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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2 T − 1 . In such case the relation

The physical property relating 1-cells to 2-cells is local mass di ff usivity, i.e., [ π 1 ] = L

3 T − 1 , since [

1 ] = L . Ultimately, the flux has units [ φ ] = M T − 1 as

between ϕ and φ is given by ˆ 1 with [ ˆ 1 ] = L

required for the balance Eq. 2. In heat conduction in structured solids, the extensive quantity is heat energy in 3-cells, represented by ψ with [ ψ ] = E , and the intensive quantity is the temperature, represented by β with [ β ] = θ . The physical property relating 0-cells to 3-cells is local heat capacity per 3-cell volume, i.e., [ π 0 ] = E Θ − 1 L − 3 ; alternatively [ π 3 ] = E − 1 θ L 3 . In such case, β is related to ψ by ˆ 0 with [ ˆ 0 ] = E θ − 1 T − 1 or by ˆ 3 with [ ˆ 3 ] = E − 1 θ T . The variation of β is ϕ = − δ 0 β with [ ϕ ] = [ β ] = θ . The physical property relating 1-cells to 2-cells is local thermal conductivity, i.e., [ π 1 ] = E θ − 1 L − 1 T − 1 . In such case the relation between ϕ and φ is given by ˆ 1 with [ ˆ 1 ] = E T − 1 , since [ 1 ] = L . Ultimately, the heat flux has units [ φ ] = E T − 1 as required for the balance Eq. 2. The two examples demonstrate the generality of the procedure. The topology provides the variations of intensive quantities along elements of internal structures, as well as the temporal variations of extensive quantities from fluxes. The concrete geometric realisation of the 3-complex and the nature of the physical problem enter only via the Hodge star operators. The procedure can be tailored to represent advective and convective (advective & di ff usive) mass, heat or charge transport, including in porous media. The formulation of boundary conditions is shown in Section 4. The mechanics of a 3-complex is not infinitesimal and the metric used in the construction of p arises from an initial (undeformed) configuration of the 3-complex. This suggests that the continuum analogue of discrete internal forces is the first Piola-Kirchho ff stress tensor, P . It is a two-point tensor, which is not symmetric and represents the tractions in the current (deformed) configuration divided by the areas in the initial (undeformed) configuration. Notably, tr ( P ) = 3 s h , where s h is the hydrostatic stress, and the balance of linear momentum is given by ∇ · P + b = 0 , (3) which is similar to the one using the familiar Cauchy stress tensor, but the body force density b and the divergence operator are taken with respect to the initial (undeformed) configuration. P is power-conjugate to the rate of deforma tion gradient, ˙ F = ∇ ˙ u = d ( ∇ u ) / dt , where u and ˙ u are the displacements and their rates (velocities), respectively. This allows for working with time increments of ∇ u , instead of ∇ ˙ u , and express constitutive relations between ∇ u and P . The rate of change of the internal energy, ˙ U , is the sum of the rates of changes of the thermal (heat) energy, ˙ Q , and the mechanical energy, ˙ W = P : ˙ F . Considering mechanical problems in isolation is an approximation that assumes negligible changes of thermal energy. When these are not neglected, the corresponding changes of temperature will result in thermal strains. The thermal strains at a point with temperature T are given by a spherical tensor with factor ε T = α T T , where α T is the coe ffi cient of thermal expansion, and they contribute to the displacements. P can be decomposed as P = P 0 + P 1 + P 2 , where P 0 i j = s h δ i j is a spherical tensor, P 1 i j = P i j (1 − δ i j ) is a non symmetric tensor with zero diagonal coe ffi cients containing the deviatoric components of P tangential to planes in the initial (undeformed) configuration, and P 2 i j = ( P i j − s h ) δ i j is a diagonal tensor containing the deviatoric components of P normal to planes in the initial (undeformed) configuration. The discrete analogue of P are internal forces given by three cochains: a 0-cochain, τ 0 , of hydrostatic components with [ τ 0 ] = F L − 2 ; a 1-cochain, τ 1 , of tangential deviatoric components with [ τ 1 ] = F L − 1 (corresponding to line-integrated continuum tangential deviatoric stresses); and a 2 cochain, τ 2 , of normal deviatoric components with [ τ 2 ] = F (corresponding to area-integrated continuum normal deviatoric stresses). In addition, the change of thermal energy is a 3-cochain, τ 3 , with [ τ 3 ] = F L , which contributes to the balance of linear momentum. The discrete analogue of vector fields are two cochains. The analogue of u is a 1-cochain, η 1 with [ η 1 ] = L 2 (corre sponding to line-integrated continuum displacement increments), and a 2-cochain, η 2 with [ η 2 ] = L 3 (corresponding to area-integrated continuum displacement increments). The analogue of b is a 1-cochain, β 1 with [ β 1 ] = F L − 2 (corresponding to line-integrated body force density), and a 2-cochain, β 2 with [ β 2 ] = F L − 1 (corresponding to area integrated body force density). The relations between external fields and cochains will be given in Section 4. The balance of momenta is given by four equations: 3 δ 2 τ 2 = 0 (at 0-cells) and δ 2 1 τ 1 = 0 (at 3-cells); (4) 2 δ 1 τ 1 + δ 0 3 τ 3 + β 1 = 0 (at 1-cells) and 1 δ 0 τ 0 + δ 1 2 τ 2 + β 2 = 0 (at 2-cells) . (5) 3.2. Balance of momenta