nsf workshop on the emerging applications and future ... · structural integrity of advanced...

A. Sáez

UNIVERSIDAD DE SEVILLA

NSF Workshop On the Emerging Applications and Future Directions of the BEM. Akron, Sep. 2010

BEM for modeling fracture mechanics problems 1

BEM for modeling fracture mechanics problems

Andrés Sáez

University of Seville (Spain)

Collaborators: J. Domínguez, Ch. Zhang, F. García-Sánchez, R. Rojas-Díaz, M. Wünsche

A. Sáez




Motivation:

Structural Integrity of Advanced Materials

Design of structural materials demands for high performance applications:we want lighter materials and new capacities

Tailored materials with good strength/weight ratio → composites

Multifield materials for control and smart structures applications →piezoelectricity…

Damage Tolerance Philosophy

Existence of a flaw (crack…) in amechanical component will noimply the end of its service life →need to evaluate its performanceand reliability

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Tools to clearly characterize how a damaged structural element behavesare thus needed in order to predicts its:

working conditions &

remaining service life

The AIM is to extend to this new class of materials the damage estimation techniques previously developed for classic materials

WE’LL TACKLE THE PROBLEM FROM A NUMERICAL POINT OF VIEW (BEM)


Motivation:


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Outline

Fracture mechanics of advanced materials:

Anisotropic materials (composites)

Multifield materials:

Piezoelectric

Magnetoelectroelastic

Why BEM for fracture?

BEM strategies for fracture mechanics

Dual or Hypersingular BEM

Statics

Dynamics

Concluding remarks

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Fracture mechanics of anisotropic materials

• Anisotropic materials in nature: Zinc, magnesium, wood, ice…

• Engineered materials: composites

2-D anisotropic behavior law

12

22

11

662616

262212

161211

12

22

11

2

aaa

aaa

aaa

http://www.km77.com/marcas/volkswagen/2005/prototipos/ecoracer/gra/05.asp

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The existence of a crack will alter the stress fields and may lead to failurefor loads much lower than the design load bearing capacity of thecomponent

• MICRO-scale

• MESO-scale

•MACRO-scale This talk

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rCfKij

/),(

rCgKuij),(

Displacement and stress fields around the crack tip(1):

Stress Intensity Factors (SIF)

(1) Sih, Paris e Irwin (1965)

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When tackling the problem numerically, the selected method should fulfill some requirements:

Adequate representation of the mechanical fields around the crack

SIF (or other fracture parameters) accurate computation

Easy crack discretization

rCfKij

/),(

rCgKuij),(

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Fracture mechanics of multifield materials

Ability to convert energy among mechanical and non-mechanical fields:

• Mechanical (elastic)

• Electric

• Magnetic Elastomagnetic

Piezoelectric (PE)

Magnetoelectroelastic (MEE)

Terfenol-D

layers PZT

Magnetic field

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Piezoelectric Materiales

Direct piezoelectric effect (1): a voltage is produced when the material is under tension or compression stress.

Inverse piezoelectric effect: when a potential difference is applied across the crystal it causes its deformation.

(1) Jacques y Pierre Curie, 1880

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PE materials

Are always anisotropic

Show larger PE coupling for artificialmaterials (e.g., PZT ceramics)

PZT unit cell: 1. PZT unit cell in the symmetric cubic state

above the Curie temperature.

2. Tetragonally distorted unit cell below theCurie temperature. Electric dipoles in domains:

1. unpoled ferroelectric ceramic2. during and

3. after poling (piezoelectric ceramic).

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Applications for PE materials:

sensors

actuators

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Applications for PE materials: Smart structures

Actuator Layer

Flexible Beam

Sensor Layer

Actuator Input

Sensor Output

Disturbance

GainControl

MEMS

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How do we model PE materials?:

Extended notation for piezoelectricity (Barnett & Lothe, 1975)

4

3,2,1

I

Iuu

i

I

Extended displacements vector

4

3,2,1

I

I

Di

ij

iJ

Extended stress tensor

PE behavior law

elastic, piezoelectric & dielectric constants

j

lmml

i

ijuu

D ,

,,

iJLmC

ijke ij

ijlmC kjie 0, iiJ

Electric potential Electric displacement

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Magnetoelectroelastic materials

d31

mode

+

-

Terfenol-D

Induced Strian

Terfenol-D

Induced Strian

3

Electrode

Electrode

Terfenol-D

E ~ 100GPa

Terfenol-D

E ~ 100GPa

PZT E ~ 90GPa

Composite materials consisting of both piezoelectric andpiezomagnetic phases may exhibit a magnetoelectric

coupling effect that is not shown by any of the material

phases alone. The magnetoelectric coupling of the resultingcomposite may be much larger that that of a single phase

magnetoelectric material (Van Suchtelen, 1972; Nan, 1994;Benveniste, 1995).

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Magnetoelectroelastic materials may be formulated in an elastic-likefashion by using the followinggeneralized notation:

5

4

3,2,1

I

I

Iu

u

i

I

Extended displacement vector

5

4

3,2,1

IB

ID

I

i

i

ij

iJ

Extended stress tensor

Electric potential Electric displacement

Magnetic potential Magnetic induction

Modeling MEE materials

Constitutive equations:

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Why study fracture in multifield materials?:

• structural integrity

• cracks alter the electric/magnetic reading: may lead to adopt wrong

decissions (the structure is no longer smart)

• the electric and magnetic fields have influence on the crack growth

processActuator Layer

Flexible Beam

Sensor Layer

Actuator Input

Sensor Output

Disturbance

GainControl

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Fracture mechanics of multifield materials

Fracture of MEE materialsSong, Z.F., Sih, G.C., Crack initiation behavior in

magnetoelectroelastic composite under in-plane deformation,

Theor. and Appl. Fract. Mechanics, 39 (2003), pp. 189-207.

Wang, B.L., Mai, Y.-W., Crack tip field in

piezoelectric/piezomagnetic media, European Journal of

Mechanics, A/Solids, 22 (2003), pp. 591-602.

rCfKij

/),(

rCgKuij),(

Generalized intensity factors:• stress intensity factors (SIF): KI and KII

• electric displacement intensity factor (EDIF): KD

• magnetic induction intensity factor (MIIF): KB

Fracture of PE materialsParton, V.Z., Fracture mechanics of piezoelectric materials,

Acta Astronautica, 3 (1976), pp. 671-683.

Pak, Y.E., Crack extension force in a piezoelectric material,

Journal of Applied Mechanics, Transactions ASME, 57

(1990), pp. 647-653.

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Why BEM for fracture?

Is BEM able to answer properly the questions we previously raised?

When tackling the problem numerically, the selected method should fulfill some requirements:

Adequate representation of the field variables around the crack

SIF (or other fracture parameters) accurate computation

Easy crack discretization

rCfKij

/),(

rCgKuij),(

The answer is YEAH !!!!

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Some BEM advantages

• Automatic satisfaction of the radiation conditions at infinity

• Ability to capture high stress gradients

• Easy implementation of elements modeling crack-tip fields in

fracture

• The mesh is reduced in one dimension

Some disadvantages

• (Need appropriate fundamental solution)

• Singular integrations

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BEM strategies for fracture mechanics

0. Use tailored fundamental solutionsC

+

-1. Subregion technique

SR 1

SR 2

SR 1

dpuxdxuxpuc JIJJIJJIJ

** )()(),()(

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** )()(),()(

dpdNdusNpc JrIJrJrIJrJIJ

**

C

+

-

Displacement Boundary Integral Equation (BIE)

Traction BIE

iJLrrCN

2. Dual or hypersingular BEM

r

)(xe

x

Collocation point

Observation point

)(ln* ru

)/1(* rp

)/1(* rd

)/1(* 2rs Extended C.O.D. u

0

0

0

B

D

p

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Dual or hypersingular BEM

Meshing strategy?

Availability of reasonable fundamental solutions for statics?

Singular and hypersingular integrations?

Computation of fracture parameters?

What about dynamics?


** )()(),()(

dpdNdusNpc JrIJrJrIJrJIJ

**

Inv - 24

1 11

1C 3C2C

N3= 1

N1= -1

N2=NC2= 0

= 1 = 3

NC3NC1

L/2L/2

N1=NC1 N2=NC2 N3=NC3= 0 = 1= -1

discontinuous quarter-point (+)

L/4 3L/4vértice

degrieta

semidiscontinuous EID (c)

N1 N2=NC2 N3NC1 NC3

= 0 = 1= -1 = 1 = 3

N1 N2=NC2NC1 N3=NC3

Meshing strategy

crack

continuous DBIE (c)

Traction BIE (TBIE) → displacement field C 1

discontinuous TBIE (+)

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5

1

* )-ln(ReR

R

x

RRIJR

S

IJ zzHAu

Observation point : ),( 2121 ξRRz

),( 2121 xxxxz R

x

R xCollocation point x:

Anisotropic (elastic): Eshelby et al. (1953)

Piezoelectric: Barnett & Lothe (1975)

MEE: Liu et al. (2001)

Static fundamental solution

5

1

21*

)-(Re

R R

x

R

RRIJR

S

IJzz

nnHLp

)()()( **xxξ

dpudupuc JIJJIJJIJ

2

21*

)-(Re

R

x

R

RS

rIJzz

nns

)-(Re 21**

R

x

R

Rr

S

rIJ

S

IJzz

NNNdd

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Evaluation of Strongly Singular and Hypersingular Integrals

122

2

1

1

nnd

dx

dx

d

d

dx

dx

d

d

dR

RRR

pIJ x, 1

LJMQMIM n1 n2

zMx zM

#

Regularization technique follows the works by García-Sánchez et al. (EABE2004,C&S2005, TAFM2008). This approach is valid for any of the 2-D fundamentalsolutions derived by Stroh’s formalism and has no restrictions on the type/shapeof the boundary elements:

)()( 2211 xxzz RRxRR

CHANGE OF VARIABLES

THE JACOBIAN IS BUILT-IN THE FUNDAMENTAL SOLUTION

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For instance, for the hypersingular integrations:

122

2

1

1

nnd

dx

dx

d

d

dx

dx

d

d

dR

RRR

Rqq

R

R

q

RqRq

R

d

d

00

0

'

)0()(

R

R

qR

R

qRRqqq

R

R dddIeee

1

'1

)'(1

'' 020002

Regular Analytic Analytic

dnnI q

R

RRe

221

1)(''

2

*

)(

1

zz

sxrIJ

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Evaluation of fracture parameters by extrapolation:

tip

crack

x2

x1

r

rμθ,f *

K→Δu Ru

at NC3

L/64r

πθ

L

49L/64

L/4L/64

N1 NC1 N2 =NC2 NC3 N3

z=-3/4 z=-1z=3/4z=1 z=0

Cracktip

1-L

r2z

KHKTG

2

1

rμθ,f *

K→Δu Ru

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Some results: BEM vs FEM & Experimental Results. Composite material.

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FEM Mesh(classic FEM, not X-FEM)

BEM Mesh

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MEF

Some results: BEM vs FEM & Analytic. Piezoelectric material.

BEM

error < 0.1 %

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Some results: BEM vs Analytic. MEE material.

BaTiO3

CoFe2O4

50%

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What about dynamics?

Anisotropic: Wang & Achenbach (1995)

Piezoelectric: Denda et al. (2004)

MEE: Rojas-Díaz et al. (2008)

For instance, in the frequency domain, the

fundamental solution is obtained as:

),,(),(),,( *** ξxξxξxR

IJS

IJIJ uuu

1

m2

m

m

IJ

2

*

IJ )(dS )-()(kc8

1),,(u

η

ηξx·ηξx

regular: ),,(u R*

IJ ξx

1

m

qq

2

m

m

IJ

2)(dS)-·(ln2

Ec8

1

η

ηξxη

1

mm

qq

2

m

m

IJ

2)(dS)-·(ln2)-()(k

Ec8

1

η

ηξxηξx·η

),(u S*

IJ ξxsingular:(statics)

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x

y

R

rr

D/B

D/B

rr

D(t)=D0H(t)

22(t)=0 H(t)

t

B(t)=B0H(t)

10 discontinuous quadratic elements for the crack (8 + 2 QP)

Some dynamic results: Combined magneto-electro-mechanical impacts.

Curved crack in infinite MEE domain

BaTiO3

CoFe2O4

50%

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Ux Uy

uy.ppt

ux.ppt

E.ppt

H.ppt

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D(t)=D0H(t)

22(t)=0 H(t)

t

B(t)=B0H(t)

Some dynamic results: Finite cracked MEE plate under combined

magneto-electro-mechanical impacts.

BaTiO3

CoFe2O4

50%

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Concluding remarks

BEM works for fracture applications!!,leading to accurate evaluation of therelevant fracture parameters

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Some additional issues

Realistic boundary conditions

Include other relevant variables (T…) and material nonlinearities

Develop adequate fracture criteria for multifield materials

Improve fundamental solutions

and much more…

A. Sáez




Thanks for your attention!!

[email protected]

A. Sáez




Some references of our group’s work:

Wünsche, M., García-Sánchez, F., Sáez, A., Zhang, Ch.

A 2D time-domain collocation-Galerkin BEM for dynamic crack analysis in piezoelectric solids

(2010) Engineering Analysis with Boundary Elements, 34 (4), pp. 377-387.

Rojas-Díaz, R., García-Sánchez, F., Sáez, A.

Analysis of cracked magnetoelectroelastic composites under time-harmonic loading

(2010) International Journal of Solids and Structures, 47 (1), pp. 71-80.

Rojas-Díaz, R., García-Sánchez, F., Sáez, A., Zhang, Ch.

Dynamic crack interactions in magnetoelectroelastic composite materials

(2009) International Journal of Fracture, 157 (1-2), pp. 119-130.

García-Sánchez, F., Zhang, C., Sáez, A.

2-D transient dynamic analysis of cracked piezoelectric solids by a time-domain BEM

(2008) Computer Methods in Applied Mechanics and Engineering, 197 (33-40), pp. 3108-3121.

Rojas-Díaz, R., Sáez, A., García-Sánchez, F., Zhang, C.

Time-harmonic Green's functions for anisotropic magnetoelectroelasticity

(2008) International Journal of Solids and Structures, 45 (1), pp. 144-158.

García-Sánchez, F., Rojas-Díaz, R., Sáez, A., Zhang, Ch.

Fracture of magnetoelectroelastic composite materials using boundary element method (BEM)

(2007) Theoretical and Applied Fracture Mechanics, 47 (3), pp. 192-204.

Sáez, A., García-Sánchez, F., Domínguez, J.

Hypersingular BEM for dynamic fracture in 2-D piezoelectric solids(2006) Computer Methods in Applied Mechanics and Engineering, 196 (1-3), pp. 235-246.

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Some references of our group’s work:

García-Sánchez, F., Sáez, A., Domínguez, J.

Two-dimensional time-harmonic BEM for cracked anisotropic solids


García-Sánchez, F., Sáez, A., Domínguez, J.

Anisotropic and piezoelectric materials fracture analysis by BEM(2005) Computers and Structures, 83 (10-11 SPEC. ISS.), pp. 804-820.

García, F., Sáez, A., Domínguez, J.

Traction boundary elements for cracks in anisotropic solids


Sáez, A., Domínguez, J.

Dynamic crack problems in three-dimensional transversely isotropic solids


Sáez, A., Domínguez, J.

Far field dynamic Green's functions for BEM in transversely isotropic solids

(2000) Wave Motion, 32 (2), pp. 113-123.

Sáez, A., Ariza, M.P., Domínguez, J.Three-dimensional fracture analysis in transversely isotropic solids


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