Some thoughts on the shear problem Jendrik Voss, Christian Thiel, - PowerPoint PPT Presentation

Some thoughts on the shear problem Jendrik Voss, Christian Thiel, Robert J. Martin and Patrizio Neff GAMM-Jahrestagung Wien February 2019 Chair for Nonlinear Analysis and Modelling Faculty of Mathematics University of Duisburg-Essen

Introduction Simple shear deformation [Thiel, Voss, Martin, and Neff 2018b] A simple shear deformation is a mapping ϕ : Ω ⊂ R 3 → R 3 of the form   1 γ 0   = ✶ + γ e 2 ⊗ e 1 ∇ ϕ = F γ = 0 1 0 0 0 1 with the amount of shear γ ∈ R . F 1 1      1 1 ϑ     � �� � 1 1 γ Shear in nonlinear elasticity J. Voss, C. Thiel, R. J. Martin and P. Neff 1 / 17

Introduction Pure shear stress A pure shear stress is a stress tensor T ∈ Sym(3) of the form   0 s 0 T s =   = s ( e 1 ⊗ e 2 + e 2 ⊗ e 1 ) s 0 0 0 0 0 with the amount of shear stress s ∈ R . upper shear force 1  e 2   e 3   1 ϑ     e 1 � �� � 1 γ lower shear force Shear in nonlinear elasticity J. Voss, C. Thiel, R. J. Martin and P. Neff 2 / 17

Introduction In isotropic nonlinear elasticity the Cauchy stress tensor is σ = β 0 ✶ + β 1 B + β − 1 B − 1 � � and B = FF T . with β i = β i I 1 ( B ) , I 2 ( B ) , I 3 ( B )     0 s 0 1 γ 0 σ = T s =  ,    Set s 0 0 F γ = 0 1 0 0 0 0 0 0 1     β 1 γ 2 0 s 0 ( β 1 − β − 1 ) γ 0  = σ = ( β 0 + β 1 + β − 1 ) ✶ +   β − 1 γ 2  s 0 0 ( β 1 − β − 1 ) γ 0 0 0 0 0 0 0 ⇒ γ 2 ( β 1 − β − 1 ) = 0 = then γ = 0 or s = 0 . Pure shear Cauchy stress never corresponds to a simple shear deformation! Shear in nonlinear elasticity J. Voss, C. Thiel, R. J. Martin and P. Neff 3 / 17

Introduction Questions: • Independent of the elasticity law, which kind of deformations do correspond to pure shear Cauchy stress? [Destrade, Murphy, and Saccomandi 2012; Moon and Truesdell 1974; Mihai and Goriely 2011] • Which of these deformations are suitable to be called ‘shear’? • Which constitutive requirements ensure that only ‘shear’ deformations correspond to pure shear Cauchy stress? Shear in nonlinear elasticity J. Voss, C. Thiel, R. J. Martin and P. Neff 4 / 17

Which kind of deformations correspond to pure shear stress? B = FF T and � σ ( B ) commute for any isotropic stress response. ⇐ ⇒ B and � σ ( B ) are simultaneously diagonalizable. σ ( B ) = T s can be diagonalized to Q diag( s , − s , 0) Q T with �   1 − 1 0 1  ∈ SO(3) .  Q := √ 1 1 0 √ 2 0 0 2 3 ) Q T with [Thiel, Voss, Martin, and Neff 2018a] Thus B = Q diag( λ 2 1 , λ 2 2 , λ 2     λ 2 1 + λ 2 λ 2 1 − λ 2 1 + γ 2 0 γ 0 2 2 B = 1  λ 2 1 − λ 2 λ 2 1 + λ 2  � = F γ F T   . 0 γ = γ 1 0 2 2 2 2 λ 2 0 0 0 0 1 3 Shear in nonlinear elasticity J. Voss, C. Thiel, R. J. Martin and P. Neff 5 / 17

Which kind of deformations correspond to pure shear stress?     λ 2 1 + λ 2 λ 2 1 − λ 2 0 0 0 s 2 2 B = 1 σ ( B ) = T s =    λ 2 1 − λ 2 λ 2 1 + λ 2  . 0 0 ⇐ ⇒ 0 � s 2 2 2 2 λ 2 0 0 0 0 0 3 Then F is uniquely determined by triaxial stretch and simple shear     1 0 a 0 0 γ     Q F = F γ diag( a , b , c ) Q = 0 1 0 0 b 0 0 0 1 0 0 c up to an arbitrary Q ∈ SO(3) with � � γ = λ 2 1 − λ 2 λ 2 1 + λ 2 2 2 2 a = λ 1 λ 2 b = c = λ 3 , , , . λ 2 1 + λ 2 λ 2 1 + λ 2 2 2 2 Shear in nonlinear elasticity J. Voss, C. Thiel, R. J. Martin and P. Neff 6 / 17

Which of these deformations are suitable to be called shear? Linear Elasticity The linear elastic Cauchy stress σ lin = 2 µ dev ε + κ tr ε with ε = sym( F − ✶ ) and dev ε = ε − 1 3 tr ε ✶ is a pure shear if and only if   γ 0 0 2 γ   F = ✶ + 0 0 + A , A ∈ so (3) . 2 0 0 0 � �� � ε ∈ Sym(3)       γ γ 1 0 0 0 0 0 γ 2 2  = ✶ + γ − γ      F γ = 0 1 0 0 0 + 0 0 . 2 2 0 0 1 0 0 0 0 0 0 � �� � � �� � ε γ ∈ Sym(3) ω γ ∈ so (3) infinitesimal pure infinitesimal rotation shear strain Shear in nonlinear elasticity J. Voss, C. Thiel, R. J. Martin and P. Neff 7 / 17

Which of these deformations are suitable to be called shear? ground parallel (deck of cards) γ ✶ + ε γ F γ = ✶ + ε γ + ω γ • The deformation F γ is infinitesimally volume preserving, tr ε γ = 0. • The deformation F γ is planar, eigenvalue 1 to eigenvector e 3 . • The deformation F γ is ground parallel, eigenvectors e 1 and e 3 . Shear in nonlinear elasticity J. Voss, C. Thiel, R. J. Martin and P. Neff 8 / 17

Which of these deformations are suitable to be called shear? Generalizing from linear elasticity to nonlinear elasticity • Pure shear Cauchy stress acts only in a plane • Leonardo da Vinci: “Nessuno effetto ` e in natura sanza ragione” (No effect is in nature without cause) Codex Atlanticus − → Nonlinear shear deformation should be planar   0 0 s γ   σ = 0 0 s 0 0 0 Shear in nonlinear elasticity J. Voss, C. Thiel, R. J. Martin and P. Neff 9 / 17

Which of these deformations are suitable to be called shear? Definition: Finite shear deformation [Thiel, Voss, Martin, and Neff 2018b] • The deformation F is volume preserving, det F = 1. • The deformation F is planar, eigenvalue 1 to eigenvector e 3 . • The deformation F is ground parallel, eigenvectors e 1 and e 3 . ⇒ there exists λ ∈ R + with λ 1 = λ , λ 2 = 1 = λ and λ 3 = 1.     0 0 λ 1 + λ 2 λ 1 − λ 2 0 s √ B = 1 σ ( B ) = T s =  =    . s 0 0 ⇒ V = λ 1 − λ 2 λ 1 + λ 2 0 � 2 0 0 0 0 0 2 λ 3 Shear in nonlinear elasticity J. Voss, C. Thiel, R. J. Martin and P. Neff 10 / 17

Which of these deformations are suitable to be called shear? Finite pure shear stretch     e α + e − α e α − e − α λ + 1 λ − 1 0 0 V = 1 λ λ  = 1 e α − e − α e α + e − α  λ − 1 λ + 1   0 0 λ λ 2 2 0 0 2 0 0 2     cosh( α ) sinh( α ) 0 0 0 α  = exp    = sinh( α ) cosh( α ) 0 0 0 =: V α , α := log λ . α . 0 0 1 0 0 0 . . � �� � matrix exponential infinitesimal pure shear strain infinitesimal pure shear strain ε γ finite pure shear stretch V α tr ε γ = 0 det V α = 1 exp γ = 2 α Shear in nonlinear elasticity J. Voss, C. Thiel, R. J. Martin and P. Neff 11 / 17

Which of these deformations are suitable to be called shear?       0 s 0 1 γ 0 a 0 0 σ ( B ) = T s =   =     Q . � s 0 0 ⇒ F = 0 1 0 0 b 0 0 0 0 0 0 1 0 0 c with λ 1 = λ , λ 2 = 1 λ , λ 3 = 1 and α = log λ : Finite simple shear deformation     1 √ 0 0 1 tanh(2 α ) 0 cosh(2 α )  �    F = 0 1 0  Q  0 cosh(2 α ) 0 0 0 1 0 0 1   1 sinh(2 α ) 0 1   Q =: F α . = 0 cosh(2 α ) 0 � � cosh(2 α ) 0 0 cosh(2 α ) Shear in nonlinear elasticity J. Voss, C. Thiel, R. J. Martin and P. Neff 12 / 17

Which of these deformations are suitable to be called shear? Finite simple shear deformation A finite simple shear deformation is a mapping ϕ : Ω ⊂ R 3 → R 3 of the form   1 sinh(2 α ) 0 1   ∇ ϕ = F α = � 0 cosh(2 α ) 0 � cosh(2 α ) 0 0 cosh(2 α ) α ≪ 1 with the linearization F α − → F γ and γ = 2 α . F α = V α R increases the height: 1 1             � cosh(2 α ) 1 ϑ ⋆            1 � �� � 1 sinh(2 α ) � cosh(2 α ) � cosh(2 α ) Shear in nonlinear elasticity J. Voss, C. Thiel, R. J. Martin and P. Neff 13 / 17

Which of these deformations are suitable to be called shear? F = F γ diag( a , b , c ) γ F γ diag( a , b , c ) V α R F = V α R Shear in nonlinear elasticity J. Voss, C. Thiel, R. J. Martin and P. Neff 14 / 17

Constitutive requirements in hyperelasticity Which constitutive requirements ensure that only finite shear deformations correspond to pure shear Cauchy stress?     cosh( α ) sinh( α ) 0 0 0 λ  = Q · · Q T with λ = e α ,   1  V α = sinh( α ) cosh( α ) 0 0 0 λ 0 0 1 0 0 1 � �� � pure shear deformation       0 s 0 s 0 0 1 − 1 0 1   = Q   Q T ,   . σ ( B ) = 0 0 0 − s 0 Q = √ 1 1 0 � s 2 0 0 0 0 0 0 0 0 1 λ 2 = 1 λ 1 = λ , λ 3 = 1 = ⇒ σ 1 = s , σ 2 = − s , σ 3 = 0 λ , . � �� � � �� � singular values of F principal Cauchy stresses Shear in nonlinear elasticity J. Voss, C. Thiel, R. J. Martin and P. Neff 15 / 17