OpenFOAMslides-01 (1185931), страница 2

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Some solids are plastic. These deformunder the action of a sufficient load and deformation continues as long asa load is applied, providing the material does not rupture. Deformationceases when the load is removed, but the plastic solid does not return toits original state.In ideal elastic solid stess vector and so stress tensor depends ondeformations: ~σ~n = ~σ i ni = σ ik ~ek niσ ik = λI1 (ε)g ik + 2µεikIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVM(22)March 10, 201617 / 20Difference between solids and fluidsAn ideal elastic solid will deform under load and, once the load is removed,will return to its original state.

Some solids are plastic. These deformunder the action of a sufficient load and deformation continues as long asa load is applied, providing the material does not rupture. Deformationceases when the load is removed, but the plastic solid does not return toits original state.In ideal elastic solid stess vector and so stress tensor depends ondeformations: ~σ~n = ~σ i ni = σ ik ~ek niσ ik = λI1 (ε)g ik + 2µεikE =µIlias Sibgatullin (Moscow University)3λ + 2µ,λ+µσ=(22)λ2(λ + µ)OpenFOAM course 1: theory of FVM(23)March 10, 201617 / 20Differentianl equations of continuum media:divergence form∂ρ+ ∇i (ρv i ) = 0∂t(24)∂(ρv k ) + ∇i (ρv k v i ) = ∇i σ ik + ρf k∂t(25)X∂(ρφ) + ∇i (ρφv i ) =fφ(26)∂tSuch a form is somtimes also called conservation form since as we will seein FV mehod integral quantities are conserved after space discretisation(decomposition to finite volumes).So this form of equations is essential for OpenFOAMIlias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 201618 / 20Divergence form of differentianl equations in tensornotation∂ρ+ div (ρ~v ) = 0∂t∂(ρ~v ) + div (ρ~v ~v ) = div σ + ρ~f∂tX∂(ρφ) + div (ρφ~v ) =fφ∂tv1 v1 v1 v2 v1 v3~v ~v means ~v ⊗ ~v = [vi vk ] =  v2 v1 v2 v2 v2 v3 v3 v1 v3 v2 v3 v3div (ρ~v ~v ) means div (ρv k ~v )ek ; div (σ) means ∇i p ik ekDo you think that engeneers who uses OpenFOAM and commercialpackages for years always understand it? You’re wrong!Ilias Sibgatullin (Moscow University)OpenFOAM course 1: theory of FVMMarch 10, 201619 / 20.

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