Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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Figure 6.15: A regular triangular lattice with a hexagonal unit cell and a central and angular interaction between the first neighbors ( α − β model). The angle of the angular spring with respect to the horizontal axis, in the reference con figuration, illustrated in Figure 6.15, is θ ( b ) = ( b − 1 ) π / 3 ( b = 1 , . . . , 6 ) . The infinitesimal change of that angle is ∆ θ ( b ) = ∆ θ ( b ) γ = e γαβ ε βδ n α n δ , ( α , β , δ = 1 , 2 and γ = 3 ) (6.42) with remark that e γαβ is the permutation tensor (with γ = 3 for 2D). With regards to Figure 6.15, the corresponding infinitesimal change of the angle between two adjacent bonds ( b and b + 1), associated with the angular spring constant β ( b ) , is ∆ φ ( b ) = ∆ θ ( b + 1 ) − ∆ θ ( b ) . Therefore, the deformation energy stored in the angular spring β ( b ) is E ( b ) β = β ( b ) | ∆ φ ( b ) | 2 = 1 2 ( b + 1 ) α n ( b + 1 ) δ − n ( b ) α n (6.43) The components of the effective stiffness tensor of the triangular α − β model can be derived by summation of the deformation energies of the central interactions (6.33) and all six angular springs ((6.43). Ostoja-Starzewski [3], [8] arrived at the following non-zero components of the effective stiffness tensor C 1111 = C 2222 = 1 2 √ 3 9 4 α + β ℓ 2 = E 1 − ν 2 C 1122 = C 2211 = 1 2 √ 3 3 4 α − 9 4 β ℓ 2 = E ν 1 − ν 2 (6.44) 1 2 β ( b ) n e γαβ ε βδ h n ( b ) δ io .

1 2 √ 3

β ℓ 2

3 4

9 4

E 2 ( 1 − ν )

α +

C 1212 =

=

With respect to Equations (6.44), it can be shown that the isotropy condition (6.36) is satisfied, so that only two independent elastic constants remain. Based on the same