Cusp Formation for a Nonlocal Evolution Equation

The paper is concerned with the asymptotic behaviour of the solutions to a nonlocal evolution equation which arises in models of phase separation. As in the Allen–Cahn equations, stationary spatially nonhomogeneous solutions exist, which represent the interface profile between stable phases. Local stability of these interface profiles is proved.

Download Full-text

Modeling the Dynamic Failure of Railroad Tank Cars Using a Physically Motivated Internal State Variable Plasticity/Damage Nonlocal Model

Modelling and Simulation in Engineering ◽

10.1155/2013/815158 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11 ◽

Cited By ~ 3

Author(s):

Fazle R. Ahad ◽

Koffi Enakoutsa ◽

Kiran N. Solanki ◽

Yustianto Tjiptowidjojo ◽

Douglas J. Bammann

Keyword(s):

Finite Element ◽

Evolution Equation ◽

Internal State ◽

Mesh Size ◽

High Rate ◽

Length Scale ◽

Internal State Variable ◽

State Variable ◽

Nonlocal Evolution Equation ◽

The Impact

We used a physically motivated internal state variable plasticity/damage model containing a mathematical length scale to idealize the material response in finite element simulations of a large-scale boundary value problem. The problem consists of a moving striker colliding against a stationary hazmat tank car. The motivations are (1) to reproduce with high fidelity finite deformation and temperature histories, damage, and high rate phenomena that may arise during the impact accident and (2) to address the material postbifurcation regime pathological mesh size issues. We introduce the mathematical length scale in the model by adopting a nonlocal evolution equation for the damage, as suggested by Pijaudier-Cabot and Bazant in the context of concrete. We implement this evolution equation into existing finite element subroutines of the plasticity/failure model. The results of the simulations, carried out with the aid of Abaqus/Explicit finite element code, show that the material model, accounting for temperature histories and nonlocal damage effects, satisfactorily predicts the damage progression during the tank car impact accident and significantly reduces the pathological mesh size effects.

Download Full-text

Stability of a suspension bridge with a localized structural damping

Discrete & Continuous Dynamical Systems - S ◽

10.3934/dcdss.2021089 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Zayd Hajjej ◽

Mohammad Al-Gharabli ◽

Salim Messaoudi

Keyword(s):

Evolution Equation ◽

Decay Rate ◽

Suspension Bridge ◽

Resonance Phenomenon ◽

Structural Damping ◽

Exponential Decay Rate ◽

Nonlocal Evolution Equation ◽

Bridge Collapse ◽

Sudden Transition ◽

Tacoma Narrows Bridge

Strong vibrations can cause lots of damage to structures and break materials apart. The main reason for the Tacoma Narrows Bridge collapse was the sudden transition from longitudinal to torsional oscillations caused by a resonance phenomenon. There exist evidences that several other bridges collapsed for the same reason. To overcome unwanted vibrations and prevent structures from resonating during earthquakes, winds, ..., features and modifications such as dampers are used to stabilize these bridges. In this work, we use a minimum amount of dissipation to establish exponential decay- rate estimates to the following nonlocal evolution equation<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{tt}(x,y,t)+\Delta^2 u(x,y,t) - \phi(u) u_{xx}- \left(\alpha(x, y) u_{xt}(x,y,t)\right)_x = 0, $\end{document} </tex-math></disp-formula>which models the deformation of the deck of either a footbridge or a suspension bridge.

Download Full-text