Group analysis and exact solutions of the dynamic equations of plane strain of an incompressible nonlinearly elastic body

A plane-strain study of steady sliding by a smooth rigid indentor at any constant speed on a class of orthotropic or transversely isotropic half-spaces is performed. Exact solutions for the full displacement fields are constructed, and applied to the case of the generic parabolic indentor. The closed-form results obtained confirm previous observations that physically acceptable solutions arise for sliding speeds below the Rayleigh speed, for a single critical transonic speed, and for all supersonic speeds. Continuity of contact zone traction is lost for the latter two cases. Calculations for five representative materials indicate that contact zone width achieves minimum values at high, but not critical, subsonic sliding speeds. A key feature of the analysis is the factorization that gives, despite anisotropy, solution expressions that are rather simple in form. In particular, a compact function of the Rayleigh-type emerges that leads to a simple exact formula for the Rayleigh speed itself.

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Symmetry operators and exact solutions of a type of time-fractional Burgers–KdV equation

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887819500324 ◽

2019 ◽

Vol 16 (02) ◽

pp. 1950032 ◽

Cited By ~ 2

Author(s):

Azadeh Naderifard ◽

S. Reza Hejazi ◽

Elham Dastranj ◽

Ahmad Motamednezhad

Keyword(s):

Exact Solutions ◽

Fourth Order ◽

Vector Fields ◽

Kdv Equation ◽

Group Analysis ◽

Invariant Subspaces ◽

Similarity Solutions ◽

Optimal System ◽

Lie Point Symmetries ◽

Symmetry Operators

In this paper, group analysis of the fourth-order time-fractional Burgers–Korteweg–de Vries (KdV) equation is considered. Geometric vector fields of Lie point symmetries of the equation are investigated and the corresponding optimal system is found. Similarity solutions of the equation are presented by using the obtained optimal system. Finally, a useful method called invariant subspaces is applied in order to find another solutions.

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The exact solutions of von mises yielding criterion for ideally plastic body under uniform pressure in case of plane strain

Applied Mathematics and Mechanics ◽

10.1007/bf01908232 ◽

1982 ◽

Vol 3 (4) ◽

pp. 573-579

Author(s):

Fan Ja-shen

Keyword(s):

Exact Solutions ◽

Plane Strain ◽

Uniform Pressure ◽

Plastic Body ◽

Von Mises

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Harmonic Wave Propagation in a Periodically Layered, Infinite Elastic Body: Plane Strain, Analytical Results

Journal of Applied Mechanics ◽

10.1115/1.3424481 ◽

1979 ◽

Vol 46 (1) ◽

pp. 113-119 ◽

Cited By ~ 39

Author(s):

T. J. Delph ◽

G. Herrmann ◽

R. K. Kaul

Keyword(s):

Wave Propagation ◽

Asymptotic Behavior ◽

Plane Strain ◽

Elastic Body ◽

Harmonic Wave ◽

Wave Number ◽

Periodic Medium ◽

Frequency Wave ◽

Wave Numbers ◽

Band Characteristic

The problem of harmonic wave propagation in an unbounded, periodically layered elastic body in a state of plane strain is examined. The dispersion spectrum is shown to be governed by the roots of an 8 × 8 determinant, and represents a surface in frequency-wave number space. The spectrum exhibits the typical stopping band characteristic of wave propagation in a periodic medium. The dispersion equation is shown to uncouple along the ends of the Brillouin zones, and also in the case of wave propagation normal to the layering. The significance of this uncoupling is examined. Also, the asymptotic behavior of the spectrum for large values of the wave numbers is investigated.

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Magnetic Fields Generated by a Mechanical Singularity in a Magnetized Elastic Half Plane

Journal of Applied Mechanics ◽

10.1115/1.3176071 ◽

1989 ◽

Vol 56 (1) ◽

pp. 89-95 ◽

Cited By ~ 18

Author(s):

Chau-Shioung Yeh

Keyword(s):

Fourier Transform ◽

Magnetic Fields ◽

Exact Solutions ◽

Closed Form ◽

Plane Strain ◽

Half Plane ◽

Elastic Solids ◽

Soft Ferromagnetic ◽

Single Force ◽

The Fourier Transform

The induced magnetic fields generated by a line mechanical singularity in a magnetized elastic half plane are investigated in this paper. One version of linear theory for soft ferromagnetic elastic solids which has been developed by Pao and Yeh (1973) is adopted to analyze the plane strain problem undertaken. By applying the Fourier transform technique, the exact solutions for the generated magnetic inductions due to various mechanical singularities such as a single force, a dipole, and single couple are obtained in a closed form. The distributions of the generated inductions on the surface are shown with figures.

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