One-Dimensional Well-Posed Nonlocal Elasticity Models for Finite Domains

2021 ◽  
pp. 149-168
Author(s):  
Mohammad Ali Maneshi ◽  
Esmaeal Ghavanloo ◽  
S. Ahmad Fazelzadeh
Keyword(s):  
Nonlocal Elasticity ◽  
One Dimensional ◽  
Finite Domains ◽  
Well Posed

Well-posed nonlocal elasticity model for finite domains and its application to the mechanical behavior of nanorods

2020 ◽  
Vol 231 (10) ◽  
pp. 4019-4033 ◽  
Author(s):  
Mohammad A. Maneshi ◽  
Esmaeal Ghavanloo ◽  
S. Ahmad Fazelzadeh
Keyword(s):  
Mechanical Behavior ◽  
Nonlocal Elasticity ◽  
Finite Domains ◽  
Well Posed

scholarly journals The generalized viscoelastic spring

2020 ◽  
Vol 476 (2236) ◽  
pp. 20190881
Author(s):  
Kostas P. Soldatos
Keyword(s):  
Viscous Flow ◽  
A Priori ◽  
Closed Form Solution ◽  
Form Solution ◽  
Viscoelastic Deformation ◽  
One Dimensional ◽  
Flow Formation ◽  
Inelastic Material ◽  
Flow Potential ◽  
Well Posed

A spring/rod model is presented that describes one-dimensional behaviour of solids susceptible to large or small viscoelastic deformation. Derivation of its constitutive equation is underpinned by the fact that the internal energy, which the elastic part of deformation stores in the spring, changes in time with the observed strain as well as with some, unknown part of the strain-rate. The latter emerges through the action of a viscous flow potential and is the source of inelastic deformation. Thus, unlike its conventional viscoelasticity counterparts, the model does not postulate a priori a rule that relates strain with viscous flow formation. Instead, it considers that such a rule, as well as other important features of combined elastic and inelastic material response, should become known a posteriori through the solution of a relevant, well-posed boundary value problem. This paper begins with considerations compatible with large viscoelastic deformations and gradually progresses through simpler modelling situations. The latter also include the case of small viscoelastic strain that underpins formulation of classical, spring-dashpot viscoelastic models. In an example application, a relevant closed-form solution is obtained for a spring undergoing small viscoelastic deformation under the influence of a viscous flow potential which is quadratic in the stress.

scholarly journals Nonlocal Effect on Stiffness Measurements of a Collagen Molecule

2015 ◽  
Vol 82 (3) ◽  
Author(s):  
Victor Birman
Keyword(s):  
Mechanical Properties ◽  
Nonlocal Elasticity ◽  
Nonlocal Elasticity Theory ◽  
Md Simulations ◽  
Experimental Methods ◽  
Nonlocal Effect ◽  
Collagen Molecule ◽  
One Dimensional ◽  
Collagen Molecules ◽  
Range Of Values

Accurate modeling of collagen molecules including their stiffness is essential for our understanding of mechanics of collagen fibers and tissues where these fibers play a prominent role. Studies of mechanical properties of collagen molecules employing various experimental methods and molecular dynamics (MD) simulations yield a broad range of values of the modulus of elasticity. The effect of nonlocal elasticity on the molecule stiffness derived from experiments and simulations is assessed in this brief. The estimate of the correction accounting for the nonlocal effect utilizes the exact solution of the nonlocal elasticity theory for one-dimensional elastic bars. It is demonstrated that the effect of nonlocal elasticity on the stiffness of collagen molecules can be neglected.

scholarly journals Well-Posed Lateral Boundary Conditions for Spectral Semi-Implicit Semi-Lagrangian Schemes: Tests in a One-Dimensional Model

2009 ◽  
Vol 137 (1) ◽  
pp. 315-330 ◽  
Author(s):  
F. Voitus ◽  
P. Termonia ◽  
P. Bénard
Keyword(s):  
Boundary Conditions ◽  
Weather Prediction ◽  
Implicit Time ◽  
Lateral Boundary ◽  
Dimensional Model ◽  
One Dimensional ◽  
Lateral Boundary Conditions ◽  
Boundary Treatment ◽  
Truncation Scheme ◽  
Well Posed

Abstract The aim of this paper is to investigate the feasibility of well-posed lateral boundary conditions in a Fourier spectral semi-implicit semi-Lagrangian one-dimensional model. Two aspects are analyzed: (i) the complication of designing well-posed boundary conditions for a spectral semi-implicit scheme and (ii) the implications of such a lateral boundary treatment for the semi-Lagrangian trajectory computations at the lateral boundaries. Straightforwardly imposing boundary conditions in the gridpoint-explicit part of the semi-implicit time-marching scheme leads to numerical instabilities for time steps that are relevant in today’s numerical weather prediction applications. It is shown that an iterative scheme is capable of curing these instabilities. This new iterative boundary treatment has been tested in the framework of the one-dimensional shallow-water equations leading to a significant improvement in terms of stability. As far as the semi-Lagrangian part of the time scheme is concerned, the use of a trajectory truncation scheme has been found to be stable in experimental tests, even for large values of the advective Courant number. It is also demonstrated that a well-posed buffer zone can be successfully applied in this spectral context. A promising (but not easily implemented) alternative to these three above-referenced schemes has been tested and is also presented here.

Asymptotic stability for a laminated beam with structural damping and infinite memory

2020 ◽  
Vol 25 (10) ◽  
pp. 1979-2004 ◽  
Author(s):  
Wenjun Liu ◽  
Xiangyu Kong ◽  
Gang Li
Keyword(s):  
Wave Propagation ◽  
Relaxation Function ◽  
Rotation Angle ◽  
Structural Damping ◽  
Laminated Beam ◽  
Infinite Memory ◽  
One Dimensional ◽  
Special Cases ◽  
Lack Of Exponential Stability ◽  
Well Posed

In this paper, we consider a one-dimensional laminated beam with structural damping and an infinite memory acting on the effective rotation angle. Under appropriate assumptions imposed on the relaxation function, we show that the system is well-posed by using the Hille–Yosida theorem, and then we establish general decay results, from which exponential and polynomial decays are only special cases, in the case of equal speeds of wave propagation as well as that of nonequal speeds. In the particular case when the wave propagation speeds are different and the relaxation function decays exponentially, we show the lack of exponential stability.

A continuum theory of surface growth

2010 ◽  
Vol 466 (2123) ◽  
pp. 3135-3152 ◽  
Author(s):  
Neil Hodge ◽  
Panayiotis Papadopoulos
Keyword(s):  
Inertial Frame ◽  
Continuum Theory ◽  
The Body ◽  
Surface Growth ◽  
Governing Equations ◽  
Predictive Capacity ◽  
One Dimensional ◽  
Deformable Bodies ◽  
Material Points ◽  
Well Posed

A continuum theory of surface growth in deformable bodies is presented. The theory employs a decomposition of the deformation and growth processes, which leads to a well-posed set of governing equations. It is argued that an evolving reference configuration is required to track the material points in the body. The balance laws are formulated with respect to a non-inertial frame of reference, which is used to track the motion of the body. A one-dimensional example problem is included to showcase the predictive capacity of the theory.

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