Computing Upper and Lower Bounds for the

J-Integral in Two-Dimensional Linear Elasticity

¢¡ ¤£ ¤¥

Z.C. Xuan , K.H. Lee , A.T. Patera , J. Peraire

Singapore-MIT Alliance ¡

Department of Mechanical Engineering, National University of Singapore £

Department of Mechanical Engineering, Massachusetts Institute of Technology ¥ Department of Aeronautics and Astronautics, Massachusetts Institute of Technology

(This paper is dedicated to the memory of Kwok Hong Lee)

Abstract— We present an a-posteriori method for computing derived from Eshelby’s [3] energy momentum tensor along the rigorous upper and lower bounds of the J-integral in two direction of the possible crack extension. An alternative form dimensional linear elasticity. The J-integral, which is typically of the J-integral in which the contour integral is transformed expressed as a contour integral, is recast as a surface integral which yields a quadratic continuous functional of the displace- into a domain integral involving a suitably defined weighting ment. By expanding the quadratic output about an approximate function is given in [6]. The expression for the energy release finite element solution, the output is expressed as a known rate given in [6] appears to be very versatile and has an easier computable quantity plus linear and quadratic functionals of and more convenient generalization to three dimensions than the solution error. The quadratic component is bounded by the the original form [13]. energy norm of the error scaled by a continuity constant, which is determined explicitly. The linear component is expressed as an Regardless of the method chosen to evaluate the stress inten- inner product of the errors in the displacement and in a computed sity factor, a good approximation to the solution of the linear adjoint solution, and bounded using standard a-posteriori error elasticity equations is required. Unfortunately, the problems estimation techniques. The method is illustrated with two fracture of interest involve singularities and this makes the task of problems in plane strain elasticity. computing accurate solutions much harder. For instance, it is well known [16] that the convergence rate of energy norm of a I. INTRODUCTION

standard finite element solution for a linear elasticity problem  ¤ The accurate prediction of stress intensity factors in crack involving a ¦¢§©¨ reentrant corner is no higher than ,

tips is essential for assessing the strength and life of structures where  is the typical mesh size. This problem was soon using linear fracture mechanics theories. A crack is assumed realized and as a consequence a number of mesh adaptive to be stable when the magnitude of the stress concentration algorithms have been proposed [4], [5], [7], [8], [14] which, at its tip is below a critical material dependent value. Stress in general, improve the situation considerably. In some cases intensity factors derived from linearly elastic solutions are [7], [8], the adaptivity is driven by errors in the energy norm widely used in the study of brittle fracture, fatigue, stress of the solution, whereas in some others [4], [5], [14], a more corrosion cracking, and to some extend for creep crack growth. sophisticated goal-oriented approach based on a linearized Since the analytical methods for solving the equations of form of the output is used. elasticity are limited to very simple cases, the finite element In this paper we present a method for computing strict method is commonly used as the alternative to treat the more upper and lower bounds for the value of the J-integral in complicated cases. The methods for extracting stress intensity two dimensional linear fracture mechanics. The J-integral is factors from computed displacement solutions fall into two written as a bounded quadratic functional of the displacement categories: displacement matching methods, and the energy and expanded into computable quantities plus additional linear based methods. In the first case, the form of the local solution and quadratic terms in the error. The linear terms are bounded is assumed, and the value of the displacement near crack tip using our previous work for linear functional outputs [9], [11], is used to determine the magnitude of the coefficients in the [12] and the quadratic term is bounded with the energy norm asymmptotic expansion. In the second case, the strength of of the error scaled by a suitably chosen continuity constant, the singular stress field is related to the energy released rate, which can be determined a priori. The bounds produced are i.e. the sensitivity of the total potential energy to the crack strict with respect to the solution that would be obtained position. An expression for calculating the energy release on a conservatively refined reference mesh. This restriction rate in two dimensional cracks was given in [13] and is however can be eliminated if the more rigorous techniques for known as the J-integral. The J-integral is a path independent bounding the outputs of the exact weak solutions are employed contour integral involving the projection of the material force [10], [15]. Also, not exploited here, but of clear practical interest, is the fact that the bound gap can be decomposed ¢ in its plane, we are interested in determining the energy

into a sum of positive elemental contributions thus naturally release rate, ‘s 84m , such that, leading to an adaptive mesh adaptive approach [12]. We think that the algorithm presented is an attractive alternative to the I€„ 84ms%W ’‘s 84b¢“y existing methods as it guarantees the certainty of the computed For a two-dimensional linear elastic body the energy release bounds. This is particularly important in critical problems x relating to structural failure. The method is illustrated for an 2 open mode and a mixed mode crack examples. G II. PROBLEM FORMULATION

We consider a linear elastic body occupying a region 

 

  . The boundary of  , , is assumed to be piecewise d l smooth, and composed of a Dirichlet portion ! , and a

x1 $&%' ( *)+ ("

Neumann portion #" , i.e. . We assume that

,.-/ 10  ("32  a traction 22 is applied on the Neumann Wc

boundary and that the Dirichlet boundary conditions are ho-

7 :-<;>=

mogeneous. The displacement field 45%6 87

?A@ @

¤9 

3T

C E-F GHI JKML %ONQPSRQ

%B 8C satisfies the  followingD9weak form of the elasticity equations Fig. 1. Crack geometry showing coordinate axes and the J-integral contour

and domain of integration.

@ @ @Z^ @

U

V%W JX ZY\[], -`;

84 (1)

9 9 9 96_ 9

rate, ‘” 4m , can be calculated as a path independent line in which

integral known as the ‘ -integral [13]. If we consider the

ced

@ @g

geometry shown in figure 1, the ‘ -integral has the following

aX b% XHf  9

9 expression,

c hSi

ceh–•˜—™š

@Z^ @1g

g

x4

[], % ,`f

‘s 84bV% H›f 9

9 (6)

9



xœ

t

XQ-j 10 GK2

 

— u

where is the body force. The bi-linear form ™

@ 

 where is any path beginning at the bottom crack face

U

k mlD;onp;rq

is given by, uˆ

%B lžwe}D~

9 and ending at the top crack face, is the

ctdvu

š š

@ g › ›Ÿ%

@ strain energy density, is the traction given as ,

U

s% 8k`mlSwx  zy

8k (2)

%¡ 

9 and is the outward unit normal to . An

D9  @

alternative expression for ‘” 4m was proposed in [6], where the  Here, w denotes the second order deformation tensor which

@ contour integral is transformed to the following area integral u

is defined as the symmetric part of the gradient tensor { .

@ @ @

@ expression,

V% a{ Yj a{ |#}D~ 

That is, wx . The stress is related

c¢d¤£p•

—™ u

g x¥

to the deformation tensor through a linear constitutive relation ¤4

|

3y J{H¥! f

‘s 4bV% (7)

u xœ

of the form ¤œ

 @ @

 

s%\rl©w 

(3)

¥  Gˆ¦¤

Here, the weighting function is any function in  that

where  is the constant fourth-order elasticity tensor. We is equal to one at the crack tip and vanishes on .

@

 

¥ ‘s 4m ‚l#;ƒq define the total potential energy functional € For a given , we observe that is a bounded quadratic

as functional of 4 . For our bounding procedure it is convenient

@ @ @ @ @(^

¦

U

s% # † JX ! M[‡,

€„ (4) to make the quadratic dependence of the output on the solu-

9 9 9

~ tion more explicit. To this end, we define the bilinear form

@

 

§

¨

4 ˆlD;on`;©q

It is straightforward to see that the solution, , to the problem 8k as,

9

@

£ ctd

(1) minimizes the total potential energy, and that u

g @



§

¨

|

 k s% J{H¥! f 8kp

¦ ¦

U

9

¤œ



4 4ˆs% LŠL‹L 43LŠLŠL y

€„ 84ˆV%‰ (5) 

ctd¤£

9 u

~ ~

@ g

¦ x¥

U

8k`ˆl©w  

(8)

LŠLŠLŒfLŠLŠL©% f fŽ 

Where denotes energy norm associated with 9

~ ¤œ

9

U



f f 

@ 

the coercive bilinear form . 

¨

9 mlD;on`;©q and its symmetric part 8k ,

In fracture mechanics we are often interested in determining 9

@ @ @

¦ §

the strength of the crack tip stress fields. A common way to §

¨ ¨ ¨

s% 8k ZY kp’y

k (9)

9 9

do that is to relate the so called stress intensity factors to 9 ~ the energy released per unit length of crack advancement (see It is clear from these definitions that,

figure 1). If the total potential energy defined by (5), decreases

¨

‘s 84bb% 84 4ˆ

by an amount I€ 4ˆ when the crack advances by a distance (10)

9 9

and that there exists ª`«­¬ such that, and therefore,

@ @ @ @

¨

¦ ¦ ¦ ¦



¨

m®Mª¯L‹LŠL LŠLŠL -`;+y

(11)

 

ÁbLŠL‹L À· LŠLŠL ® · 4 m® ÁˆL‹LŠL À·ÂY LŠL‹L y

°

9 _

9 (18)

¿ 9 ¿

À À III. BOUNDING PROCEDURE B. Quadratic term

Our objective is to compute upper and lower bounds, for 4 ‘” 4m , where satisfies problem (1). Let us consider a finite In the appendix we show that for two dimensional linear

element approximation 4ˆ°-`;“° satisfying elasticity, a suitable value for the continuity constant in ex-

@ @Z^ @

@ pression (11) is given by

U

84m° s%W aX ¯YQ[‡, -`;°±y

(12) Á©Í

9 9 9 96_

GËŚY AL {Î¥mL



ª©¦‚%ÅēÆIÇ

;“°/'; ;

Here, is a finite dimensional subspace of . For •

9

|IÈVÉSÊ¢È

Á¤Ï Í ÍÎÐ¶Ñ Á©Í ÐÕÑ

¦ ¦

 

simplicity, we shall assume that ;‚° is the space of piecewise

Ë ÌY  Ë YQ GËŚY 

Ñ¢Ò¤ÓDÔ Ñ¢Ò¢Ö©Ô

linear continuous functions defined over a triangulation, ²¤° , 

of  which satisfies the Dirichlet boundary conditions. An (19)

Í

%Qׄ}¢ G~˜ ¦YµØ˜ Ì ‘ ° approximation to ‘s 84ˆ , , can be obtained as where is the elastic shear modulus, is the

elastic bulk modulus which is given by ÌÙ%†×Ã}t ¦©Y~©Ø¢}t ˤ ¦e

¨

‘S°³% 84V° 4m°3

Ø  ÌÙ%†×Ã}t ˤ ¦s +~SØ¢

 9

9 for plane stress, and for plain strain.

× Ø where, for convenience, ¥ in (7) is chosen to be piecewise In these expressions, is Young’s elastic modulus and is

linear over the elements ´¯°-µ²t° . Exploiting the bi-linearity the Poisson’s ratio. Therefore, we write

@

¨

8k 

of , we can write ¨

9



¦

· ·˜m®Mª L‹LŠL ·(LŠL‹L y

9

‘s 84ˆ! j‘ °

¨

· ·˜ ¨

¨ The computation of a bound for is straightforward

9

% 84 4b! 84m° 4b°¶

LŠL‹L ·ZLŠL‹L 9

9 once a bound for the error in the energy norm has been

¨ ¨ ¨

841 4V° 41 E4V°3(Yj~ 84 4V°’! ~ 84V° 4m°¶

% obtained.

9 9 9

¨ ¨

· ·˜ZY*~ · 4V°’

% (13)

9 9

9 IV. BOUNDS FOR ENERGY NORM OF THE ERROR 4m° where ·+%‰4* E4m° is the error in the approximation . It

An essential ingredient for the computation of the bounds ¹Kº is clear that if we are able to compute bounds ¸ and for

for our output of interest, ‘s 84m , is the calculation of upper

the quadratic and linear error terms,

À#·`Ú }DÀ

bounds for the energy norm of · , and . There is

¿

¨

· ·˜»L¢®M¸

L an extensive body of literature on this subject (see [1], [11]), 9 and a number of methods an approaches have been proposed and

¨ which, in principle, would be applicable here. These methods

0

¹ ® · 4V°’ˆ®*¹b¼ 9

9 typically compute an upper bound of the energy norm of ‘(º then, the bounds for ‘s 84ˆ , , follow as, the error measured with respect to a reference solution. This

reference solution can be a finite element solution obtained

0 0

‘ =‘ :¸†Yj~©¹ ®Q‘s 84ˆˆ®Q‘ Y*¸­Y*~D¹b¼E=‘¯¼³y ° ° on a very fine mesh or a solution obtained using a high A. Linear term degree interpolation polynomial. More recently, a method was In order to derive upper an lower bounds for the linear proposed in [15] for Poisson’s equation which is based on

¨ the use of the complementary energy principle and gives 4V°’ term · , we introduce the following adjoint problem: 9 bounds for the error relative to the exact weak solution. The

find ½¾-µ; such that

@ @

@ method has been since generalized to the linear elasticity

U ¨

½„s% 4  -`;

° (14)

9 96_ 9 9 equations in [10]. Here, we will use essentially the method

proposed in [2] for Poisson’s equation, extended to the linear

½ - and the corresponding finite element approximation, °

elasticity problem. This approach is easy to implement but

; j;

° , such that

@ @

@ has the disadvantage that the error in the solution is measured

U ¨

½ s% 4  -µ; y

° °

° (15)

9 96_

9 with respect to the finite element solution obtained on a

@ @

U conservatively refined mesh. Future work will incorporate the

G· b%†¨ -µ;

From (1) and (12), it follows that for all ° . 9

U more rigorous bounding procedures described in [10].

G· ½ ’%<¨ In particular, ° . This, combined with the above equations (14) and9 (15) gives the following representation for We start by combining equations (1) and (12) to write an

the linear error term, equation for the error,

@ @ @Z^ @ @ @

U U

¨ U

· s%& JX ¯YQ[‡, 84b° s=†ÛÎ  -`;

· 4 s% · 

° (16)

9 9 9 9 9Ü_

92¿ 9

9 (20)

@

  %Q½‰ H½

where ° is the error in the adjoint solution. Now,

ÛÎ mlD;Ýq

  ¿ where is the residual. Next, we introduce the

using the parallelogram identity, we have that for all À:- ,

;ß*; ²˜°

“broken” space Þ , supported by the triangulation ,

¦ ¦ ¦ ¦

?¢@ @ @

U

 

G· s% ÁbLŠL‹L À„·ÂY L‹LŠL ÁbLŠLŠL Àֈ L‹LŠL

 

(17) 3T

;Ý%Þ L -E G ´$°¶ ´$°³-–²t° %\N

¿ ¿ 9

92¿ on (21)

À À

9#_ 9 9 ²¢°

where ´$° denotes a typical triangle of with boundary shall assume here that the discretization is sufficiently rich so

@

U U U

ë ë

¤´$° 8k  · ·  · ·˜

Þ Þ Þ

. We can then extend the bi-linear form and that the difference between Þ and is negligible.

@

9 9 9 

the residual ÛÎ to admit functions which are discontinuous Summing the elemental equations (29) over all the elements

@ %Q·

across triangles. We define in ²t° , and letting leads to,

c

u

@ @ g

U

U

· ·˜s%ÕÛÎ G·¤# HÛÎ ïê ·˜

°

Þ

k s% 8k`ml w¤  

u

(22) 9

|IÈ

9

^

§

È

|

c c h

i

Y á [ 84m°3”f ·Ã µê$°’· y

Ñ (30)

@ @vg @vg

|¤È

9

È

| àeä»å

Û s% XHf Y ,µf

È È

| ÉSÊ

|IÈ

Ñ

|¤È |IÈ!à

@

꯰¶·µ-`;“° ÛÎ ïê$°’·˜b%†¨

U Since , , and

84m° ’y

u

(23) u

|¤È

9

§ §

%W 84V°’!f

84b°’!f (31) |IÈ

and write, |xð

È

@ @

U U

x´ %'¤´¶ñ -–ç

° è

°

s% á 8k 

k (24) on the common edge of the two neighbor

|IÈ 9

9 elements, the last term in the right hand side is also zero. This

È È

| ÉSÊ @

@ yields,

ÛÎ s% Û vy

á U

(25) U

È

|

· ·˜s%†ÛÎ ·˜b% · ·¢ãy

Þ (32)

9 9

|IÈVÉSÊAÈ

@

u

%†·

4j%Q·VYÙ4 ´ °

Substituting ° on element into the governing The last equality follows from (20) with . Finally since

U

·3 · ·3 ·¢mò­¨

¶{rf 4mˆ%WX ´ ·

° Þ

equations, , we find that, on , the error Þ , from the above expression, it follows the 9

satisfies the differential equation desired result,

u u

U U

· ·¢ˆ® · ·¢ L‹LŠL ·(L‹LŠLe®L‹LŠL ·ZL‹LŠL$y

¶{©f ·¤s%âXµY*{rf 4b°’ãy

Þ Þ

(26) Þ or (33)

9 9 9

Multiplying both sides of the local error equation (26) by If we now start with the adjoint error equation,

@

-ÅÞ;

@ @ @

, and performing integration by parts using Green’s @

U ¨ U

V% 4b°3! ½ b=MÛ`óS 

° (34)

9 9 9 formula leads to the weak form of the error residual differential 9¿

equation over a triangle

u and repeat the same process outlined above, we can determine

@ @ @Z^ @

U

s%†Û ZY\[ 87$!f -5Þ;1y Þ

G· a reconstructed adjoint error satisfying,

Ñ

¿

|IÈ |IÈ |IÈ

9 9 9ˆ_

È

| àeäæå U

(27) U

 LŠL‹LZy K® LŠLŠL LŠL‹L¢®âLŠL‹L

Þ Þ

Þ or (35)

¿t9 ¿ 9 ¿ ¿e92¿ ¿

²¢°

Here, ç¢è is the set of all edges in the triangulation

U

"

fŽ

excluding those on , and is the outward unit normal Finally, we note that due to the symmetry of f , equations

|IÈ

9

u x´

to ° . (20) and (34) can be combined into the following equation, u

By approximating the normal traction 84ˆmf by the

@ @

¦ ¦

È

|

U

À#·Ú b%†ÀZÛÎ ZÚ Û Ce

ó

4 

traction computed by averaging the traction ° on the two (36)

¿e9 9 À

neighboring elements [1], that is, À

u

u and an upper bound for the norm of the combined error is

§ 84V°’!f

84ˆ!f thus,

|IÈjé |IÈ

u u

Ð

¦

¦ ¦

0

¼

U

4b°’»L Y 4b°’»L f %

Ñ Ñ

Ô



L‹LŠL$y L‹LŠL À(·Ú LŠLŠLt®*À L‹LŠL ·(L‹LŠLAÚ*~ · ZY L‹LŠL

|¤È

9

È È

| |

Þ Þ Þ Þ

~ (37)

¿ ¿ 9 ¿

À À 

we can write the following problem for the approximate local

À À %

Since in equation (18) is arbitrary we choose 

·p-5Þ; ´¯°

error Þ over each element ,

u

LŠL‹L L‹LŠL }¢LŠL‹L ·ZLŠL‹L Þ

Þ to obtain,

@ @ @Z^ @

¿

§

U

· s%­Û $Y­[ 84 !f -oÞ;1y

°

Ñ

Þ

U ¨ U

È È È

| 9 | | 9 9m_

|IÈ#àeä

å

~ · A 3~¤L‹LŠL ·(L‹LŠL»LŠL‹L LŠL‹Lt® G· 4 ˆ®­~ · ãYÂ~˜LŠLŠL ·ZLŠLŠL»LŠL‹L L‹LŠLSy

°

Þ Þ Þ Þ Þ Þ Þ Þ

¿ ¿ 9 9 ¿ ¿ (28) 9 In general, the above local problems are not solvable. A (38) common way to fix this problem is to solve instead the V. SUMMARY OF THE BOUNDS PROCEDURE modified local problems [2],

We summarize here the steps involved in the implementation

@ @ @

U

· s%†Û µê$° 

u

Þ of the bounds procedure.

|¤È |¤È

9

@ @Z^ @

§

Y\[ 4 !f µê -oÞ;‚ë ° ° : Solve a global problem for the approximate displace-

Ñ STEP 1

È

| 9 9Œ_ 9

|IÈ#àeä

å

4 4 -p;

° °

(29) ment field ° : find such that

@ @ @Z^ @

@

U

4V° V%W JX ZY\[], -`;“°

ê Âl1Þ;ìq ; Þ

° °

9 9 9ˆ_ 9

in which denotes the nodal interpolation 9

@ @

-íÞ; ê -îÞ; °

operator. Thus, we have that if , then, ° ,

@ @ @

and determine ‘ °³%‰‘s 84b°’ .

- ; Þ ê % ° and if , then, ° , . Problem (29) is local over STEP 2: Solve a global problem for the approximate adjoint

each element but is still infinite dimensional. In practice, we

½ ½ -p;“° ° solution ° : find such that

discretize (29) over a conservatively fine mesh and compute an

@ @ @

U ¨

ë

· · ½ s% 4V°’ -`;“°

° Þ

approximation, Þ , to (see [1], [2], or [11] for details). We

9 9 9ˆ_

STEP 3: Solve for each element ´¯°.-*²t° a local problem Due to the symmetry of the problem, we only use one

ë ë

· · -5Þ; Þ

for the reconstructed displacement Þ : find such that quarter of the plate for the finite element analysis. We use

@ @

@ a 5 by 5 square area surrounding the crack tip as the support,

U

ë

· V%­Û µê$° 

Þ

u

¦

 ¥

|IÈ |¤È

9 , of the weighting function (see figure 3). Figure 4

@ @Z^ @

§

Y\[ 4 !f µê -ßÞ;“ë+y

° °

Ñ shows three of the linear triangular meshes used for the global

È

| 9 9ˆ_ |IÈ(àeä å computations together with an illustration of the broken mesh

STEP 4: Solve for each element ´¯°.-*²t° a local problem used for the local Neumman problems. For all the calculations

ë ë

-5Þ; Þ

for the reconstructed adjoint error Þ : find such that

ËS~ ¿

¿ the reference mesh is taken to be a refinement of the

@ @ @

² °

U original coarse mesh . The value of the output on the

ë s%†Û µê 

°

ó Á Á

Þ

u

È

| 9 ¿

È

|

¦Aþ˜y ~ ü

@Z^ @

@ reference mesh is which still has a relative error with

§

¤

Y\[ ½ ”f µê -ßÞ;“ë–y

°

Ñ

°

¨¢yŽ÷

È

9 9ˆ_

| respect to the exact solution of about .

|¤È!àeä

å ‘e°

U Table I shows the results for the output , the norms of

LŠL‹L ·¤ë¤LŠL‹L% ô ·˜ë ·˜ë aõ L‹LŠL ë¤LŠL‹L%



Þ Þ Þ

STEP 5: Calculate Þ ,

9 ¿ U

U the reconstructed errors, and the computed upper and lower

ô ë ë aõ ·˜ë ë  ‘ ‘!0

22

¼

Þ Þ Þ

Þ , and , and determine and as,

¿ 9 ¿ 9 ¿ ‘

bounds, ‘(º , for . The observed convergence rate of the

U

¦



ë ë ë ë ë

· LŠLŠL Y · ¯Y³L‹LŠL · LŠLŠL»LŠL‹L L‹LŠL

‘¯¼%‘S°*Y:ª¯L‹LŠL bound gap is somewhat higher than the expected value of .

Þ Þ Þ

Þ (39)

¿ ¿

9 We note that in the table I, all the results shown have been

U

0



ë ë ë ë ë

%‘S°† ª¯L‹LŠL · LŠL‹L Y · ! †L‹LŠL · LŠL‹LAL‹LŠL LŠL‹L¯y

‘ multiplied by two since, due to symmetry, only half of the

Þ Þ Þ

Þ (40)

¿ ¿ 9 domain is considered. VI. EXAMPLES In the first example we consider a plate with two edge cracks subjected to a uniformly distributed tensile stress as shown in 5 figure 2. The plate is assumed to be in plane strain. The value

of the tensile force acting on the two ends of the plate is

ö U %r÷

%5¦ and the dimension of the crack is . The non- 5 ¨

dimensionalized Young’s modulus is ¦©y and the Poisson’s Ë ratio is ¨˜y . The analytical value of the mode-I normalized Wc

stress intensity factor ø’ù for the problem has been determined

öú û¯U s%¦Sy‹¦¢üS~SýDþ in [7] to be ø3ù}t . Therefore, the exact value Cracktip

™ 5

Ò¤ÿ¡ £¢

‘ % ¦Â \Ø Œø }I×O%

 

of the J-integral is obtained as ù ËS~ ýI¨ ¦Aþ¢y . p

Fig. 3. Support of weighting function ¥ for the evaluation of the J-integral.

TABLE I

BOUND RESULTS

¦¨§ © ¦ § ¦ §  ¦ § Coarse mesh size ¦

 15.3666 17.4156 18.4982 19.0412 19.2982

¡ 

 9.3386 3.7972 1.3066 0.4358 0.1242

      

 6.3484 3.5576 1.9112 1.0523 0.5470

       2.1082 1.6198 1.0937 0.6672 0.3738 60 "! -6.8999 6.1726 14.4087 17.6771 18.9033 5 $# 65.3107 36.8182 25.3829 21.3569 19.9695 Cracktip In the second example we consider a plate with an inclined 30 crack subjected to a uniformly distributed tensile stress as shown in figure 5. The plate is assumed to be in plane strain.

The value of the tensile force acting on the two ends of the ö

20 plate is %ì¦ . The non-dimensionalized Young’s modulus

¨ ¨¢y Ë is ¦©y and the Poisson’s ratio is . The analytical value of

the mode-I and node-II normalized stress intensity factors, ø3ù8ù

p ø3ù and , for the problem have been determined in [7]

Á

öú û¯U öú û¯U

­% ¨¢y ü©ü˜¦©¦ ø3ù8ù}¢ †% ¨˜y ÷©ü ý to be ø3ù}t and , respectively. Therefore, the exact value of the J-integral is

Fig. 2. Geometry of a double edge-cracked plate subjected to a uniform ™

Ò¤ÿ¡ £¢

‘ %& ¦K 1Ø æ 8ø Y:ø }I×W%\ü¢y Ë¢¦A§SË

   ù8ù tensile stress. obtained as ù .

(a) (b) (c) (d)



'&)( '&+*-,

% %

Fig. 4. Finite element meshes used for open crack problem: (a) coarse mesh % , (b) medium mesh , (c) fine mesh , and (d) illustration of

 ¦¨§ .+©

broken mesh used for local problem solution associated to coarse mesh % (actual mesh is twice as fine e.g. ).

u

0 1 0 1 ?0 ?1

T T

| ~ | w %

We use a 3 by 3 square area surrounding the crack tip as the vectors % and ,

9 9 9 9

     2

u ¥

the support, K¦ , of the weighting function (see figure 6). then, we have that

32:w Figure 7 shows three of the linear triangular meshes used for % the global computations together with an illustration of the 9

where 2 is the matrix of elastic coefficients

45 :; Í

broken mesh used for the local Neumman problems. For all the Í



́Y76 8 ̂ 8 ¨

ËS~ Í

calculations the reference mesh is taken to be a refinement Í



Ì 8 ÌÃY96 8 ¨

2o% ²¢°

of the original coarse mesh . The value of the output on Í

¨ ¨

the reference mesh is ü¢yŽ~DËS¨¢¦ which has a relative error with

¤

Í

Á Ì

respect to the exact solution of about ¦©y . where is the shear modulus and is the bulk modulus as ‘

Table II shows the results for the output ° , the norms of defined in section III. Let

| <

the reconstructed errors, and the computed upper and lower @

¤C ¤C ¤C ¤C

   

%>= ‘

bounds, ‘Zº , for .

9 9 9 9

xœ xœ xœ xœ

   @? @

TABLE II then, for a given displacement field -p; , we have

4 : ;

BOUND RESULTS 5

¦î¨ ¨¡¨

<

@

¨>¨ ¨ ¦

w %

¦¨§ © ¦ § ¦¨§  ¦¨§

Coarse mesh size ¦

9



¦ ¦í¨

4.1722 5.3889 5.9313 6.1325 6.2034 ¨

/¡ 

4 :

5 ; Í

1.8579 0.2197 0.0393 0.0114 0.0020 Í

     

 

u



̄Y96 ¨ ¨ ̓ 8

6.4310 3.6156 1.7524 0.8373 0.3971 8

@ <

     

Í Í





̓ 8 ¨ ¨ ̄Y76 8

2.3234 1.8924 1.2509 0.7153 0.3738 %

! (41)



Í 9

-7.8728 -0.1657 4.0848 5.6162 6.1221 Í

¨ ¨

$# 25.7264 13.9579 8.5476 6.8368 6.4229 u

and —™

@ <

¦ ¦

| |

fAw % y

% (42)

~ ~

2 A

APPENDIX Here, 2 is given by

4C :ED

Í Í

C D



6

8 ¨ ¨ Ì 8

We prove here the expression (11) and provide an upper ̄Y

5 ;

Í Í

A

ª ¨

bound for the value of the constant . For the two dimensional ¨

Í Í

2o% y ¨

case of interest here, the stress and deformation tensors only ¨

Í Í



8 8

̓ ¨ ́Y96 have three independent components. Therefore, if we define ¨

(a) (b) (c) (d)



'&)( '&+*-,

% %

Fig. 7. Finite element meshes used for open crack problem: (a) coarse mesh % , (b) medium mesh , (c) fine mesh , and (d) illustration of

 ¦¨§ .+© broken mesh used for local problem solution associated to coarse mesh % (actual mesh is twice as fine e.g. ).

p

Cracktip

3

O 5 45 1. 20 6 Wc .5 3 1 Cracktip

Fig. 6. Support of weighting function ¥ for the evaluation of the J-integral.

10

and —™

@ < A @ <

x¥ ¦ x¥

|

% = 2 y

~ ¤œ

p xœ

  ?

@ @

¨ 

Then, can be written as

9 d¤£

Fig. 5. Geometry of plate with an inclined crack subjected to a uniform c

@ @ @ A @ <ˆg ¦

tensile stress. ¨

|<7I

F

V% 2KJ 

x¥

9 9

~ xœ



@

L‹LŠL LŠLŠL

The quadratic terms in (7) can now be expressed as and  , is given by

ced ced¤£

@

@ @ @

u

U

@ @

<

 ¦ |

s% ³ò &y

|

9

@

|

Œ{Î¥# f  %

2 2

9

xœ ~  If we consider now the symmetric generalized eigenvalue

where problem,

•

:ED 4C

Í Í Í Í

Ñ Ñ Ñ Ñ

¦ < ¦ ¦ ¦ A A @

F

x¥

D C

 

~¢ ḠY 8  ḠY 8  G̓ 8 

6

ÑAÒ ÑH ÑH Ñ¢Ò

%\N 2/ 1~NM 2 D

C (43)

9

ÍÕÑ Í Ñ ÍÕÑ Í Ñ

¦ ¦ ¦ ¦

¤œ

; 5







F 8 8

6

ḠY ~ ḠY  

ÑH Ñ¢Ò Ñ¢Ò ÑH

?

Í Í

Ñ Ñ %

¦ ¦

8

T

ª%QēÆIÇ M M M M

9 it is clear that if we choose then,

¨ ¨

ÑH Ñ¢Ò

¤9 S9 9

6

Í Ñ Í Ñ

@ @ @ @

¦ ¦

¨ U



8 6 8

ḠY ¨ ¨ G̓  

m®*ª 

ÑH

Ñ¢Ò (44)

9 9 9 as required. The eigenvalues of (43) can be found explicitly

with the help of a symbolic manipulation program and the final ª

expression for ª is given in (19). We note that the value of ¥

thus computed depends on {Î¥ . In our context, is chosen {H¥ to be piecewise linear on ²¢° , in which case is piecewise

constant. In order to determine the appropriate value for ª we ² simply take the maximum over all the elements in ° .

REFERENCES [1] M. Ainsworth, J.T. Oden, A posteriori error estimation in finite element analysis, Computer Methods in Applied Mechanics and Engineering, 142 (1997) 1-88. [2] R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Mathematics of Computation, 44 (1985), 283-301. [3] J.D. Eshelby, The energy momentum tensor in continuum mechanics, In Inelastic Behavior of Solids (Edited by M.F. Kanninen et al.), 77-114, McGraw-Hill, New York (1970). [4] P. Heintz, F. Larsson, P. hansbo and K. Runesson, Adaptive strategies and error control for computing material forces in facture mechanics, Preprint 2002-18, Chalmers Finite Element Center, Chalmers Univeristy of Technology, 2002. [5] P. Heintz, K. Samuelsson, On adaptive strategies and error control in fracture mechanics, Preprint 2002-14, Chalmers Finite Element Center, Chalmers Univeristy of Technology, 2002. [6] F.Z. Li, C.F. Shih and A. Needleman, A comparison of methods for calculating energy release rates, Engineering Fracture Mechanics, 21 (1985) 405-421. [7] S.H. Lo, C.K. Lee, Solving crack problems by an adaptive refinement procedure, Engineering Fracture Mechanics, 43 (1992) 147-163. [8] K.S.R.K. Murthy and M. Mukhopadhyay, Adaptive finite element anal- ysisi of mixed-mode fracture problems containing multiple crack-tips with an automatic mesh generator, Interantional Journal of Fracture 108: 251-274, 2001. [9] M. Paraschivoiu, J. Peraire, A.T. Patera, A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equa- tions, Computer Methods in Applied Mechanics and Engineering. 150 (1997) 289-312. [10] N. Pares, J. Bonet, A. Huerta and J. Peraire, Guaranteed bounds for outputs of interest in linear elasticity, submitted to Comp. Meth. for Appl. Mech. and Engrg., 2003. [11] A.T. Patera and J. Peraire, A general Lagrangian formulation for the computation of a-posteriori finite element bounds’, chapter in ”Error Estimation and Adaptive Discretization Methods in CFD”, Eds: T. Barth & H. Deconinck, Lecture Notes in Computational Science and Engineering, Vol. 25, 2002. [12] J. Peraire, A.T. Patera, Bounds for linear-functional outputs of coercive partial differential equations: Local indicators and adaptive refinement, Proceedings of the Workshop On New Advances in Adaptive Compu- tational Methods in Mechanics, edited by P. Ladeveze and J.T. Oden, Elsevier 1998. [13] J.R. Rice, A path independent integral and approximate analysis of strain concentration by notches and cracks, Journal of Applied Mechanics, 35 (1968) 379-386. [14] M. Ruter, E. Stein, Goal-oriented a posteriori error estimates in elastic fracture mechanics, Fifth World Congress on Computational mechanics, Vienna, Austria, July, 2002. [15] A.M. Sauer-Budge, J. Bonet, A. Huerta and J. Peraire, Computing bounds for linear outputs of exact weak solutions to Poisson’s equation, submitted to SINUM, March 2003.

[16] B.A. Szabo, Estimation and control of error based on O convergence, chapter 3 in Accuracy Estimates and Adaptive Refinements in Finite

Element Computations, edited by I. Babuska,P O.C. Zienkiewicz, J. Gago and E.R. de A. Oliveira, 1986, John Wiley.