Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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Triangular Bernoulli-Euler beam lattice Bernoulli-Euler beams that transmit axial and shear forces and bending moments are commonly used in beam lattice models to simulate crack propagation and fracture (Figure 6.16). In 2D micropolar continuum ( u 3 = 0, ϕ 1 = ϕ 2 = 0), kinematics of such beam lattice is described with three functions: a nodal displacement ( u 1 , u 2 ), and a nodal rotation ( ϕ 3 = ϕ ). The additional kinematic function ϕ = ϕ 3 is completely independent of the displacement field (i.e., it is independent and different from the antisymmetric rotation ( u j , i − u i , j ) / 2 of the classical continuum theory). Adoption of the linearity assumption of the three kinematic functions leads to the expressions for the local asymmetric strain ( γ ) and the torsional strain (curvature, κ ) γ αβ = u β , α + e βα 3 ϕ , κ δ = ϕ , δ (6.46) that fully describe the micropolar deformation. With regards to (6.46), recall that e βα 3 stands for the permutation tensor, and the repeated Greek indices imply summation. Thus, the nodal rotation ϕ does not contribute to the normal strains (i.e., the elongation of the generic material fiber), which implies that there is no difference between the normal strains in the micropolar and the classical continuum theories ( γ 11 = ε 11 , γ 22 = ε 22 ). The “micro-polar strain” defined by (6.46) 1 has the same form as the one used in Cosserat model [92]. The average normal (axial) strain in the half-beam of the unit cell is γ ( b ) = n ( b ) α n ( b ) β γ αβ , (6.47) which is to say that γ ( b ) L ( b ) is the average change of beam length. The difference between the angle of rotation of the beam chord and the rotation of its end nodes is ˜ γ ( b ) = n ( b ) α ˜ n ( b ) β γ αβ = n ( b ) α ˜ n ( b ) β u α , β − ϕ . (6.48) Where n ( b ) and ˜ n ( b ) designate, respectively, the unit vectors in the longitudinal and lateral directions (Figure 6.16c). Therefore, the difference between the angles of rotation of the beam ends (correspond ing to the lattice nodes) is κ ( b ) ≡ n ( b ) δ κ δ . (6.49) The Bernoulli-Euler beam theory implies that in each beam the relations between forces and displacements, and moments and angles of rotation, are of the following form

12 E ( b ) I ( b ) ( L ( b ) ) 2

F ( b ) = E ( b ) A ( b ) γ ( b ) , Q ( b ) =

˜ γ ( b ) , M ( b ) = E ( b ) I ( b ) κ ( b ) .

(6.50)

In Equations (6.50) the familiar relations for the area, A ( b ) = t ( b ) · h ( b ) , and axial moment of inertia, I ( b ) = t ( b ) [ h ( b ) ] 3 / 12, of the beam cross-section are used (Figure 6.30b). For the triangular beam lattice with a hexagonal unit cell and the spacing of the mesh L ( b ) (Figure 6.30), the deformation energy of the unit cell is

6 ∑ b = 1 h

F ( b ) γ ( b ) + Q ( b ) ˜ γ ( b ) + M ( b ) κ ( b ) i

L ( b ) 2

1 2

U cell =

(6.51)

.