Internal Variables and Dynamic Degrees of Freedom

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

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“…The accepted functional dependence (16) and the equations of state (17) lead to the representation of the internal force (10) in the form…”

Section: Dual Internal Variablesmentioning

confidence: 99%

Generalized thermomechanics with dual internal variables

2010

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“…Considering the limit case e ≡ 0 and a simple linear hardening rule, R ref (p) = Hp, H being the plastic modulus, the cumulative plastic strain is solution of the differential equation 27) which delivers two types of solutions depending on the sign of the plastic modulus.…”

Section: (B) Scalar Microstrainmentioning

confidence: 99%

“…The proposed procedure is especially useful to derive the form of the regularization operators under anisothermal or multiphysics conditions [21]. The thermodynamic foundations of phase field models are well established [26] and a unified thermomechanical formulation is possible to reconcile the gradient, micromorphic and phase-field model classes [21,[27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage

Forest¹

2016

Proc. R. Soc. A.

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International audienceThe construction of regularization operators presented in this work is based on the introduction of strain or damage micromorphic degrees of freedom in addition to the displacement vector and of their gradients into the Helmholtz free energy function of the constitutive material model. The combination of a new balance equation for generalized stresses and of the micromorphic constitutive equations generates the regularization operator. Within the small strain framework, the choice of a quadratic potential w.r.t. the gradient term provides the widely used Helmholtz operator whose regularization properties are well known: smoothing of discontinuities at interfaces and boundary layers in hardening materials, and finite width localization bands in softening materials. The objective is to review and propose nonlinear extensions of micromorphic and strain/damage gradient models along two lines: the first one introducing nonlinear relations between generalized stresses and strains; the second one envisaging several classes of finite deformation model formulations. The generic approach is applicable to a large class of elastoviscoplastic and damage models including anisothermal and multiphysics coupling. Two standard procedures of extension of classical constitutive laws to large strains are combined with the micromorphic approach: additive split of some Lagrangian strain measure or choice of a local objective rotating frame. Three distinct operators are finally derived using the multiplicative decomposition of the deformation gradient. A new feature is that a free energy function depending solely on variables defined in the intermediate isoclinic configuration leads to the existence of additional kinematic hardening induced by the gradient of a scalar micromorphic variable

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“…Different types of arguments leading to the same conclusion have been introduced in the context of the Grad hierarchy in [35] and in the context of continuum mechanics in [36], [37].…”

Section: Fundamental Thermodynamic Relation Conjugate State Variablesmentioning

confidence: 99%

Role of thermodynamics in extensions of mesoscopic dynamical theories

Grmela

2016

Communications in Applied and Industrial Mathematics

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Complex macroscopic systems (like for instance those encountered in nanotechnology and biology) need to be investigated in a family of mesoscopic theories involving varying amount of details. In this paper we formulate a general thermodynamics providing a universal framework for such multiscale viewpoint of mesoscopic dynamics. We then discuss its role in making extensions (i.e. in lifting a mesoscopic theory to a more microscopic level that involves more details).

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Internal Variables and Dynamic Degrees of Freedom

Cited by 90 publications

References 34 publications

Generalized thermomechanics with dual internal variables

Generalized thermomechanics with dual internal variables

Nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage

Role of thermodynamics in extensions of mesoscopic dynamical theories

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