Hookean elasticity (ideal elasticity) n. Stress-strain behavior in which stress and strain are directly proportional, in accordance with Hooke s law. Serway RA, Faugh JS, Bennett CV (2005) College physics. Thomas, New York. 

Figure 5.9. DMA sinusoidal stress-strain response curves for ideal elastic, ideal viscous, and viscoelastic materials illustrating that the phase shift between stress and strain for a viscoelastic material lies between those for the ideal materials (courtesy of TA Instruments).

If one or both of the materials are modeled as being elastic-ideally plastic, then the methods adopted for study of elastic response can be extended to establish conditions for onset of plastic deformation and the progression... 

Figure 7.15 schematically shows the transitions in the onset and spread of plastic yielding in a bilayer comprising an elastic-ideally plastic film on an elastic substrate, in response to an increase in temperature from a reference state, presuming that Og < af and that T > 0. The temperature increase Ty at which plastic yielding commences in the film and the temperature change Tpi at which the entire film material first becomes fully plastic are marked on this figure. Note that the substrate curvature varies linearly with temperature when both the film and substrate are elastic, that is, for T < Ty beyond this point, the slope of the curvature versus temperature... 

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... 

If the substrate is also an elastic-ideally plastic material with yield strength magnitude ay, it follows from (7.37) through (7.40) that plastic yielding will always commence at the interface within the film material and that it there will be no yielding in the substrate material if... 

Now consider the system comprising the thin elastic-ideally plastic film on a relatively thick elastic substrate which is subjected to a temperature history as shown in the inset of Figure 7.16. Starting from the reference temperature T = T — To = 0, the temperature is first increased to Tj ax, then decreased to zero, and then again increased to Tmax- This history is plotted against time but, because no aspect of the response is sensitive to rate of deformation, the detailed shape or waveform of the curve of temperature versus time is unimportant except for the general property that it varies monotonically from zero to T oax, decreases from T ax to zero, and again increases to Tmax-... 

Fig. 7.16. Stress-temperature history for a thin film of elastic-ideally plastic material on a relatively thick substrate during a temperature excursion. Tte tensile yield stress of the film material depends on temperature according to (Ty T). After one temperature cycle, the response adopts a cyclic behavior for fixed temperature limits.

Going back to the Goodman line we may say that the safety domain of Fig. 5.32 is not totally accessible, meaning that not really all mean stress values may have an effect on fatigue. As matter of fact, not always falls down to zero as goes to stress amplitude value Of ax exists below which, under particular loading conditions, it is not possible to go, whatever the mean stress may be. To understand it, suppose to be dealing with an elastic-ideally plastic material whose characteristic is that of Fig. 5.42. This... 

Fig. 5.42 Schematic of Shake-down effect in an elastic-ideally plastic material subjected to cyclic stress amplitude and mean stress

The next step is to cope with the unrealistic high pressure values above material strength. Due to these high pressure values, the material will deform plastically. For this reason a linear-elastic, ideal-plastic material behaviour was introduced. This is done by the implementation of a new variable into the elasticity equation (4). The value of ftpiast has to be calculated iteratively ... 

Fig. 6. Film thickness and pressure distribution with linear-elastic, ideal-plastic material behaviour...

Since the pressure drops down at die fictitious solid contact spots, there will be no plastic deformation at these spots themselves. High pressure values can only be found directly upstream of the bump. Hence, the implementation of the linear-elastic, ideal-plastic material behaviour without Paum leads to an increasing plastic deformation upstream of the fictitious solid contact spot during the iterations and not at the contact itself. So no useful solution can be obtained. 

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