Towards a Gradient Truss Model. Part I: Bar Element Displacement-Force Relations

Abstract

Recently the author presented a note on gradient truss model by assuming that the bars can support strain gradients along their length. Hence, higher-order strain gradient terms and an internal length scale parameter (the gradient elastic coefficient) were introduced into the constitutive stress-strain relation of the truss bar element. However, in order to solve the governing higher order differential equation, extra non-classical boundary conditions are required and these need to be investigated in order to optimize proper combinations. In this paper, extra non-classical boundary conditions are studied by considering the support conditions of the bar based on imposed strain and strain gradient dominated deformation mechanisms. The quantitative response and applicability of the model under different combinations of extra non-classical boundary conditions are examined. Consequently, twelve non-homogeneous displacement-force relations are presented and form the basis for the gradient truss element stiffness matrices presented in Part II. With these, a relation between the macro-scale of the bar element and an internal length scale in the micro-scale is established thus micro-scale dependent behaviour in the material being modelled can be directly captured using this new truss bar elements. Moreover, the weakening and the strengthening effect of strain and strain gradients imposed at the bar nodes can also be revealed. As an illustration of the model a numerical example is presented
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