Extensions of Kronecker product algebra with applications in continuum and computational mechanics

    Authors

    D. W. Nicholson;B. Lin

    Comments

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    Abbreviated Journal Title

    Acta Mech.

    Keywords

    NEAR-INCOMPRESSIBLE ELASTOMERS; THERMOMECHANICAL RESPONSE; Mechanics

    Abstract

    Kronecker product algebra is widely applied in control theory. However, it does not appear to have been commonly applied to continuum and computational mechanics (CCM). In broad terms the goal of the current investigation is to extend Kronecker product algebra so that it can be broadly applied to CCM. Many CCM quantities, such as the tangent compliance tensor in finite strain plasticity, are very elaborate or difficult to derive when expressed in terms of tensor indicial or conventional matrix notation. However, as shown in the current article, with some extensions Kronecker product algebra can be used to derive compact expressions for such quantities. In the following, Kronecker product algebra is reviewed and there are given several extensions, and applications of the extensions are presented in continuum mechanics, computational mechanics and dynamics. In particular, Kronecker counterparts of quadratic products and of tensor outer products are presented. Kronecker operations on block matrices are introduced. Kronecker product algebra is extended to third and fourth order tensors. The tensorial nature of Kronecker products of tensors is established. A compact expression is given for the differential of an isotropic function of a second-order tensor. The extensions are used to derive compact expressions in continuum mechanics, for example the transformation relating the tangent compliance tensor in finite strain plasticity in undeformed to that in deformed coordinates. A compact expression is obtained in the nonlinear finite element method for the tangent stiffness matrix in undeformed coordinates, including the effect of boundary conditions prescribed in the current configuration. The aforementioned differential is used to derive the tangent modulus tensor in hyperelastic materials whose strain energy density is a function of stretch ratios. Finally, block operations are used to derive a simple asymptotic stability criterion for a damped linear mechanical system in which the constituent matrices appear explicitly.

    Journal Title

    Acta Mechanica

    Volume

    136

    Issue/Number

    3-4

    Publication Date

    1-1-1999

    Document Type

    Article

    Language

    English

    First Page

    223

    Last Page

    241

    WOS Identifier

    WOS:000081701500008

    ISSN

    0001-5970

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