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In this work, we study lattice structures that exhibit a bistable behavior, i. e., they can snap from one stable state to another, and are also completely reversible, capable of reverting back to its original state through a heat treatment. We design this behavior by constructing lattice structures using networks of nonlinear springs that display tension-compression asymmetry and have different thermal expansion coefficients. The mismatch in the thermal expansion coefficients induces residual stresses in the springs which results in the lattice structure exhibiting bistability at low temperatures and monostability at high temperatures. This behavior mimics the crystallographic phase transformations of shape memory alloys, but here artificially introduced in a structural lattice. By analyzing a representative unit cell, we quantify the effect that the stiffness and the thermal expansion coefficient of the springs have on the stability of the structural lattice. In addition, for simple 2D lattices, using the concept of universal unfoldings of singularity theory, we perform a perturbation analysis to identify the key variables of the structure where controlling defects is important, as they lead to drastic changes in the bifurcation behavior of the lattice. Finally, we verify numerically our analytical predictions in both 2D and 3D simulations using continuation techniques. The examples proposed confirm that the bistable and reversible features of the unit cell carry on to the macroscale, opening the route for the design of lattice structures for energy absorption applications that can hea} with a heat treatment., 37 pages, 20 figures

Recent theoretical and computational progress has led to unprecedented understanding of symmetry-breaking instabilities in 2D dynamic fracture. At the heart of this progress resides the identification of two intrinsic, near crack tip length scales -- a nonlinear elastic length scale $\ell$ and a dissipation length scale $\xi$ -- that do not exist in the classical theory of cracks. In particular, it has been shown that at a high propagation velocity $v$, cracks in 2D brittle materials undergo an oscillatory instability whose wavelength varies linearly with $\ell$, and at yet higher propagation velocities and larger loading levels, a tip-splitting instability emerges, both in agreements with experiments. In this paper, using phase-field models of brittle fracture, we demonstrate the following properties of the oscillatory instability: (i) It exists also in the absence of near-tip elastic nonlinearity, i.e. in the limit $\ell\!\to\!0$, with a wavelength determined by the dissipation length scale $\xi$. This result shows that the instability crucially depends on the existence of an intrinsic length scale associated with the breakdown of linear elasticity near crack tips, independently of whether the latter is related to nonlinear elasticity or to dissipation. (ii) It is a supercritical Hopf bifurcation, featuring a vanishing oscillations amplitude at onset. (iii) It is largely independent of the fracture energy $\Gamma(v)$ that is controlled by a dissipation time scale. These results substantiate the universal nature of the oscillatory instability of ultra-high speed cracks in 2D. In addition, we provide evidence indicating that the ultra-high velocity tip-splitting instability is controlled by the limiting rate of elastic energy transport inside the crack tip region. Finally, we describe in detail the numerical implementation scheme of the employed phase-field fracture approach., Comment: Main changes: a new Appendix C (and Fig. C.1) and new data on tip-splitting angles

In earlier papers (Leblond et.al., 2011, 2019), we presented linear stability analyses of the coplanar propagation of a crack loaded in mixed-mode I+III, based on a "double'' propagation criterion combining Griffith (1920)'s energetic condition and Goldstein and Salganik (1974)'s principle of local symmetry. The difference between the two papers was that in the more recent one, the local value of the critical energy-release-rate was no longer considered as a constant, but heuristically allowed to depend upon the ratio of the local mode III to mode I stress intensity factors. This led to a much improved, qualitatively acceptable agreement of theory and experiments, for the "threshold'' value of the ratio of the unperturbed mode III to mode I stress intensity factors, above which coplanar propagation becomes unstable. In this paper, the analysis is extended to the case where a small additional mode II loading component is present in the initially planar configuration of the crack, generating a small, general kink of this crack from the moment it is applied. The main new effect resulting from presence of such a loading component is that the instability modes present above the threshold must drift along the crack front during its propagation. This prediction may be useful for future theoretical interpretations of a number of experiments where such a drifting motion was indeed observed., Comment: 33 pages, 10 figures

In a previous paper (Leblond et al., 2011), we proposed a theoretical interpretation of the experimentally well known instability of coplanar crack propagation in mode I+III. The interpretation relied on a stability analysis based on analytical expressions of the stress intensity factors for a crack slightly perturbed both within and out of its original plane, due to Gao and Rice (1986) and Movchan et al. (1998), coupled with a double propagation criterion combining Griffith's energetic condition and principle of local symmetry. Under such assumptions instability modes were indeed evidenced for values of the mode mixity ratio of the mode III to mode I stress intensity factors applied remotely larger than some threshold depending only on Poisson's ratio. Unfortunately, the predicted thresholds were much larger than those generally observed for typical values of this material parameter. While the subcritical character of the nonlinear bifurcation from coplanar to fragmented fronts has been proposed as a possible explanation for this discrepancy (Chen et al., 2015), we propose here an alternative explanation based on the introduction of a constitutive relationship between the fracture energy and the mode mixity ratio, which is motivated by experimental observations. By reexamining the linear stability analysis of a planar propagating front, we show that such a relationship suffices, provided that it is strong enough, to lower significantly the threshold value of the mode mixity ratio for instability so as to bring it in a range more consistent with experiments. Interesting formulae are also derived for the distributions of the perturbed stress intensity factors and energy release rate, in the special case of perturbations of the crack surface and front obeying the principle of local symmetry and having reached a stationary state., Comment: 30 pages and 7 figures

In some previous papers, we presented some linear stability analyses of the coplanar propagation of a crack loaded in mixed-mode I + III, using a propagation criterion combining a Griffith-type energetic condition and Goldstein and Salganik’s “principle of local symmetry”. In the last one, the local value of the fracture energy was no longer considered as a constant but heuristically permitted to depend upon the ratio of the local mode III to mode I stress intensity factors. As a result, a much improved agreement of theory and experimental observations was obtained for the “threshold” value of the ratio of the unperturbed mode III to mode I stress intensity factors, above which coplanar propagation becomes unstable. This analysis is extended here to the situation, of considerable practical significance, where a small additional mode II loading component is present in the initially planar configuration of the crack. This component induces a small, general kink of this crack from the moment it is applied. The main novelty resulting from its application is that the instability modes, present above the threshold, must drift along the crack front during its propagation. It is hoped that this prediction will be useful to theoretically interpret a number of experiments where such a drifting motion was indeed observed but left unexplained.