Micromorphic media Samuel Forest Mines ParisTech / CNRS Centre des - PowerPoint PPT Presentation

Micromorphic media Samuel Forest Mines ParisTech / CNRS Centre des Mat´ eriaux/UMR 7633 BP 87, 91003 Evry, France Samuel.Forest@mines-paristech.fr

Plan Introduction 1 Mechanics of generalized continua Kinematics of micromorphic media Method of virtual power 2 A hierarchy of higher order continua 3 Continuum thermodynamics and hyperelasticity 4 Linearization 5 Linearized strain measures Linear Cosserat elasticity Exercise 1 Elastoviscoplasticity of micromorphic media 6 Decomposition of strain measures Constitutive equations Elastoviscoplasticity of strain gradient media Exercise 2

Notations Cartesian bases: reference basis ( E K ) K =1 , 2 , 3 , current basis ( e i ) i =1 , 2 , 3 A = A i e i , ∼ = A ij e i ⊗ e j , ∼ = A ∼ = A = A ijk e i ⊗ e j ⊗ e k , A A A ≈ s + A a symmetric and skew–symmetric parts A ∼ = A ∼ ∼ tensor products a ⊗ b = a i b j e i ⊗ e j , ∼ ⊗ B ∼ = A ij B kl e i ⊗ e j ⊗ e k ⊗ e l A A ∼ ⊠ B ∼ = A ik B jl e i ⊗ e j ⊗ e k ⊗ e l contractions . . A · B = A i B i , ∼ : B ∼ = A ij B ij , . B ∼ = A ijk B ijk A A ∼ nabla operators ∇ x = , i e i , ∇ X = , K E K u ⊗ ∇ X = u i , J e i ⊗ E J , ∼ · ∇ x = σ ij , j e i σ 3/68

Notations Cartesian bases: reference basis ( E K ) K =1 , 2 , 3 , current basis ( e i ) i =1 , 2 , 3 A = A i e i , ∼ = A ij e i ⊗ e j , ∼ = A ∼ = A = A ijk e i ⊗ e j ⊗ e k , A A A ≈ s + A a symmetric and skew–symmetric parts A ∼ = A ∼ ∼ tensor products a ⊗ b = a i b j e i ⊗ e j , ∼ ⊗ B ∼ = A ij B kl e i ⊗ e j ⊗ e k ⊗ e l A A ∼ ⊠ B ∼ = A ik B jl e i ⊗ e j ⊗ e k ⊗ e l contractions . . A · B = A i B i , ∼ : B ∼ = A ij B ij , . B ∼ = A ijk B ijk A A ∼ nabla operators ∇ x = , i e i , ∇ X = , K E K u ⊗ ∇ X = u i , J e i ⊗ E J , ∼ · ∇ x = σ ij , j e i σ 3/68

Notations Cartesian bases: reference basis ( E K ) K =1 , 2 , 3 , current basis ( e i ) i =1 , 2 , 3 A = A i e i , ∼ = A ij e i ⊗ e j , ∼ = A ∼ = A = A ijk e i ⊗ e j ⊗ e k , A A A ≈ s + A a symmetric and skew–symmetric parts A ∼ = A ∼ ∼ tensor products a ⊗ b = a i b j e i ⊗ e j , ∼ ⊗ B ∼ = A ij B kl e i ⊗ e j ⊗ e k ⊗ e l A A ∼ ⊠ B ∼ = A ik B jl e i ⊗ e j ⊗ e k ⊗ e l contractions . . A · B = A i B i , ∼ : B ∼ = A ij B ij , . B ∼ = A ijk B ijk A A ∼ nabla operators ∇ x = , i e i , ∇ X = , K E K u ⊗ ∇ X = u i , J e i ⊗ E J , ∼ · ∇ x = σ ij , j e i σ 3/68

Notations Cartesian bases: reference basis ( E K ) K =1 , 2 , 3 , current basis ( e i ) i =1 , 2 , 3 A = A i e i , ∼ = A ij e i ⊗ e j , ∼ = A ∼ = A = A ijk e i ⊗ e j ⊗ e k , A A A ≈ s + A a symmetric and skew–symmetric parts A ∼ = A ∼ ∼ tensor products a ⊗ b = a i b j e i ⊗ e j , ∼ ⊗ B ∼ = A ij B kl e i ⊗ e j ⊗ e k ⊗ e l A A ∼ ⊠ B ∼ = A ik B jl e i ⊗ e j ⊗ e k ⊗ e l contractions . . A · B = A i B i , ∼ : B ∼ = A ij B ij , . B ∼ = A ijk B ijk A A ∼ nabla operators ∇ x = , i e i , ∇ X = , K E K u ⊗ ∇ X = u i , J e i ⊗ E J , ∼ · ∇ x = σ ij , j e i σ 3/68

Notations Cartesian bases: reference basis ( E K ) K =1 , 2 , 3 , current basis ( e i ) i =1 , 2 , 3 A = A i e i , ∼ = A ij e i ⊗ e j , ∼ = A ∼ = A = A ijk e i ⊗ e j ⊗ e k , A A A ≈ s + A a symmetric and skew–symmetric parts A ∼ = A ∼ ∼ tensor products a ⊗ b = a i b j e i ⊗ e j , ∼ ⊗ B ∼ = A ij B kl e i ⊗ e j ⊗ e k ⊗ e l A A ∼ ⊠ B ∼ = A ik B jl e i ⊗ e j ⊗ e k ⊗ e l contractions . . A · B = A i B i , ∼ : B ∼ = A ij B ij , . B ∼ = A ijk B ijk A A ∼ nabla operators ∇ x = , i e i , ∇ X = , K E K u ⊗ ∇ X = u i , J e i ⊗ E J , ∼ · ∇ x = σ ij , j e i σ 3/68

Plan Introduction 1 Mechanics of generalized continua Kinematics of micromorphic media Method of virtual power 2 A hierarchy of higher order continua 3 Continuum thermodynamics and hyperelasticity 4 Linearization 5 Linearized strain measures Linear Cosserat elasticity Exercise 1 Elastoviscoplasticity of micromorphic media 6 Decomposition of strain measures Constitutive equations Elastoviscoplasticity of strain gradient media Exercise 2

Plan Introduction 1 Mechanics of generalized continua Kinematics of micromorphic media Method of virtual power 2 A hierarchy of higher order continua 3 Continuum thermodynamics and hyperelasticity 4 Linearization 5 Linearized strain measures Linear Cosserat elasticity Exercise 1 Elastoviscoplasticity of micromorphic media 6 Decomposition of strain measures Constitutive equations Elastoviscoplasticity of strain gradient media Exercise 2

Mechanics of generalized continua Principle of local action: the stress state at a point X depends on variables defined at this point only [Truesdell, Toupin, 1960] [Truesdell, Noll, 1965] local action Continuous Medium nonlocal nonlocal theory: integral formulation [Eringen, 1972] action Introduction 6/68

Mechanics of generalized continua Simple material: A material is simple at the particle X if and only if its response to deformations homogeneous in a neighborhood of X determines uniquely its response to every deformation at X . [Truesdell, Toupin, 1960] [Truesdell, Noll, 1965] simple Cauchy medium (1823) material F (classical / Boltzmann) ∼ local action non simple Continuous material Medium nonlocal nonlocal theory: integral formulation [Eringen, 1972] action Introduction 7/68

Mechanics of generalized continua Simple material: A material is simple at the particle X if and only if its response to deformations homogeneous in a neighborhood of X determines uniquely its response to every deformation at X . [Truesdell, Toupin, 1960] [Truesdell, Noll, 1965] simple Cauchy continuum (1823) material F (classical / Boltzmann) ∼ Cosserat (1909) u , R local medium ∼ of order n micromorphic action [Eringen, Mindlin 1964] non simple u , χ Continuous material ∼ second gradient [Mindlin, 1965] medium ∼ ⊗ ∇ F ∼ , F of grade n Medium gradient of internal variable [Maugin, 1990] u , α nonlocal nonlocal theory: integral formulation [Eringen, 1972] action Introduction 8/68

Plan Introduction 1 Mechanics of generalized continua Kinematics of micromorphic media Method of virtual power 2 A hierarchy of higher order continua 3 Continuum thermodynamics and hyperelasticity 4 Linearization 5 Linearized strain measures Linear Cosserat elasticity Exercise 1 Elastoviscoplasticity of micromorphic media 6 Decomposition of strain measures Constitutive equations Elastoviscoplasticity of strain gradient media Exercise 2

Kinematics of micromorphic media • Degrees of freedom of the theory DOF := { u , χ } ∼ ⋆ displacement u ( X , t ) and microdeformation χ ∼ ( X , t ) of the material point X ⋆ current position of the material point x = Φ( X , t ) = X + u ( X , t ) ⋆ deformation of a triad of directors attached to the material ξ i ( X ) = χ ∼ ( X ) · Ξ i point • Polar decomposition of the generally incompatible microdeformation field χ ∼ ( X , t ) ♯ · U ♯ χ ∼ = R ∼ ∼ internal constraints ♯ ⋆ Cosserat medium ∼ ≡ R χ ∼ ♯ ⋆ Microstrain medium ∼ ≡ U χ ∼ ⋆ Second gradient medium ∼ ≡ F χ ∼ Introduction 10/68

Kinematics of micromorphic media • Degrees of freedom of the theory DOF := { u , χ } ∼ ⋆ displacement u ( X , t ) and microdeformation χ ∼ ( X , t ) of the material point X ⋆ current position of the material point x = Φ( X , t ) = X + u ( X , t ) ⋆ deformation of a triad of directors attached to the material ξ i ( X ) = χ ∼ ( X ) · Ξ i point • Polar decomposition of the generally incompatible microdeformation field χ ∼ ( X , t ) ♯ · U ♯ χ ∼ = R ∼ ∼ internal constraints ♯ ⋆ Cosserat medium ∼ ≡ R χ ∼ ♯ ⋆ Microstrain medium ∼ ≡ U χ ∼ ⋆ Second gradient medium ∼ ≡ F χ ∼ Introduction 10/68

Directors in materials tri` edre directeur in a single crystal: 3 lattice vectors “Les directeurs ne subissent pas la mˆ eme transformation que les lignes mat´ erielles. C’est en cela que le milieu plastique diff` ere du milieu continu classique. On doit le concevoir un peu comme un milieu de Cosserat.” [Mandel, 1973] Introduction 11/68

Kinematics of micromorphic media u (Φ − 1 ( x , t ) , t ) • velocity field v ( x , t ) := ˙ • deformation gradient ∼ = 1 ∼ + u ⊗ ∇ X F v ⊗ ∇ x = ˙ − 1 • velocity gradient F ∼ · F ∼ − 1 • microdeformation rate ∼ · χ χ ˙ ∼ − 1 · χ • Lagrangian microdeformation gradient ∼ := χ ∼ ⊗ ∇ X K ∼ • gradient of the microdeformation rate tensor − 1 ⊠ F − 1 ) ⊗ ∇ x = χ ∼ · ˙ − 1 ) ( ˙ ∼ · χ ∼ : ( χ χ K ∼ ∼ ∼ Lj ) , k = χ iP ˙ χ iL χ − 1 K PQR χ − 1 Qj F − 1 ( ˙ Rk [Eringen, 1999] Introduction 12/68