scholarly journals Compression of a Polar Orthotropic Wedge between Rotating Plates: Distinguished Features of the Solution

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 270
Author(s):  
Sergei Alexandrov ◽  
Elena Lyamina ◽  
Pham Chinh ◽  
Lihui Lang
Keyword(s):  
Boundary Value Problem ◽  
Orthotropic Material ◽  
Friction Surface ◽  
Friction Stress ◽  
Critical Value ◽  
Velocity Fields ◽  
Approach Infinity ◽  
Space Stress ◽  
Polar Orthotropic ◽  
Infinite Wedge

An infinite wedge of orthotropic material is confined between two rotating planar rough plates, which are inclined at an angle 2α. An instantaneous boundary value problem for the flow of the material is formulated and solved for the stress and the velocity fields, the solution being in closed form. The solution may exhibit the regimes of sliding or sticking at the plates. It is shown that the overall structure of the solution significantly depends on the friction stress at sliding. This stress is postulated by the friction law. Solutions, which exhibit sticking, may exist only if the postulated friction stress at sliding satisfies a certain condition. These solutions have a rigid rotating zone in the region adjacent to the plates, unless the angle α is equal to a certain critical value. Solutions which exhibit sliding may be singular. In particular, some space stress and velocity derivatives approach infinity in the vicinity of the friction surface.

Singular perturbation of a nonlinear problem with multiple solutions

1985 ◽  
Vol 100 (3-4) ◽  
pp. 327-341
Author(s):  
Anne-Marie Lefevere
Keyword(s):  
Free Boundary ◽  
Boundary Value Problem ◽  
Multiple Solutions ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Critical Value ◽  
Free Boundary Value Problem ◽  
Singular Limits ◽  
Natural Extensions ◽  
Barrier Parameter

SynopsisA nonlinear boundary value problem (P) having positive parameters L and a is considered. We associate with it a family of perturbed problems () affected by the presence of a barrier parameter γ related to L and a. There is a critical value L*(a) of the parameter L such that for L >L*(a), (P) has no regular solution. Then some natural extensions of (P), solutions of a free boundary value problem, arise as singular limits of ().

scholarly journals Solution Behavior near Envelopes of Characteristics for Certain Constitutive Equations Used in the Mechanics of Polymers

Materials ◽  
2019 ◽  
Vol 12 (17) ◽  
pp. 2725 ◽  
Author(s):  
Sergei Alexandrov ◽  
Lihui Lang ◽  
Elena Lyamina ◽  
Prashant P. Date
Keyword(s):  
Friction Surface ◽  
Yield Criterion ◽  
System Of Equations ◽  
Singular Behavior ◽  
Material Models ◽  
Metal Plasticity ◽  
Plane Strain Deformation ◽  
Stress Components ◽  
Approach Infinity ◽  
Derivatives Of

The present paper deals with plane strain deformation of incompressible polymers that obey quite a general pressure-dependent yield criterion. In general, the system of equations can be hyperbolic, parabolic, or elliptic. However, attention is concentrated on the hyperbolic regime and on the behavior of solutions near frictional interfaces, assuming that the regime of sliding occurs only if the friction surface coincides with an envelope of stress characteristics. The main reason for studying the behavior of solutions in the vicinity of envelopes of characteristics is that the solution cannot be extended beyond the envelope. This research is also motivated by available results in metal plasticity that the velocity field is singular near envelopes of characteristics (some space derivatives of velocity components approach infinity). In contrast to metal plasticity, it is shown that in the case of the material models adopted, all derivatives of velocity components are bounded but some derivatives of stress components approach infinity near the envelopes of stress characteristics. The exact asymptotic expansion of stress components is found. It is believed that this result is useful for developing numerical codes that should account for the singular behavior of the stress field.

Milling Behavior of Gum Elastomers: Experiment and Theory

1967 ◽  
Vol 40 (4) ◽  
pp. 1126-1138 ◽  
Author(s):  
Noboru Tokita ◽  
James L. White
Keyword(s):  
Chemical Composition ◽  
Mathematical Analysis ◽  
Viscoelastic Medium ◽  
Polymer Melt ◽  
Critical Value ◽  
Velocity Fields ◽  
Unstable Flow ◽  
Viscoelastic Media ◽  
Viscous Forces ◽  
Roll Mill

Abstract When an uncompounded elastomer is processed on a two roll mill, four different regions of mechanical behavior are observed, depending upon the temperature and the severity of the nip deformation. This behavior is observed on materials with a wide variety in chemical composition, though the severity varies. The flow at high temperatures is typical of melt or polymer solution behavior. At lower temperatures unstable flow and elastic solidlike regions are observed. By presuming the elastomer to be an isotropic viscoelastic medium, stress and velocity fields were computed in the polymer melt region. The unstable regime was found to correspond to a critical value of the ratio of viscoelastic to viscous forces. The mathematical analysis, done in terms of the Green-Rivlin-Noll theory of viscoelastic media, extends earlier studies of deformation in this geometry by Gaskell and Bergen.

scholarly journals On vibrations in Green-Naghdi thermoelasticity of dipolar bodies

2019 ◽  
Vol 27 (1) ◽  
pp. 125-140 ◽  
Author(s):  
M. Marin ◽  
A. Chirilă ◽  
L. Codarcea ◽  
S. Vlase
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Critical Value ◽  
Spatial Behavior ◽  
Type Iii ◽  
Dipolar Bodies ◽  
Dipolar Structure

Abstract This study is concerned with the theory of thermoelasticity of type III proposed by Green and Naghdi, which is extended to cover the bodies with dipolar structure. In this context we construct a boundary value problem for a prismatic bar which is subjected to some harmonic in time vibrations. For the oscillations whose amplitudes have the frequency lower than a critical value, we deduce some estimates for describing the spatial behavior.

scholarly journals Crevasse Deformation and Examples from Ice Stream B, Antarctica

1990 ◽  
Vol 36 (122) ◽  
pp. 3-10 ◽  
Author(s):  
P.L. Vornberger ◽  
I.M. Whillans
Keyword(s):  
Velocity Field ◽  
Strain Rate ◽  
Critical Value ◽  
Velocity Fields ◽  
The Third ◽  
Velocity Gradients ◽  
Ice Stream ◽  
Lateral Shearing ◽  
Extensional Strain ◽  
Ice Stream B

AbstractCrevasses, once formed, are subject to rotation and bending according to the velocity field through which they travel. Because of this, crevasse shapes can be used to infer something about the velocity field of a glacier. This is done using a model in which each crevasse opens perpendicularly to the principal extensional strain-rate, when that strain-rate exceeds some specified critical value, and is then deformed according to the same velocity gradients that formed the crevasse. This model describes how crevasses are formed, translated, rotated, bent, and lengthened.Velocity fields are sought for which calculations produce crevasses approximating those found in three example areas on Ice Stream B, Antarctica. The first example is the hook-shaped crevasses that occur just outside the chaotic shear zone at the ice-stream margin. They are used to infer a rate of lateral shearing, and side drag. The second example, a pattern of splaying crevasses, is satisfactorily simulated by a model with side-drag stress varying linearly across the ice stream. This confirms that this region is restrained almost entirely by side drag. The third example is transverse crevasses and their change in orientation, but many different velocity fields can produce the observed pattern. Of these three examples, the shapes of hook-shaped marginal crevasses and splaying crevasses can provide useful information whereas transverse crevasses are less helpful.

scholarly journals Solution Behavior Near Very Rough Walls under Axial Symmetry: An Exact Solution for Anisotropic Rigid/Plastic Material

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 184
Author(s):  
Sergei Alexandrov ◽  
Elena Lyamina ◽  
Pierre-Yves Manach
Keyword(s):  
Boundary Value Problem ◽  
Axial Symmetry ◽  
Friction Surface ◽  
Plastic Material ◽  
Boundary Value ◽  
System Of Equations ◽  
Rigid Plastic ◽  
Solution Behavior ◽  
Rigid Plastic Material ◽  
Friction Surfaces

Rigid plastic material models are suitable for modeling metal forming processes at large strains where elastic effects are negligible. A distinguished feature of many models of this class is that the velocity field is describable by non-differentiable functions in the vicinity of certain friction surfaces. Such solution behavior causes difficulty with numerical solutions. On the other hand, it is useful for describing some material behavior near the friction surfaces. The exact asymptotic representation of singular solution behavior near the friction surface depends on constitutive equations and certain conditions at the friction surface. The present paper focuses on a particular boundary value problem for anisotropic material obeying Hill’s quadratic yield criterion under axial symmetry. This boundary value problem represents the deformation mode that appears in the vicinity of frictional interfaces in a class of problems. In this respect, the applied aspect of the boundary value problem is not essential, but the exact mathematical analysis can occur without relaxing the original system of equations and boundary conditions. We show that some strain rate and spin components follow an inverse square rule near the friction surface. An essential difference from the available analysis under plane strain conditions is that the system of equations is not hyperbolic.

scholarly journals Crevasse Deformation and Examples from Ice Stream B, Antarctica

1990 ◽  
Vol 36 (122) ◽  
pp. 3-10 ◽  
Author(s):  
P.L. Vornberger ◽  
I.M. Whillans
Keyword(s):  
Velocity Field ◽  
Strain Rate ◽  
Critical Value ◽  
Velocity Fields ◽  
The Third ◽  
Velocity Gradients ◽  
Ice Stream ◽  
Lateral Shearing ◽  
Extensional Strain ◽  
Ice Stream B

AbstractCrevasses, once formed, are subject to rotation and bending according to the velocity field through which they travel. Because of this, crevasse shapes can be used to infer something about the velocity field of a glacier. This is done using a model in which each crevasse opens perpendicularly to the principal extensional strain-rate, when that strain-rate exceeds some specified critical value, and is then deformed according to the same velocity gradients that formed the crevasse. This model describes how crevasses are formed, translated, rotated, bent, and lengthened.Velocity fields are sought for which calculations produce crevasses approximating those found in three example areas on Ice Stream B, Antarctica. The first example is the hook-shaped crevasses that occur just outside the chaotic shear zone at the ice-stream margin. They are used to infer a rate of lateral shearing, and side drag. The second example, a pattern of splaying crevasses, is satisfactorily simulated by a model with side-drag stress varying linearly across the ice stream. This confirms that this region is restrained almost entirely by side drag. The third example is transverse crevasses and their change in orientation, but many different velocity fields can produce the observed pattern. Of these three examples, the shapes of hook-shaped marginal crevasses and splaying crevasses can provide useful information whereas transverse crevasses are less helpful.

scholarly journals Eigenfunctions of plane elastostatics III the Wedge

1965 ◽  
Vol 5 (2) ◽  
pp. 241-257 ◽  
Author(s):  
V. T. Buchwald
Keyword(s):  
Difference Equation ◽  
Boundary Value Problem ◽  
Eigenfunction Expansion ◽  
Boundary Value ◽  
Fourier Integral ◽  
Complementary Function ◽  
Differential Difference Equation ◽  
Plane Elastostatics ◽  
Infinite Wedge ◽  
Residue Theory

SummaryThe boundary value problem of the infinite wedge in plane elastostatics is reduced to the solution of a differential-difference equation. The complementary function of this equation is determined in the form of a Fourier integral, which, on expansion by residue theory, gives the complete eigenfunction expansion for the wedge. The properties of the eigenfunctions are discussed in some detail, and orthogonality property is derived.

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