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Article

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

1
School of Mechanical Engineering and Automation, Beihang University, Beijing 100191, China
2
Ishlinsky Institute for Problems in Mechanics RAS, Moscow 119526, Russia
3
Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam
4
Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam
5
Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India
*
Author to whom correspondence should be addressed.
Materials 2019, 12(17), 2725; https://doi.org/10.3390/ma12172725
Received: 9 July 2019 / Revised: 4 August 2019 / Accepted: 19 August 2019 / Published: 26 August 2019
(This article belongs to the Special Issue Advanced Polymer Design and Manufacturing)
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. View Full-Text
Keywords: constitutive equations; interface; friction; singularity; envelope of characteristics; continuum mechanics constitutive equations; interface; friction; singularity; envelope of characteristics; continuum mechanics
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MDPI and ACS Style

Alexandrov, S.; Lang, L.; Lyamina, E.; Date, P.P. Solution Behavior near Envelopes of Characteristics for Certain Constitutive Equations Used in the Mechanics of Polymers. Materials 2019, 12, 2725. https://doi.org/10.3390/ma12172725

AMA Style

Alexandrov S, Lang L, Lyamina E, Date PP. Solution Behavior near Envelopes of Characteristics for Certain Constitutive Equations Used in the Mechanics of Polymers. Materials. 2019; 12(17):2725. https://doi.org/10.3390/ma12172725

Chicago/Turabian Style

Alexandrov, Sergei, Lihui Lang, Elena Lyamina, and Prashant P. Date. 2019. "Solution Behavior near Envelopes of Characteristics for Certain Constitutive Equations Used in the Mechanics of Polymers" Materials 12, no. 17: 2725. https://doi.org/10.3390/ma12172725

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