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 ⊂ R3 → R3 of the form ∇ϕ = Fγ =   1 γ 1 1   = ✶ + γe2 ⊗ e1 with the amount of shear γ ∈ R. 1 1 1 1 1          1

γ

F

ϑ

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• J. Voss, C. Thiel, R. J. Martin and P. Neff

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Introduction

Pure shear stress

A pure shear stress is a stress tensor T ∈ Sym(3) of the form T s =   s s   = s (e1 ⊗ e2 + e2 ⊗ e1) with the amount of shear stress s ∈ R. upper shear force lower shear force e1 e2 e3 1 1          1

γ

ϑ

Shear in nonlinear elasticity

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Introduction

In isotropic nonlinear elasticity the Cauchy stress tensor is σ = β0✶ + β1B + β−1B−1 with βi = βi

• I1(B), I2(B), I3(B)
• and B = FF T.

Set σ = T s =   s s   , Fγ =   1 γ 1 1     s s   = σ = (β0 + β1 + β−1)✶ +   β1γ2 (β1 − β−1)γ (β1 − β−1)γ β−1γ2   = ⇒ γ2 (β1 − β−1) = 0 then γ = 0

• r

s = 0 . Pure shear Cauchy stress never corresponds to a simple shear deformation!

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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

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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)QT with

Q := 1 √ 2   1 −1 1 1 √ 2   ∈ SO(3) . Thus B = Q diag(λ2

1, λ2 2, λ2 3)QT with [Thiel, Voss, Martin, and Neff 2018a]

B = 1 2   λ2

1 + λ2 2

λ2

1 − λ2 2

λ2

1 − λ2 2

λ2

1 + λ2 2

2λ2

3

  = FγF T

γ =

  1 + γ2 γ γ 1 1   .

Shear in nonlinear elasticity

• J. Voss, C. Thiel, R. J. Martin and P. Neff

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Which kind of deformations correspond to pure shear stress?

• σ(B) = T s =

  s s   ⇐ ⇒ B = 1 2   λ2

1 + λ2 2

λ2

1 − λ2 2

λ2

1 − λ2 2

λ2

1 + λ2 2

2λ2

3

  . Then F is uniquely determined by triaxial stretch and simple shear F = Fγ diag(a, b, c)Q =   1 γ 1 1     a b c   Q up to an arbitrary Q ∈ SO(3) with a = λ1λ2

• 2

λ2

1 + λ2 2

, b =

• λ2

1 + λ2 2

2 , c = λ3 , γ = λ2

1 − λ2 2

λ2

1 + λ2 2

.

Shear in nonlinear elasticity

• J. Voss, C. Thiel, R. J. Martin and P. Neff

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Which of these deformations are suitable to be called shear?

Linear Elasticity

The linear elastic Cauchy stress σlin = 2µ 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

F = ✶ +  

γ 2 γ 2

 

• ε∈Sym(3)

+A , A ∈ so(3) . Fγ =   1 γ 1 1   = ✶ +  

γ 2 γ 2

 

• εγ∈Sym(3)

infinitesimal pure shear strain

+  

γ 2

− γ

2

 

• ωγ∈so(3)

infinitesimal rotation

.

Shear in nonlinear elasticity

• J. Voss, C. Thiel, R. J. Martin and P. Neff

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Which of these deformations are suitable to be called shear?

γ

✶ + εγ Fγ = ✶ + εγ + ωγ

ground parallel (deck of cards)

• The deformation Fγ is infinitesimally volume preserving, tr εγ = 0.
• The deformation Fγ is planar, eigenvalue 1 to eigenvector e3.
• The deformation Fγ is ground parallel, eigenvectors e1 and e3.

Shear in nonlinear elasticity

• J. Voss, C. Thiel, R. J. Martin and P. Neff

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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 γ σ =   s s  

Shear in nonlinear elasticity

• J. Voss, C. Thiel, R. J. Martin and P. Neff

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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 e3.
• The deformation F is ground parallel, eigenvectors e1 and e3.

= ⇒ there exists λ ∈ R+ with λ1 = λ, λ2 = 1

λ and λ3 = 1.

• σ(B) = T s =

  s s   = ⇒ V = √ B = 1 2   λ1 + λ2 λ1 − λ2 λ1 − λ2 λ1 + λ2 2λ3   .

Shear in nonlinear elasticity

• J. Voss, C. Thiel, R. J. Martin and P. Neff

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Which of these deformations are suitable to be called shear?

Finite pure shear stretch

V = 1 2   λ + 1

λ

λ − 1

λ

λ − 1

λ

λ + 1

λ

2   = 1 2   eα + e−α eα − e−α eα − e−α eα + e−α 2   =   cosh(α) sinh(α) sinh(α) cosh(α) 1   = exp

. . .

matrix exponential

  α α  

• infinitesimal pure shear strain

=: Vα , α := log λ . infinitesimal pure shear strain εγ tr εγ = 0 finite pure shear stretch Vα det Vα = 1

exp

γ = 2α

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?

• σ(B) = T s =

  s s   = ⇒ F =   1 γ 1 1     a b c   Q. with λ1 = λ, λ2 = 1

λ, λ3 = 1 and α = log λ:

Finite simple shear deformation

F =   1 tanh(2α) 1 1     

1

√

cosh(2α)

• cosh(2α)

1    Q = 1

• cosh(2α)

  1 sinh(2α) cosh(2α)

• cosh(2α)

  Q =: Fα .

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 ⊂ R3 → R3 of the form ∇ϕ = Fα = 1

• cosh(2α)

  1 sinh(2α) cosh(2α)

• cosh(2α)

  with the linearization Fα

α≪1

− → Fγ and γ = 2α. increases the height: 1 1 1 1

1

• cosh(2α)

                      

• cosh(2α)
• sinh(2α)
• cosh(2α)

Fα = VαR

ϑ⋆

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 = Fγ diag(a, b, c) γ

R Vα diag(a, b, c) Fγ

F = VαR

Shear in nonlinear elasticity

• J. Voss, C. Thiel, R. J. Martin and P. Neff

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Constitutive requirements in hyperelasticity

Which constitutive requirements ensure that only finite shear deformations correspond to pure shear Cauchy stress? Vα =   cosh(α) sinh(α) sinh(α) cosh(α) 1   = Q ·   λ

1 λ

1  

• pure shear deformation

· QT with λ = eα,

• σ(B) =

  s s   = Q   s −s   QT, Q = 1 √ 2   1 −1 1 1 1   . λ1 = λ , λ2 = 1 λ , λ3 = 1

• singular values of F

= ⇒ σ1 = s , σ2 = −s , σ3 = 0

• principal Cauchy stresses

.

Shear in nonlinear elasticity

• J. Voss, C. Thiel, R. J. Martin and P. Neff

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Constitutive requirements in hyperelasticity

I1 = tr B = λ2

1 + λ2 2 + λ2 3 ,

I2 = tr(Cof B) = λ2

1λ2 2 + λ2 1λ2 3 + λ2 2λ2 3 ,

I3 = det B = λ2

1λ2 2λ2 3

σ = β0✶ + β1B + β−1B−1 with βi = βi

• I1(B), I2(B), I3(B)
• =

⇒ β1 + β−1 = 0 and β0 = 0 ∀ λ ∈ R+ with λ1 = λ , λ2 = 1 λ , λ3 = 1 . β0 = 2 √I3

• I2

∂W ∂I2 + I3 ∂W ∂I3

• ,

β1 = 2 √I3 ∂W ∂I1 , β−1 = −2

• I3

∂W ∂I2 , = ⇒ ∂W ∂I1 = ∂W ∂I2 and I2 ∂W ∂I2 + ∂W ∂I3 = 0 ∀ I1 = I2 ≥ 3 , I3 = 1 . σi = λi λ1 λ2 λ3 ∂W ∂λi (λ1, λ2, λ2) = ⇒ λ ∂W ∂λ1 + 1 λ ∂W ∂λ2 = 0 and ∂W ∂λ3 = 0 ∀ λ1 = λ , λ2 = 1 λ , λ3 = 1 .

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• J. Voss, C. Thiel, R. J. Martin and P. Neff

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Constitutive requirements in hyperelasticity

Tension-compression symmetry [Voss, Baaser, Martin, and Neff 2018]

Elastic energy W : GL

+(3) → R of the form

W (F) = Wtc(F) + f (det F) , where Wtc is tension-compression symmetric, i.e. Wtc(F −1) = Wtc(F) and f ′(1) = 0.

Hencky-type [Neff, Ghiba, and Lankeit 2015; Neff, Lankeit, Ghiba, Martin, and Steigmann 2015]

Elastic energy W : GL

+(3) → R of the form

W (F) = ψ

• dev log V 2, |tr log V |2

for arbitrary functions ψ : R2

+ → R. Shear in nonlinear elasticity

• J. Voss, C. Thiel, R. J. Martin and P. Neff

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References I

Destrade, M., J. G. Murphy, and G. Saccomandi (2012). “Simple shear is not so simple”. International Journal of Non-Linear Mechanics 47.2. Pp. 210–214. issn: 0020-7462. Mihai, L. A. and A. Goriely (2011). “Positive or negative Poynting effect? The role of adscititious inequalities in hyperelastic materials”. Proc. R. Soc. A 467.2136. Pp. 3633–3646. Moon, H. and C. Truesdell (1974). “Interpretation of adscititious inequalities through the effects pure shear stress produces upon an isotropic elastic solid”. Archive for Rational Mechanics and Analysis 55.1. Pp. 1–17. Neff, P., I.-D. Ghiba, and J. Lankeit (2015). “The exponentiated Hencky-logarithmic strain energy. Part I: Constitutive issues and rank-one convexity”. Journal of Elasticity 121.2. Pp. 143–234. doi: 10.1007/s10659-015-9524-7. Neff, P., J. Lankeit, I.-D. Ghiba, R. J. Martin, and D. J. Steigmann (2015). “The exponentiated Hencky-logarithmic strain energy. Part II: coercivity, planar polyconvexity and existence of minimizers”. Zeitschrift f¨ ur angewandte Mathematik und Physik 66.4. Pp. 1671–1693. doi: 10.1007/s00033-015-0495-0. Thiel, C., J. Voss, R. J. Martin, and P. Neff (2018a). “Do we need Truesdell’s empirical inequalities? On the coaxiality of stress and stretch”. to appear in International Journal of Non-Linear Mechanics. available at arXiv:1812.03053. Thiel, C., J. Voss, R. J. Martin, and P. Neff (2018b). “Shear, pure and simple”. in press: International Journal of Non-Linear Mechanics. Voss, J., H. Baaser, R. J. Martin, and P. Neff (2018). “More on anti-plane shear”. Journal of Optimization Theory and Applications. Pp. 1–24.

References II

Thank you for your attention

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