Keywords: generalized continua, meshfree methods, strain gradient theory, size-scale effects, oriented material behaviour.

Within the last decades the development of generalized continuum models have attracted much interest. The range was spanned from theories designed for very specific physical phenomena to frameworks which had unifying ambitions and were a basis for various approaches which were derived from it. In this work a generalized continuum formulation is introduced which is based on a theoretical framework of a generalized deformation approach proposed by Sansour [2]. That is the deformation field is composed by a macro- and micro-component according to the consideration that the generalized continuum consists of a macro- and micro-continuum. It is demonstrated that by specific definition of the topology of the micro-space this generalized deformation formulation allows for the derivation of a generalized variational principle together with corresponding strain measures and underlying equilibrium equations. The approach leads to a strain gradient formulation which incorporates in a very natural manner intrinsic length scale parameters and does not involve additional degrees of freedom than those required in a classical continuum. The approach considers a geometrically exact description of finite deformation within the macro-continuum, but as a first step linearizes the deformation within the micro-continuum. Furthermore, it is assumed that the deformation field can only be varied within the macro-continuum so that the balance equations are established for the macro-space. The constitutive law however, is defined at the microscopic level and the geometrical specification of the micro-continuum is the only material input which goes beyond that needed in a classical description. It is shown that conventional constitutive laws can incorporated in a very straightforward manner. This fact is demonstrated using two different hyperelastic materials, firstly, the linear Saint-Venant-Kirchhoff model and secondly, a non-linear statistically based one.

The application of the proposed generalized deformation formulation to a moving least square-based meshfree method [1] is shown to provide the flexibility in terms of the continuity and consistency requirements required by this approach.

Various computations demonstrate that this model is able to address fundamental physical phenomena which are related to the underlying microstructure of the material, in particular scale-effects and oriented material behaviour. Clear differences are revealed between a classical Green strain tensor-based [3] and the proposed non-classical formulation.

1

P. Lancaster, K. Salkauskas, "Surface generated by moving least square methods", Mathematics of Computations, 37(155), 141-158, 1981. doi:10.2307/2007507

2

Sansour C., "A unified concept of elastic-viscoplastic Cosserat and micromorphic continua", Journal de Physique IV Proceedings, 8, 341-348, 1998. doi:10.1051/jp4:1998842

3

S. Skatulla and C. Sansour, "On meshfree computations of shells", Proceedings, Third MIT Conference, Boston, USA, 2005.

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