Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling

2010 ◽  
Vol 40 (1-2) ◽  
pp. 125-181 ◽  
Author(s):  
Gianni Dal Maso ◽  
Antonio DeSimone ◽  
Francesco Solombrino
Keyword(s):  
Weak Formulation ◽  
Quasistatic Evolution ◽  
Cam Clay

scholarly journals Cohesive fracture with irreversibility: Quasistatic evolution for a model subject to fatigue

2018 ◽  
Vol 28 (07) ◽  
pp. 1371-1412 ◽  
Author(s):  
Vito Crismale ◽  
Giuliano Lazzaroni ◽  
Gianluca Orlando
Keyword(s):  
Crack Surface ◽  
Crack Opening ◽  
Young Measure ◽  
Small Strain ◽  
Weak Formulation ◽  
Cohesive Fracture ◽  
Quasistatic Evolution ◽  
Antiplane Elasticity ◽  
Energy Dissipated ◽  
Model Subject

In this paper we prove the existence of quasistatic evolutions for a cohesive fracture on a prescribed crack surface, in small-strain antiplane elasticity. The main feature of the model is that the density of the energy dissipated in the fracture process depends on the total variation of the amplitude of the jump. Thus, any change in the crack opening entails a loss of energy, until the crack is complete. In particular this implies a fatigue phenomenon, i.e. a complete fracture may be produced by oscillation of small jumps. The first step of the existence proof is the construction of approximate evolutions obtained by solving discrete-time incremental minimum problems. The main difficulty in the passage to the continuous-time limit is that we lack of controls on the variations of the jump of the approximate evolutions. Therefore we resort to a weak formulation where the variation of the jump is replaced by a Young measure. Eventually, after proving the existence in this weak formulation, we improve the result by showing that the Young measure is concentrated on a function and coincides with the variation of the jump of the displacement.

scholarly journals Quasistatic evolution for Cam-Clay plasticity: The spatially homogeneous case

2010 ◽  
Vol 5 (1) ◽  
pp. 97-132 ◽  
Author(s):  
Gianni Dal Maso ◽  
◽  
Francesco Solombrino ◽  
Keyword(s):  
Homogeneous Case ◽  
Quasistatic Evolution ◽  
Spatially Homogeneous ◽  
Cam Clay

Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution

2011 ◽  
Vol 44 (3-4) ◽  
pp. 495-541 ◽  
Author(s):  
Gianni Dal Maso ◽  
Antonio DeSimone ◽  
Francesco Solombrino
Keyword(s):  
Viscosity Solution ◽  
Quasistatic Evolution ◽  
Cam Clay

QUASISTATIC EVOLUTION FOR CAM-CLAY PLASTICITY: EXAMPLES OF SPATIALLY HOMOGENEOUS SOLUTIONS

2009 ◽  
Vol 19 (09) ◽  
pp. 1643-1711 ◽  
Author(s):  
GIANNI DAL MASO ◽  
ANTONIO DESIMONE
Keyword(s):  
Initial Conditions ◽  
Evolution Problem ◽  
Quasistatic Evolution ◽  
Homogeneous Solutions ◽  
Softening Behavior ◽  
Spatially Homogeneous ◽  
Cam Clay

We study a quasistatic evolution problem for Cam-Clay plasticity under a special loading program which leads to spatially homogeneous solutions. Under some initial conditions, the solutions exhibit a softening behavior and time discontinuities. The behavior of the solutions at the jump times is studied by a viscous approximation.

scholarly journals A derivation of the Liouville equation for hard particle dynamics with non-conservative interactions

2020 ◽  
pp. 1-35
Author(s):  
Benjamin D. Goddard ◽  
Tim D. Hurst ◽  
Mark Wilkinson
Keyword(s):  
Liouville Equation ◽  
Particle Dynamics ◽  
Fundamental Importance ◽  
Continuum Models ◽  
Hard Particle ◽  
Interacting Particles ◽  
Weak Formulation ◽  
Physical Systems ◽  
Physical Particle

The Liouville equation is of fundamental importance in the derivation of continuum models for physical systems which are approximated by interacting particles. However, when particles undergo instantaneous interactions such as collisions, the derivation of the Liouville equation must be adapted to exclude non-physical particle positions, and include the effect of instantaneous interactions. We present the weak formulation of the Liouville equation for interacting particles with general particle dynamics and interactions, and discuss the results using two examples.

Geometrically exact, intrinsic mixed variational formulation for smart, slender multilink composite structures

2021 ◽  
pp. 1045389X2110322
Author(s):  
P M G Bashir Asdaque ◽  
Sitikantha Roy
Keyword(s):  
Composite Structures ◽  
Turbine Blades ◽  
Space Structures ◽  
Wind Turbine Blades ◽  
Weak Formulation ◽  
Helicopter Blades ◽  
Static Mechanical ◽  
Computational Platform ◽  
Swept Wings ◽  
Variational Statement

Flexible links are often part of massive aerospace structures like helicopter or wind turbine blades, satellite bae, airplane wings, and space stations. In the present work, a mixed variational statement based on intrinsic variables is derived for multilinked smart slender structures. Equations involved in the derivation do not involve approximations of kinematical variables to describe the deformation of the reference line or the rotation of the deformed cross-section of the slender links resulting in a geometrically exact formulation. Finite element equations are derived from weak formulation, which can analyze large geometrically non-linear problems. The weakest possible variational statement provides greater flexibility in the choice of shape functions, therefore reducing the associated numerical complexities. The present work focuses on developing a single integrated computational platform which can study multibody, multilink, lightweight composite, structural system built with both embedded actuations, sensing, as well as passive links. Validation of static mechanical and electrical outputs from 3D FE simulation and literature proves the efficacy of the computational platform. Dynamic results will be communicated in future correspondence. The computational platform developed here can be applied for monitoring and active control applications of flexible smart multilink structures like swept wings, multi-bae space structures, and helicopter blades.

scholarly journals Non-Isothermal Creeping Flows in a Pipeline Network: Existence Results

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1300
Author(s):  
Evgenii S. Baranovskii ◽  
Vyacheslav V. Provotorov ◽  
Mikhail A. Artemov ◽  
Alexey P. Zhabko
Keyword(s):  
Nonlinear Boundary ◽  
Galerkin Approximation ◽  
Large Data ◽  
Temperature Dependent ◽  
Weak Formulation ◽  
Temperature Dependent Viscosity ◽  
Pipeline Network ◽  
Flux Boundary Conditions ◽  
Creeping Flows ◽  
Dependent Viscosity

This paper deals with a 3D mathematical model for the non-isothermal steady-state flow of an incompressible fluid with temperature-dependent viscosity in a pipeline network. Using the pressure and heat flux boundary conditions, as well as the conjugation conditions to satisfy the mass balance in interior junctions of the network, we propose the weak formulation of the nonlinear boundary value problem that arises in the framework of this model. The main result of our work is an existence theorem (in the class of weak solutions) for large data. The proof of this theorem is based on a combination of the Galerkin approximation scheme with one result from the field of topological degrees for odd mappings defined on symmetric domains.

scholarly journals A Banach space mixed formulation for the unsteady Brinkman–Forchheimer equations

2020 ◽  
Author(s):  
Sergio Caucao ◽  
Ivan Yotov
Keyword(s):  
Banach Space ◽  
Time Discretization ◽  
Monotone Operators ◽  
Rates Of Convergence ◽  
Mixed Formulation ◽  
Model Parameters ◽  
Weak Formulation ◽  
Finite Element Approximations ◽  
Backward Euler ◽  
Discrete Finite Element

Abstract We propose and analyse a mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Our approach is based on the introduction of a pseudostress tensor related to the velocity gradient and pressure, leading to a mixed formulation where the pseudostress tensor and the velocity are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation in a Banach space setting, employing classical results on nonlinear monotone operators and a regularization technique. We then present well posedness and error analysis for semidiscrete continuous-in-time and fully discrete finite element approximations on simplicial grids with spatial discretization based on the Raviart–Thomas spaces of degree $k$ for the pseudostress tensor and discontinuous piecewise polynomial elements of degree $k$ for the velocity and backward Euler time discretization. We provide several numerical results to confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method for a range of model parameters.

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