Sensitivity Analysis of Linear Elastic Cracked Structures Using Generalized Finite Element Method

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Sensitivity Analysis of Linear Elastic Cracked StructuresUsing Generalized Finite Element MethodMahendra Kumar Pala & Amirtham Rajagopalba Department of Civil Engineering, University of Tokyo, Tokyo, Japanb Department of Civil Engineering, Indian Institute of Technology Hyderabad, IndiaPublished online: 19 Aug 2014.

To cite this article: Mahendra Kumar Pal & Amirtham Rajagopal (2014) Sensitivity Analysis of Linear Elastic Cracked StructuresUsing Generalized Finite Element Method, International Journal for Computational Methods in Engineering Science andMechanics, 15:5, 422-437, DOI: 10.1080/15502287.2014.917740

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International Journal for Computational Methods in Engineering Science and Mechanics, 15:422–437, 2014Copyright c© Taylor & Francis Group, LLCISSN: 1550-2287 print / 1550-2295 onlineDOI: 10.1080/15502287.2014.917740

Sensitivity Analysis of Linear Elastic Cracked StructuresUsing Generalized Finite Element Method

Mahendra Kumar Pal1 and Amirtham Rajagopal2

1Department of Civil Engineering, University of Tokyo, Tokyo, Japan2Department of Civil Engineering, Indian Institute of Technology Hyderabad, India

In this work, a sensitivity analysis of linear elastic cracked struc-tures using two–scale Generalized Finite Element Method (GFEM)is presented. The method is based on computation of materialderivatives, mutual potential energies, and direct differentiation.In a computational setting, the discrete form of the mutual poten-tial energy release rate is simple and easy to calculate, as it onlyrequires the multiplication of the displacement vectors and stiff-ness sensitivity matrices. By judiciously choosing the velocity field,the method only requires displacement response in a sub-domainclose to the crack tip, thus making the method computationallyefficient. The method thus requires an exact computation of dis-placement response in a sub-domain close to the crack tip. To thisend, in this study we have used a two-scale GFEM for sensitiv-ity analysis. GFEM is based on the enrichment of the classicalfinite element approximation. These enrichment functions incor-porate the discontinuity response in the domain. Three numericalexamples which comprise mode-I and mixed mode deformationsare presented to evaluate the accuracy of the fracture parameterscalculated by the proposed method.

Keywords Generalized finite element, Shape sensitivity, Fracturemechanics, Enrichment functions

1. INTRODUCTIONShape sensitivity analysis procedures that are frequently em-

ployed for structural design optimization [1] offer an excel-lent procedure for direct, analytical evaluation of first–orderand higher-order derivatives of potential energy. Such proce-dures also find applications in adaptive finite element strategies[2] in computing the nodal sensitivities. In linear elastic frac-ture mechanics, the structural components containing cracksare parametrized in terms of stress intensity factors and strain

Address correspodence to Dr. Amirtham Rajagopal, Indian Instituteof Technology Hyderabad, ODF Campus, Yeddumailaram, Hyderabad502205, India. E-mail: [email protected]

Color versions of one or more of the figures in the article can befound online at www.tandfonline.com/ucme.

energy release rates. The stress and strain fields ahead of thecrack tip are expressed in terms of these parameters and pro-vide a mechanistic relationship between the residual strengthof a structural component to the size and location of a crackin that component. Sensitivity analysis of a crack driving forceplays an important role in many fracture mechanics applica-tions involving the stability and arrest of crack propagation [3],reliability analysis [4], parameter identification, and other con-siderations. For instance, the derivatives of the stress intensityfactors or other fracture parameters are often required to pre-dict the probability of fracture initiation and/or instability incracked structures [5]. In all these situations, the evaluation ofresponse derivatives with respect to crack size is a challeng-ing task and requires shape sensitivity analysis [6] and mustbe based on variational principles [7]. There have been severalstudies on shape sensitivity analysis for mode-I conditions usinga virtual crack extension technique (for instance, see [8, 9]) orby introducing a direct integration approach for virtual crackextension [10] for single crack or multiple crack systems. Therehave been many applications of concepts of shape sensitivity forevaluation of first-order derivatives of potential energy [11–14].These approaches have also been used for energy release ratewithout any mesh perturbation. There have also been exten-sions of these methods to formulate the second-order sensitivityof the potential energy [5] in order to compute the first-orderderivative of energy release rate. However, since the method iscomplex, no attempts have been made to present the numericalsolutions. To overcome such complexities, the domain integralrepresentation of the J− integral [15] that required only the first-order sensitivity of the displacement field was used for linearelastic cracked structures in mode I and mixed modes [3, 16].The methods have also been extended to functionally gradedmaterials [17]. Most of these approaches for shape sensitivityanalysis are based on the variational formulation of equilibriumequation and are solved using the finite element method. Therehave been many works on improving or enriching the classi-cal FEM approximations and have been used for studies on

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SENSITIVITY ANALYSIS USING GFEM 423

FIG. 1. Shape function with intrinsic crack tip enrichment functions (a)√

rsin(θ/2), (b)√

rcos(θ/2), (c)√

rsin(θ )sin(θ/2), (d)√

rsin(θ )cos(θ/2).

sensitivity analysis using the so-called extended finite elementmethods [18].

This paper presents a methodology of predicting first-ordersensitivity of the J−integral for a crack in a homogeneous,isotropic, and linear-elastic structure subjected to mode-I load-ing conditions using the generalized finite element method(GFEM) [19]. The method involves the material derivative con-cept of continuum mechanics, domain integral representationof the J−integral and direct differentiation. In the GFEM, [20],the approximations are constructed by combining the local baseswith partition of unity functions [21]. The main objective of theenrichment functions is to empower the approximations so asto capture accurately the fields ahead of the crack tip. Hencejudicious choice of enrichment functions is important [22]. Thechoice of the enrichment functions[23] is made by consideringthe interactions between the local (near crack tip)[24] and theglobal (structure) behavior[25]. Thus, the GFEM with global-local enrichment used a two-scale decomposition of the solutionof a fracture mechanics problem: a smooth coarse scale com-ponent and a singular fine scale component. There have beenseveral works reported in the literature [26] on applications ofgeneralized finite element method for solving problems[27].Such a procedure allows us to compute the sensitivity of thefracture parameters accurately because of the improved dis-placement approximations ahead of the crack tip.

The paper is organized as follows: Section 2 presents shapefunction used in GFEM like crack tip enrichment and modifiedHeaviside shape function. Section 3 explains the discretizationprocess using GFEM. This section also illustrates the differencebetween global and local approaches. Section 4 illustrates thevelocity field that is adopted in this study. Sections 5 and 6 ex-plains the J-integral and the sensitivity of J-integral formulation.Section 7 present several numerical examples to demonstrate theperformance of the proposed sensitivity analysis method usingGFEM. Finally, Section 8 and 9 present results and conclusionsconcerning the proposed formulation.

2. GENERALIZED FINITE ELEMENT METHODThe GFEM [19, 20–22, 25] uses the concept of partition

of unity method (PUM) to build new approximation functions[23, 26, 28]. The GFEM approximation space consists of threecomponents: (a) patches or clouds ωα is a union of the finite el-ements sharing a node α of the finite element mesh covering thedomain �; i.e., � = ∪N

α=1ωα; (b) partition of unity subordinateto cover {ωN

α=1}: the Lagrangian finite element shape functionsϕα, α = 1, ....N , of the finite element mesh covering the domainof interest � constitute a partition of unity; i.e.,

∑Nα=1 ϕα(x) = 1

for all x in �; (c) the patch or cloud approximation spaces χα:to each cloud or patch we associate a DL(α)-dimensional space

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424 M.K. PAL AND A. RAJAGOPAL

FIG. 2. Shape function with crack tip enrichment functions.

χα of functions defined on ωα , namely

χα = span{Lαi, 1 ≤ i ≤ DL(α), Lαi ∈ H 1(ωα)} (1)

The basis functions Lαi above are also known asenrichment functions . In the GFEM framework, a shape func-tion is built by combining a standard finite element shape func-tion ϕα and an enrichment function Lαi[29].

φiα(x) = ϕα(x)Lαi(x) (2)

In the context of a discretized finite element domain, theGFEM approximation uhp(x) of a displacement field u(x) isdefined as

uhp(x) =N∑

α=1

DL∑i=1

uαiϕα(x)Lαi(x) (3)

where uαi are the nodal degrees of freedom at the element nodeα, N is the number of nodes, and DL is the number of enrichmentfunctions assigned to node α. uhp(x) is the local approximationdefined in the support domain ωα , which is the union of the finiteelement sharing the same node xα [30]. Appropriate choice ofthe enrichment functions is required to be made based on specificapplications [31]. In fracture mechanics applications, specialfunctions can be used to simulate the presence of cracks withoutthe need for remeshing or adaptive mesh refinement [32]. Therehave been several works giving details on choice of enrichment

functions like, for instance, harmonic functions [20], polynomialfunctions [19, 33], global-local approximations [30, 34], enric-ment functions based on proper and orthogonal decomposition[24] amongst others. The set of polynomial enrichment func-tions [33] at a support domain ωα associated with a node Xα isgiven by

{Lαi}DL

i=1 ={

1,{X1 − X1α}

hα

,{X2 − X2α}

hα

,{X3 − X3α}

hα

....

}(4)

where X1α,X2α,X3α are the coordinates for the node xα;X1, X2, X3 are the coordinates of the point where the en-richment is computed; and hα is a scaling factor. There-fore the GFEM approximation of the continoius field uαi isgiven by

uhpα (x) =

DL∑i=1

uαiLp

αi(x) (5)

if Lαi represent an approximation set containing classicalfinite element approximation LFE

αi and local approximationLlocal

αi like, Lαi = {LFEαi , Llocal

αi }. Then, there exist two possi-ble modes for defining local approximation to enrich a element:enriching the vector coefficient, intrinsic enrichment, and en-riching approximation, extrinsic enrichment. A simple modeldescribing the intrinsic way of enriching the approximation is

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FIG. 3. Shape function with modified Heaviside function.

as follows in Figure 1.

LFEαi =

{LFE

αi − 1 for x ∈ ωα

LFEαi for x /∈ ωα

}(6)

The crack tip enrichment functions are shown in Figure 2.The modified shape functions are shown in Figure 3.

Another example includes Mode I and Mode II crack prob-lems, the local approximation Llocal

αi are assumed by con-sidering the crack tip parameter in polar coordinate form.The approximation field variable in intrinsic form is asfollows,

Llocalαi (r, θ ) = {√

rsin(θ/2),√

rcos(θ/2),√

rsin(θ )sin(θ/2),√rsin(θ )cos(θ/2)

}(7)

Intrinsic enrichment techniques have serious drawback, likesimilar strain field across the discontinuity and lower numberof degree of freedom for enriching the approximation affectingquality of solution.

The approximation of field variable expressed in ex-trinsic form with a heaviside function can be written asfollows,

H (r) ={

1 for r > 00 for r < 0

}(8)

Modified Heaviside functions often include a shifting of thefunction to make the field variable interpolate at the prescribedlocation. The modified shape functions are shown in Figure 2.

Level set functions are used for defining the presence of crackin an element. A level set function is defined based on a distancemetric. A sinusoidal level set function used as an enrichment anddefined over a patch is shown in Figure 4 a. The classical finiteelement shape functions are shown in Figure 4 b. The enrichedshape functions are shown in Figure 4 c.

3. GLOBAL-LOCAL PROBLEMA linear elastic body having domain �Global and bound-

ary Global is as shown in Figure 5. Let t represent pre-scribed traction over unit surface area t

Global and u repre-sent prescribed displacement over surface u

Global , such thatt

Global ∪ uGlobal ∈ Global . Finally, f represent body force over

per unit volume of the domain �. Then, the state of stressrepresented by stress tensor σ is resultant of body force f, sur-face traction t, and surface displacement u acting on the body�Global . For the equilibrium of the body �, the resultant equa-tion is written as,

∇ · σ + f = 0 in �Global (9)

σ · n = t at tGlobal (10)

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FIG. 4. Generalized finite element method shape function: (a) Classical finiteelement linear shape functions: (b) Enriched shape functions based on level sets:(c) Enriched shape functions.

u = u at uGlobal (11)

σ = C : ε in �Global (12)

ε = ∇su (13)

where n is the outward normal vector on tGlobal , σ is the cauchy

stress tensor, C is the hookean constitutive tensor, and ε isthe linear strain tensor. Let uGlobal be the generalized finiteelement solution of the initial static problem (see Figure 5)for the field displacement u. The weak form of the problem isstated as Find uGlobal ∈ XGlobal(�Global) ⊂ H 1(�Global) suchthat ∀ vGlobal ∈ XGlobal(�Global), which satisfies the governing

equation.∫�Global

σ (uGlobal) : ε(vGlobal) dx + β

∫u

Global

uGlobal · vGlobal ds

=∫

tGlobal

t · vGlobal ds + β

∫u

Global

u · vGlobal ds (14)

where vGlobal are the virtual displacements, β is the penaltyparameter used to assign displacement boundary conditions de-fined on u

Global , and XGlobal(�Global) is a discretization of theHilbert space H 1(�Global) defined in �Global . In the presenceof cracks, the space XLocal(�Local) has GFEM shape functionsbuilt with discontinous and singular enrichments. Equation (14)is solved in the same way as in the global problem with thedifference that the boundary conditions can now be prescribedusing the far-field smooth global solution. The local solutionuLocal thus obtained is used as an enrichment function in theglobal coarse mesh through global-local generalized finite ele-ment shape functions. The enriched shape functions defined onthe coarse-scale (global) mesh are defined as

φα(x) := ϕα(x)uLocal(x) (15)

where the partition of unity function ϕα is provided by a global,coarse, finite element mesh and uLocal has the role of enrich-ment or basis function for the cloud space χα(ωα). uLocal at anystage of analysis can thus be interpreted as a global-local en-richment function and the function defined above is denoted asa global-local GFEM shape function. The global generalized fi-nite element space containing the shape functions φα is denotedas XGlobal(�Global) and is given by

XGlobal(�) ={

uhp

Global =N∑

α=1

ϕα(x)uhpα (x)

+∑

β∈Jglobal

ϕβ(x)uglobal

β (x)

}(16)

where Jglobal is the index set of nodes. The first term of the sumin the above expression represents coarse-scale approximationand the second term represents the fine scale approximation.

4. VELOCITY FIELDConsider a 3D body with domain �, boundary , and body

material point with position vector X ∈ �. Assume that at timeT = τ body will have domain �τ and boundary τ , as shownin Figure 5. Translation from one place to another place can beexpressed as

T = X → Xτ ∀ X ∈ � (17)

where X and Xτ correspond to position vectors in reference andperturbed configurations. If T is the transformation mapping,and τ is the scalar time with the parameters of shape change,

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FIG. 5. Generalized finite element method: global and local problems.

then there exists a relation as

Xτ = T(X, τ ) (18)

�τ = T(�, τ ) (19)

�τ = T(�, τ ) (20)

A velocity field can be defined as time rate of change of positionvector and it is given as,

V(Xτ , τ ) = d Xτ

dτ= d T(X, τ )

dτ= ∂ T(X, τ )

∂τ(21)

Using Taylors expansion and ignoring the higher–orderterms, the transformation mapping can be approximated by

T(X, τ ) = T(X, 0) + τ∂ T(X, 0)

∂τ+ HOT

T(X, τ ) ∼= X + τ V(X, 0) (22)

4.1 Velocity Field DefinitionThe velocity field used in this study has been defined as

V(X) =⎧⎨⎩

V1(X)

V2(X)

⎫⎬⎭ = V1,tip =

⎧⎨⎩

A1(X1)A2(X2)

0

⎫⎬⎭ (23)

where

A1(X1) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

1 if |X1| ≤ b1

X1 − B + 2a

b1 − B + 2aif X1 > b1

X1 + 2a

b1 + 2aif X1 > −b1

(24)

A2(X2) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

1 if |X2| ≤ l1

X2 − (L/2)

l1 − (L/2)if X2 > l1

X2 + (L/2)

l1 + (L/2)if X2 > −l1

(25)

The velocity field is assumed constant over the region wheredomain integral and sensitivity are evaluated. The velocity fieldis zero at the boundary and varying bilinear at the other domainexcluding the crack face and essential boundary. At the crackface and essential boundary it is assumed a linear variation. Theassumption is depicted in Figure 6.

5. J INTEGRALJ−integral is a characterization parameter of the fracture

problem. It characterizes the state of affairs in the region ahead

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P Q

RS

B

Crack Width

2a

L 2l

2b

X1

X1

V1 = 0

V1 = 0

V1 = 0 V1 = 0V1(x)

FIG. 6. Schematic diagram of velocity field in the domain.

of the crack tip. Energy release density (G) is a special case of theJ−integral. G is limited to the linear elastic material, whereasJ-integral is applicable to all kinds of material behavior; i.e.linear, nonlinear, hyper-elastic and many more. J−integral isthe most suitable parameter to be determined as the crack tipexhibits elastic-plastic behavior. For a plane problem, considera path around the crack tip, which starts from one surface ofthe crack and ends at the other surface of the crack. This pathis arbitrary in nature. It can be smooth or it may have somesharp corner, but the path should be continuous and should bewithin the material. The J-integral was first applied to fractureproblems for the plane problem. J-Integral is defined as

J =∫

(Wdx2 − Ti

∂ui

∂x1d

)(26)

where W = ∫ εσij dεij is the strain energy density and Ti is

the component of traction vector, and ui is the component ofdisplacement vector. For the two dimensional case, W can bemore explicitly expressed as

W =∫ ε11

σ11dε11 +∫ ε12

σ12dε12 +∫ ε21

σ21dε21 +∫ ε22

σ22dε22

(27)

For, the symmetric and elastic problem σ12 = σ21, hence

W =∫ ε11

σ11dε11 + 2∫ ε12

σ12dε12 +∫ ε22

σ22dε22 (28)

Traction components Ti at the points on the path are expressedthrough the well-known Cauchys relation Ti = σijnj . Thus, by

Start

Solve the bilinear form using GFEMand get the displacement u

bA(u, u) = LA(u)

Solve the bilinear form using GFEMand get the material derivative u

bA(u, u) = LV (u) − bV (u, u)

Compute the function

G1, G2, G3, ..., G6

Compute the sensitivity of J

J =∑6

i=1

∫Gi dA

End

FIG. 7. Flowchart for sensitivity analysis.

knowing the stress field and direction of the normal at the pointon path , we can determine the components of traction vector(Ti). Finally, we can come up with the value of the J-Integral.The flowchart for sensitivity analysis is presented in Figure 7and is discussed in detail in the next section.

6. SENSITIVITY ANALYSISThe variation form of the governing equation can be written

as [19]

bA(u, v) = lA(v) ∀ v ∈ u (29)

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Crack Width

a = 7.5 unit

B = 15 unit

L = 35 unit

δ = 1unit

FIG. 8. Numerical Example I: SE(T) specimen with top displacement.

where u and v are actual displacement and virtual displacementof the structures, and

bA(u, v) =∫

σij (u)εij (v) d� (30)

lA(v) =∫

f bi vi d� +

∫

Ti vi d (31)

Crack Width

a = 7.5 unit

B = 15 unit

L = 35 unit

δ = 1unit

δ = 1unit

FIG. 9. Numerical Example II: SE(T) specimen with symmetric displacement.

Crack Width

a = 7.5 unit

B = 15 unit

L = 35 unit

δ = 1unit

E1 and ν1

E2 and ν2

FIG. 10. Numerical Example III: Bi-material of SE(T) specimen with topdisplacement.

6.1 Material DerivativeFor a given point X ∈ �, material derivative is expressed as

[19]

u = limτ→0

[uτ (X + τV(X)) − u(X)

τ

](32)

If uτ is continuous, then it can be extended in the neighborhoodof στ .

u = u′(X) + ∇uT V(X) (33)

where,

u′(x) = limτ→0

[uτ (X) − u(X)

τ

](34)

TABLE 1Sensitivity analysis for edge crack problem: Plain strain

condition (Numerical Example I)

Normalized Sensitivity

a/BSIF of J-Integral Difference

Ratio GFEM FEM GFEM Direct Method %

0.1 0.4083 0.3931 0.3273 0.3666 -0.03930.2 1.4639 1.4888 0.2842 0.2833 0.00080.3 1.5171 1.7160 0.3689 0.3433 0.02550.5 1.2660 1.0628 0.5852 0.5633 0.0218

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TABLE 2Sensitivity analysis for edge crack problem: Plain stress

condition (Numerical Example I)




0.1 0.4058 0.4101 0.2580 0.2433 0.01460.2 1.4639 1.3831 1.3460 1.3366 0.00930.3 1.5168 1.3247 1.5030 1.4733 0.02960.5 1.2658 1.3160 1.9540 2.1533 -0.1993

Material derivative has commutative properties; i.e.,(∂u∂Xx

)′= ∂(u)′

∂Xx

(35)

6.2 Variational FormulationLet’s assume a domain functional �1 defined over the domain

�τ . In the integral form, it can be expressed as

�1 =∫

�τ

fτ (Xτ ) d�τ (36)

where fτ is regular function over domain �τ and domain � isCk regular. For the assumed functional, the material derivative[19] can be written as

�1 =∫

�

[f ′(X) + div(f (X)V(X))] d� (37)

For the functional form of:

�2 =∫

�τ

g(uτ ,∇uτ ) d�τ (38)

The material derivative of �2 in domain � can be written as

�2 =∫

�

[g,ui

ui − g,ui(ui,jVj ) + g,ui,j

ui,j

− g,ui,j(ui,jVj ),j + div(g V)

]d� (39)

TABLE 3Sensitivity analysis for edge crack problem: Plain strain

condition (Numerical Example II)




0.1 0.4721 0.4921 0.5110 0.4973 0.01370.2 0.6363 0.7168 0.1566 0.1366 0.01990.3 1.0766 1.0642 0.1092 0.1101 -0.00080.5 1.0060 1.0628 0.3175 0.3333 -0.0158


condition (Numerical Example II)




0.1 0.4721 0.4401 0.0939 0.1005 -0.00650.2 0.6356 0.6838 0.4212 0.4333 -0.01210.3 1.0766 1.0247 0.4693 0.4700 -0.00070.5 1.0062 1.0100 1.6345 1.6066 0.0279

If no body force is equal to zero, then the variational equation(17) can be written as

bA(u, u) ≡∫

σij (u) εij (u) d� = lA(u)

≡∫

Ti zi d (40)

Taking the material derivative of Equation (28) and using Equa-tions (20) and (21)

bA(u, u) = l′V (u) − b′V(u, u)∀u ∈ U (41)

where V denotes the velocity field dependency

l′V(u) =∫

{ − Ti(ui,jVj ) + [(Tiui),j nj

+κ(Tiui)](Vi ni)

}d (42)

b′V (u, u) =

∫�

[σij (u)(ui,kVk,j ) + σij (u)(ui,kVk,j )

− σij (u) εij (u) ∇V]

(43)

where nj is the j th component of unit normal n to domain, andκ is the curvature of the boundary.


condition (Numerical Example III) E1/E2 = 0.1




0.1 1.2960 1.3248 3.9800 4.1100 5.10000.3 1.5090 1.5283 93.1100 87.7500 -5.7300

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TABLE 6Sensitivity analysis for edge crack problem: (Numerical Example IV) E2/E1 = 0.1, fixed grip loading (constant strain ε0)

a/B Normalized SIF Sensitivity of J-Integral DifferenceRatio GFEM Analytical [35] FEM[17] GFEM Direct Method %

0.1 1.1625 1.1648 1.1577 8.0746 8.0745 0.00020.2 1.2962 1.2963 1.2892 18.0526 18.0527 0.00050.3 1.5062 1.5083 1.4965 41.587 41.589 0.00110.4 1.8137 1.8246 1.8009 99.947 99.946 0.00130.5 2.294 2.3140 2.2583 259.918 259.923 0.00210.6 3.1542 3.1544 2.9997 773.549 773.619 0.0026

6.3 Sensitivity of J-IntegralIn the absence of the body force, equilibrium (14) can be

rewritten as

J =∫

�

[σij

∂ui

∂x1− W δ1j

]∂q

∂xj

d� (44)

where δ1j is kronecker delta, q is arbitrary weight function,which varies linearly for zero to one. � is the enclosed area bythe contour . Equation (32) can be expanded further as

J =∫

�

[(σ11

∂u1

∂x1+ σ12

∂u2

∂x1

)∂q

∂x1

+(

σ12∂u1

∂x1+ σ22

∂u2

∂x1

)∂q

∂x2− W

∂q

∂x1

]d� (45)

For the 2D problem (i.e., plane stress and plain strain),Equation (33) can be written as

J =∫

�

g d� (46)

where g = g1 +g2 +g3 +g4 −g5 −g6. The explicit expressionsof gi can be found in Appendix I. Material derivative of theJ-integral cab be written as

J =∫

�

[g′ + ∇(gV)] d� (47)

where g′ = g′1 + g′

2 + g′3 + g′

4 − g′5 − g′

6. V is the velocity field.It has been described in the above section and the schematicdiagram of the velocity variation is shown in Fig. 4. Assumingthe crack length as the variable increasing in the direction x1,the expression of Equation (34) can be rewritten as

J =∫

�

[G1 + G2 + G3 + G4 − G5 − G6] d� (48)

where

G = g′i + ∇(giV)i = 1, 2, 3, ...6 (49)

The formula for g1 and G1 is given below

g1 = E

2(1 − ν2)

[∑Ni,xui +

∑Ni,x

∑�kbik

+∑

Ni

∑�k,xbik

]2 ∂q

∂x1(50)

The explicit expression of Gi can be find in Appendix 2.

G1 = E

1 − ν2

(∑Ni,xui +

∑Ni,x

∑�kbik

+∑

Ni

∑�k,xbik

)2 ∂u1

∂x1

∂q

∂x1− E

1 − ν2

(∑Ni,xui

+∑

Ni,x

∑�kbik +

∑Ni

∑�k,xbik

)2 ∂V1

∂x1

∂q

∂x1

(51)

TABLE 7Sensitivity analysis for edge crack problem: (Numerical Example IV) E2/E1 = 0.1, membrane loading (constant tensile stress σt )


0.1 0.8119 0.8129 0.8096 5.4723 5.4748 0.00110.2 1.2964 1.2965 1.2925 23.0412 23.0411 0.00310.3 1.8580 1.8581 1.8488 71.7512 71.7514 0.00060.4 2.5695 2.5699 2.5454 205.412 205.224 0.00130.5 3.5681 3.5701 3.5001 601.867 601.8302 0.00240.6 5.1878 5.1880 4.9645 1939.687 1938.654 0.0029

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TABLE 8Sensitivity analysis for edge crack problem: (Numerical Example IV) E2/E1 = 0.1, pure bending loading (linear stress σb)


0.1 2.0423 2.0427 2.0289 5.4723 5.4748 0.00010.2 1.9039 1.9040 1.8907 23.0412 23.0411 0.00040.3 1.8853 1.8859 1.8662 71.7512 71.7514 0.00030.4 1.9776 1.9778 1.9437 205.412 205.224 0.00120.5 2.2150 2.2151 2.1468 601.867 601.8302 0.00240.6 2.7169 2.7170 2.5544 1939.687 1938.654 0.0029

7. NUMERICAL EXAMPLES

7.1 Example I: SE(T) Specimen with Top DisplacementConsider a single-edged-tension [SE(T)] specimen with

width W = 15 units, length L = 35 units, and a crack lengtha, subjected to a far-field remote displacement δ = 1 unit. Theplate with four a

Wratios, namely 0.1, 0.2, 0.3 and 0.5, has been

considered for analysis. The elastic modulus, shear modulus andpoissons ratio are considered as E = 35 units, G = 26 units,and ν = 0.33, respectively and both plane stress and plane strainconditions were studied.

The top surface of the SE(T) specimen was subjected touniform displacement of 1 unit, as shown in Figure 8. The modelconsisted of 78 and 112 elements for a

B= 0.1 and 0.2, 0.3 and

0.5. SIF and sensitivity of J-integral have been calculated nearthe crack tip for all the given cases. A domain of 2l1 ∗ 2b1 hasbeen considered for calculating the J−integral and its sensitivityfor a

B= 0.1, 0.2, 0.3, and 0.5.

7.2 Example II: SE(T) Specimen with SymmetricDisplacement

The objective of numerical example II is to compute the sen-sitivity of the J−integral around the crack tip for symmetricaldisplacement (displacement at the top and bottom surface) keep-ing the geometry of SE(T) the same as numerical example I, asshown in Figure 9.

7.3 Example III: Bi-material of SE(T) Specimen with TopDisplacement

Two different materials with the modulus ration E1E2

= 0.1are considered for the sensitivity analysis of the J−integralaround the crack tip. The placement of crack and the inter-face of the two materials considered are as shown in Figure 10.The geometry of materials, crack geometry, and discretizationare maintained identical to numerical example I. A unit dis-placement is provided only at the top surface of the SE(T)specimen and the example is solved only for the plane stresscondition.

7.4 Example IV: SE(T) Specimen with Varying Propertiesunder Symmetric Tensile and Linear Loading

An edge-cracked plate with length L = 8 units, widthB = 1 unit , and crack length a is considered. Three loadingconditions were considered, including uniform fixed grip load-ing (constant strain ε0), membrane loading (constant tensilestress σt ), and pure bending (linear stress σb). A varying mate-rial property was taken into consideration. The elastic moduluswas assumed to follow an exponential function, given by

E(x1) = E1exp(βx1), 0 ≤ x1 ≤ W (52)

where E1 = E(0), E2 = E(W ), and β = ln(E2E1

) are twoindependent material parameters that characterize the elasticmodulus variation. The following numerical values were used:E1 = 1 units, E2/E1 = exp(β) = 0.1, 0.2, 5 and 10, anda/B = 0.1, 0.2, 0.3, 0.4, 0.5 and 0.7 was used. Poissions ra-tio was constant with ν = 0.3. A plane strain condition wasassumed. The results of the analysis were compared with thetheoretical solution given by Erdogan and Wu[35] and with thenumerical results presented in [17].

8. RESULTS AND DISCUSSIONTables 1–4 contain the results for numerical examples I and

II by considering two methods. The first method is based onvariational formulation of GFEM and the second is a directmethod is based on the first principle of derivative. Since no an-alytical solution was available to validate, we have adopted thedifferentiation technique. The first principle of differentiation isdefined as the change in the value of function with respect tothe small change (tending to zero) in independent variable, say4–6% in each case. Results show that the predicted, normalizedSIF obtained in the present study agrees very well with the directdifferentiation method and for various a/W ratios.The results inTables 1–4 demonstrate that continuum shape sensitivity analy-sis provides accurate estimates of ∂J/∂a when compared withcorresponding results using the finite-difference method. Unlikethe virtual crack extension technique, no mesh perturbation isrequired using the proposed method.

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The results tabulated in Table 5 correspond to Numerical Ex-ample III. The J−integral depends on the various independentvariables such as a

Wratio, loading, and boundary conditions.

Results show that the J-integral is very sensitive to the appliedload, the geometry of the model, and the analysis type. Numer-ical results show that the variational form of GFEM providesa more accurate estimate than the direct method. Unlike thedirect method, no finer meshing required for J-integral com-putation. The difference in results between the two approachesis 3%.

Tables 6-8 shows the normalized mode-I stress intensity fac-tors K1/σ0

√πa, K1/σt

√πa, K1/σb

√πa and ∂J/∂a under

fixed grip, membrane loading, and bending, respectively, forvarious a/B ratios and for E2/E1 = 0.1. It is observed that thepredicted, normalized SIF obtained in the present study agreesvery well with the analytical results [35] and with the finiteelement results [17] for all types of loading and for variousa/B ratios. Tables 6-8 also show the results for ∂J/∂a, the firstcomputed using the proposed method and the second calculatedusing the finite difference method. A pertubation of 10−5 timesthe crack length was used in the finite difference calculation.The results indicate that continuum shape sensitivity analysisusing generalized finite element method provides accurate esti-mates of ∂J/∂a when compared with the corresponding resultsusing the finite difference method and finite element method.

9. CONCLUSIONA new method has been developed for sensitivity analysis of

the discontinuity in elastic, isotropic, and homogeneous materialusing the GFEM as the tool for fracture analysis. The proposedmethod involves the material derivatives, displacement deriva-tives, and J−-Integral calculation. All these steps are based onthe variational form, which required the discretization of thedomain. The GFEM with enrichment functions has shown to bea very advanced tool for fracture analysis. Because no pertur-bation in mesh is involed in the present method, the errors dueto the discretization are minimized. The difference in resultsbetween the two approaches is 3%.

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5. E. Taroco, Shape Sensitivity Analysis in Linear Elastic Fracture Mechanics,Comp. Meth. in Appl. Mech. and Eng., vol. 188, pp. 697–712, 2000.

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7. K. Washizu, Variational Methods in Elasticity and Plasticity, PergamonPress, New York, 1968.

8. H. G. deLorenzi, On the Energy Release Rate and the J-integral for 3DCrack Configurations, Int. J. Fracture, vol. 19, pp. 183–193, 1982.

9. H. M. Koh, R. B. Haber, Explicit Expressions for Energy Release RatesUsing Virtual Crack Extensions, Int. J. Num. Meth. in Eng., vol. 21, pp.301–315, 1985.

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11. J. N. Reddy, E. J. Barbero, The Jacobian Derivative Method for ThreeDimensional Fracture Mechanics, Com. in App. Num. Meth., vol. 6, pp.507–518, 1990.

12. R. Saliba, E. Taroco, M. J. Venere, R. A. Feijoo, C. Padra, Shape SensitivityAnalysis for Energy Release Rate Evaluation and its Application to theStudy of Three Dimensional Cracked Bodies, Comp. Meth. in Appl. Mech.and Eng., vol. 188(4), pp. 507–518, 1990.

13. K. K. Choi, K. H. Chang, A Study of Design Velocity Field Computationfor Shape Optimal Design, Finite Elements in Analysis and Design, vol.15(4), pp. 317–341, 1994.

14. V. Komkov, E. J. Haug, K. K. Choio, Design Sensitivity Analysis ofStrucutres Systems, Academic Press, New York, 1986.

15. Y. H. Park, G. Chen, S. Rahman, Shape Sensitivity Analysis of LinearElastic Cracked Structures under Mode-I Loading, J. Pressure Vessel Tech.,vol. 124, pp. 476–482, 2002.

16. Y. H. Park, G. Chen, S. Rahman, Shape Sensitivity Analysis in Mixed ModeFracture Mechanics, Comp. Mech., vol. 27, pp. 282–291, 2001.

17. B. N. Rao, S. Rahman, Continuum Shape Sensitivity Analysis of theMode-I Fracture in Functionally Graded Materials, Comp. Mech., vol. 36,pp. 62–75, 2005.

18. K. H. Chang, M. S. Edke, Shape Sensitivity Analysis for 2d Mixed ModeFractures Using XFEM and Level Set Method, Mech. Based Design ofStructures and Machines., vol. 38, pp. 328–347, 2010.

19. K. Copps, I. Babuska, T. Strouboulis, The Design and Analysis of theGeneralized Finite Element Method, Comp. Meth. in Appl. Mech. andEng., vol. 181(1), pp. 43–69, 2000.

20. I. Babuska, T. Strouboulis, K. Copps, The Generalized Finite ElementMethod, Comp. Meth. in Appl. Mech. and Eng., vol. 190, pp. 4081–4193,2001.

21. I. Babuska, J. M. Melnek, The Partition of Unity Finite Element Method,Comp. Meth. in Appl. Mech. and Eng., vol. 139, pp. 289–314, 1996.

22. J. E. Osborn, I. Babuska, U. Banerjee, On the Principle for Selection ofShape Functions for the Generalized Finite Element Method, Comp. Meth.in Appl. Mech. and Eng., vol. 191, pp. 5595–5629, 2002.

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25. C. A. Duarte and D. J. Kim, Analysis and Applications of a GeneralizedFinite Element Method with Global Local Enrichment Functions, Comp.Meth. in App. Mech. and Eng., vol. 197(6–8), pp. 487–504, 1998.

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33. O. C. Zeinkiewicz, J. T. Oden, C. A. Durate, A New Cloud Based HP FiniteElement Method, Comp. Meth. in App. Mech. and Eng., vol. 153(1-2), pp.117–126, 1998.

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APPENDIX I

Plane Stress Condition

g1 = E

1 − ν2

{1

2

I∑i=1

Ni,xu2i−1 +{

I∑i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}b(2i−1) k

}2∂q

∂x1(53)

g2 = E

1 + ν

{I∑

i=1

Ni,xu2i−1

{+

I∑i=1

Ni,x

∑k

φk +I∑

i=1

Ni

}

×∑

k

φk,xb(2i−1)k

}∂u2

∂x1

∂q

∂x1(54)

g3 = E

1 + ν

{1

2

I∑i=1

Ni,yu2i−1 +{

I∑i=1

Ni,y

∑k

φk +I∑

i=1

Ni

×∑

k

φk,y

}bik +

I∑i=1

Ni,x

∑k

φk +I∑

i=1

Ni

∑k

φk,x

}

×{

I∑i=1

Ni,xu2i−1 +{

I∑i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}∂q

∂x1(55)

g4 = E

1 − ν2

{I∑

i=1

Ni,yu2i +{

I∑i=1

Ni,y

∑k

φk +I∑

i=1

Ni

×∑

k

φk,y

}+ ν

{I∑

i=1

Ni,xu2i−1 +I∑

i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}}b(2i−1)k

∂u2

∂x1

∂q

∂x2(56)

g5 = E

1 + ν

{1

2

{I∑

i=1

Ni,yu2i−1 +{

I∑i=1

Ni,y

∑k

φk, y

}

× bik +{

I∑i=1

Ni,x

∑k

φk +I∑

i=1

Ni

∑k

φk,x

}b2ik

}}2

× ∂q

∂x1(57)

g6 = E

1 − ν2

{1

2

{I∑

i=1

Ni,yu2i +{

I∑i=1

Ni,y

∑k

φk

+I∑

i=1

Ni

∑k

φk,y

}b2ik

}2

+ ν

{I∑

i=1

Ni,xu2i−1

+{

I∑i=1

Ni,x

∑k

φk +I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}

×{

1

2

{I∑

i=1

Ni,yu2i−1 +{

I∑i=1

Ni,y

∑k

φk

+I∑

i=1

Ni

∑k

φk,y

}bik +

I∑i=1

Ni,xu2i +{

I∑i=1

Ni,x

×∑

k

φk +I∑

i=1

Ni

∑k

φk,x

}b2ik

}}}∂q

∂x1(58)

Plain Strain Condition

g1 = E(1 − ν)

(1 − 2ν)(1 + ν)

1

2

{I∑

i=1

Ni,xu2i−1

{+

I∑i=1

Ni,x

×∑

k

φk +I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}2∂q

∂x1(59)

g2 = E

1 + ν

1

2

{I∑

i=1

Ni,yu2i−1 +{

I∑i=1

Ni,y

∑k

φk

+I∑

i=1

Ni

∑k

φk,y

}b2ik +

I∑i=1

Ni,xu2i +{

I∑i=1

Ni,x

×∑

k

φk +I∑

i=1

Ni

∑k

φk,x

}b2ik

}∂u2

∂x1

∂q

∂x1(60)

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g3 = E

1 + ν

{1

2

I∑i=1

Ni,yu2i−1 +{

I∑i=1

Ni,y

∑k

φk

+I∑

i=1

Ni

∑k

φk,y

}bik +

I∑i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}{I∑

i=1

Ni,xu2i−1 +{

I∑i=1

Ni,x

×∑

k

φk +I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}∂q

∂x1(61)

g4 = E

(1 + ν)(1 − 2ν)

{(1 − ν)

I∑i=1

Ni,yu2i−1

+{

I∑i=1

Ni,y

∑k

φk +I∑

i=1

Ni

∑k

φk,y

}bik

+ ν

{I∑

i=1

Ni,xu2i−1 +{

I∑i=1

Ni,x

∑k

φk +I∑

i=1

Ni

×∑

k

φk,x

}b(2i−1)k

}}∂u2

∂x1

∂q

∂x2(62)

g5 = E

1 + ν

{1

2

{I∑

i=1

Ni,yu2i−1 +{

I∑i=1

Ni,y

×∑

k

φk,y

}bik +

{I∑

i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}b2ik

}}2∂q

∂x1(63)

g6 = E

1 − ν2

{1

2(1 − ν)

{I∑

i=1

Ni,yu2i +{

I∑i=1

Ni,y

×∑

k

φk +I∑

i=1

Ni

∑k

φk,y

}b2ik

}2

+ ν

{I∑

i=1

Ni,xu2i−1 +{

I∑i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}{1

2

{I∑

i=1

Ni,y

× u2i−1 +{

I∑i=1

Ni,y

∑k

φk +I∑

i=1

Ni

∑k

φk,y

}

× bik +I∑

i=1

Ni,xu2i +{

I∑i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}b2ik

}}}∂q

∂x1(64)

APPENDIX II

Plane Stress Condition

G1 = E

1 − ν2

1

2

{{I∑

i=1

Ni,xu2i−1 +{

I∑i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}∂u1

∂x1−

{I∑

i=1

Ni,xu2i−1

+{

I∑i=1

Ni,x

∑k

φk +I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}2

× ∂V1

∂x1

}(65)

G2 = E

1 + ν

∂q

∂x1

{{∂u2

∂x1

{∂u2

∂x1+ 1

2

∂u1

∂x1− 1

2

{I∑

i=1

Ni,x

×∑

k

φk +I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}

× ∂V1

∂x2− ∂u2

∂x1

∂V1

∂x1

}+ 1

2

∂u1

∂x1

{∂u1

∂x1

}}(66)

G3 = 1

2

E

(1 + ν)

∂q

∂x1

{{I∑

i=1

Ni,xu2i−1 +{

I∑i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

{∂u2

∂x1+ ∂u1

∂x2

}

+ 2

{1

2

{I∑

i=1

Ni,yu2i−1 +I∑

i=1

Ni,y

∑k

φk

}bik

+I∑

i=1

Ni,x u2i +{

I∑i=1

Ni,x

∑k

φ9k +I∑

i=1

Ni

∑k

φk,x

}

× b2ik

}}∂u1

∂x1−

{I∑

i=1

Ni,x

∑k

φk +I∑

i=1

Ni

∑k

φk,x

}

× b(2i−1)k

}2∂V1

∂x2−

{I∑

i=1

Ni,xu2i−1 +{

I∑i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}∂u2

∂x1

∂V1

∂x1−

{I∑

i=1

Ni,xu2i−1

+{

I∑i=1

Ni,x

∑k

φk +I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}

×{

1

2

{I∑

i=1

Ni,yu2i−1

{I∑

i=1

Ni,y

∑k

φk +I∑

i=1

Ni

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×∑

k

φk,y

}bik +

I∑i=1

Ni,xu2i +{

I∑i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}b2ik

}}E

1 + ν

∂q

∂x1

∂V1

∂x2(67)

G4 = E

1 − ν2

∂q

∂x2

∂u2

∂x1

{∂u2

∂x2+ ν

∂u1

∂x1

}− E

1 − ν2

∂q

∂x2

×{{

∂u2

∂x1

}2∂V1

∂x2+ ν

{k∑

i=1

Ni,xu2i−1 +{

I∑i=1

Ni,x

×∑

k

φk +I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}∂u2

∂x1

∂V1

∂x1

}

+ E

1 − ν2

{{I∑

i=1

Ni,y

∑k

φk +I∑

i=1

Ni

∑k

φk,y

}

× b2ik

{+ ν

{I∑

i=1

Ni,x

∑k

φk +I∑

i=1

Ni

∑k

φk,x

}

× b(2i−1)k

}}{∂q

∂x1

∂u2

∂x1− ∂q

∂x1

∂u2

∂x1

∂V1

∂x2

}(68)

G5 = E

1 + ν

∂q

∂x1

{1

2

{I∑

i=1

Ni,yu2i−1 +{

I∑i=1

Ni,y

∑k

φk

+I∑

i=1

Ni

∑k

φk,y

}bik +

I∑i=1

Ni,xu2i +{

I∑i=1

Ni,x

×∑

k

φk +I∑

i=1

Ni

∑k

φk,x

}b2ik

}}{∂u1

∂x2+ ∂y2

∂x1

−∂u2

∂x1

∂V1

∂x1−

{I∑

i=1

Ni,xu2i−1 +{

I∑i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}∂V1

∂x2

}(69)

G6 = E

1 − ν2

∂q

∂x1

{{{I∑

i=1

Ni,yu2i +{

I∑i=1

Ni,y

∑k

φk

+I∑

i=1

Ni

∑k

φk,y

}b2ik

}+ ν

{I∑

i=1

Ni,xu2i−1

+{

I∑i=1

Ni,x

∑k

φk +I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}}

×{

∂u2

∂x2− ∂u2

∂x1

∂V1

∂x2

}+ ν

{I∑

i=1

Ni,yu2i

+{

I∑i=1

Ni,y

∑k

φk +I∑

i=1

Ni

∑k

φk,y

}b2ik

}

×{

∂u2

∂x1−

{I∑

i=1

Ni,xu2i−1 +{

I∑i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}∂V1

∂x1

}}(70)

Plain Strain Condition

G1 = E(1 − ν)

(1 + ν)(1 − 2ν)

1

2

{{I∑

i=1

Ni,xu2i−1

+{

I∑i=1

Ni,x

∑k

φk +I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}∂u1

∂x1

−{

I∑i=1

Ni,xu2i−1 +{

I∑i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}2∂V1

∂x1

}(71)

G2 = E

1 + ν

∂q

∂x1

{∂u2

∂x1

{∂u2

∂x1+ 1

2

∂u1

∂x1− 1

2

{I∑

i=1

Ni,xu2i−1

+{

I∑i=1

Ni,x

∑k

φk +I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}∂V1

∂x2

−∂u2

∂x1

∂V1

∂x1

}+ 1

2

∂u1

∂x1

{∂u1

∂x1

}}(72)

G3 = 1

2

E

(1 + ν)

∂q

∂x1

{{I∑

i=1

Ni,xu2i−1 +{

I∑i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}{∂u2

∂x1+ ∂u1

∂x2

}

+2

{1

2

{I∑

i=1

Ni,yu2i−1 +I∑

i=1

Ni,y

∑k

φk

}bik

+I∑

i=1

Ni,xu2i +{

I∑i=1

Ni,x

∑k

φk +I∑

i=1

Ni

×∑

k

φk,x

}b2ik

}{∂u1

∂x1−

{I∑

i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}2∂V1

∂x2

+{

I∑i=1

Ni,x

∑k

φk+I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}∂u2

∂x1

∂V1

∂x1

}

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] at

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−{

I∑i=1

Ni,xu2i−1 +{

I∑i=1

Ni,x

∑k

φk +I∑

i=1

Ni

∑k

φk,x

}

× b(2i−1)k

}{1

2

{I∑

i=1

Ni,yu2i−1

{I∑

i=1

Ni,y

∑k

φk

+I∑

i=1

Ni

∑k

φk,y

}bik +

I∑i=1

Ni,xu2i +{

I∑i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}b2ik

}}E

1 + ν

∂q

∂x1

∂V1

∂x2(73)

G4 = E

(1 + ν)(1 − 2ν)

∂q

∂x2

∂u2

∂x1

{ν

{∂u1

∂x1−

{I∑

i=1

Ni,xu2i−1

+{

I∑i=1

Ni,x

∑k

φk +I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}∂V1

∂x1

}

+ (1 − ν)

{∂u2

∂x2− ∂u2

∂x2

∂V1

∂x1

}}+ E

(1 + ν)(1 − 2ν)

×{

(1 − ν)

{I∑

i=1

Ni,y

∑k

φk +I∑

i=1

Ni

∑k

φk,y

}b2ik

+{

ν

{I∑

i=1

Ni,x

∑k

φk +I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}}

×{

∂q

∂x1

∂u2

∂x1− ∂q

∂x1

∂u2

∂x1

∂V1

∂x2

}(74)

G5 = E

1 + ν

∂q

∂x1

{1

2

{I∑

i=1

Ni,yu2i−1 +{

I∑i=1

Ni,y

∑k

φk

+I∑

i=1

Ni

∑k

φk,y

}bik +

I∑i=1

Ni,xu2i +{

I∑i=1

Ni,x

×∑

k

φk +I∑

i=1

Ni

∑k

φk,x

}b2ik

}}{∂u1

∂x2+ ∂y2

∂x1

− ∂u2

∂x1

∂V1

∂x1−

{I∑

i=1

Ni,xu2i−1 +{

I∑i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}∂V1

∂x2

}(75)

G6 = E

(1 + ν)(1 − 2ν)

∂q

∂x1

{{(1 − ν)

{I∑

i=1

Ni,yu2i

+{

I∑i=1

Ni,y

∑k

φk +I∑

i=1

Ni

∑k

φk,y

}b2ik

}

+ ν

{I∑

i=1

Ni,xu2i−1 +{

I∑i=1

Ni,x

∑k

φk

+I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}}{∂u2

∂x2− ∂u2

∂x1

∂V1

∂x2

}

+ ν

{I∑

i=1

Ni,yu2i +{

I∑i=1

Ni,y

∑k

φk

+I∑

i=1

Ni

∑k

φk,y

}b2ik

}{∂u2

∂x1−

{I∑

i=1

Ni,xu2i−1

+{

I∑i=1

Ni,x

∑k

φk +I∑

i=1

Ni

∑k

φk,x

}b(2i−1)k

}

× ∂V1

∂x1

}}(76)

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ity o

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] at

21:

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9 A

ugus

t 201

4

Sensitivity Analysis of Linear Elastic Cracked Structures Using Generalized Finite Element Method

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Transcript of Sensitivity Analysis of Linear Elastic Cracked Structures Using Generalized Finite Element Method