Manual Microscopic and Macroscopic Simulation: Towards Predictive Modelling of the Earthquake Process

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The description of the tool set-up is shown in Figure 6. The diameter of the punch is taken equal to mm. For the friction between the punch and the blank, the associated Coulomb friction coefficient is assumed to be equal to 0. The plane geometries and the dimensional characteristics of the used specimens are the same as those defined in Kami et al. However, the thickness is taken equal to 1 mm for all of the studied specimens. Three values for the width parameter w are used in the FE simulations: 30 mm, 90 mm, and mm the latter corresponds to a circular specimen.

The different tools die, punch, and blank holder are modeled as analytical rigid surfaces, while the specimens are meshed using C3D8R linear solid finite elements. In order to capture the bending effects and hence to improve the computation accuracy, three layers of elements are used in the thickness direction.

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This choice seems to be a pragmatic compromise in terms of good accuracy of the results and reasonable computational times. The mesh density is increased in the contact zone between the punch and the specimen. To carry out the forming process, the punch is displaced in the vertical direction up to a maximal displacement of 60 mm.

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Different punch force versus displacement curves are shown in Figure 8. As expected, it is observed from the curves of Figure 8 that as the width w increases, there is an increase in the load requirement. The dot reported on the different curves indicates the occurrence of macroscopic instability, as predicted by the loss of ellipticity criterion within the associated specimens.

It is clearly shown that the onset of such plastic instability does not necessarily coincide with the maximum of the force punch for the various specimens. Figure 9 shows the deformed shapes for the various Nakajima specimens, with the associated von Mises equivalent stress distribution, at the onset of plastic instability.

Microscopic and Macroscopic Simulation: Towards Predictive Modelling of the Earthquake Process

One may notice that the value of equivalent stress is relatively low in the central area of the specimens. However, this von Mises equivalent stress reaches its maximal values in narrow zones near the dome apex. In order to illustrate the occurrence and development of macroscopic instability, the distribution of the loss of ellipticity indicator at the onset of instability is displayed in Figure This indicator takes as value 0 resp. Red resp. The plastic instability, which corresponds to the nullity of the loss of ellipticity indicator, is first reached in the dome apex region, where the von Mises equivalent stress is maximal see Figure 9.

For more clarity, the corresponding elements in red color are surrounded by more visible red circles in Figure It is observed from this figure that the instability zone has extended to the entire dome apex region. In order to better understand the initiation and evolution of the instability, three particular elements are selected in each specimen for the following analysis: Element A Elem.

A , which corresponds to the central element of the specimen, Element B Elem. B , which is representative of the zone near the localization area, and Element C Elem. C , which corresponds to the element where the instability is first triggered see Figure The evolution of the von Mises equivalent plastic strain rate D e q p as a function of the punch displacement for the different analyzed elements and the different specimens is shown in Figure It is clearly observed from this figure that Elem.

A resp.

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C has the lower resp. Consequently, Elem. A exhibits the slower deformation, while Elem. C displays the more rapid plastic deformation, which ultimately tends to localize. This trend is common to the three specimens. Note however that the different elements do not have the same evolution for D e q p. Indeed, for Elem. A and Elem. B, D e q p increases before the onset of instability in the specimen, while it decreases afterwards, which typically corresponds to elastic unloading in these elements.

On the other hand, D e q p increases monotonically for Elem. C, and reaches very high values when plastic instability occurs. Consequently, the plastic deformation localizes in this element. These trends may be correlated to the results obtained with the application of the Marciniak-Kuczynski approach Ben Bettaieb and Abed-Meraim, Indeed, elements A and B play the role of the homogeneous zone, whereas element C plays the role of the imperfection band.

In contrast to the Marciniak-Kuczynski analysis, strain localization can obviously be detected in the FE simulations without the need for introducing any initial geometric or material imperfection. One of the advantages of using a multiscale model at each integration point of the FE simulations compared to phenomenological modeling is the capability of accurately accounting for the effect of initial and induced plastic anisotropy during the deformation.

This anisotropy has a significant influence on the mechanical behavior at the different integration points, and consequently on the overall response of the specimen. It is clearly revealed from this figure that the final texture of Elem.

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A for all specimen geometries is not significantly different from the initial texture shown in Figure 1 b. This result is quite expectable considering the fact that the deformation in this element is the most moderate, as compared to the other analyzed elements Elem. B and Elem. Therefore, the influence of deformation-induced anisotropy due to grain rotation during plastic deformation may not be quite important at the end of the simulation. On the other hand, the final textures corresponding to Elem. C are significantly different from the initial texture shown in Figure 1 b.

This result is also quite expectable considering the significant evolution of the strain rate in this element, as observed in Figure In order to accurately simulate sheet forming processes, a suitable constitutive framework is required. In this physically-based model, the link between the microscopic and macroscopic scales is achieved by using the self-consistent scale-transition scheme. The proposed model accounts for the initial and induced plastic anisotropy through the modeling of the texture evolution. The implemented constitutive model has been coupled with an indicator of loss of ellipticity in order to predict the occurrence of macroscopic plastic instability.

This coupling has been used mainly in the following two applications:. To investigate the effect of the number of grains, which constitute the polycrystalline aggregate, on the predicted forming limit diagram. To predict the onset and the evolution of plastic instability within sheet specimens undergoing a deep drawing process. Akpama, H.

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  8. Influence of the yield surface curvature on the forming limit diagrams predicted by crystal plasticity theory. Latin American Journal of Solids and Structures Beaudoin, A. Application of polycrystal plasticity to sheet forming. Computer Methods in Applied Mechanics and Engineering Ben Bettaieb, M. Investigation of localized necking in substrate-supported metal layers: comparison of bifurcation and imperfection analyses. International Journal of Plasticity On the numerical integration of rate independent single crystal behavior at large strain.

    Eshelby, J.

    Microscopic and Macroscopic Simulation: Towards Predictive Modelling of the Earthquake Process

    The inset shows that the slider motion is not oscillatory, and that the block sticks as the shear stress increases linearly and then fails rapidly when the shear stress drops. Figure: Phase diagram for slider motion as a function of spring stiffness and driving velocity.

    Above each curve, the block slides stably, and below stick-slip occurs. Our analytical predictions agree very well with results from numerical integration of the STZ equations. The upper curve is for strain localization, which shows that localization leads to stick-slip for a larger range of parameters.

    Plot shows shear stress as a function of load point displacement. Groups of events usually include several small events followed by a large event, but successive larger events are also possible. Figure: Plot of effective temperature evolution during a dynamic rupture simulation. Paper Type Research article.

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