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Non-classical continuum theory for solids incorporating internal rotations and rotations of Cosserat theories

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Abstract

This paper presents a non-classical continuum theory in Lagrangian description for solids in which the conservation and the balance laws are derived by incorporating both the internal rotations arising from the Jacobian of deformation and the rotations of Cosserat theories at a material point. In particular, in this non-classical continuum theory, we have (i) the usual displacements \(( \pmb {\varvec{u }})\) and (ii) three internal rotations \(({}_i \pmb {\varvec{{\varTheta } }})\) about the axes of a triad whose axes are parallel to the x-frame arising from the Jacobian of deformation (which are completely defined by the skew-symmetric part of the Jacobian of deformation), and (iii) three additional rotations \(({}_e \pmb {\varvec{{\varTheta } }})\) about the axes of the same triad located at each material point as additional three degrees of freedom referred to as Cosserat rotations. This gives rise to \( \pmb {\varvec{u }}\) and \({}_e \pmb {\varvec{{\varTheta } }}\) as six degrees of freedom at a material point. The internal rotations \(({}_i \pmb {\varvec{{\varTheta } }})\), often neglected in classical continuum mechanics, exist in all deforming solid continua as these are due to Jacobian of deformation. When the internal rotations \({}_i \pmb {\varvec{{\varTheta } }}\) are resisted by the deforming matter, conjugate moment tensor arises that together with \({}_i \pmb {\varvec{{\varTheta } }}\) may result in energy storage and/or dissipation, which must be accounted for in the conservation and the balance laws. The Cosserat rotations \({}_e \pmb {\varvec{{\varTheta } }}\) also result in conjugate moment tensor which, together with \({}_e \pmb {\varvec{{\varTheta } }}\), may also result in energy storage and/or dissipation. The main focus of the paper is a consistent derivation of conservation and balance laws that incorporate aforementioned physics and associated constitutive theories for thermoelastic solids. The mathematical model derived here has closure, and the constitutive theories derived using two alternate approaches are in agreement with each other as well as with the condition resulting from the entropy inequality. Material coefficients introduced in the constitutive theories are clearly defined and discussed.

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  1. Mechanical Engineering, University of Kansas, Lawrence, KS, USA

    K. S. Surana & A. D. Joy

  2. Mechanical Engineering, Texas A&M University, College Station, TX, USA

    J. N. Reddy

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  1. K. S. Surana
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Correspondence to K. S. Surana.

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Communicated by Andreas Öchsner.

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Surana, K.S., Joy, A.D. & Reddy, J.N. Non-classical continuum theory for solids incorporating internal rotations and rotations of Cosserat theories. Continuum Mech. Thermodyn. 29, 665–698 (2017). https://doi.org/10.1007/s00161-017-0554-1

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