Experimental investigations on superelastic shape-memory alloys (SMAs) show a dependence of the stress-strain relationship on the loading-unloading rate. This feature is of particular importance when utilizing SMA materials for seismic applications, since the loading rate may affect the structural response.Motivated by this observation and by the fact that there exist relatively few studies on the material modelling of SMAs in earthquake engineering, the present work addresses a uniaxial constitutive equation able to describe the rate-dependent behaviour of superelastic SMAs.The formulation of the model is based on two scalar internal variables, the static martensite fraction and the dynamic martensite fraction, for which three different types of evolutionary equations in rate form are proposed. Moreover, the model takes into account the different elastic properties between austenite and martensite. Finally, after discussing two possible approaches for the solution of the corresponding time-discrete framework, the ability of the model to simulate experimental data obtained from uniaxial tests performed on SMA wires and bars at frequency levels of excitation typical of earthquake engineering is assessed. reversible micromechanical phase transitions by changing the crystallographic structure from an austenitic phase to a martensitic phase. This capacity results in two major features at the macroscopic level, which are the superelastic effect and the shape-memory effect [9].Due to these unique characteristics, SMAs lend themselves to innovative applications in many scientific fields,
We generalize simplicial minisuperspace models associated with restricting the topology of the universe to be that of a cone over a closed connected combinatorial 3-manifold by considering the presence of a massive scalar field. By restricting all the interior edge lengths and all the boundary edge lengths to be equivalent and the scalar field to be homogeneous on the 3-space, we obtain a family of two-dimensional models that includes some of the most relevant triangulations of the spatial universe. After studying the analytic properties of the action in the space of complex edge lengths we determine its classical extrema. We then obtain steepest-descent contours of constant imaginary action passing through Lorentzian classical geometries yielding a convergent wavefunction of the universe, dominated by the contributions coming from these extrema. By considering these contours we justify semiclassical approximations based on those classical solutions, clearly predicting classical spacetime in the late universe. These wavefunctions are then evaluated numerically. For all of the models examined we find wavefunctions predicting Lorentzian oscillatory behaviour in the late universe.
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