[MUSIC] Hello. In this sequence, we shall analyze the soil behavior for large excitations. We have already considered a soil specimen in the lab. We may apply loading/unloading cycles at small, for weak quakes, as well as large amplitudes, for strong quakes. Such experiments are called cyclic triaxial tests. They lead to small or large loops depending on the loading amplitude. We have already defined the damping ratio between the dissipated energy Delta E and the elastic energy W. How can we define small and large amplitudes? We first display the shear stress, tau, with respect to the shear strain, gamma, for monotonic loading. At very low amplitudes, the shear modulus is Gmax. At the current shear strain gamma, we define the tangent modulus G(gamma). If we now consider a loading-unloading cycle at very small amplitude in red, the modulus is very close to Gmax and the dissipated energy is negligible. Increasing the amplitude, we get a larger cycle, but the modulus is still close to Gmax and the dissipated energy maybe modeled through the previous linear viscoelastic approach. Increasing the amplitude a bit more, the modulus decreases and the dissipation increases. Finally, for the largest amplitude, the soil properties drastically change, leading to a second modulus in red, which is much smaller than Gmax. Such loading and loading cycles represent the hysteretic or non-linear behaviour of soils at large amplitudes. What happens during an earthquake, inducing numerous such exciting cycles? If we consider the first cycle, we may define the secant modulus Gsec1, for the first cycle. At the origin. The maximum modulus Gmax1 is defined and is the same at the beginning of each unloading and reloading phase. After several loading cycles, we observe a new secant modulus GsecN and maximum modulus GmaxN, for cycle number N. An increasing number of cycles made thus lead to a significant modulus degradation of soils. An alternative to the cyclic triaxial test is the resonant column test involving torsional vibrations. As in previous sequences, we may identify the resonance frequency and thus the shear modulus but using various excitation levels. Each resonance curve gives a specific modulus at a given level. Gathering all the shear moduli for various resonant column tests in a single plot with respect to shear strain, we get the modulus reduction at various levels. Similarly, we may plot the damping ratio progressively increasing with the excitation level. Finally, we may gather various data from different tests as proposed by Darendeli. The shear modulus decreases with respect to the shear strain in black for sands, in colors for clays. Similarly, for the damping ratio, it is nearly constant until 0.01% and fastly increases for larger excitations. The plasticity index PI is zero for sands in black and ranges from 15 to 100% for clays. From the lab, we may move to the field either to take a sample or to perform one of the numerous field tests. The standard penetration test, the pressuremeter test, the cone penetration test, among others. Various correlations have been proposed between the lab tests and the various field tests. How can we use such data to assess seismic risk? We may consider constitutive models to predict the seismic motion for large quakes. In the early 70s, the hyperbolic model was proposed by Hardin and Drnevich. It gives the variations of the shear modulus through a hyperbolic function. gamma_r is the reference strain as shown in the plot. Moving from the modulus to the damping, we may easily define its variations through an increasing hyperbolic function. With thus managed to model the variations of both variables in a simple way. More complex modes are available. Such a rheological model involves an elastic spring in blue and a friction body in red. If the loading amplitude is very small, only the spring is activated, and we get a linear stress strain curve. The shear strain gamma equals the elastic shear strain gamma_e. Above a given stress tau_y, the friction body starts to slide, and the stress remains constant. It characterizes a perfectly plastic behavior above the elastic limit tau_y. The shear strain, gamma, now equals the elastic shear strain gamma_e plus the plastic shear strain gamma_p. If the system is unloaded, the friction body stops and the spring is activated until the -tau_y stress is reached. The sliding phase then restarts. The so-called plasticity criterion is simply written as modulus of tau less or equal to tau_y. It is relevant for both the loading and unloading phase. The next reloading phase is then purely elastic. When reaching tau_y once again, we finally close the loop. In this complete cycle, we may identify the secant modulus, as well as the dissipated energy in blue. The shape is a bit different from the hysteretic curve observed in the experiments. Is it possible to consider more complex models? We may think about assembling several spring-friction bodies in this way. At very low strains, only the first spring is activated. The loading curve on the left is a straight line. The shear modulus on the right is Gmax. Reaching the first yielding level Y1, the first friction body slides thus activating its spring. We get a gentle slope on the left and the lower modulus on the right. The principle is the same when reaching yielding level Y2, and yielding level Y3: moderate slope, smaller shear modulus. In the framework of cyclic plasticity, we may thus consider various complex models leading to realistic modulus reduction curves as well as loading-unloading cycles. Let's sum up this sequence during seismic excitation soils are submitted to numerous cyclic loadings, smaller ones first, and larger ones afterwards. In the lab we may observe such hysteretic loops and assess the soil nonlinear behavior. Through elastic plastic models, we may predict the soil cyclic response. In the case of the 5.9 Emilia earthquake, in Italy, we can follow the smaller and then larger stress-strain loops, changing with the seismic shaking: smaller modulus, but much larger dissipation!
