scholarly journals Formulation and existence of weak solutions for a problem of adhesive contact with elastoplasticity and hardening

2021 ◽  
Vol 13 (8) ◽  
pp. 168781402110391
Author(s):  
Ramiro Peñas Galezo
Keyword(s):  
Existence Of Solutions ◽  
Monotone Operators ◽  
Viscosity Coefficient ◽  
Adhesive Contact ◽  
Weak Formulation ◽  
Linear Partial Differential Equations ◽  
Rate Dependent ◽  
Multivalued Operators ◽  
Non Linear ◽  
Deformable Bodies

This paper presents the weak formulation of a quasi-static evolution model for two deformable bodies with uni-directional adhesive unilateral contact on which external loads act. Small deformations and linearized elastoplasticity with hardening are assumed. The adhesion component is rate-dependent or rate-independent according to the choice of the viscosity coefficient of the glue; elastoplasticity is considered rate-independent. The weak formulation is expressed as a doubly non-linear problem with unbounded multivalued operators, as a function of internal and boundary displacements, plastic and symmetric strain tensors, and the bonding field and its gradient. This paper differs from other formulations by coupling the equations defined inside and on the boundary of the solids in functional form. In addition to this novelty, we verify the existence of solutions by a path other than that displayed in similar articles. The existence of solutions is demonstrated after considering a succession of doubly non-linear problems with an unbounded operator, and verifying that the solution of one of the problems is also a solution to the objective problem. The proof is supported by previous results from non-linear Partial differential equations theory with monotone operators.

scholarly journals From adhesive to brittle delamination in visco-elastodynamics

2017 ◽  
Vol 27 (08) ◽  
pp. 1489-1546 ◽  
Author(s):  
Riccarda Rossi ◽  
Marita Thomas
Keyword(s):  
Existence Of Solutions ◽  
Adhesive Contact ◽  
Momentum Balance ◽  
Flow Rule ◽  
Inertial Term ◽  
Rate Dependent ◽  
Elastic Bodies ◽  
Dependent Rate ◽  
Model Features ◽  
Energetic Solution

In this paper, we analyze a system for brittle delamination between two visco-elastic bodies, also subject to inertia, which can be interpreted as a model for dynamic fracture. The rate-independent flow rule for the delamination parameter is coupled with the momentum balance for the displacement, including inertia. This model features a nonsmooth constraint ensuring the continuity of the displacements outside the crack set, which is marked by the support of the delamination parameter. A weak solvabi- lity concept, generalizing the notion of energetic solution for rate-independent systems to the present mixed rate-dependent/rate-independent frame, is proposed. Via refined variational convergence techniques, existence of solutions is proved by passing to the limit in approximating systems which regularize the nonsmooth constraint by conditions for adhesive contact. The presence of the inertial term requires the design of suitable recovery spaces small enough to provide compactness but large enough to recover the information on the crack set in the limit.

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.

Attractive forces slow contact formation between deformable bodies underwater

2021 ◽  
Vol 118 (41) ◽  
pp. e2104975118
Author(s):  
Mengyue Sun ◽  
Nityanshu Kumar ◽  
Ali Dhinojwala ◽  
Hunter King
Keyword(s):  
Three Dimensional ◽  
Adhesive Contact ◽  
Final State ◽  
Underwater Adhesion ◽  
Frustrated Total Internal Reflection ◽  
Solid Deformation ◽  
Hydrophilic Surfaces ◽  
Deformable Bodies ◽  
Adhesive Interactions

Thermodynamics tells us to expect underwater contact between two hydrophobic surfaces to result in stronger adhesion compared to two hydrophilic surfaces. However, the presence of water changes not only energetics but also the dynamic process of reaching a final state, which couples solid deformation and liquid evacuation. These dynamics can create challenges for achieving strong underwater adhesion/friction, which affects diverse fields including soft robotics, biolocomotion, and tire traction. Closer investigation, requiring sufficiently precise resolution of film evacuation while simultaneously controlling surface wettability, has been lacking. We perform high-resolution in situ frustrated total internal reflection imaging to track underwater contact evolution between soft-elastic hemispheres of varying stiffness and smooth–hard surfaces of varying wettability. Surprisingly, we find the exponential rate of water evacuation from hydrophobic–hydrophobic (adhesive) contact is three orders of magnitude lower than that from hydrophobic–hydrophilic (nonadhesive) contact. The trend of decreasing rate with decreasing wettability of glass sharply changes about a point where thermodynamic adhesion crosses zero, suggesting a transition in mode of evacuation, which is illuminated by three-dimensional spatiotemporal height maps. Adhesive contact is characterized by the early localization of sealed puddles, whereas nonadhesive contact remains smooth, with film-wise evacuation from one central puddle. Measurements with a human thumb and alternatively hydrophobic/hydrophilic glass surface demonstrate practical consequences of the same dynamics: adhesive interactions cause instability in valleys and lead to a state of more trapped water and less intimate solid–solid contact. These findings offer interpretation of patterned texture seen in underwater biolocomotive adaptations as well as insight toward technological implementation.

On the Dirichlet problem for weakly non-linear elliptic partial differential equations

1977 ◽  
Vol 76 (4) ◽  
pp. 283-300 ◽  
Author(s):  
E. N. Dancer
Keyword(s):  
Dirichlet Problem ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Existence Of Solutions ◽  
Elliptic Partial Differential Equations ◽  
Partial Differential ◽  
Weakly Nonlinear ◽  
Partial Derivatives ◽  
Nonlinear Elliptic ◽  
Non Linear

SynopsisWe study the existence of solutions of the Dirichlet problem for weakly nonlinear elliptic partial differential equations. We only consider cases where the nonlinearities do not depend on any partial derivatives. For these cases, we prove the existence of solutions for a wide variety of nonlinearities.

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