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Constitutive models
Published in Paulo B. Lourenço, Angelo Gaetani, Finite Element Analysis for Building Assessment, 2022
Paulo B. Lourenço, Angelo Gaetani
This approach is intended for applications in which the material is essentially subjected to monotonic loading under fairly low confining pressures. In this case, cracking is assumed to be the most crucial aspect of the material behaviour, which controls the structural response. Being a smeared crack approach, the presence of cracks enters the calculation by modifying the constitutive relationships in correspondence of the integration points, introducing, consequently, anisotropy. In particular, cracking is assumed when the stresses reach a limiting condition: in this case the definition crack detection surface results rather appropriate. With this aim, a Rankine-type criterion is often employed, sometimes referred to as constant and linear tension cut-off. These are shown in Figure 3.47, where ft and fc are the tensile and compressive strengths, respectively.
Heat Treatment by Induction
Published in Valery Rudnev, Don Loveless, Raymond L. Cook, Handbook of Induction Heating, 2017
Valery Rudnev, Don Loveless, Raymond L. Cook
Regardless of the fact that an increase in tempering temperature results in a monotonic reduction of the hardness and strength in the majority of applications, a change in impact toughness may not be monotonic with an increase of tempering temperature. Embrittlement phenomena can occur after tempering at certain temperature ranges, leading to a drop in impact toughness. There are several types of embrittlement that can be associated with as-tempered structures. This includes but is not limited to the following [30]:
Function-on-Function Kriging, With Applications to Three-Dimensional Printing of Aortic Tissues
Published in Technometrics, 2021
Jialei Chen, Simon Mak, V. Roshan Joseph, Chuck Zhang
From the underlying physics of the stress–strain relationship, it is known that (i) the stress O(s) is always positive, (ii) the stress is zero when the strain is zero (this is known as the free-standing state, see Malvern 1969), and (iii) stress–strain curves are typically monotone and non-decreasing, since a larger force is needed to stretch further. To account for (i), a standard log-transformation of stress O(s) is performed prior to modeling and parameter estimation, and the final results are transformed back to ensure the predicted stress is always positive. To account for (ii) and (iii), we choose the basis functions in (12) to be , along with an additional constraint of to ensure the mean function is monotone and nondecreasing. This is equivalent to assuming the mean stress–strain curve takes the following form , which is a typical parameterization in biomedical literature (Rengier et al. 2010; Chen, Wang et al. 2018). This provides a simple and effective way to encourage monotonicity via the mean function specification; one can also extend the shape-constrained GP model in Wang and Berger (2016) to impose sample path monotonicity, but this is beyond the scope of this work.