On Purely Mechanical Simple Kinematic Internal Constraints

Abstract

The classic purely mechanical approach to materials with simple kinematic internal constraints is supplemented. A right Cauchy-Green tensor which locally represents the kinematically admissible restricted domain of a finite hyperelastic stress response function is constructed explicitly. It satisfies all imposed constraints identically. It is obtained by a procedure which annihilates the banned modes of deformation in the actual placement. A unique direct sum based stress decomposition is obtained. Further, a procedure is provided, to seamlessly relax constraints, or reversibly, that allows constraints to smoothly develop under loading starting from an unconstrained description. The involved relaxation of internal constraints is briefly illustrated herein. References to published full feathered applications are given where the method is used and verified in finite element form. A. Zdunek internal constraints is laid down. Notably, it leaves the mechanical interpretation of the introduced Lagrange multipliers undetermined. Later, the so-called constraint manifold theory by Podio-Guidugli [3], Gurtin and Podio-Guidugli [5], Podio-Guidugli and Vianello [4] and recently Vianello [6], and by Carlson et al. [7], to mention a few, updated the classic theory using geometric-algebraic analogies from linear-algebra. The additive stress decomposition emerges from spanning the workless stress on the normals to the set of constraints. The work-performing stress is isolated as the orthogonal complement, spanned by the tangent set.The normalisation condition removing the indeterminate part from the extra stress, which makes the stress decomposition unique, is only mentioned as an option in [1, Sect. 30]. It is invoked firmly in the constraint manifold theory (see, e.g., [7]), where orthogonality is assumed between the workless and the work-performing stresses. The constraint manifold theory concludes making the direct sum based stress decomposition unique by constructing the orthogonal projection to the set of constraint normals. It is concluded that the normalisation condition as presented in [1, Sect. 30] and in [7] is an auxiliary assumption to make the stress decomposition unique.According to Steigmann [11] the subject of materials with internal constraints is not especially well treated in text and monographs. The necessity and usefulness of a normalisation condition, introduced in [1, Sect. 30], has been questioned, see for example Bertram and Glüge [9]. Negahban [10] discusses different assumed stress decompositions and normalisations. Normalised work performing stresses are more seldom used in analytic works. See, for example, the presentation of the classic families of controllable non-homogeneous solutions of arbitrary isotropic incompressible materials in Truesdell and Noll [1, Sects. 56 and 57]. In computational mechanics, on the other hand, the advantage of using the normalised deviatoric description of the work-performing stresses in nearly incompressible hyperelasticity was early recognised by Simo, Taylor and ...

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