49
Vincent Chiaruttini, Georges Cailletaud Vincent Chiaruttini, Georges Cailletaud [email protected] [email protected] Non Linear Computational Mechanics Athens MP06/2012 Non Linear Computational Mechanics Athens MP06/2012 The Finite Element method for The Finite Element method for nonlinear structural problems nonlinear structural problems

The Finite Element method for nonlinear structural …mms2.ensmp.fr/msi_paris/archives-transparents/FE_NL.pdfNon Linear Computational Mechanics Athens MP06/2012– The Finite Element

Embed Size (px)

Citation preview

Vincent Chiaruttini, Georges CailletaudVincent Chiaruttini, Georges [email protected]@onera.fr

Non Linear Computational Mechanics Athens MP06/2012–Non Linear Computational Mechanics Athens MP06/2012–

The Finite Element method forThe Finite Element method fornonlinear structural problemsnonlinear structural problems

2 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

OutlineOutline

Generalities on computational strategies for nonlinear problemsExamples (contact, crack propagation, non-linear behaviour, geometrical non-linearities)

Classical algorithm for nonlinear or time dependant problems

Local numerical aspects of plasticityElastic-plastic behaviour

Local integration of non-linear models

Global numerical aspects of plasticitySolution process

Consistent tangent matrix

Examples of solution process

Presentation of Z-mat

3 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Non-linearities in structural problemsNon-linearities in structural problems

Contact

Due to the non-penetration condition

Crack propagation problem under time dependant loading

When crack propagates the solution becomes non-linearly time dependant

Geometrical nonlinearities

For large deformation, instabilities can also occur (buckling)

Nonlinear constitutive relationship

Non linear behaviour: elastoplasticity, damage, viscosity

4 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Solution processSolution process

Iterative algorithmScalar example

For any kind of regular function, no direct process exists

Iterative algorithmStop when a convergence criterion is satisfied

Newton methodBuilt on the linear verification of the first order Taylor development nullity

¯nding u j f(u) = 0

building un ! u j f(u) = 0rank k j jf(uk)j < "crit

f(uk+1) ¼ f(uk) + (uk+1 ¡ uk) f0(uk) = 0

u0 u2u1

f(u)

u

) uk+1 = uk ¡ f(uk)=f0(uk)

5 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Solution processSolution process

Newton methodBuilt on the linear verification of the first order Taylor development nullity

Convergence depends on u0

When convergesRank k errorRecurrence on error relationshipTaylor expansion closely to the exact solution

Quadratic convergenceClose enough to the solution each iteration produces twice more significant new digits

f(uk+1) ¼ f(uk) + (uk+1 ¡ uk) f0(uk) = 0

) uk+1 = uk ¡ f(uk)=f0(uk)

ek = uk ¡ uek+1 ¡ ek = uk+1 ¡ uk = ¡f(uk)=f 0(uk)

f (uk) = f 0(u)ek +1

2f 00(u)e2k + o(e2k)

f 0(uk) = f 0(u) + f 00(u)ek + f 000(u)e2k + o(e2k)

ek+1 = ek ¡2f 0ek + f 00e2k

2f 0 + f 00ek + f 000e2k+ o(e2k) =

f 00(u)2f 0(u)

e2k + o(e2k) = O(e2k)

6 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Solution processSolution process

Newton methodQuadratic convergence

Require to update the derivative at each iteration

Modified Newton methodsConstant direction

Linear convergence ek+1 = (1¡ f 0(u)=K)ek + o(jekj) = O(jekj)

f 0(uk) ¼ K = cst = f 0(u0)

u0 u2u1

f(u)

u

uk+1 = uk ¡ f(uk)=K

7 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Solution processSolution process

Newton methodQuadratic convergence

Require to update the derivative at each iteration

Modified Newton methodsConstant direction Linear convergenceSecant update

Golden ratio convergence order

u0 u2u1

f(u)

u

f 0(uk) ¼ K = cst = f 0(u0)

uk+1 = uk + f(uk)uk ¡ uk¡1

f(uk)¡ f(uk¡1)ek+1 = O(jekj

1+p5

2 )

8 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Solution processSolution process

Newton method for a set of equationsVectorial function

At each iteration a linear system is solved

where the operator constitutes the rigidity matrix at

Quadratic convergence when close to solutionThe neighbourhood where such kind of convergence is observed depends on the vectorial function properties

F (U) = 0

0 = F (Uk) +

·@F i

@U j(Uk)

¸U jk

Uk

Convergence is insuredConvergence is insuredfor convex functionsfor convex functions

9 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Examples of convergence using a Newton algorithmExamples of convergence using a Newton algorithm

10 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Examples of convergence using a Newton algorithmExamples of convergence using a Newton algorithm

11 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

ODE integrationODE integration

Time dependant problem are ruled by differential equationsReduce high order differential systems to first order

General formulation

Euler-type integration schemesFinite difference time discretization

θ-method (method-B)

Explicit Forward Euler (θ=0) Crank-Nicholson (θ=0.5) Implicit Euler (θ=1)Conditionally stable Unconditionally stable Unconditionally stable1st order accurate 2nd order accurate 1st order accurate

d2y

dt2+ g(t)

dy

dt= r(t) ,

½dydt = z(t)dzdt = r(t) ¡ g(t) z(t)

[ _Y ] = [F (t; [Y ])] [ _Y (t = t0)] = [Y0]

_Y (t) = F (t; Y )

Y n+1 ¡ Y n¢t

= µFn+1(Y n+1) + (1¡ µ)Fn(Y n)

forward methods _Y (tn+1) ¼Y n+1 ¡ Y n

¢t

Y (tn) = Yn; tn+1 = tn +¢t

12 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

ODE integrationODE integration

Time dependant problem are ruled by differential equationsRunge-Kutta explicit integration

Minimal multiple evaluations of on a given time increment to insure a specific order accuracy, based on Taylor expansionRK1 is the forward explicit Euler schemeRK2 using one mid-point sub calculation at

2nd order method

RK4 popular method using 4 sub calculations

4th order method

F (t; y(t))

tn+ 12

yn+1 = yn +¢t f¡tn+1=2; yn+1=2

¢yn+1=2 = yn +

¢t

2f (tn; yn)

yn+1 = yn +¢t=6 (k1 + 2k2 + 2k3 + k4)

k1 = f(tn; yn)

k4 = f(tn+1=2; yn +¢tk3)

k2 = f(tn+1=2; yn + (¢t=2)k1)k3 = f(tn+1=2; yn + (¢t=2)k2)

Initial slope on the interval

Mid interval slope using k1

Mid interval slope using k2

Final slope using k3 on the interval

yn+1 = yn +¢ty0n + (¢t=2)y00n + o(¢t2)

13 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

OutlineOutline

Generalities on computational strategies for nonlinear problemsExamples (contact, crack propagation, non-linear behaviour, geometrical non-linearities)

Classical algorithm for nonlinear or time dependant problems

Local numerical aspects of plasticityElastic-plastic behaviour

Local integration of non-linear models

Global numerical aspects of plasticitySolution process

Consistent tangent matrix

Examples of solution process

Presentation of Z-mat

14 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Local numerical aspects of plasticityLocal numerical aspects of plasticity

AssumptionsSmall strain: linearised kinematicQuasi-static loadings: no dynamic effects

Small strain elastoplastic problemElastic domain

A usual elastic behaviour is applied

Yield surface

Irreversible/dissipative phenomena can occur

Example Von Mises criterion

f(¾) < 0

f(¾) = 0

f(¾) =q3=2¾

D: ¾

D¡R with ¾

D= ¾ ¡ 1

3Tr(¾) I

15 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Local numerical aspects of plasticityLocal numerical aspects of plasticity

Small strain elastoplastic problemPlastic strain

Strain partition

Rigidity

Yield surface evolution (convex), hardeningWhen the yield surface evolves

Isotropic hardening yield surface increase→Kinematic hardening yield surface translation→

Flow rulesPlasticity (normality rule)

Cumulated plastic strain rate

" = "e + "in = "e + "p

¾ = A : "e = A : (" ¡ "p)

_"p = _ @f

@¾; _ ¸ 0; f(¾) · 0; _ f(¾) = 0

p(t) =

Z t

0

r3

2

¡_"p(¿) : _"p(¿)

¢d¿³= _ for a Von Mises criterion

´

f = 0

16 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Local numerical aspects of plasticityLocal numerical aspects of plasticity

Common formalism for viscoplasticity/multipotentials/large strainStrain partition

Small strain Large strain

Elasticity

Flow rulesPlasticity (state must stay on the yield surface at the end of the increment)

Viscoplasticity (state expected to be on the relevant equipotential at the end of the increment)

Hardening rules

" = "e + "in = "e + "p + "vp F = E P

¾ = A : "e

_"p =X

s

_ sns

_"vp =X

s

_vsms

_ s from _fs = 0

_vs =

¿f s

K

Àn

_YI = fct(YI ; "p; "vp)

17 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Numerical solution processNumerical solution process

Elastoplastic exampleRelations

Equilibrium and strain definition

Behaviour

Boundary conditions (initial equilibrium and no plasticity)

r¾ + f = 0 " =1

2

¡r u+rT u

¢

t 2 [0; T ]

_p ¸ 0 ¾eq ¡R(p) · 0_"p = _p3

2¾eq¾D

u = ud on @u­

¾ ¢ n = F on @f­

²p(t = 0) = 0

¾ = A : ("¡ "p)

_p (¾eq ¡R(p)) = 0

¾eq =

r3

2k¾

Dk

18 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Numerical solution processNumerical solution process

Elastoplastic exampleMechanical stateTemporal discretization

Incremental temporal approach using a regular gridThe solution is search incrementaly at each time step tn+1 (all previous step states being known)

Spatial discretizationFE method in displacement

Nonlinearity comes from the non linear relation between stress and strain relation

Local integration of the mechanical behaviourAt each point of the structure process is defined by the following process

Global equilibriumVerified by the solution of the nonlinear variational formulation

tn+1 = tn +¢t

8u¤ 2 U0ad;Z

­

¾n+1

: "¤d­ =

Z

­

fn+1

:u¤d­+

Z

@F­

Fn+1:u¤dS

S(x; t) =©u(x; t); "(x; t); "p(x; t); ¾(x; t)

ª

(un+1;Sn) ! ¾n+1

= F(un+1;Sn)

R(un+1;u¤;Sn) = 0;8u¤ 2 U0ad

19 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Local behaviour integration processLocal behaviour integration process

Generic interface for any constitutive equationGauss point process (where the element variational formulation is integrated)Definition

External parameters (ep) imposed as inputIntegrated variables (vint)Auxiliary variables (vaux) just for outputCoefficients (coef) material parametersPrimal and dual variables prescribed variables and associated fluxes

Primal: strain increment on the current time interval (input to the local integration process)Dual: stress obtained useful for variational formulation (as output of the local integration process)

Local GP integration ODE integration to get

vint, vaux, dual variables

Dual variablesPrimal variables

"p; "

p

¾

F(un+1;Sn)

20 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Local behaviour integration processLocal behaviour integration process

Time integrationNormality rule as an ODE

Explicit integration Runge-Kutta (RK4 usually, beware of stability condition)→

Implicit integration → θ-methods (stable but requires a local Jacobian computation)

_"p = _p3

2¾eq¾D

, "pn+1

= "pn+

Z tn+1

tn

_"p(¿) d¿

21 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Local behaviour integration processLocal behaviour integration process

Elastic prediction and correction algorithmSolution depends on the test (is the evolution purely elastic on ∆t ?)

Elastic prediction

Convexity of the yield surface gives

purely elastic evolution, prediction is correct

Else a plastic evolution is observedA correction must be applied on the elastic prediction

where is the outward unit normal to the final yeld surface

¾eq ¡R(p) · 0

¾en+1

= ¾n+A : ¢"e

n

f(¾en+1

; pn) ¸ f(¾n+1

; pn +¢pn)

f(¾en+1

; pn) · 0

8<:

¾n+1

= ¾en+1

"pn+1

= "pn

pn+1 = pn

¢pn > 0; ¢pn f(¾n+1; pn +¢pn) = 0 ) f(¾n+1

; pn +¢pn) = 0

¾n+1

= ¾en+1

¡A : ¢"pn

¢"pn= ¢pn

r3

2Nn+1

¢pn > 0

Nn+1

22 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Local behaviour integration processLocal behaviour integration process

Elastic prediction and correction algorithmSolution depends on the test (is the evolution purely elastic on ∆t ?)

Elastic prediction

Convexity of the yield surface gives

purely elastic evolution, prediction is correct

Else a plastic evolution is observedA correction must be applied on the elastic prediction

where is the outward unit normal to the final yeld surface

¾eq ¡R(p) · 0

¾en+1

= ¾n+A : ¢"e

n

f(¾en+1

; pn) ¸ f(¾n+1

; pn +¢pn)

f(¾en+1

; pn) · 0

8<:

¾n+1

= ¾en+1

"pn+1

= "pn

pn+1 = pn

¢pn > 0; ¢pn f(¾n+1; pn +¢pn) = 0 ) f(¾n+1

; pn +¢pn) = 0

¾n+1

= ¾en+1

¡A : ¢"pn

¢"pn= ¢pn

r3

2Nn+1

¢pn > 0

Nn+1

23 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Radial return algorithmRadial return algorithm

For von Mises based criteria or isotropic or linear kinematic hardeningThe corrective term is oriented by the final normal that can be computed a priori

For von Mises criterion the final normal is collinear to the elastic predictor deviator

Lamé coefficient

Final state on the yeld surface gives to verify

that allows to find

A : ¢"pn

Nn+1

=¾eDn+1

k¾eDn+1

k ¾n+1

= ¾en+1

¡A : ¢pn

r3

2Nn+1

¹¾eqn+1 = ¾e;eqn+1 ¡ 3¹¢pn

¾e;eqn+1 ¡ 3¹¢pn ¡R(pn +¢pn) = 0

¢pn

f(¾n+1; pn+1) = 0

f(¾n; pn) = 0

¾Dn

¾Dn+1

¾eDn+1

24 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Generalized radial return algorithmGeneralized radial return algorithm

Closest point projection techniqueFor a generalized normality rule

Fluxes

f(¾n+1; pn+1) = 0

f(¾n; pn) = 0

¾Dn

¾Dn+1

¾eDn+1

¢"p =@f

@¾¢p ¢®I =

@f

@YI¢p

¢¾ = A :¡¢"¡¢"P

¢

¢YI =MI ¢®I

¢§ =

·¢¾¢YI

¸=

"A : ¢"

0

#¡"

A 0

0 MI

#"@f@¾@f@YI

#¢p

25 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Generic formulation of the local integration processGeneric formulation of the local integration process

26 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

OutlineOutline

Generalities on computational strategies for nonlinear problemsExamples (contact, crack propagation, non-linear behaviour, geometrical non-linearities)

Classical algorithm for nonlinear or time dependant problems

Local numerical aspects of plasticityElastic-plastic behaviour

Local integration of non-linear models

Global numerical aspects of plasticitySolution process

Consistent tangent matrix

Examples of solution process

Presentation of Z-mat

27 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Global aspectsGlobal aspects

Solving the global discrete problemGlobal problem

with

Global Newton algorithmAt each time step the solution is searched iteratively starting from the last converged solution leading to the following linear system at iteration i

To obtain the residual evaluation, it is necessary to perform the local integration to get

To calculate the global consistent tangent operator, it is necessary to obtain a local consistent tangent operator (that can be kept constant if a modified Newton algorithm with constant operator is involved)

When the residual is small enough the solution is reached

R(un+1;u¤;Sn) = 0;8u¤ 2 U0adZ

­

F(un+1;Sn) : "¤d­¡Z

­

fn+1

:u¤d­¡Z

@F­

Fn+1:u¤dS

R(un+1;u¤;Sn) =

F(uin+1;Sn)

R(uin+1;u¤;Sn) +R0(uin+1;u¤;Sn)[ui+1n+1 ¡ uin+1] = 0; 8u¤ 2 U0ad

R0(uin+1; u¤;Sn) =Z

­

@F@u

(un+1;Sn) : "¤d­°°R(ui+1n+1;u¤;Sn)

°° < ´NL

28 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Local consistent tangent operatorLocal consistent tangent operator

In the local integration process

29 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

OutlineOutline

Generalities on computational strategies for nonlinear problemsExamples (contact, crack propagation, non-linear behaviour, geometrical non-linearities)

Classical algorithm for nonlinear or time dependant problems

Local numerical aspects of plasticityElastic-plastic behaviour

Local integration of non-linear models

Global numerical aspects of plasticitySolution process

Consistent tangent matrix

Examples of solution process

Presentation of Z-mat

30 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Viscoelastic with isotropic hardeningViscoelastic with isotropic hardening

31 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Viscoelastic with isotropic hardeningViscoelastic with isotropic hardeningImplementationImplementation

32 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Viscoelastic with isotropic hardeningViscoelastic with isotropic hardeningImplementationImplementation

33 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Viscoelastic with isotropic hardeningViscoelastic with isotropic hardeningImplementationImplementation

34 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Large strain viscoelastic with isotropic hardening Large strain viscoelastic with isotropic hardening

35 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

System of residualsSystem of residuals

36 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

AlgorithmAlgorithm

37 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Tangent matrixTangent matrix

38 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

OutlineOutline

Generalities on computational strategies for nonlinear problemsExamples (contact, crack propagation, non-linear behaviour, geometrical non-linearities)

Classical algorithm for nonlinear or time dependant problems

Local numerical aspects of plasticityElastic-plastic behaviour

Local integration of non-linear models

Global numerical aspects of plasticitySolution process

Consistent tangent matrix

Examples of solution process

Presentation of Z-mat

39 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Presentation of the Z-mat libraryPresentation of the Z-mat library

40 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Inside Z-matInside Z-mat

41 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Object oriented designObject oriented design

42 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Isotropic and non-linear kinematic modelIsotropic and non-linear kinematic model

43 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Datafile examplesDatafile examples

44 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

ZebFront interface layer for efficientZebFront interface layer for efficientbehaviour developmentbehaviour development

45 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Explicit model implementation - ZebFrontExplicit model implementation - ZebFront

46 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Implicit model implementation - ZebFrontImplicit model implementation - ZebFront

47 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Multimat capabilitiesMultimat capabilities

48 FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

Multimat capabilitiesMultimat capabilities

FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–FE analysis for non linear mechanics Athens MP06 - V.Chiaruttini–

ReferencesReferences

Bonnet M., Frangi A. (2006)Analyse des solides déformables par la méthode des éléments finis. Editions Ecole Polytechnique.Belytschko, T., Liu, W., and Moran, B. (2000).Nonlinear Finite Elements for Continua and Structures.Besson, J., Cailletaud, G., Chaboche, J.-L., and Forest, S. (2001).Mecanique non linéaire des matériaux. Hermes.–Ciarlet, P. and Lions, J. (1995).Handbook of Numerical Analysis : Finite Element Methods (P.1), Numerical Methods for Solids (P.2). North Holland.Dhatt, G. and Touzot, G. (1981).Une présentation de la méthode des élements finis. Maloine.Hughes, T. (1987).The finite element method: Linear static and dynamic finite element analysis. Prentice Hall –Inc.Simo, J. and Hughes, T. (1997).Computational Inelasticity. Springer Verlag.Zienkiewicz, O. and Taylor, R. (2000).The finite element method, Vol. I-III (Vol.1: The Basis, Vol.2: Solid Mechanics, Vol. 3: Fluid dynamics). Butterworth Heinemann.–