PSI - Issue 74

Mitra Delshadmanesh et al. / Procedia Structural Integrity 74 (2025) 9–16 Mitra Delshadmanesh / Structural Integrity Procedia 00 (2025) 000–000

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with a dimension of a force. Expression (4) represents the elastic potential of the model, and the correlated equilibrium conditions are obtained by minimizing the overall potential energy. Nowadays, it is generally accepted that strain gradient theory can describe mechanical size effects. Nevertheless, the choice of material parameters is still challenging. Material parameters derived from numerical fits of beam-bending experiments are higher than those obtained from ab initio computer simulations of the phonon dispersion relation. In the present study, we attempt to avoid this discrepancy by choosing material parameters , and (6) This combination of parameters is related to a deformation mode characterized by the expectation value < η 211 2 > obtained by averaging η 211 2 over all possible orientations of coordinate systems. Indeed, component η 211 of the second gradient of displacements describes a change of material orientation along the x-direction and, at the same time, it describes an increase of volumetric strain along the y-direction, as depicted in Fig. 5 schematically.

Fig. 5. Schematic illustration of the deformation generated by η 211 . There is a rotation of the material orientation along the x-direction and a volume change along the y-direction.

This deformation contributes to the deformation mode of beam bending, but it is different from longitudinal or transversal phonon excitation. Longitudinal phonons do not show any material rotation, while transversal phonons are purely deviatoric deformations without volume change. To derive a rotational invariant quadratic form related to η 211 , we here express the value of this component in a rotated coordinate system by the components of the unrotated system according to tensor transformation rules for third-rank tensors:

(7) where, for simplicity, this expression is given for the 2-dimensional case. Now, the expectation value for < η 211 2 > averaged over all possible rotation angles 0 < α < 2 π becomes (8) This quadratic form is invariant with respect to rotations of the coordinate system, and it can be expressed as a linear combination of the quadratic forms appearing in equation (4). Consequently, assigning an internal energy density to < η 211 2 > leads to the relation (6) for the material parameters. 3.2 Finite Element simulation based on strain gradient elasticity The constitutive equations of strain gradient elasticity were implemented as a Galerkin method in ABAQUS through user subroutine UEL. Tetrahedral elements with 4 nodes were defined, where every node was equipped with generalized Degrees of Freedom (DOF) in addition to ordinary displacement DOF. Here, components of a finite strain