Linear Cosserat elasticity, conformal curvature and bounded stiffness

Transcription

1 1 Linear Cosserat elasticity, conformal curvature and bounded stiffness Patrizio Neff, Jena Jeong Chair of Nonlinear Analysis & Modelling, Uni Dui.-Essen Ecole Speciale des Travaux Publics, Cachan, Paris Euromech 510: Mechanics of Generalized Continua: A hundred years after the Cosserats, U Paris 6, May 13-16, 2009

2 2 Plan of the lecture Notation The infinitesimal Cosserat model The Cosserat balance equations and the Cosserat curvature energy A new coercive inequality for conformal curvature Conformal maps A beam network versus the conformal Cosserat model Circular torsion revisited: Gauthiers formula The idea behind bounded stiffness Relations to Lakes experimental evidence Open problems and Summary

3 3 Notation u : Ω R 3 R 3, the displacement, Ω reference configuration. u, the displacement gradient, 11 the identity. W ( u, A) elastic free energy density. A, A, B, tr [A] norm, scalar product and trace. A : Ω so(3) infinitesimal Cosserat-rotation, skew-symmetric matrix. θ = axl A R 3 rotation vector, anti(θ) = A : A.ξ = axl A ξ. ε = u A, non-symmetric micropolar-stretch tensor.

4 4 The Cosserat approach Idea: relax constraint on the rotations to coincide with continuum rotations skew( u). Allow for independent (micro) rotations A so(3) (Cosserat 1909, motivated by triedre caché of surface theory). Aero, Capriz, Eringen, Grioli, Kunin, Maugin, Nowatzki, Palmov, Suhubi... Attribute energy to spatial variations of rotations (curvature): W curv ( axl A) µ L c 2 axl A 2. Internal length scale L c, size-effects. Look for independent fields u, A in the minimization problem. Suited for material with granular substructure (individual rotations of particles), spin, magnetization, homogenisation of heterogeneous materials. Explore lower energy paths, avoid stability problems, shear-band regularization. Major open problem: determine Cosserat parameter values for specific materials. Linear, isotropic case has six parameters: µ, λ, µ c, α, β, γ. Lamé µ, λ size-independent, is Cosserat couple modulus µ c size-independent?

5 5 The linear isotropic Cosserat problem in variational form Reference domain Ω R 3 I(u, A) = W mp (ε) + W curv ( axl A) dv min. w.r.t. (u, A) Ω u Γ = u d, A Γ free, θ = axl A, ε = u A, W mp (ε) = µ sym ε 2 + λ 2 tr [sym ε]2 + µ c skew ε 2 = µ sym u 2 + λ 2 tr [ u]2 + µ c skew ( u A) 2 W curv ( θ) = γ + β 2 dev sym θ 2 + γ β 2 skew θ 2 + k c 2 tr [ θ]2 Existence: γ β > 0, k c 0 Jeong/Neff (MMS08), general micromorphic model: A so(3) p gl(3), Neff/Forest (JEL07). Infinitesimal elasto-plastic extension, Neff/Che lmiński (PRSE05). γ = β, k c = 0 Conformally invariant model Neff/Jeong (ZAMM09) Conformal Simulation: Jeong/Ramezani/Münch/Neff (subm. ZAMM)

6 6 The Cosserat balance equations E-L: linear momentum+angular momentum: linear second order PDE-system Div σ = f, Div m = 4 µ c axl skew ε, balance of linear momentum balance of angular momentum σ = 2µ sym ε + 2µ c skew ε + λ tr [ε] 11 = (µ + µ c ) ε + (µ µ c ) ε T + λ tr [ε] 11 = 2µ dev sym ε + 2µ c skew ε + K tr [ε] 11, m = γ θ + β θ T + α tr [ θ] 11 = (γ + β) dev sym θ + (γ β) skew θ + φ = axl A, u Γ = u d, k c := 3α + (γ + β) 3. 3α + (γ + β) 3 tr [ θ] 11, m couple stress tensor, linear function of curvature θ = axl A. Conformally invariant case m = (γ + β) dev sym θ.

7 7 The Cosserat curvature energy The curvature term has the form W curv ( φ) = γ + β 2 dev sym θ 2 + γ β skew θ 2 + k c 2 2 pointwise coercivity of curvature (apply Poincaré s inequality, H 1 -norm) tr [ θ]2 c + > 0 K gl(3) : W curv (K) c + K 2 γ > β > 0, k c > 0 symmetric coercivity of curvature (apply Korn s second inequality) c + > 0 K gl(3) : W curv (K) c + sym K 2 γ β > 0, k c > 0 conformal curvature (apply???) c + > 0 K gl(3) : W curv (K) c + dev sym K 2 γ = β > 0, k c = 0

8 8 A new coercive inequality for conformal curvature Let φ : Ω R n R n and n 3 and dev X = X 1 n tr [X] 11 n. Then c + > 0 φ H 1 (Ω) : dev sym φ 2 M n n + φ 2 R n dv c+ φ H 1 (Ω,R n ). Proof: Jeong/Neff (MMS08) Ω Counterexample for n = 2 (all analytic maps in the plane are in the kernel). Infinitesimal conformal maps for n = 3: constants Â, Ŵ so(3), p R, b R 3 dev sym φ = 0 φ(x) = 1 ( 2 axl(ŵ 2 ), x x axl(ŵ ) x 2) + [ p 11 + Â].x + b Conformal maps preserve infinitesimal shapes and angles. They do not generate couple stresses for conformal curvature.

9 9 Conformal maps Conformal maps for n = 3: constants Â, Ŵ so(3), p R, b R 3 u C (x) = 1 2 ( 2 axl(ŵ ), x x axl(ŵ ) x 2) + [ p 11 + Â].x + b Conformal maps preserve infinitesimal shapes and angles. They do not generate couple stresses for conformal curvature Infinitesimal conformal mappings u C : R 3 R 3 locally shape preserving: a prototype elastic deformation, stress deviator dev σ( u C ) = 0 (von Mises Plasticity).

10 10 Universal solutions for conformal Cosserat curvature For conformal curvature and constant bulk modulus K the solution of Div σ( u, A) = 3K axl Ŵ, Div m = 4 µ c axl skew( u A), u Γ = u C is uniquely given by u(x) = u C (x), independent of inhomogeneous µ(x), µ c (x), γ(x). z y y x x z y z x The deformed cubes are curvature free because of conformal curvature!

11 11 A beam network versus the conformal Cosserat model Often, Cosserat models are motivated with a beam model in mind. But a conformal Cosserat model is not a homogenized beam model: Consider a planar beam structure with Euler-Bernoulli theory. Apply conformal displacements u C to all beams nodes: u C (x) = 1 2 ( 2 axl(ŵ ), x x axl(ŵ ) x 2) + [ p 11 + Â].x + b, where Ŵ, Â so(3), b R 3, p R are given constant parameters. In our comparison we use b = 0, Â = 0 and some generic values for p and Ŵ.

12 12 The response of a beam network Left: Initial structure with boundary conditions, displacement vectors and deformed mesh. All beam nodes are conformally displaced. Right: Trend of curves (plotted on undeformed mesh) indicates the curvature of beams. As a result, the beam structure does respond with curvature contrary to a conformal Cosserat model, whose response would be curvature free. The beams response is always with curvature. Thus the conformal Cosserat model cannot be identified with a homogenized beam model. We rather expect a homogenized beam model to give rise to a uniform positive definite curvature expression W curv ( θ) µ L 2 c θ 2 (γ > β > 0, k c > 0).

13 13 Circular torsion revisited Consider circular cylinder with radius a > 0, length L and applied resultant end moments. Classical size-independent response: relation between torque Q [N m] and twist per unit length θ L [1/m] is given by Q = µ J Ω t θ L, Ω t 1, where µ > 0 is the classical shear modulus, J = π a4 2 polar moment of inertia of cross section. Classical rigidity: a 4, normalized torsional rigidity Ω t 1.

14 14 Gauthiers Cosserat solution Cosserat size-dependent response: relation between torque Q [N m] and twist per unit length θ L [1/m] is given by Q = µ J Ω t θ L, Ω t = ( ) ( 2 lt 1 4 a 3 Ψ χ (p a) 1 Ψ χ (p a) ), where Ψ := β + γ α + β + γ, non-dimensional polar ratio, 0 Ψ 3 2, l 2 t := β + γ 2 χ (ξ) := 1 µ, characteristic length for torsion, I 1 (ξ) ξ I 0 (ξ), p2 := 4 µ c α + β + γ = 2 µ c Ψ µ l 2 t I 1 (ξ), I 0 (ξ), modified Bessel functions of first kind.,

15 15 The idea of bounded stiffness Circular wire with radius a > 0. Classically: rigidity in tension a 2, rigidity in torsion and bending a 4. Cosserat: increased rigidity in torsion and bending (size effects): thinner samples of the same material respond stiffer. For certain parameter ranges of the Cosserat solid this effect is dramatic: rigidity in torsion proportional to a 2. Normalised torsional rigidity Ω t (normalised against the classical value) has a singularity proportional to 1 a 2. However,..., infinite stiffening effects are unphysical. Lakes95 Bounded stiffness: require stiffness increase for thinner and thinner samples (normalised against the classical stiffness) is bounded independent of wire radius a, i.e. a singularity free response.

16 16 Bounded normalized rigidity in torsion Ω t Ω t versus wire radius a and length scale l t for µ c /µ = 5, polar ratio Ψ = 3 2. Here, Ω t is a continuous function, increasing for increasing l t if a > 0 and assuming the constant value Ω t = µ c µ for a = 0, independent of l t, allowing to obtain a size-independent, stable identification of µ c.

17 17 Singular normalized rigidity in torsion Ω t Ω t versus wire radius a and length scale l t for µ c /µ = 5, polar ratio Ψ = 2 3 (α = β = γ). Ω t is discontinuous at a = 0 and there is no possibility to influence the behaviour near a = 0 by varying µ c /µ.

18 18 Experimental evidence: Bounded stiffness in torsion: polar ratio Ψ = 3 2 α = 1 3 (β + γ) k c = 0 Bounded stiffness in bending: symmetry β = γ. Bending and torsion of circular Cosserat wire: bounded stiffness conformal curvature µ L2 c 2 dev sym axl A 2. Conformal curvature still shows size-effects. Lakes : Identification of Cosserat parameters through series of size-experiments for foams: polar ratio Ψ 3 2, sometimes symmetry β = γ, but not always satisfied. A priori β γ singular stiffening in bending for small wire radius. Interesting: repeat bending experiments with smaller diameters to see singularity.

19 19 Open problems Indeterminate couple stress problem: bounded stiffness in bending is satisfied (symmetric curvature contribution sym curl u 2 ), but impossible to satisfy in torsion! Linear elasticity is conformally invariant for bulk modulus K = 0 (ν = 1), so is Cosserat for conformal curvature. Should any extended continuum model inherit this invariance of linear elasticity? What about anisotropic solids? Is conformal invariance a more fundamental concept than bounded stiffness? Does bounded stiffness always imply conformal invariance for K = 0? Is bending and torsion of circular wires by resultant end moments sufficient to claim bounded stiffness for any boundary value problem? Extension of conformal curvature to finite strain case? What type of invariance?

20 20 Summary Conformal curvature dev sym θ 2 is only positive semi-definite. Conformal curvature case is nevertheless well-posed: it is the weakest curvature. Conformal curvature case is not a homogenized beam model, no grid model. Singular stiffening effects are problematic: Bounded stiffness in bending and torsion of circular wires implies conformal curvature. Need to determine 6 constants for the Cosserat problem. Reduced to 4: two Lamé constants µ, λ, couple modulus µ c and one length scale parameter L c. For conformal curvature Cosserat couple modulus µ c bending and torsion. is size-independent in Numerical simulation shows that for conformal curvature, the internal length scale parameter L c can be chosen orders of magnitude larger for the same material than compared to classical Cosserat model.

21 21 References P. Neff, The Cosserat couple modulus for continuous solids is zero viz the linearized Cauchy-stress tensor is symmetric, ZAMM 86, , 2006 P. Neff and S. Forest, A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results, JEL, 87, , 2007 J. Jeong and P. Neff, Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions, Math. Mech. Solids, online 2008 P. Neff and J. Jeong and I. Münch and H. Ramezani, Mean field modeling of isotropic random Cauchy elasticity versus microstretch elasticity, ZAMP 60(3), , 2009 P. Neff and J. Jeong, A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy, ZAMM 89(2), ,2009 J. Jeong and H. Ramezani and I. Münch and P. Neff, A numerical study for linear isotropic Cosserat elasticity with conformally invariant curvature, submitted to ZAMM P. Neff and J. Jeong and H. Ramezani, Subgrid interaction and micro-randomness - novel invariance requirements in infinitesimal gradient elasticity, submitted to IJSS P. Neff and J. Jeong and A. Fischle, Stable identification of linear isotropic Cosserat parameters: bounded stiffness in bending and torsion implies conformal invariance of curvature, submitted to Acta Mech. P. Neff and Kwon-Il Hong and J. Jeong, The Reissner-Mindlin plate is the Γ-limit of Cosserat elasticity, to appear in M3AS, 2009, P. Neff and K. Che lmiński, Hloc 1 -stress and strain regularity in Cosserat-Plasticity, ZAMM 89(4), , 2009, Palmov-anniversary issue. see also neff/