Stress amplitude

At loading stresses between the HEL and the strong shock threshold, a two-wave structure is observed with an elastic precursor followed by a viscoplastic wave. The region between the two waves is in transition between the elastic and the viscoplastic states. The risetime of the trailing wave is strongly dependent on the loading stress amplitude [5]. 

Figure 4.11. Diagrammatic sketches of atomic lattice rearrangements as a result of dynamic compression, which give rise to (a) elastic shock, (b) deformational shock, and (c) shock-induced phase change. In the case of an elastic shock in an isotropic medium, the lateral stress is a factor v/(l — v) less than the stress in the shock propagation direction. Here v is Poisson s ratio. In cases (b) and (c) stresses are assumed equal in all directions if the shock stress amplitude is much greater than the material strength.

Meir and Clifton [12] study shocked <100) LiF (high purity) with peak longitudinal stress amplitudes 0.5 GPa. A series of experiments is reported in which surface damage is gradually eliminated. They find that, while at low-impact velocities the dislocations in subgrain boundaries are immobile and do not affect the dislocation concentration in their vicinity, at high-impact velocities ( 0.1 km/s) dislocations emitted from subgrain boundaries appear to account for most of the mobile dislocations. 

A.R. Champion and R.W. Rohde, Hugoniot Equation of State and the Effect of Short Stress Amplitude and Duration on the Hardness of Hadfield Steel, J. Appl. Phys. 41,2213-2223 (1970). 

Hugoniot. At low-stress amplitudes pot /o C S c,o, where c,o is the adiabatic longitudinal elastic sound speed at p = pg. 

Assuming that the cyclic waveform used in the previous section was sinusoidal then the effect of using a square wave is to reduce, at any frequency, the level of stress amplitude at which thermal softening failures start to occur. This is because there is a greater energy dissipation per cycle when a square wave is used. If a ramp waveform is applied, then there is less energy dissipation per cycle and so higher stresses are possible before thermal runaway occurs. 

For convenience, in the previous sections it has been arranged so that the mean stress is zero. However, in many cases of practical interest the fluctuating stresses may be always in tension (or at least biased towards tension) so that the mean stress is not zero. The result is that the stress system is effectively a constant mean stress, a superimposed on a fluctuating stress a a- Since the plastic will creep under the action of the steady mean stress, this adds to the complexity because if the mean stress is large then a creep rupture failure may occur before any fatigue failure. The interaction of mean stress and stress amplitude is usually presented as a graph of as shown in Fig. 2.76. 

This represents the locus of all the combinations of Ca and Om which cause fatigue failure in a particular number of cycles, N. For plastics the picture is slightly different from that observed in metals. Over the region WX the behaviour is similar in that as the mean stress increases, the stress amplitude must be decreased to cause failure in the same number of cycles. Over the region YZ, however, the mean stress is so large that creep rupture failures are dominant. Point Z may be obtained from creep rupture data at a time equal to that necessary to give (V cycles at the test frequency. It should be realised that, depending on the level of mean stress, different phenomena may be the cause of failure. 

The level of mean stress also has an effect on the occurrence of thermal failures. Typically, for any particular sUess amplitude the stable temperature rise will increase as the mean stress increases. This may be to the extent that a stress amplitude which causes a stable temperature rise when the mean stress is zero, can result in a thermal runaway failure if a mean stress is superimposed. 

Fig. 2.76 Relationships between stress amplitude and mean stress...

During fatigue the stress amplitude usually remains constant and brittle failure occurs as a result of crack growth from a sub-critical to a critical size. Clearly the rate at which these cracks grow is the determining factor in the life of the component. It has been shown quite conclusively for many polymeric materials that the rate at which cracks grow is related to the stress intensity factor by a relation of the form... 

Stress amplitude = mean stress = To allow for mean stress use... 

Typical stress-time profiles for the various materials (28.5-at. % Ni, fee and bcc) and various stress regions are shown in Fig. 5.12. The leading part of the profile results from the transition from elastic to plastic deformation. The extraordinarily sharp rise in stress for the second wave in Fig. 5.12(a) and the faster arrival time compared with that in Fig. 5.12(b) is that expected if the input stress is above the transition, whereas the slower rise in Fig. 5.12(b) is that expected if the stress input to the sample is below the transition. The profile in Fig. 5.12(c) for the bcc alloy was obtained for an input particle velocity approximately equal to that in Fig. 5.12(a) for the fee alloy. The bcc alloy shows a poorly defined precursor region, but, in any event, much faster arrival times are observed for all stress amplitudes, as is indicative of lower compressibility. 

Examples of fatigue curves for unreinforced (top) and reinforced (bottom) plastics are shown in Fig. 2-44. The values for stress amplitude and the number of load cycles to failure are plotted on a diagram with logarithmically divided abscissa and English or metrically divided ordinates. 

For a given grade of SAN, stress amplitude measured in accordance with DIN 53 442 is about 25 MPa for a failure at 10 cycles. 

The analytical solutions derived in Sections 4.3 and 4.4 for the stress distributions in the monotonic fiber pull-out and fiber push-out loadings are further extended to cyclic loading (Zhou et al., 1993) and the progressive damage processes of the interface are characterized. It is assumed that the cyclic fatigue of uniform stress amplitude causes the frictional properties at the debonded interface to degrade... 

Figure 5.41 Stress amplitude versus cycles to failure illustrating (a) fatigue limit and (b) fatigue life. Reprinted, by permission, from W. Callister, Materials Science and Engineering An Introduction, 5th ed., p. 212. Copyright 2000 by John Wiley Sons, Inc.

The material for an acoustic lens should have a low attenuation, and a high velocity to minimize aberrations. Sapphire is an excellent material in both these respects. But the high velocity has a less desirable consequence. An acoustic impedance can be defined, which is equal to the product of the velocity and the density. The impedance of sapphire for longitudinal waves travelling parallel to the c-axis is thus 44.3 Mrayl, compared with the impedance of water which at room temperature is about 1.5 Mrayl, rising to 1.525 Mrayl at 60°C. When sound is transmitted across an interface between two materials of different impedance, the stress amplitude transmission coefficient is ( 6.4.1 Auld 1973 Brekhovskikh and Godin 1990)... 

When a wave is incident upon an interface, there is a requirement of continuity of traction and displacement across the interface. Traction is the force per unit area acting on an interface it is a vector with three components. If the stress amplitude (or pressure amplitude if the medium is a fluid) of a wave incident normally on an interface is unity, and the stress amplitudes of the reflected and transmitted waves are R and T, then continuity of traction requires that... 

Fig. 6.3. Reflected pressure amplitude and transmitted longitudinal and shear stress amplitudes at a fluid-solid interface (a) water-PMMA (b) water-fused silica (i) reflected wave in the fluid (ii) transmitted longitudinal wave (iii) transmitted shear...

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