An existence result for the fractional Kelvin–Voigt’s model on time-dependent cracked domains

We prove an existence result for the fractional Kelvin–Voigt’s model involving Caputo’s derivative on time-dependent cracked domains. We first show the existence of a solution to a regularized version of this problem. Then, we use a compactness argument to derive that the fractional Kelvin–Voigt’s model admits a solution which satisfies an energy-dissipation inequality. Finally, we prove that when the crack is not moving, the solution is unique.

Wave propagation dynamics in a fractional Zener model with stochastic excitation

Equations of motion for a Zener model describing a viscoelastic rod are investigated and conditions ensuring the existence, uniqueness and regularity properties of solutions are obtained. Restrictions on the coefficients in the constitutive equation are determined by a weak form of the dissipation inequality. Various stochastic processes related to the Karhunen-Loéve expansion theorem are presented as a model for random perturbances. Results show that displacement disturbances propagate with an infinite speed. Some corrections of already published results for a non-stochastic model are also provided.

DAMAGE MODELING AND ASSESSMENT FOR BRITTLE MATERIALS

Observed nonlinearities in frictional and brittle solids such as concrete, rocks, ceramics, and some composites arise mainly due to the nucleation and propagation of microvoids and microcracks. Microcrack formation, propagation, and coalescence damage the material and renders it more compliant. Microdefects and cracks are also usually irreversible and cause strong anisotropy in the response to loads. This paper presents a damage mechanics model to capture material anisotropy and damage under multiaxial stress states for proportional and fatigue type loadings. The theory is cast within the generally accepted principles of thermodynamics with internal variables where the dissipation inequality is invoked to develop loading surfaces. Flow rules for the onset of inelastic deformations is provided and specific damage laws are proposed. The model extension to the cyclic and fatigue type loading is presented and numerical results are provided with comparison to the available experimental data.

Control of the bridge crane by constructing a Lyapunov function: theoretical design and experimental verification

For the control problem of bridge cranes, it is challenging to realize fast transportation and efficient swing suppression simultaneously. Motivated by this observation, in this paper, we aim to propose a nonlinear controller achieving these objectives by constructing a desired Lyapunov function. In particular, a constructive Lyapunov function is introduced in a segmented manner. Based on that, a nonlinear control method rendering the dissipation inequality with respect to the constructed Lyapunov function is proposed straightforwardly, which achieves precise trolley positioning along with efficient payload swing elimination. The corresponding stability and convergence analysis is guaranteed by Lyapunov techniques and LaSalle’s invariance principle. Simulation and experimental results are provided to demonstrate the effectiveness and feasibility of the proposed method.

Safety Verification of Interconnected Hybrid Systems Using Barrier Certificates

Safety verification determines whether any trajectory starting from admissible initial states would intersect with a set of unsafe states. In this paper, we propose a numerical method for verifying safety of a network of interconnected hybrid dynamical systems with a state constraint based on bilinear sum-of-squares programming. The safety verification is conducted by the construction of a function of states called barrier certificate. We consider a finite number of interconnected hybrid systems satisfying the input-to-state property and the networked interconnections satisfying a dissipativity property. Through constructing a barrier certificate for each subsystem and imposing dissipation-inequality-like constraints on the interconnections, safety verification is formulated as a bilinear sum-of-squares feasibility problem. As a result, safety of the interconnected hybrid systems could be determined by solving an optimization problem, rather than solving differential equations. The proposed method makes it possible to verify the safety of interconnected hybrid systems, which is demonstrated by a numerical example.