Cracks in random brittle solids: From fiber bundles to continuum mechanics

Statistical models are essential to get a better understanding of the role of disorder in brittle disordered solids. Fiber bundle models play a special role as a paradigm, with a very good balance of simplicity and non-trivial effects. We introduce here a variant of the fiber bundle model where the load is transferred among the fibers through a very compliant membrane. This Soft Membrane fiber bundle mode reduces to the classical Local Load Sharing fiber bundle model in 1D. Highlighting the continuum limit of the model allows to compute an equivalent toughness for the fiber bundle and hence discuss nucleation of a critical defect. The computation of the toughness allows for drawing a simple connection with crack front propagation (depinning) models.Comment: The European Physical Journal Special Topics Special Topics, 201

Compliant, Large-Strain, and Self-Sensing Twisted String Actuators with Applications to Soft Robots

The twisted string actuator (TSA) is a rotary-to-linear transmission system that has been implemented in robots for high force output and efficiency. The basic components of a TSA are a motor, strings, and a load (to keep the strings in tension). The twisting of the strings shortens their length to generate linear contraction. Due to their high force output, energy efficiency, and compact form factor, TSAs hold the potential to improve the performance of soft robots. Currently, it is challenging to realize high-performance soft robots because many existing soft or compliant actuators exhibit limitations such as fabrication complexity, high power consumption, slow actuation, or low force generation. The applications of TSAs in soft robots have hitherto been limited, mainly for two reasons. Firstly, the conventional strings of TSAs are stiff and strong, but not compliant. Secondly, precise control of TSAs predominantly relies on external position or force sensors. For these reasons, TSA-driven robots are often rigid or bulky.To make TSAs more suitable for actuating soft robots, compliant, large-strain, and self-sensing TSAs are developed and applied to various soft robots in this work. The design was realized by replacing conventional inelastic strings with compliant, thermally-activated, and conductive supercoiled polymer (SCP) strings. Self-sensing was realized by correlating the electrical resistance of the strings with their length. Large strains are realized by heating the strings in addition to twisting them. The quasi-static actuation and self-sensing properties are accurately captured by Preisach hysteresis operators. Next, a data-driven mathematical model was proposed and experimentally validated to capture the transient decay, creep, and hysteretic effects in the electrical resistance. This model was then used to predict the length of the TSA, given its resistance. Furthermore, three TSA-driven soft robots were designed and fabricated: a three-fingered gripper, a soft manipulator, and an anthropomorphic gripper. For the three-fingered gripper, its fingers were compliant and designed to exploit the Fin Ray Effect for improved grasping. The soft manipulator was driven by three TSAs that allowed it to bend with arbitrary magnitude and direction. A physics-based modeling strategy was developed to predict this multi-degree-of-freedom motion. The proposed modeling approaches were experimentally verified to be effective. For example, the proposed model predicted bending angle and bending velocity with mean errors of 1.58 degrees (2.63%) and 0.405 degrees/sec (4.31%), respectively. The anthropomorphic gripper contained 11 TSAs; two TSAs were embedded in each of the four fingers and three TSAs were embedded in the thumb. Furthermore, the anthropomorphic gripper achieved tunable stiffness and a wide range of grasps

Isolation Schemes for Problems on Decomposable Graphs

The Isolation Lemma of Mulmuley, Vazirani and Vazirani [Combinatorica'87] provides a self-reduction scheme that allows one to assume that a given instance of a problem has a unique solution, provided a solution exists at all. Since its introduction, much effort has been dedicated towards derandomization of the Isolation Lemma for specific classes of problems. So far, the focus was mainly on problems solvable in polynomial time. In this paper, we study a setting that is more typical for $\mathsf{NP}$-complete problems, and obtain partial derandomizations in the form of significantly decreasing the number of required random bits. In particular, motivated by the advances in parameterized algorithms, we focus on problems on decomposable graphs. For example, for the problem of detecting a Hamiltonian cycle, we build upon the rank-based approach from [Bodlaender et al., Inf. Comput.'15] and design isolation schemes that use - $O(t\log n + \log^2{n})$ random bits on graphs of treewidth at most $t$; - $O(\sqrt{n})$ random bits on planar or $H$-minor free graphs; and - $O(n)$-random bits on general graphs. In all these schemes, the weights are bounded exponentially in the number of random bits used. As a corollary, for every fixed $H$ we obtain an algorithm for detecting a Hamiltonian cycle in an $H$-minor-free graph that runs in deterministic time $2^{O(\sqrt{n})}$ and uses polynomial space; this is the first algorithm to achieve such complexity guarantees. For problems of more local nature, such as finding an independent set of maximum size, we obtain isolation schemes on graphs of treedepth at most $d$ that use $O(d)$ random bits and assign polynomially-bounded weights. We also complement our findings with several unconditional and conditional lower bounds, which show that many of the results cannot be significantly improved

Well-posedness and Robust Preconditioners for the Discretized Fluid-Structure Interaction Systems

In this paper we develop a family of preconditioners for the linear algebraic systems arising from the arbitrary Lagrangian-Eulerian discretization of some fluid-structure interaction models. After the time discretization, we formulate the fluid-structure interaction equations as saddle point problems and prove the uniform well-posedness. Then we discretize the space dimension by finite element methods and prove their uniform well-posedness by two different approaches under appropriate assumptions. The uniform well-posedness makes it possible to design robust preconditioners for the discretized fluid-structure interaction systems. Numerical examples are presented to show the robustness and efficiency of these preconditioners.Comment: 1. Added two preconditioners into the analysis and implementation 2. Rerun all the numerical tests 3. changed title, abstract and corrected lots of typos and inconsistencies 4. added reference