Chapter V Extension to Linear Elastic Fracture Mechanics

Chapter V  Page 109 

Chapter V

Extension to Linear Elastic Fracture Mechanics

Content

Summary 110

V.1 Introduction                     112

V.2. State of the art                     113

V.2.1. Finite element method 113

V.2.2. Boundary element method 113

V.2.3. Meshless method 113

V.2.4. Extended finite element method 114

V.3. Domain decomposition                   114

V.4. Discretization                     116

V.5. Solution of the equation system 124

V.6. Applications 127

V.6.1. Patch tests 127

V.6.2. Translation tests 132

V.6.3. Mode 1 tests 135

V.6.4. Mode 2 tests 137

V.6.5. Bar with a single edge crack 139

V.6.6. Nearly incompressible material 141

V.7. Remark 141

V.8. Conclusion 142

Chapter V  Page 110 

Summary

In this chapter, the natural neighbours method is extended to the domain of 2D Linear Elastic Fracture Mechanics (LEFM) using an approach based on the FRAEIJS de VEUBEKE variational principle which allows independent assumptions on the displacement field, the stresses filed, the strain field and the surface distributed support reactions field.

The 2D domain contains N nodes (including nodes on the domain contour and nodes located at each crack tip) and the N Voronoi cells corresponding these nodes are built.

The cells corresponding to the crack tip nodes are called Linear Fracture Mechanics Voronoi Cells (LFMVC). The other ones are called Ordinary Cells (OVC).

A node is located at the crack tip and, in the LFMVC containing this node, the stress and the strain discretizations include not only a constant term but also additional terms corresponding to the solutions of LEFM for modes 1 and 2 [WESTERGAARD, H.M. (1939)].

The following discretization hypotheses are admitted:

1. The assumed displacements are interpolated between the nodes with the Laplace interpolation function.

2. The assumed support reactions are constant over each edge K of Voronoi cells on which displacements are imposed

3. The assumed stresses are constant over each OVC 4. The assumed strains are constant over each OVC

5. The assumed stresses are assumed to have the distribution corresponding to modes 1 and 2 in each LFMVC

6. The assumed strains are assumed to have the distribution corresponding to modes 1 and 2 in each LFMVC

Introducing these hypotheses in the FdV variational principle produces the set of equations governing the discretized solid.

In this approach, the stress intensity coefficients are obtained as primary variables of the solution.

These equations are recast in matrix form and it is shown that the discretization parameters associated with the assumptions on the stresses and on the strains can be eliminated at the Voronoi cell level so that the final system of equations only involves the nodal displacements, the assumed support reactions and the stress intensity coefficients.

These support reactions can be further eliminated from the equation system if the imposed support conditions only involve displacements imposed as constant (in particular displacements imposed to zero) on a part of the solid contour.

The present approach has also the following properties.

In the OVCs

1. In the absence of body forces, the calculation of integrals over the area of the domain is avoided: only integrations on the edges of the Voronoi cells are required.

2. The derivatives of the Laplace interpolation functions are not required.

Chapter V  Page 111  In the LFMVCs,

1. Some integrations on the area of the LFMVCs are required but they can be calculated analytically.

2. The other integrals are integrals on the edges of the LFMVCs

3. The derivatives of the Laplace interpolation functions are not required Several applications are used to evaluate the method.

Patch test, translation test, mode 1 tests, mode 2 tests and single edge crack test confirm the

validity of this approach.

Chapter V  Page 112 

V.1. Introduction

In chapter III, starting from the Fraeijs de Veubeke (FdV) variational principle, a new approach of the natural neighbours method has been developed for problems of linear elasticity.

It has been extended, in chapter IV, to materially non linear problems.

In the absence of body forces ( F _i = 0 ), it has been shown that for both cases:

• the calculation of integrals of the type dA

A ∫ ^• can be avoided; only numerical integrations on the edges of the Voronoi cells are performed;

• the derivatives of the Laplace interpolation functions are unnecessary;

• boundary conditions of the type u _i = u~ _i on S _u can be imposed in the average sense in general and exactly if u~ _i is linear between two contour nodes, which is obviously the case for u~ _i = 0 ;

• incompressibility locking is avoided.

The motivation of the present study is to explore if this approach can be extended to solve problems of Linear Elastic Fracture Mechanics (LEFM).

Indeed, in Linear Elastic Fracture Mechanics, the stress intensity factors play a central role and, since the FdV variational principle allows discretizing the stresses and the strains independently of the assumptions on the displacement field, it seems logical and elegant to take advantage of this flexibility and to introduce, near the crack tips, stress and strain assumptions inspired from the analytical solution provided by the Theory of Elasticity [WESTERGAARD, H.M. (1939)].

However, the properties that the numerical calculation of integrals of the type dA

A ∫ ^{• could be}

avoided, that the derivatives of the Laplace interpolation functions were unnecessary and that incompressibility locking was also avoided resulted from the assumption of a constant stress field and a constant strain field in each of the Voronoi cells.

It is not obvious that these properties still hold with the non constant assumptions on the stresses and the strains near the crack tips.

These points are carefully examined in the present chapter.

On the other hand, thanks to the stress and strain assumptions used near the crack tips, the stress intensity factors corresponding to fracture modes 1 and 2 become primary variables of the numerical solution which is normally an advantage. However, it must be verified that good precision and convergence are obtained.

This is also considered in the present chapter.

It is worth recalling that the basic notions of LEFM used in the present chapter were briefly

introduced in chapter II, section II.4.

Chapter V  Page 113 

V.2. State of the art

There are many numerical approaches currently available to solve problems of LEFM and, in particular, to calculate the stress intensity factors.

Research in this domain has produced a very large number of papers and it would be extremely difficult, and perhaps useless in the frame of this thesis, to extensively review the literature on this subject.

Therefore, the following state of the art only briefly mentions some of the basic references and proposes a classification of the methods in 4 categories.

V.2.1. Finite element method

Usually, displacement-type finite elements (based on the virtual work principle) are used.

Two main categories of approaches are used [INGRAFFEA A.R. and WAWRZYNEK P.

(2003)]

In the “direct approach”, the stress intensity factors are deduced from the displacement field.

This is the case of the Crack Opening Displacement method [CHAN S.K. et al. (1970)].

In some cases, the formulation of finite elements at the crack tip can be adapted to improve the displacement field [BARSOUM R.S. (1977)].

In the “energy approach” which is generally more precise, the stress intensity factors are deduced from the energy distribution in the vicinity of the crack tip, either from the energy release as in the method of the Virtual Crack Extension [HELLEN T.K. and BLACKBURN W.S. (1975)] or from the J-integral as in the Equivalent Domain Integral Method [MORAN B. and SHIH C.F. (1987-a) & (1987-b)].

V.2.2. Boundary elements method

In this method, only the boundaries of the solid are discretized. The partial differential equations of the Theory of Elasticity are transformed into integral equations on the boundaries of the domain. Basically, the primary unknowns of the numerical problem remain the displacements.

This is the case for the “crack Green’s function method” [SNYDER M.D. and CRUSE T.A.

(1975)], the “displacement discontinuity method” [CROUCH S.L. (1976)] and the

“subregions method” [BLANFORD G.E. et al. (1991)]

But a dual method using also the surface tractions as primary unknowns has also been developed [PORTELA A. et al.(1991)].

V.2.3. Meshless method

This method has been applied to fracture mechanics since 1994 [BELYTCHKO T. et al.

(1994-b)] and, subsequently, different improvements have been introduced, for example to couple it with the finite element method [BELYTCHKO T. et al. (1996-b)], to ensure the continuity of displacements in the vicinity of the crack [ORGAN D. et al.(1996)], to improve the representation of the singularity at the crack tip [ FLEMING M. et al. (1997)], to use an arbitrary Lagrangian-Eulerian formulation [PONTHOT J.P. and BELYTCHKO T. (1998)], to enrich the displacement approximation near the crack tip [LEE S.H. and YOON Y.C. (2003)]

or to enrich the weighting functions [DUFLOT M. and NGUYEN D.H. (2004-a) and (2004- b)].

These approaches share the usual advantages and drawbacks of the meshless methods already

mentioned in chapter II, section II.2.

Chapter V  Page 114  V.2.4. Extended finite element methods

The extended finite element method (XFEM) [MOES N. et al. (1999)] allows discontinuities in the assumed displacement field. The discontinuities can be due to the presence of cracks and do not have to coincide with the finite element edges: they can be located anywhere in the domain independently of the finite element mesh.

Since chapter VI of this thesis will deal with a an extended natural neighbours method (XNEM) which is somehow inspired from the XFEM, we will not go further in the presentation of the XFEM here and come back on this subject in chapter VI.

V.3. Domain decomposition

In the natural neighbours method, the domain contains N nodes and the N Voronoi cells corresponding to these nodes are built.

At each crack tip, a node is located (figure V.1).

The cells corresponding to the crack tip nodes are called Linear Fracture Mechanics Voronoi Cells (LFMVC). The other ones are called Ordinary Voronoi Cells (OVC).

Figure V.1. A Linear Elastic Fracture Mechanics Voronoi cell (LFMVC) at a crack tip.

X

2

X

1

α

Y

1

Y

2

LFMVC

Chapter V  Page 115  The area of the domain is:

∑

=

= ^N

I

A I

A

1

(V.1) with A _I the area of Voronoi cell I.

We denote C _I the contour of Voronoi cell I.

We start from the FdV variational principle introduced in chapter II and we recall its expression for completeness (we keep the equation numbers of chapter II).

dA X )

u X ( u dA

u ) X F

( dA ) (

A

i ij j j ij i

i A

j i ji A

ij ij

ij ∫ ∫

∫ _⎥ ^⎥

⎦

⎤

⎢ ⎢

⎣

⎡ − ε

∂ + ∂

∂ Σ ∂

δ + δ

∂ + Σ

− ∂ δε Σ

− σ

= Π

δ 2

1

∫

∫ ^{Σ − δ + Σ − δ}

+

t

u

S

i i ji j S

i i ji

j r ) u dS ( N T ) u dS

N

( =0 (II.36)

or

6

0

5 4 3 2

1

+ Π + Π + Π + Π + Π =

Π

=

Π δ δ δ δ δ δ

δ (II.31)

with the different terms

= ∫

= ∫

A ij ij A δ W ( ε ij ) dA σ δε dA Π

δ ₁ (II.25)

dA X )

u X

( u

A ij

i j j

ij i

∫ ⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ −

∂ + ∂

∂

= ∂ δ δ δε

2 Σ 1 Π

δ

₂

(II.26)

dA X )

u X ( u

A ij

i j j ij i

∫ ⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ −

∂ + ∂

∂

= ∂ ε

2 Σ 1 δ Π

δ ₃ (II.27)

dA u F

A

∫

i i

−

= δ

Π

δ

₄

(II.28)

− ∫

=

St

T

i

δ u

i

dS Π

δ

₅

(II.29)

dS u r dS ) u u~

(

r _i

S i S

i i i

u u

δ

−

− δ

= Π

δ ⁶ ∫ ∫ (II.30)

For the linear elastic case, the stresses are given by:

ij ij

ij

ε

ε σ W

∂

= ∂ ( )

(II.34)

and

kl ij ijkl

ij

C ε ε

ε

W 2

) 1

( = (II.35)

where C

ijkl

is the classical Hooke’s tensor.

In (II.29) and (II.30), the integrals are computed along the domain contour. This contour is the

union of some of the edges of the exterior Voronoi cells. These edges are denoted by S _K and

we have

Chapter V  Page 116 

∑

=

= ^M

K

S K

S

1

; ∑

=

= ^M

^u

K K

u S

S

1

; ∑

=

= ^M

^t

K K

t S

S

1

; S = S _u ∪ S _t ⇒ M = M _u + M _t (V.2) where M is the number of edges composing the contour, M

u

the number of edges on which displacements u~ _i are imposed and M

t

the number of edges on which surface tractions T _i are imposed.

Using the above domain decomposition, these terms become:

∑ ∫

=

=

Π

^N

I A

I ij ij

I

dA

1

1

σ δε

δ (V. 3)

∑ ∫ ⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ −

∂ + ∂

∂

= ∂

= N

I I

A ij

i j j

ij i ) δε dA

X u δ X

u ( δ δ

1

I

2 2

Σ 1

Π (V. 4)

∑ ∫ ⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ −

∂ + ∂

∂

= ∂

= N

I I

A ij

i j j

ij i ) ε dA

X u X

( u δ

δ

1

I

3 2

Σ 1

Π (V. 5)

∑ ∫

=

−

=

Π

^N

I

I A

i

i

u dA

F

1 I

4

δ

δ (V. 6)

∑ ∫

−

=

= _K

t

S i i K

M

K

T δ u dS

Π δ

5 1

(V. 7)

∑ ∫ ∫

=

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ δ − − δ

= Π

δ

^u

K K

M K

K i S

i S

K i i

i ( u~ u ) dS r u dS

r

1

6 (V. 8)

V.4. Discretization

We make the following discretization hypotheses in the OVCs:

1. The assumed strains ε

_ij

are constant over each Voronoi cell I:

I ij

ij

ε

ε = (V. 9)

2. The asumed stresses Σ

_ij

are constant over each Voronoi cell I:

I ij

ij

= Σ

Σ (V.10)

3. The assumed support reactions r

_i

are constant over each edge K of Voronoi cells on which displacements are imposed:

K i

i

r

r = (V. 11)

4. The assumed displacements u

_i

are interpolated by Laplace interpolation functions:

∑

=

Φ

=

^N

J

J i J

i

u

u

1

(V. 12)

where u

_i^J

is the displacement of node J (corresponding to the Voronoi cell J).

For the LFMVCs we use the results of Linear Elastic Fracture Mechanics.

Chapter V  Page 117  The stress fields near the crack tip (figure V.2, r <<< ) are given in the local reference system

) ,

( Y

₁

Y

₂

by Westergaard [WESTERGAARD, H.M. (1939)].

for mode 1:

⎪ ⎪

⎪

⎭

⎪⎪

⎪

⎬

⎫

⎪ ⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

θ θ

θ + θ

θ

− θ θ

= π

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧ τ τ τ

2 3 2

2 3 1 2

2 3 1 2

2 2

1 12

22 11

cos sin

sin sin

sin sin r cos

K (V.13)

for mode 2:

⎪ ⎪

⎪

⎭

⎪⎪

⎪

⎬

⎫

⎪ ⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

θ

− θ θ

θ θ

θ

θ + θ

− θ

= π

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧ τ τ τ

) sin sin ( cos

cos sin cos

) cos cos (

sin

r K

2 3 1 2

2

2 3 2 2

2 3 2 2

2 2

2 12

22 11

(V.14)

K 1 and K ₂ are the stress intensity factors of modes 1 and 2 respectively.

In the simple loading cases of figure V.2, they are given by:

a

K ₁ = σ

_∞

π (V.15)

a

K ₂ = τ

_∞

π (V.16)

These stresses are valid for both plane stress and plane strain conditions.

a. Mode 1: simple tension b. Mode 2: pure shear Figure V.2. Modes 1 and 2 of fracture mechanics.

Crack tip coordinates (one quarter of the plate represented) σ ∞

Y

1

Y

2

a r

θ

τ ∞

a r

θ Y

2

Y

1

Chapter V  Page 118  The LFMVCs are discretized as follows:

The assumed displacements v

_i

are interpolated in the local reference system ( Y

₁

, Y

₂

) by the Laplace interpolant as in the OVCs:

∑

=

Φ

= ^N

J

i J J

i v

v

1

(V.17) where v ₁ , v ₂ are the components of the displacement at a point in the LFMVC with respect to

) Y , Y

( _{1 2} .

The assumed stresses are interpolated in the local reference system ( Y ₁ , Y ₂ ) shown on figure V.1. by:

{ } { }

[ ] { } { } { } ^{O F}

O O O

P P

P )

( g K ) ( g K ) r ( f P P P

P P P

+

=

⇔ θ

+ θ +

⎪ ⎪

⎭

⎪⎪ ⎬

⎫

⎪ ⎪

⎩

⎪⎪ ⎨

⎧

⎪ =

⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

Σ

Σ

_{1 1 2 2}

12 22 11

12 22 11

(V.18)

with

{ } { } 1 2 { } ²

1

Σ

Σ Σ

Σ

+

= K H K H

P ^F (V.19)

{ } ^{f ( r ) { g ( ) }}

H H H

H = θ

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

=

Σ Σ Σ

Σ

1

12 1 22 1 11 1

1 (V.20)

{ } ^{f ( r ) { g ( ) }}

H H H

H = θ

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

=

Σ Σ Σ

Σ

2

12 2 22 2 11 2

2 (V.21)

{ }

2 3 2

2 3 1 2

2 3 1 2

1 2

⎪ ⎪

⎪

⎭

⎪⎪

⎪

⎬

⎫

⎪ ⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

θ θ

θ + θ

θ

− θ

= θ θ

cos sin

sin sin

sin sin cos

) (

g (V.22)

{ }

⎪ ⎪

⎪

⎭

⎪⎪

⎪

⎬

⎫

⎪ ⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

θ

− θ θ

θ θ

θ

θ + θ

− θ

= θ

) sin sin ( cos

cos sin

cos

) cos cos

( sin )

( g

2 3 1 2

2

2 3 2 2

2 3 2 2

2

2 (V.23)

) r r (

f = π

2

1 (V.24)

Chapter V  Page 119 

Σ

1

K and K

_Σ

₂ can be considered as generalized stresses (or stress parameters) associated with the considered LFMVC.

The assumed strains are interpolated in the local reference system ( Y ₁ , Y ₂ ) by:

{ } { } ^{O [ ]} { } ^F

F F F

*

*

*

* O

O O

P D P

P P )

E ( ⇔ γ = γ +

⎪ ⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

⎥ ⎥

⎥

⎦

⎤

⎢ ⎢

⎢

⎣

⎡

ν + ν

−

ν

− +

⎪ ⎪

⎭

⎪⎪ ⎬

⎫

⎪ ⎪

⎩

⎪⎪ ⎨

⎧

γ γ γ

⎪ =

⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧ γ γ γ

1

2 0 0

0 1

0 1 1

2 2 ₁₂

22 11

12 22 11

12 22 11

(V.25)

with

state stress plane in

state strain plane in 1 ²

⎪ ⎪

⎩

⎪ ⎪

⎨

⎧ ν

= − E

E

E ^*

state stress plane in

state strain plane in 1

⎪ ⎪

⎩

⎪ ⎪

⎨

⎧

ν ν

− ν

* = v

The assumed support tractions r _i are constant over each edge K of Voronoi cells on which displacements are imposed:

i K

i r

r = (V.26)

Usually, the lips of the fracture are not submitted to imposed displacements. In such a case, (V.26) is useless in a LFMVC.

The assumption (V.25) implies a priori that, in the LFMVC, σ _ij = Σ _ij . In a given problem, there could be N _C crack tips.

For the sake of simplicity, the LFMVCs will be numbered from 1 to N _C and the OVCs are numbered from N _C + 1 to N .

The introduction of these assumptions in the FdV variational principle (II.31) leads to the equations of table V.2 with the notations of table V.1.

The details of the calculations are given in annex 2.

Table V.1. Matrix notations for the Linear Elastic Fracture Mechanics case

Notations and symbols Comments

[ ] [ ]

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧

⎪⎭ =

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

α α

−

α

= α

2 1 2

1

u R u v

; v cos sin

sin

R ^I cos ^I _I ^I _I ^I Rotation matrix between

) X , X

( _{1 2} and ( Y ₁ , Y ₂ ) ^I (figure 1)

{ }

⎪ ⎪

⎭

⎪⎪ ⎬

⎫

⎪ ⎪

⎩

⎪⎪ ⎨

⎧

ε ε ε

= ε

I I

I I

12 22 11

2

I I I I

12 22

11 ε 2ε

ε

=

ε Strains in OVC

(I = N _C + 1 , N )

Chapter V  Page 120  { }

⎪ ⎪

⎭

⎪⎪ ⎬

⎫

⎪ ⎪

⎩

⎪⎪ ⎨

⎧

γ γ

γ

= γ

I I I I

12 22 11

2

I I I I

12 22

11 γ 2γ

γ

= γ

Strains in LFMVC

( I = 1 , N _C ) with respect to its local reference system ( Y ₁ , Y ₂ ) ^I

{ }

⎪ ⎪

⎭

⎪⎪ ⎬

⎫

⎪ ⎪

⎩

⎪⎪ ⎨

⎧

σ σ σ

= σ

I I I I

12 22 11

I I I I

12 22

11 σ 2σ

σ

=

σ Stresses in OVC

(I = N _C + 1 , N )

{ }

⎪ ⎪

⎭

⎪⎪ ⎬

⎫

⎪ ⎪

⎩

⎪⎪ ⎨

⎧

= I I I I

P P P P

12 22 11

I I

I P I P P

P = _{11 22} 2 ₁₂

Stresses in LFMVC , ( I = 1 , N _C ) with respect to its local

reference system ( Y ₁ , Y ₂ ) ^I

{ } ⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

= ⎧

Σ

Σ Σ

I

I I K K K

2 1

I

I K I K

K

_Σ

=

_Σ

₁

_Σ

₂

Generalized stresses or stress parameters in the LFMVC

( I = 1 , N _C ) { } ⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

= ⎧ _I ^I I

u u u

2 1

I

I u I u

u = _{1 2}

Displacements of node I belonging to cell I

(I = N _C + 1 , N )

{ } ⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

= ⎧ _I ^I I

v v v

2 1

I

I v I v

v = _{1 2}

Displacements of node I belonging to cell I

( I = 1 , N _C ) with respect to the local reference system ( Y ₁ , Y ₂ ) ^I A I ; C _I Area and contour of cell I

) I (

K Edge K of cell I

Φ J Laplace function (J=1,N)

I A

J IJ i

i F dA

F ~

I

∫ ^Φ

= ; { }

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

= ⎧ _IJ IJ IJ

F ~ F ~ F ~

2 1

{ } ∑ { }

=

= N I

IJ

J F ~

F ~

1

{ } ^{F ~ J} is the nodal force at node J equivalent to the body forces F _i

K S

J KJ i

i T dS

T ~

K

∫ ^Φ

= ; { }

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

= ⎧ _KJ ^KJ KJ

T ~ T ~ T ~

2

1 ; { } ∑ { }

=

=

^t

M K

KJ

J T ~

T ~

1

{ } T ~ ^J is the nodal force at node J equivalent to the surface tractions T _i

{ } ∑

=

=

^u

M K

KJ i K

J r B

r

1

{ } ^{r J} is the nodal force at node J equivalent to the assumed surface

support reactions on S _u

∫

= S

K

K K i

i u~ dS

U ~

; { }

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

= ⎧ _K ^K K

U ~ U ~ U ~

1

2 { } U ~ ^K is a generalized displacement taking account of imposed displacements u~ _i on edge K

∫ ^Φ

= S

K

K

KJ J dS

B Integration over the edge K of a cell

Chapter V  Page 121 

∫ ^Φ

= C

I

I I J IJ j

j N dC

A ; [ ]

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

= ⎡ ^IJ _{IJ IJ} ^IJ IJ

A A

A A A

1 2

2 1

0

0 A ^IJ _j can also be computed by

J ) I ( K ) I ( K all

) I ( K j

IJ j N B

A ⁼ ∑

[ ]

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎥

⎦

⎤

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎢

⎣

⎡

∂ Φ

∂

∂ Φ

∂ ∂

Φ

∂ ∂ Φ

∂

= Φ

∂

1 2

2 1

0

0

Y Y

Y Y

J J

J J

J Derivatives of Laplace function

(J=1,N)

{ } ^H

^Σ

1 ^{= f ( r ) { g} 1 ^{( θ ) }} ; { } ^H

^Σ

2 ^{= f ( r ) { g} 2 ^{( θ ) }} Stress functions in a LFMVC

{ } [ ] { } I

A T , J

IJ H dA

w

I

1

Σ

1

∫ ^{∂ Φ}

= ;

{ } [ ] { } I

A T , J

IJ H dA

w

I

2 ⁼ ∫ ^{∂ Φ}

Σ

2 ^;

[ ] ^{W IJ} = [ { } ^w 1 ^IJ { } ^w 2 ^IJ ]

[ ] W ^IJ is an influence matrix such that [ ] { } ^{W IJ u J} gives the contribution of the

displacements at node J on the generalized strains in the LFMVCs { } σ ^I = [ ] { } ^{C I} ε ^I ⇔ { } ε ^I = [ ] { } ^{D I} σ ^{I , I} = ^{1 , N} Hooke’s law for cell I

[ ]

[ ] { } ^{[ ]} { }

[ ] { } ^{[ ]} { }

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

⎤

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

⎡

=

∫

∫

∫

∫

Σ Σ

Σ Σ

Σ Σ

Σ Σ

I I

I I

A

I I A

I I

A

I I A

I I I

dA H D H dA

H D H

dA H D H dA

H D H V

2 2

1 2

2 1

1 1

{ }

{ }

{ } ^{[ ] I} { } ^I

A

I A

I I

I IH

dA H

dA H

e e e

I

I

0

0 2

0 1

2

1 = γ

⎪ ⎪

⎪

⎭

⎪ ⎪

⎪

⎬

⎫

⎪ ⎪

⎪

⎩

⎪ ⎪

⎪

⎨

⎧

γ γ

⎪⎭ =

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

= ⎧

∫

∫

Σ Σ

γ

γ γ

[ ] _C

A

I A

I

I I , N

dA H

dA H IH

I

I

1

2 1

=

⎪ ⎪

⎪

⎭

⎪⎪ ⎪

⎬

⎫

⎪ ⎪

⎪

⎩

⎪⎪ ⎪

⎨

⎧

= ∫

∫

Σ Σ

{ }

[ ] { }

[ ] { } ^{[ ] [ ] I I} { } ^I

A

I I I A

I I I I I

P D IH dA

H D P

dA H D P

e e e

I

I

0

2 0

1 0

2 0 1 0

0 =

⎪ ⎪

⎪

⎭

⎪ ⎪

⎪

⎬

⎫

⎪ ⎪

⎪

⎩

⎪ ⎪

⎪

⎨

⎧

⎪⎭ =

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

= ⎧

∫

∫

Σ Σ

{ }

[ ] { }

[ ] { } ^{[ ] [ ] I I { } I}

A

I I I A

I I I I

I IH D

dA H D

dA H D

e e e

I

I

= τ

⎪ ⎪

⎪

⎭

⎪ ⎪

⎪

⎬

⎫

⎪ ⎪

⎪

⎩

⎪ ⎪

⎪

⎨

⎧

τ τ

⎪⎭ =

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧

=

∫

∫

Σ Σ

2 1

2

1

Chapter V  Page 122  Table V.2. Discretized equations in matrix form for the linear elastic fracture mechanics

case.

Equations Comments

{ } { } { } ^{2 [ ] { } [ ] [ ] { } J ,}

1

0 e V K W R u

e

e ^I

N J

T , IJ I I

I I ∑

=

γ

+

Σ

−

+

=

N C , I 1 =

Compatibility equations for the

LFMVCs (V.27)

{ } { } σ ^I = Σ ^I , I = N _C + 1 , N Constitutive stresses

in OVCs (V.28) { } { } τ ^I = ^{P I , I} = ^{1 , N} _C Constitutive stresses

in LFMVCs (V.29) { } [ ] A { } u , I N , N

A ^J _C

N J

T , IJ

I I 1

1

+

=

=

ε ∑

=

Compatibility equation for the

OVCs (V.30)

[ ] [ ] { } [ ] [ ] { }

∑ ∫

=

Σ

⎪

⎭

⎪ ⎬

⎫

⎪ ⎩

⎪ ⎨

⎧

Φ

∂

C

+

I

N I

I A

T I , J T , I I

IJ T ,

I W K R P dA

R 1

0

[ ] ^A { } { } { } ^{F ~ T ~ B { } r , J , N}

N N I

M K K J KJ I J

IJ

C

u

1 0

1 1

=

=

−

−

− Σ

+ ∑ ∑

+

= =

Equilibrium equation

for each Voronoi cell (V.31)

{ } ^{γ +} _∫ [

_Σ

^{[ ]} { }

^Σ

⁺

_Σ

^{[ ]} { }

^Σ

]

A

I

I I I I

I I

I K D H K D H dA

A ⁰ ₁ ¹ ₂ ²

[ ] [ ] { } _C

N J

J I A

J dA I R u , I , N

I

1 1

= Φ

∂

= ∑ ∫

=

Average strain of LFM for the

LFMVCs (V.32)

{ } { } ^K _u

N J

J

KJ U ~ , K , M

u

B 1

1

=

∑ =

=

Compatibility

equation on edge K (V.33)

It is seen that, in the absence of body forces, the only terms that require an integration over the area of a LFMVC are { } w ₁ ^IJ , { } w ₂ ^IJ , [ ] ^{V I ,} [ ] ^{IH and} ∑ ∫ [ ] [ ] { }

=

Φ

N ∂

J

J I A

J dA I R u

1

I

in (V.32).

In order to avoid the calculation of the derivatives of the Laplace interpolation function in

{ } ^w ₁ ^IJ and { } ^w ₂ ^IJ , an integration by part is performed.

{ } ^w [ ] { } ^{H dA [ ] M} { } ^{H dC} { } ^{H dA}

I I

I

C A

J T I

, J I A

T I , J

IJ ⁼ ∫ ^{∂ Φ}

^Σ

^{1 =} ∫ ^Φ

^Σ

^{1 −} ∫ ^{Φ ∂}

^Σ

¹

1 (V.34)

Chapter V  Page 123 

{ } [ ] { } ^{[ ]} { } { } I

C A

J T I

, J I A

T I , J

IJ H dA M H dC H dA

w

I I

I

∫ ∫

∫ ^{∂ Φ}

^Σ

^{= Φ}

^Σ

^{− Φ ∂}

^Σ

= ^{2 2 2}

2 (V.35)

with

[ ] ⎥ ⎥ ⎥

⎦

⎤

⎢ ⎢

⎢

⎣

⎡

=

I I

I I I

M M

M M M

1 2

2 1

0 0

; { }

⎪ ⎪

⎪

⎭

⎪⎪

⎪

⎬

⎫

⎪ ⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

∂ + ∂

∂

∂

∂ + ∂

∂

∂

=

∂

Σ Σ

Σ Σ

Σ

2 22 1 1

12 1

2 12 1 1

11 1

1

Y H Y

H

Y H Y

H

H ; { }

⎪ ⎪

⎪

⎭

⎪⎪

⎪

⎬

⎫

⎪ ⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

∂ + ∂

∂

∂

∂ + ∂

∂

∂

=

∂

Σ Σ

Σ Σ

Σ

2 22 2 1

12 2

2 12 2 1

11 2

2

Y H Y

H

Y H Y

H H

(V.36)

where M ₁ ^I and M ₂ ^I are the 2 components of the outside normal to the contour of the LFMVC with respect to the local reference system ( Y ₁ , Y ₂ ) ,.

However, since the stresses (V.13, V.14) introduced in the assumption (V.18) constitute a solution of the Theory of Elasticity, they satisfy the equilibrium equations:

0 0

2 12 1 1

11 1 2

12 1

11 =

∂ + ∂

∂

⇒ ∂

∂ = τ + ∂

∂ τ

∂

^{Σ Σ}

Y H Y

H Y

Y (V.37)

0 0

2 22 1 1

12 1 2

22 1

12 =

∂ + ∂

∂

⇒ ∂

∂ = τ + ∂

∂ τ

∂

^{Σ Σ}

Y H Y

H Y

Y (V.38)

Consequently

{ } ⁼ _∫ [ ] ^{∂ Φ} { }

^Σ

⁼ _∫ ^{Φ [ ]} { }

^Σ

I

I

C

T I , J I A

T I , J

IJ H dA M H dC

w ₁ ^{1 1} (V.39)

{ } ⁼ _∫ [ ] ^{∂ Φ} { }

^Σ

⁼ _∫ ^{Φ [ ]} { }

^Σ

I

I

C

T I , J I A

T I , J

IJ H dA M H dC

w ₂ ^{2 2} (V.40)

This eliminates the integration over the area of the LFMVC as well as the calculation of the derivatives of the Laplace function.

Similarly, an integration by parts in (V.32) gives:

{ } N ^{[ ] [ ] { } [ ] [ ]} I I , T ^{{ }} I C J

J I T , I IJ

I LA R u D IH K I , N

A 1

1

0 = − =

γ

_Σ

∑

=

(V.41)

with

[ ] { }

{ } ⎪ ⎪

⎭

⎪ ⎪

⎬

⎫

⎪ ⎪

⎩

⎪ ⎪

⎨

⎧

=

⎪ ⎪

⎭

⎪ ⎪

⎬

⎫

⎪ ⎪

⎩

⎪ ⎪

⎨

⎧

= ∫

∫

∫

∫

Σ Σ

Σ Σ

I I

I I

A

I A

I

A

I T A

I T I

dA H

dA H

dA H

dA H

IH 2

1

2 1

(V.42)

and

Chapter V  Page 124 

[ ] ^{I , N , J , N}

dC M

dC M

dC M

dC M

LA _C

C

J I I C

J I I

C

J I I C

J I I IJ

I I

I

I

1 1

0

0

1 2

2 1

=

=

⎥ ⎥

⎥ ⎥

⎦

⎤

⎢ ⎢

⎢ ⎢

⎣

⎡

Φ Φ

Φ Φ

= ∫ ∫

∫

∫

(V.43)

Using the definition of [ ] ^{A IJ} in table V.2 and introducing the angle α _I between the global and the local frames (figure V.1), we get:

[ ] ⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

α + α

α + α

−

α + α

− α

+

= α _IJ

IJ I IJ I

IJ I I

I IJ I IJ

I IJ I IJ

IJ

A sin A

cos A

cos A

sin

A cos A

sin A

sin A

LA cos

2 1

2 1

2 1

2 1

0

0

(V.44) Hence, the only terms requiring integration over a cell are [ ] ^{V I} and [ ] ^IH .

However, these terms can be integrated analytically as explained in annex 3.

Consequently, in the present theory, none of the terms requires numerical integration over the area of a cell. The only numerical integrations required are integrations over edges of the Voronoi cells.

This shows that the benefits obtained by this approach in the linear elastic domain and in the case of materially non linear problems remain valid for applications in Linear Elastic Fracture Mechanics.

V.5. Solution of the equation system

Since we consider a linear elastic material, the constitutive equation for a Voronoi cell J is:

{ } ^{σ J =} [ ] { } ^{C J ε J} (V.45)

where [ ] C ^J is the Hooke compliance matrix of the elastic material composing cell J.

Introducing this in (V.31), we get:

[ ] [ ] { } [ ] [ ] { }

∑ ∫

= Σ

+

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧ + ∂ Φ

C

I

N I

I A

T I , J T , I I

IJ T ,

I W K R P dA

R

1

0

[ ] ^A [ ] ^{C A { } { } { } F ~ T ~ B { } r , J { } , N}

N N I

M K K J KJ I J

I

* I IJ

C

u

1 0

1 1

⊂

=

−

−

−

∑ ε ∑

+

= =

(V.46) with

[ ] ^{[ ] I}

I

* I C

C = A 1 (V.47)

The term [ ] [ ] { } I A

T I , J T ,

I P dA

R

I

∫ ^{∂ Φ 0} of (V.46) is integrated by parts and then, the term ^A _I { } ^{ε I}

given by (V.30) is introduced in (V.46). This gives: