Generalized thermomechanics with dual internal variables

Abstract

The formal structure of generalized continuum theories is recovered by means of the extension of canonical thermomechanics with dual weakly non-local internal variables. The canonical thermomechanics provides the best framework for such generalization. The Cosserat, micromorphic, and second gradient elasticity theory are considered as examples of the obtained formalization.Keywords Generalized continua · Canonical thermomechanics · Internal variables · Microstructure · Cosserat medium · Micromorphic medium MotivationGeneralized continuum theories extend conventional continuum mechanics by incorporating intrinsic microstructural effects in the mechanical behavior of materials [1][2][3][4][5]. Internal variable approach was always an alternative framework for the continuum modeling of such effects in materials [6][7][8][9][10][11][12]. However, the wellestablished theory of internal variables of state [13,14] cannot completely describe a generalized medium because an internal variable of state has no inertia, but it dissipates. If inertia is introduced, the internal variable must be treated as an actual degree of freedom [15]. Accordingly, the variable is not "internal" any more but can be controlled for instance at the boundary of a body.A more general thermodynamic framework of the internal variable theory presented recently [16] is based on a duality between internal variables, which make possible to derive evolution equations both for internal variables of state and internal degrees of freedom. A natural question relates to the ability of this duality concept to comprise inertial effects. To answer this question, we show how the dual internal variables can be introduced into continuum mechanics and how certain generalized continuum theories can be interpreted in terms of the dual internal variables.The most suitable framework for the generalization of continuum theory by weakly non-local dual internal variables enriched by an extra entropy flux is the material formulation of continuum thermomechanics [17,18]. Therefore, basic definitions of the canonical thermomechanics [17] are recalled in Sect. 2 of the paper. Then dual variables are introduced in Sect. 3 and evolution equations for both dissipative and non-dissipative processes are derived in Sect. 3.2. Linear Cosserat, micromorphic, and second gradient elasticity theories are

Entropy Principle and Recent Results in Non-Equilibrium Theories

“…The thermodynamic approach can substitute variational considerations. For example evolution equations of generalized continua can be obtained with the help of the entropy principle also in the ideal, nondissipative limit [73,74].…”

“…The 7 ω real numbers in Equations (72) and (73) can be arbitrarily chosen, changing the global initial conditions (70), and the 6 ω of them in (73), (the higher derivatives), can be changed without influencing the coefficients of the higher derivatives in the system (65), since these quantities depend on the elements of the state space only. This is enough to prove your assertion…”

Internal structures and internal variables in solids

“…Moreover, the limitation of governing equations to the first order is avoided by the concept of dual internal variables [Ván et al 2008]. In what follows, the application of dual internal variables to thermomechanics [Berezovski et al 2011b] is recalled shortly. Thus, in the framework of the phenomenological continuum theory, it is assumed that the influence of a microstructure on the overall macroscopic motion of a body can be taken into account by the introduction of internal variables, which we associate with the integral distributed effect of the microstructure.…”

“…At the same time, the recently developed concept of dual internal variables [Ván et al 2008] permits retrieving Mindlin micromorphic theory [Berezovski et al 2011b] on the basis of the material formulation of continuum mechanics [Maugin 1993;. This means that the dual internal variable approach is at least as powerful as the widely accepted micromorphic description.…”

Untitled

“…of momentum and energy keep their forms [Berezovski et al 2011b]: (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) with the modified Eshelby stress tensor (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) and the intrinsic source terms…”