Projects in Fracture Simulations

1st Cenaero Workshop

Projects in Fracture Simulations

Authors: G. Becker L. Wu

V.-D. Nguyen X. Wang

L. Noels

• Different projects going on

– Fracture of composites • ERA-NET

• CENAERO, e-Xstream, IMDEA, Tudor – Mean-field homogenization with damage

• ERA-NET

• CENAERO, e-Xstream, IMDEA, Tudor – Fracture or MEMS

• UCL

– Fracture of thin structures • MS3, GDTech

– Multiscale • ARC

• Different methods

– Need of common computational tools – Need of maximum flexibility

GMSH (elasticity) Solver Linear System Dof Manager (Bi)linear terms Function Space Quadrature Rule Geo tools

Tools for linear finite element analysis

2011 1st _{GMSH Workshop: solid mechanics project 4} GMSH (elasticity) Solver Linear System Dof Manager (Bi)linear terms Function Space NonLinear MechSolver NLDofManager QS & Explicit Quadrature Rule NLSystem QS & Explicit Material Law IPField Geo tools NLTerms Part Domain

Structure for non-linear finite element analysis

•Pure virtual classes

•Definition of classical material laws •Time integration

•Parallel implementation •…

2011 1st _{GMSH Workshop: solid mechanics project 5} GMSH (elasticity) Solver Linear System Dof Manager (Bi)linear terms Function Space NonLinear MechSolver NLDofManager QS & Explicit Quadrature Rule NLSystem QS & Explicit Material Law dgshells Function Space Material Law IPVariable Interface Element Terms PartDomain Interface Solver – projects IPField Geo tools NLTerms Applications Part Domain

2011 1st _{GMSH Workshop: solid mechanics project 6} GMSH (elasticity) Solver Linear System Dof Manager (Bi)linear terms Function Space NonLinear MechSolver NLDofManager QS & Explicit Quadrature Rule NLSystem QS & Explicit Material Law dgshells Function Space Material Law IPVariable Interface Element Terms PartDomain Interface Solver – projects IPField Geo tools NLTerms Applications Part Domain dG3D NonLocalDam FE2

• Discontinuous Galerkin formulation

– Finite-element discretization

– Same discontinuous polynomial approximations for the • Test functions _h and

• Trial functions d

– 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)+ x (a+1)- Field (a+1)+ (a)- (a-1)-

• Discontinuous Galerkin formulation

– // & fracture

– Formulation in terms of the first Piola stress tensor P

&

– Weak formulation obtained by integration by parts on each element e

• 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 , for all mesh sizes hs

– Those terms can also be explicitly derived from a variational formulation (Hu-Washizu-de Veubeke functional) [Noels & Radovitzky, IJNME 2006 & JAM 2006]

• Cohesive Zone Method for fracture

– Based on the use of cohesive elements • Inserted between bulk elements

– Intrinsic Law

• Cohesive elements inserted from the beginning • Drawbacks:

– Efficient if a priori knowledge of the crack path – Mesh dependency [Xu & Needelman, 1994]

– Initial slope 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

• New DG/extrinsic method

– Interface elements inserted from the beginning – Interface law initially the DG interface forces – Impact of alumina plate

• MATERA project: SIMUCOMP

– CENAERO, e-Xstream, IMDEA Materials, Tudor, ULg

– Application to composite – Representative nature? – First results

• Thin bodies

– FRIA (MS3, GDTech) – C1 _{continuity required}

– Test functions

• New DG interface terms

– Consistency – Compatibility – Stability

2011 CFRAC 2011: Advances in Cohesive-element Modeling of Material Failure 13

(a-1) (a) x (a+1) Field (a-1)+ _(a)+ x (a+1)- Field (a+1)+ (a)- (a-1)- Extension for fracture C0_{/DG formulation}

[Noels & Radovitzky, CMAME 2008]

DG formulation [Becker & Noels, IJNME 2011]

• New cohesive law for thin bodies

– Should take into account a through the thickness fracture

• Problem : no element on the thickness • Very difficult to separate fractured and not fractured parts

– Solution:

• Application of cohesive law on

– Resultant stress

– Resultant bending stress

• In terms of a resultant opening

D_x D_r D_c= N, M N M N₀ M₀ D* 2G_c s_max

• Application

– Notched elasto-plastic cylinder submitted to blast

– Future application: Rupture of MEMS • UCL

• Extension to damage

– Transition from damage to crack

• At lost of ellipticity or at lost of convergence [Huespe 2009]

• Fracture energy = Damage energy remaining

2011 1st _{GMSH Workshop: solid mechanics project 16}

σ_c G_c Δ_c G_c = energy to dissipate from D_c to 1 σ σ_y0 ε Damage theory Cohesive zone σ_c D ε 1 D_c

• Extension to damage

– First results

• Elastic damage • Loading too fast!!!

• MATERA project: SIMUCOMP

– CENAERO, e-Xstream, IMDEA Materials, Tudor, ULg

• Mean Field Homogenization

– 2-phase composite

– Mori-Tanaka assumption

• Extension to damage?

2011 1st _{GMSH Workshop: solid mechanics project 18}

Macro Macro Macro Micro MFH n



_D

_

C

s

ε

ε

ε



v



v

• Damage

• Implicit non-local approach

– New equation on an internal variable

2011 1st _{GMSH Workshop: solid mechanics project 19}

The numerical results change with the size of mesh and direction of mesh

Homogenous unique solution

Lose of uniqueness

Strain localized

The numerical results change without convergence a a c a  2 



Green function as weight functions w

• Non-local damage

– Lemaitre-chaboche

• S₀ and n are the material parameters • Y is the strain energy release rate • p is the accumulated plastic strain

– New equation in the system

2011 1st _{GMSH Workshop: solid mechanics project 20}

p p c p 2  p S Y p c p c p S Y D n n      ) ( ...) ( ) ( 0 4 2 2 1 0                                 p p p p u p p u uu d d F F F F p u K K K K _{ext int}

• MFH with Non-local damage

– Based on Linear Composite Comparison

&

– Finite elements with 4dofs/node

for homogenized material related to matrix only

2011 1st _{GMSH Workshop: solid mechanics project 21}

0 0 1 1 σ σ σ d  d d   D D d d d alg _{0 0} 0 0 (1 )C : ε σˆ σ    σˆ₀ σ₀ /(1D) p p D D a d  d d     lg _{0 ˆ0} : ε σ C σ Mori-Tanaka D D a d  d  d  d alg _{0 0 0} 0 0 I lg I 1C ε (1 )C : ε σˆ σ              only material matrix for material comopsite for 2 2 p p l p 0 f σ                          p p p p u p p u uu d d F F F F p u K K K K _{ext int}

• MFH with Non-local damage

– Epoxy-CF (30%)

2011 1st _{GMSH Workshop: solid mechanics project 22}

-150 -100 -50 0 50 100 150 200 0 0.02 0.04 0.06 0.08 0.1 Strain Stress (MPa) Random Cells Periodical Cell MFH

• Application

– Epoxy-CF (30%) – Transverse loading – Mesh independent

2011 1st _{GMSH Workshop: solid mechanics project 23}

0 50 100 150 200 250 300 0 0.0002 0.0004 0.0006 0.0008 0.001 Displacement (m) Force (N/mm ) Mesh size 0.43 mm Mesh size: 0.3 mm Mesh size: 0.15 mm

• Computational Multiscale

2011 1st _{GMSH Workshop: solid mechanics project 24}

F_M

P_M

• Comparison

• NonLinearMechSolver

– Generic tool to solve mechanic problems – // implementation based on DG

• Applications

– Different projects which include the solver • Projects are independent

– First results

– More work coming …

• Material library

– Mechanics: get stresses from deformations

– Generic law are defined in nonLinearMechSolver • Can be derived in applications (specificities)

– Basic class

– IPStateBase allows to save data at integration points

2011 1st _{GMSH Workshop: solid mechanics project 27}

class materialLaw {

public:

enum matname{…} protected :

int _num; // law number (must be unique !)

bool _initialized; // to initialize law

double _timeStep; // for law which works on increment. (same for all)

double _currentTime; // time of simulation (same for all)

public:

// constructors, destructor, set & get data functions

virtual void createIPState(IPStateBase* &ips, const bool* state_=NULL,

const MElement* ele=NULL, const int nbFF_=0) const=0; };

• Non linear laws have to store history

– Elasto-plastic law • Plastic deformation • …..

• IPStateBase regroups IPVariables for same point but at different times

– The contain depends on the law

• For each law there is a specific IPVariable (inheritance tree are the same)

2011 1st _{GMSH Workshop: solid mechanics project 28}

class IPStateBase{ public:

// constructor & destructor

enum whichState{initial, previous, current}; virtual IPVariable* getState( const whichState wst=IPStateBase::current) const=0; };

class IPVariable{ public :

// constructor & destructor

virtual double get(const int i) const{ return 0.;} };

• Same than (elasticity)Solver

– dofManager assembles linear and bilinear terms by • Linking a Dof to a unique system position

– The systems are implemented in different formats • Taucs, PETSc,Gmm for quasi-statics

• Blas and PETSc for dynamics (only vector operations)

• In parallel

– Based on DG method