PSI - Issue 43

Andrey P. Jivkov et al. / Procedia Structural Integrity 43 (2023) 15–22 Author name / Structural Integrity Procedia 00 (2023) 000–000

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Note, that Eqs. 5 are a discrete analogue of Eq. 3, while Eqs. 4 set additional restrictions on the internal forces, corresponding to the symmetry of Cauchy stress tensor, which is missing for the continuum P . The discrete displacements are related to four deformation cochains 0 = 3 δ 2 η 2 , [ 0 ] = L 0 ; 1 = 2 δ 1 η 1 , [ 1 ] = L 1 ; 2 = δ 1 2 η 2 ; [ 2 ] = L 2 , 3 = δ 2 1 η 1 , [ 3 ] = L 3 , (6) where, 1 is work-conjugate to the normal deviatoric stress components τ 2 , 2 is work-conjugate to the tangential deviatoric stress components τ 1 , 3 represents volume change and is work-congugate to the hydrostatic stress τ 0 , and 0 is the thermal strain, which is related to the thermal energy rate, τ 3 . The increment of the mechanical energy is a 3-cochain, as required, calculated by ˙ W = τ 0 3 + τ 1 2 + τ 2 1 . The constitutive relations between deformation and force cochains are given by: 3 = ˆ 0 τ 0 , 0 = ˆ 3 τ 3 , τ 2 = ˆ 1 1 , τ 1 = ˆ 2 2 . (7) where ˆ p contain appropriate material parameters. Specifically for linear thermo-elasticity π 0 contains coe ffi cients of the form 1 /κ , where κ are local bulk moduli, π 1 and π 2 contain coe ffi cients of the form 2 µ , where µ are local shear moduli, and π 3 contains coe ffi cients of the form α T /ζ , where ζ is the local heat capacity per 3-cell volume. Let the boundary of the domain represented by K is ∂ K . For problems involving balance of a scalar quantity: Neumann boundary conditions (BC) are prescribed by providing fluxes, φ , normal to the 2-cells on ∂ K ; Dirichlet BC are imposed on the intensive quantity, β , i.e., they are prescribed at the 0-cells on ∂ K . The formulation is thus: ∂ ( ˆ 0 β ) /∂ t = − δ 2 φ + χ β = β 0 initial in K β ( t ) = ¯ β ( t ) Dirichlet on ∂ K φ ( t ) = ¯ φ ( t ) Neumann on ∂ K , (8) where the BC are formulated as constraints on β and φ cochains. Problems involving balance of momenta require a connection between intrinsic quantities and the external world. The connection is provided by a choice of a coordinate system. With respect to this, let u˙ ( σ 0 ) and b ( σ 0 ) be the displacement rate and prescribed body force density at σ 0 , respectively, t ( σ 2 ) the prescribed traction vector at σ 2 , l ( σ 1 ) the geometric vectors along σ 1 , a ( σ 2 ) the area vectors of σ 2 , with consistent orientations. Neumann BC are applied using only this information. If a 2-cell σ 2 with boundary 1-cells σ k 1 has a prescribed t ( σ 2 ), its action is represented by the inner product of t and a (normal traction), so that τ 2 ( σ 2 ) = t · a ( σ 2 ), and the inner product of t and l (tangential traction), so that τ 1 ( σ k 1 ) = t · l ( σ k 1 ). Every 0-cell has n ≥ 3 adjacent 1-cells and m ≥ 3 adjacent 2-cells, and two associated matrices: A ( σ 0 ) with n rows and 3 columns, where each row contains the coordinates of the vector along one adjacent 1-cell; B ( σ 0 ) with m rows and 3 columns, where each row contains the coordinates of the area vector normal to one adjacent 2-cell. A ( σ 0 ) maps external vectors at σ 0 to scalars on its adjacent 1-cells, increasing their dimension by L . B ( σ 0 ) maps external vectors at σ 0 to scalars on its adjacent 2-cells, increasing their dimension by L 2 . These maps are used to calculate the intrinsic representation of b , so that β 1 = A · b , and β 2 = B · b enter Eq. 5. The left-inverses (Moore-Penrose inverses) of these maps are given by A ∗ ( σ 0 ) = [ A T ( σ 0 ) A ( σ 0 )] − 1 A T ( σ 0 ) and B ∗ ( σ 0 ) = [ B T ( σ 0 ) B ( σ 0 )] − 1 B T ( σ 0 ) (9) A ∗ ( σ 0 ) has 3 rows and n columns and maps scalars associated with adjacent 1-cells to external vectors at σ 0 , re ducing their dimension by L . B ∗ ( σ 0 ) has 3 rows and m columns and maps scalars associated with adjacent 2-cells to external vectors at σ 0 , reducing their dimension by L 2 . These maps are used to relate the intrinsic cochains, β 1 and β 2 , to externally applied displacement rates, u˙ ( σ 0 ), i.e., to apply Dirichlet boundary conditions. Specifically, u˙ ( σ 0 ) = A ∗ ( σ 0 ) η 1 ( σ 1 ) + B ∗ ( σ 0 ) η 2 ( σ 2 ). As in the scalar case, Dirichlet and Neumann BC are applied as constraints on η 1 and η 2 , τ 1 and τ 2 cochains. In a problem involving balance of momenta, the recommended unknown cochains are τ 0 , η 1 , η 2 , and τ 3 , and the formulation is constructed using Eqs. 4–7. 4. Boundary value problems