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Dynamical systems approach of internal length in fractional calculus

Beda P. B.

Engineering Transactions

2017

Vol. 65, iss. 1

209--215

Conventionally, non-local properties are included in the constitutive equations in the form of strain gradient-dependent terms. In case of the second gradient dependence an internal material length can be obtained from the critical eigenmodes in instability problems. When non-locality is included by using fractional calculus, a generalized strain can be defined. Stability investigation can be also performed and internal length effects can be studied by analysing the critical eigenspace. Such an approach leads to classical results for second gradient, but new phenomena appear in the first gradient case.

Constitutive relations for dynamic material instability at finite deformation

Beda P. B.

Archives of Mechanics

2001

Vol. 53, nr 4-5

319-331

This paper aims to present a mathematically consistent formulation of the second gradient dependence in the constitutive equations for material instability phenomena in case of finite deformations. Thus the set of fundamental equations of the solid continuum (the kinematic equations, the Cauchy equations of motion and the constitutive equations) should also be written for finite deformations. Two basic properties are required: the existence and regular propagation of waves and the generic behavior at the loss of stability. Firstly, the wave dynamics is studied. To encounter the second gradient effects, we should use the third order waves here. Secondly, the system of fundamental equations completed with initial and boundary value conditions forms a dynamical system. Then, identifying material stability with Lapunov stability of a state of the continuus body, the loss of stability should be one of the two basic types of instabilities of dynamical systems: a static or a dynamic bifurcation. These instability modes should be strictly different for a generic dynamical system.