On objective surface rates

Abstract

Abstract. In this paper we derive objective, in the sense of surface, rates of tensors and we give a correct formulation of a two-dimensional continuum. Furthermore we present objective rates of tensors on the boundary line enclosing the surface under consideration. It can be considered as a first step to derive objective rates for generalized continua as Cosserat continua and Kirchhoff-Love type nonlinear shell theories.1. Introduction. Objective time derivatives were introduced into the constitutive equations primarily to describe properly the time change of stress and strain tensors. Starting from the pioneering papers of Zaremba [36] and Jaumann [12], the concept of objective rates has been the subject of several works. Among them the fruitful concept of "convective derivatives" was given by Oldroyd [22], It can be considered as a method of time differentiation of any tensor field attached to the deformed body, but performed after some mapping into a place where a comparison of different tensors can be made. Since Oldroyd has used the gradient of deformation for the mapping operation, all objective derivatives based on the deformation gradient or its parts are called "convective rates." Special cases are presently known as ZarembaJaumann, Cotter-Rivlin [4] or Rivlin-Ericksen derivatives related to strain rates, and the Truesdell derivative [31] related to stress rates. Additional arguments in favor of the convective derivatives were given by Sedov [25] and Masur [17]. The latter has shown how the Zaremba-Jaumann derivative is connected with that of Oldroyd.Recently, great efforts have been made for a correct formulation of two-dimensional thermomechanics of a curved surface. For many phenomena, like dynamics of a phase interfaces layer [34], viscoelastic 2-dimensional flow of the products of friction and wear [37], capillary flow [9], bubble dynamics [11], dynamics of a crack [27], elastoviscoplasticity of thin shells, etc., it is required to define useful constitutive equations. Frequently, the constitutive equations of differential or rate type are used, since they are easier to implement in computer codes than equations of integral type. These phenomenological relations should contain information about the surface geometry as well as objective surface rates both of stresses and strains.The aim of this paper is to consider a correct formulation of the two-dimensional continuum and to derive objective, but in the sense of surface, rates of tensors.

On the angular velocity slip in nano-flows

“…The procedure starts from the Lagrangian deformation measures which, after material time differentiation, are "pushed forward" to the Eulerian "picture". Adopting here threedimensional operations of "pull back" and the "push forward" discovered by Kafadar and Eringen (1971), to a twodimensional shell-like layer, we find the following surface rate of deformation introduced by Rey (2006), Stumpf and Badur (1993):…”

“…where , µ are two translational viscosity coefficients and γ, η, τ, θ are four rotational viscosity coefficients by Rey (2006), Stumpf and Badur (1993 …”

On the mass and momentum transport in the Navier–Stokes slip layer

“…We are based here on a general surface kinematics elaborated by Pietraszkiewicz (1977) and Stumpf and Badur (1993). The general form of the surface balances of mass, momentum, angular momentum, energy, and entropy, etc.…”

“…Rate of surface deformation, in analogy to the three-dimensional case, is defined as a symmetric part of the surface gradient of velocity, (Stumpf and Badur 1993):…”

Theory of surface excess Miesowicz viscosities of planar nematic liquid crystal-isotropic fluid interfaces

“…The kinematic vector needed to describe director dynamics is the surface Jaumann derivative of the tangential component of the director, N s [28]:…”