Discontinuous Galerkin methods for solid mecha nics: Application to fracture, shells & strain gradient elasticity

Discontinuous Galerkin methods

for solid mechanics:

Application to fracture, shells &

strain gradient elasticity

Ludovic Noels

Computational & Multiscale Mechanics of Materials, ULg Chemin des Chevreuils 1, B4000 Liège, Belgium

L.Noels@ulg.ac.be

Universidade Federal de Santa Catarina - August 2009 University of Liège

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Discontinuous Galerkin methods for solid

mechanics:

Application to fracture, shells & strain gradient

elasticity

Ludovic Noels

Computational & Multiscale Mechanics of Materials, ULg Chemin des Chevreuils 1, B4000 Liège, Belgium

L.Noels@ulg.ac.be

Universidade Federal de Santa Catarina - August 2009 University of Liège

Discontinuous Galerkin Methods

• Main idea

– Finite-element discretization

– Same discontinuous polynomial approximations for the

• Test functions ϕ_h and • Trial functions δϕ

– Definition of operators on the interface trace:

• Jump operator: • Mean operator:

– Continuity is weakly enforced, such that the method

• Is consistent • Is stable

• Has the optimal convergence rate

(a-1)- (a-1)+ (a)-

(a)+(a+1)-X

(a+1)+

Discontinuous Galerkin Methods

• Discontinuous Galerkin methods vs Continuous

– More expensive (more degrees of freedom) – More difficult to implement

– …

• So why discontinuous Galerkin methods?

– Weak enforcement of C1 continuity for high-order equations

• Strain-gradient effect

• Shells with complex material behaviors

• Toward high-order computational homogenization

– Exploitation of the discontinuous mesh to simulate dynamic

fracture [Seagraves, Jérusalem, Noels, Radovitzky, col. ULg-MIT]:

• Correct wave propagation before fracture • Easy to parallelize & scalable

Discontinuous Galerkin Methods

• Continuous field / discontinuous derivative

– No new nodes – Weak enforcement of C1 continuity – Displacement formulations of high-order differential equations

– Usual shape functions in 3D (no new requirement) – Applications to

• Beams, plates [Engel et al., CMAME 2002; Hansbo & Larson, CALCOLO 2002; Wells & Dung, CMAME 2007]

• Linear & non-linear shells [Noels & Radovitzky, CMAME 2008; Noels IJNME 2009]

• Damage & Strain Gradient [Wells et al., CMAME 2004; Molari, CMAME 2006; Bala-Chandran et al. 2008]

(a-1) (a)

X

(a+1)

Topics

• Key principles of DG methods

– Illustration on volume FE

• Discontinuous Mesh & Dynamic Fracture

• Kirchhoff-Love shells

– Kinematics

– Non-Linear shells – Numerical examples

• Strain gradient elasticity

Key principles of DG methods

• Application to non-linear mechanics

– Formulation in terms of the first Piola stress tensor P &

– New weak formulation obtained by integration by parts on

Key principles of DG methods

• Interface term rewritten as the sum of 3 terms

– Introduction of the numerical flux h

• Has to be consistent: • One possible choice:

– Weak enforcement of the compatibility

– Stabilization controlled by parameter β, for all mesh sizes hs

– Those terms can also be explicitly derived from a variational

Key principles of DG methods

• Numerical applications

– Properties for a polynomial approximation of order k

• Consistent, stable for β >Ck_{, convergence in the e-norm in k}

• Explicit time integration with conditional stability • High scalability

– Examples

Taylor’s impact Wave propagation

Key principles of DG methods

• Numerical applications

– Application to oligo-crystal plasticity

Aluminum oligo crystal sample

Grain profile from EBSD measurement Oligo crystal sample preparation: MIT + Alcoa.

Sample cutting, polishing & Electron Backscatter Diffraction (EBSD): Caltech + MIT (Z. Zhao).

Tensile test & Digital image correlation (DIC): Rutgers (S. Kuchnicki, A. Cuitino) + MIT.

Key principles of DG methods

• Numerical applications

– Application to oligo-crystal plasticity

Discontinuous Mesh & Dynamic Fracture

• Dynamic fracture

– Fracture: a gradual process of separation which occurs in small regions of material adjacent to the tip of a forming

crack: the cohesive zone [Dugdale 1960, Barrenblatt 1962, …]

– Separation is resisted to by a cohesive traction – 2-parameter cohesive law

• Peak cohesive traction σ_max (spall strength) • Fracture energy G_c

• Automatically accounts for time scale [Camacho & Ortiz, 1996]

• Intrinsic law vs Extrinsic law

Failure criterion external to the cohesive law

Failure criterion incorporated within the cohesive law

Discontinuous Mesh & Dynamic Fracture

• Finite element discretization & interface elements

– The cohesive law is integrated on an interface element

inserted between two adjacent tetrahedra [Ortiz & Pandolfi 1999]

– Potential structure of the cohesive law:

[Ortiz & Pandolfi 1999]

• Effective opening in terms of β_c the ratio between the shear and normal critical tractions:

• Definition of a potential: • Interface traction:

Discontinuous Mesh & Dynamic Fracture

• Two methods

– Intrinsic Law

• Cohesive elements inserted from the beginning • Drawbacks:

– Requires a priori knowledge of the crack path to be efficient – Mesh dependency [Xu & Needelman, 1994]

– Initial slope that modifies the effective elastic modulus – This slope should tend to infinity [Klein et al. 2001]:

» Alteration of a wave propagation » Critical time step is reduced

– Extrinsic Law

• Cohesive elements inserted on the fly when failure criterion is verified [Ortiz & Pandolfi 1999]

• Drawback

– Complex implementation in 3D especially for parallelization

• New DG/extrinsic method

[Seagraves, Jerusalem, Radovitzky, Noels]

– Interface elements inserted from the beginning

Discontinuous Mesh & Dynamic Fracture

• New DG/extrinsic method:

[Seagraves, Jerusalem, Radovitzky, Noels, col. MIT & ULg]

– Numerical application: the spall test

• Two opposite waves interact at the center of the specimen • The interaction leads to stresses higher than the spall stress • The specimen breaks exactly at its middle

Discontinuous Galerkin Methods

• Continuous field / discontinuous derivative

– No new nodes – Weak enforcement of C1 continuity – Displacement formulations of high-order differential equations

– Usual shape functions in 3D (no new requirement) – Applications to

• Beams, plates [Engel et al., CMAME 2002; Hansbo & Larson, CALCOLO 2002; Wells & Dung, CMAME 2007]

• Linear & non-linear shells [Noels & Radovitzky, CMAME 2008; Noels IJNME 2009]

• Damage & Strain Gradient

(a-1) (a)

X

(a+1)

Kirchhoff-Love Shells

• Description of the thin body

• Deformation mapping

• Shearing is neglected

& the gradient of thickness stretch neglected

E₁ E₂ E₃ ξ1 ξ2 A ξ2_=cst ξ1=cst Φ0=ϕ0(ξ1, ξ2)+ξ3t0(ξ1, ξ2) t₀ ξ1_=cst ξ 2 = cst t S S₀ χ = Φ_°Φ₀-1 Φ=ϕ(ξ1_{, ξ}2_)+ξ3_t(ξ1_{, ξ}2₎ with & & Mapping of the mid-surface

Mapping of the normal to the mid-surface

Kirchhoff-Love Shells

• Resultant equilibrium equations:

– Linear momentum – Angular momentum

– In terms of resultant stresses:

of resultant applied tension and torque and of the mid-surface Jacobian

Kirchhoff-Love Shells

• Non-linear material behavior

– Through the thickness integration by Simpson’s rule – At each Simpson point

• Internal energy W(C=FT_{F) with}

• Iteration on the thickness ratio in order to reach

the plane stress assumption σ33₌₀

– Simpson’s rule leads to the resultant stresses:

Kirchhoff-Love Shells

• Non-linear discontinuous Galerkin formulation

– New weak form obtained from the momentum equations

– Integration by parts on each element _Ae

– Across 2 elements δ t is discontinuous

A e-A_h NAh UA e-A_e+ NAe+ S ν− S_e- Se+ I A e -UAh IAh Φ=ϕ(ξ1, ξ2)+ξ3t(ξ1, ξ2)

Kirchhoff-Love Shells

• Interface terms rewritten as the sum of 3 terms

– Introduction of the numerical flux h

• Has to be consistent: • One possible choice:

– Weak enforcement of the compatibility

– Stabilization controlled by parameter β, for all mesh sizes hs

Linearization leads to the material tangent modulii H_m

Kirchhoff-Love Shells

• New weak formulation

• Implementation

– Shell elements

• Membrane and bending responses

• 2x2 (4x4) Gauss points for bi-quadratic (bi-cubic) quadrangles

– Interface elements

• 3 contributions

• 2 (4) Gauss points for quadratic (cubic) meshes

S

e-Se+

-Kirchhoff-Love Shells

• Pinched open hemisphere

– Properties:

• 18-degree hole

• Thickness 0.04 m; Radius 10 m • Young 68.25 MPa; Poisson 0.3

– Comparison of the DG methods

• Quadratic, cubic & distorted el.

with literature A B 0 200 400 600 800 0 5 10 15 20 P (N) δ (m ) δ x A=-δ yB, linear -δ y_B, 12 bi-quad. el. δ x_A, 12 bi-quad. el. -δ y_B, 8 bi-cubic el. δ x_A, 8 bi-cubic el. -δ y

B, 8 bi-cubic el. dist. δ x_A, 8 bi-cubic el. dist. -δ y_B, Areias et al. 2005 δ x

Kirchhoff-Love Shells

• Pinched open hemisphere

Influence of the stabilization Influence of the mesh size parameter

– Stability if β > 10

– Order of convergence in the L2-norm in k+1

100 101 102 103 104 0 2 4 6 8 10 β δ (m ) -δ y_B, 12 bi-quad. el. δ x A, 12 bi-quad. el. -δ y_B, 8 bi-cubic el. δ x_A, 8 bi-cubic el. 10-2 10-1 10-3 10-2 10-1 100 1 4 1 3 hs/ R Er ror on δ

0 2000 4000 6000 8000 10000 12000 0 5 10 15 20 25 30 q (N/m) δ (m )

Kirchhoff-Love Shells

• Plate ring

– Properties: • Radii 6 -10 m • Thickness 0.03 m

• Young 12 GPa; Poisson 0

– Comparison of DG methods • Quadratic elements with literature A B δ z_A, 16x3 bi-quad. el. δ z_B, 16x3 bi-quad. el. δ z_A, Sansour, Kollmann 2000 δ z_B, Sansour, Kollmann 2000 δ z_A, Areias et al. 2005 δ z_B, Areias et al. 2005

Kirchhoff-Love Shells

• Clamped cylinder

– Properties:

• Radius 1.016 m; Length 3.048 m; Thickness 0.03 m • Young 20.685 MPa; Poisson

0.3

– Comparison of DG methods

• Quadratic & cubic elements

with literature A 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 δ (m ) δ z_A, 12 bi-quad. el. δ z_A, 8 bi-cubic el. δ z_A, Ibrahimbegovic et al. 2001

Strain-gradient elasticity

• Strain-gradient effect

– Length scales in modern technology are now of the order of the micrometer or nanometer

– At these scales, material laws depend on the strain but also on the strain-gradient

– Example:

• Bi-material tensile test:

• E₁/E₂ = 4 • Characteristic length l • Differential equation: 1 E2 E L 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 x/L Nor m aliz ed ε Linear elasticity l/L = 0.001 l/L = 0.1 l/L = 0.5

Strain-gradient elasticity

• Strain-gradient theory of linear elasticity

– At a material point stress is a function of the strain and of the

gradient of strain [Toupin 1962, Mindlin 1964]:

• Strain energy depends on the strain and its gradient • Low and high order stresses introduced as the work conjugate of

low and high order strains: and

– Governing PDE obtained from the virtual work statement:

& &

Strain-gradient elasticity

• Discontinuous Galerkin formulation for strain-gradient theory

– Test functions u_h and trial functions δu are C0

– New weak formulation obtained by repeated integrations by

parts on each element Ωe :

Strain-gradient elasticity

• Interface term rewritten as the sum of 3 terms

– Introduction of the numerical flux h

• Has to be consistent:

• One possible choice:

– Weak enforcement of the compatibility

• Numerical applications

– Properties for polynomial approximation of order k

• Consistent, stable for β >Ck

• Convergence in the e-norm in k-1, but in k+1 in the L2_-norm

– Examples

Bi-material tensile test Stress concentration

Strain-gradient elasticity

0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 3 x/a ε YY /ε YY ∞ l/a=0 l/a=1/10 l/a=1/2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 x/L Nor m aliz ed ε Classical elasticity Analytical; l/L = 0.1 Numerical; l/L = 0.1; 6 el. Numerical; l/L = 0.1; 18 el. 1 E2 E L

Conclusions & Perspectives

• Development of a discontinuous Galerkin formulation

– A formulation has been proposed for non-linear dynamics – Application to high-order differential equations

• Strain gradient elasticity • Shells

– Application to dynamic fracture

• Works in progress

– Fracture of thin structures