Stress concentration analysis in functionally graded plates with elliptic holes under biaxial loadings

8/12/2019 Stress concentration analysis in functionally graded plates with elliptic holes under biaxial loadings

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MECHANICAL ENGINEERING

Stress concentration analysis in functionally gradedplates with elliptic holes under biaxial loadings

Tawakol A. Enab *

Production Engineering and Mechanical Design Department, Faculty of Engineering, Mansoura University, P.O. 35516,Mansoura, Egypt

Received 9 October 2013; revised 25 February 2014; accepted 13 March 2014

KEYWORDS

Functionally graded material(FGM);Stress concentration factor(SCF);

Elliptic hole;Finite element method(FEM);Uniaxial tension;Biaxial loading

Abstract Stress concentration factors (SCFs) at the root of an elliptic hole in unidirectional func-tionally graded material (UDFGM) plates under uniaxial and biaxial loads are predicted. ANSYSParametric Design Language (APDL) was used to build the nite element models for the plates andto run the analysis. A parametric study is performed for several geometric and material parameterssuch as the elliptic hole major axis to plate width ratio, the elliptical shape factor, the gradation

direction of UDFGM. It is shown that, SCF in the nite plate can be signicantly reduced bychoosing the proper distribution of the functionally graded materials. The present study may pro-vide designers an efcient way to estimate the hole effect on plate structures made of functionallygraded materials.

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1. Introduction

Functionally graded material (FGM) is a new kind of inhom-ogeneous composite. It possesses continuously varying micro-structure and mechanical properties. The main advantage of FGM is that no internal boundaries exist and the interfacial

stress concentrations can be avoided. Furthermore, function-ally graded materials (FGMs) can be designed to achieve par-ticular desired properties and the gradation in properties of thematerial can optimize stress distribution. Nowadays, therehave been increasingly many modern engineering applications

of FGMs, such as space shuttle, rocking-motor casings, andpackaging materials in microelectronic industry [1].

Stress concentrations around cutouts have great practicalimportance because they are normally the cause of failure.Stress concentration factor (SCF), K t , dened as the ratio of the maximum stress in the presence of a geometric irregularity

or discontinuity to the stress away from the effect of such irreg-ularity, i.e. applied stress. Majority of the studies performedfor the SCF have treated isotropic, orthotropic or compositeplates. However, due to the abrupt changes in the materialproperties of the laminated composite structures in the trans-verse direction and subsequently, possibility of local failureoccurrence, functionally graded materials were used as alterna-tive materials in some applications.

Wu and Mu [2] proposed a simple computation methodbased on the scale factors (SFs) to estimate the stress concen-tration factors (SCFs) of nite-width isotropic/orthotropicplates/cylinders with a circular cutout and under uniaxial orbiaxial tension. Ukadgaonker and Kakhandki [3] analyzed

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Please cite this article in press as: Enab TA, Stress concentration analysis in functionally graded plates with elliptic holes under biaxial loadings,Ain Shams Eng J (2014), http://dx.doi.org/10.1016/j.asej.2014.03.002
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the stresses in an orthotropic plate with an irregular shapedhole under different in-plane loading conditions and had wellmatching with the FEM solutions. Rao et al. [4] gave analyti-cal solution to get the stress distribution around square andrectangular cutouts in symmetric laminates as well as in isotro-pic plates. Darwish et al. [5,6] investigated the in-plane SCF incountersunk rivet holes in orthotropic laminated plates underuniaxial tension load using nite element analysis. They for-

mulated a general parametric equation for the maximumSCF. The general stress functions for determining the stressconcentration around circular, elliptical and triangular cutoutsin laminated composite innite plate subjected to arbitrarybiaxial loading at innity using Muskhelishvilis complex var-iable method obtained by Sharma [7].

Toubal et al. [8] compared their experimental results of thetensile strain eld around circular hole in a composites platewith the predictions of a theoretical model previously devel-oped by Lekhnitskiis and a nite element study. For a platecontaining a hole subjected to uniaxial tension or out-of-planebending, Yang et al. [9,10] and Yang [11] examined the sensi-tivity of the stress and strain concentration factors to platethickness as well as the Poissons ratio or moment ratio. More-over, the stress concentration and the inuence of Poissons ra-tio on the thickness-dependent SCF along the root of ellipticholes in elastic plates subjected to tension investigated byShe and Guo [12] and Yu et al. [13] using 3D FEM and someempirical formulae obtained by tting the numerical results.Sitzer and Stavsky [14] found that the SCFs were to be quitesensitive to changes in plate anisotropy and heterogeneity,direction of external tensile force and form of hole in symmet-rically laminated anisotropic plates under tension.

Arjyal et al. [15] employed the remote laser Raman spec-troscopy to measure the stress concentration arising in a com-posite Kevlar 49 ber/epoxy composite plate containing acircular hole under different strain levels until fracture. They

found that both analytical and numerical models predictedmaximum values of stress concentration that were very closeto that determined experimentally. Additionally, extensiveexperimental and numerical studies were done by Rhee et al.[16,17] for optimizing the size and location of auxiliary holesto minimize stress concentrations in uniaxially-loaded ortho-tropic materials.

Kubair and Bhanu-Chandar [18] performed a parametricstudy on functionally graded panels with circular hole underuniaxial tension by varying the functional form and the direc-tion of the material property gradation and showed that theSCF was reduced when Youngs modulus progressively in-creased away from the hole. Mohammadi et al. [19] analyzedthe effect of nonhomogeneous stiffness and varying Poissonsratio upon the SCFs at circular hole in an innite plate madeof a functionally graded material subjected to uniform biaxialtension and pure shear.

From the above discussion, it can be noted that, limited re-searches have been performed to analyze the stress concentra-tion due to circular hole in a functionally graded plate.Moreover; to our knowledge; there are no studies carried outto obtain the stress concentration due to elliptic hole in a func-tionally graded plate. Therefore, the main objective of the pres-ent research is to perform stress analysis of a nite functionallygraded material (FGM) plate with an elliptic hole under uniax-ial and biaxial loading conditions. The two-dimensional distri-butions of stresses near the elliptic hole are analyzed. The

sensitivity of the SCFs to FGM properties, such as gradationdirection and composition parameter, and the geometricparameters is examined.

2. Functionally graded material (FGM)

Functionally graded material (FGM) characterized bysmoothly variation in properties with spatial position to ef-ciently responds to the surrounding mechanical loads [20,21].It comprises a multi-phase material with volume fractions of the constituents varying gradually in a pre-determined and de-signed prole. The concept of a FGM tries to overcome thedrawbacks of composites by resolving the problem of thematerial property mismatch especially at the interfaces[2224] . The advances in material synthesis technologies haveencouraged the development of FGM with promising applica-tions in aerospace, transportation, energy, electronics and bio-medical engineering [25]. Signicant advances in fabricationand processing techniques have made it possible to produceFGMs with complex properties and shapes using computer-aided manufacturing techniques [26]. FGM manufacturing

processes include exposure to ultraviolet radiation, controlledsuspension of particles in polymer matrices, high-speed centrif-ugal casting, depositing layers on a substrate, etc. [27,28].

2.1. Volume fractions and rule of mixtures of one-dimensional FGM

Consider a plate of FGM with porosity that functionallygraded from two materials. The subscripts 1 and 2 denotematerial 1 and material 2 of the basic constituents, respectively.V 1 and V 2 are the volume fractions of material 1 and material 2 respectively. The volume fractions are distributed over thex-direction (horizontal distribution, Fig. 1 a), according tothe following relations [29,30]:

V 1 xW

m1

V 2 1 V 1 2

where W and x are the plate width and the horizontal positionof different points along it respectively. Moreover, m is aparameter that controls the composition variation throughthe plate. For material 1 rich composition m < 1, while formaterial 2 rich composition m > 1. The variation of material 1 and material 2 composition is linear if m = 1. The porosity p of the FGM may be represented for horizontal distributionmodel by [29]:

p A xW

n

1 xW

z

h i 3

where 0 A n z=nn

1 n=n zz 4

A, n and z are arbitrary parameters that control the porosityand equal to 0.1, 1 and 1 respectively [30].

FGM effective properties, with porosity and continuouslygraded prole, are determined by employing the suspendedspherical grain model. It was derived on the base of theassumption that the granular phase is in a matrix phase. Sub-sequent relations give the rules of mixture for the elastic mod-ulus [29].

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E E 01 p

1 p5 8v37 8v81 v23 8v5

where E 0 E 2E 2 E 1 E 2V

2=31

E 2 E 1 E 2 V 2=31 V 1

24

35

6

v v1V 1 v2V 2 7

E 0 is the elastic modulus when the porosity equals zero, E 1 , v1and E 2 , v2 are the elastic moduli and Poissons ratio of material 1 and material 2 , respectively.

Different functionally graded material congurations canbe obtained by varying the gradation direction of the constit-uent materials. Therefore, the volume fractions can be distrib-uted over the x-direction (horizontal distribution, x-FGM model, Fig. 1 a), the y-direction (vertical distribution, y-FGM model, Fig. 1 b), the radial direction ( r-FGM model, Fig. 1 c)and over the angular direction ( h-FGM model, Fig. 1 d). Thesedifferent models can be obtained simply by replacing the term(x/W ) in Eqs. (1) and (3) by:

( y/H ) for y-FGM model,

(R/R0

) for r-FGM model, where R ffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2

y 2

p ; R0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W 2 H 2p and (h/h0 ) for h-FGM model, where h tan 1 y = x, h0 = 90due to quarter symmetry.

3. Model formulation

3.1. Plate geometry and parameters

The conguration of the plate containing an elliptic hole andsubjected to a uniaxial tensile load or biaxial load is shownin Fig. 2 . Cartesian coordinate system is used with origin

locating at the center of the elliptic hole in the plate. The x-axisis parallel to the major axis of the elliptic hole and the y-axis isparallel the minor axis. The width of the plate is 2 W and itsheight is 2 H . The major axis of the elliptic hole is 2 a, the minoraxis is 2 b, the elliptical shape factor for the hole is t = b/a,while the corresponding curvature radius q at the root of themajor axis is: q = b2 /a = at 2 . Plates with ten different majoraxes and four different elliptical shape factors are analyzed.The geometrical parameters of the plate are chosen as follows:

2W = 2 H = 200 mm;a/W = 0.050.5 with a step of 0.05;t = b/a = 0.25, 0.33, 0. 5, 1;q = b2 /a = at 2 ranged from 0.3125 to 50 depending on thevalues of the major axis of the elliptic hole (2 a) and theelliptical shape factor ( t).

3.2. Mechanical properties of materials

In this investigation, a study of stress concentration factors forunidirectional functionally graded materials (UDFGM) plateswith central elliptical hole is carried out. The basic constituentsof the UDFGM are commercially pure titanium (CP Ti) ( mate-rial 1 ) and titanium monoboride (TiB) ( material 2 ). The elasticproperties of the base materials are [31,32]: E Ti = 107 GPa,vTi = 0.34 and E TiB = 375 GPa, vTiB = 0.14. The gradedmaterial is incorporated with an exponential material variationas discussed above. Thus Youngs modulus and Poissons ratioare functions of the coordinates.

3.3. Finite element modeling

The nite element method (FEM) is widely used for numericalsimulation and optimization of structural geometry most nota-bly when dealing with stress raisers or concentrators [33].ANSYS Parametric Design Language (APDL) was used to

x

y

x

W

V 1 V 2

(a)

x

y

y H

V 1

V 2

(b)

y

x

W

R H

V 1

V 2

(c)

y

y

x

x

W

H

V 1

V 2

(d)

y

Figure 1 The gradation direction and volume fraction distribution of unidirectional functionally graded materials at (a) x-, (b) y-,(c) r - and (d) h-directions. The origin of the coordinates coincides with the center of the elliptic hole.

Stress concentration analysis in functionally graded plates loadings 3



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build the FE models and to run the analysis. Owing to thesymmetry, only one-quarter of each plate is modeled. The plate

was modeled in two dimensions using isoparametric quadrilat-eral Plane 183 elements, which have eight nodes: four vertexnodes and four midside nodes. Each node has two degrees of freedom: translations in the nodal x- and y-directions. Thedeveloped models ensure that sufcient control can be main-tained over the mesh. An example of the mesh used is pre-sented in Fig. 3 . To improve solution accuracy, the meshwas rened around the elliptic hole. The number of the elementis related to the curvature radius at the root of the major axisq. The less of the q is, the more of the number of the element is.The total numbers of elements of the models are different fromeach other due to different q .

The boundary conditions of the quarter model were im-posed by constraining the x-displacement ( u

x) at x = 0 and

the y-displacement ( u y) at y = 0 to account for the planes of symmetry of the full model. The plate was analyzed for uniax-ial loading by applying a uniform tensile stress of r 0 in y-axisdirection. Fig. 3 shows the details of the boundary and loadingconditions of the quarter model. While, for biaxial loading theuniform stresses of r 0 and kr 0 were applied to the plate ends in y- and x -directions respectively.

3.3.1. Application of material gradient

Simulation of the FGM structure should reect material prop-erties distribution. While signicant efforts have been made to

accurately incorporate continuous property variation into -nite element formulations, many experimentally produced

FGMs exhibit a stepwise variation in properties and shouldbe modeled as such, bearing in mind the effective propertysmoothing which may be applicable [34]. Actually, there isno specied material module for the direct analysis of func-tionally graded plate in ANSYS [35]. Consequently, in thepresent work a simplest method involves the assignment of properties to each element individually was used. In fact, thisleads to a discontinuous step-type variation in properties; how-ever, this may provide a more appropriate representation of areal graded component than an ideal smooth gradient.

Therefore, the unidirectional functionally graded mediumto be analyzed is usually divided into a number of layers.Each layer has a different material denition depending onthe distribution direction (i.e. x-, y-, r-, or h-direction) andthe m parameter which controls the composition variationthrough the material. Material properties assigned to each ele-ment not to layers individually due to free meshing of theplate. The elastic parameters for each element are taken fromthe values at the element centroid. Consequently, a step-likematerial property is assumed. Theoretically, when the numberof material elements increases, the real curve of UDFGMproperty can be accurately represented. Thus using the com-mercial nite element software ANSYS, a rened nite meshemployed to achieve results with acceptable accuracy forUDFGM.

Figure 3 Representative meshing for nite element model, showing (a) entire plate and (b) close-up of meshing at elliptic hole.

x

y

2a

2W

2 b

0

2 H

0

(i)

2W

0

x

y

2a

2W

2 b

0

0

0

0 2

H

(ii) y

Figure 2 Geometry of a plate has an elliptic hole under (i) uniaxial and (ii) biaxial loadings.

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4. Results and discussion

Since the analysis of the present investigation is entirely basedon nite element modeling and its results, the verication of the developed nite element models is considered very essen-tial. Consequently, the developed nite element models wereinitially veried and then used to predict the maximum stressconcentration factors in functionally graded materials plateswith a central elliptical hole.

4.1. Verication analysis

The verication analysis was performed by using the developedFE models for certain cases in the literature and reproducingtheir results. Therefore, steel with modulus of elasticityE = 210 GPa and Poissons ratio v = 0.3 is considered forthe modeling of the isotropic material plate. However, differentisotropic materials with different mechanical properties shouldnot affect the results of the stress concentration factor [6].

2

3

4

5

6

7

Normalized major radius (a / W)

S t r e s s c o n c e n t r a t i o n f a c t o r ( S C F )

H = W H = 1.5 W H = 2 W H = 2.5 W H = 3 W H = 5 W Heywood Isida

(a) t = b/a = 1

4

5

6

7

8

9

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6



H = W H = 1.5 W H = 2 W H = 2.5 W H = 3 W H = 5 W Isida

(b) t = b/a = 0.5

Figure 4 Comparison between the FE results and Heywoods and Isida equations [36] for (a) circular hole ( t = 1) and (b) elliptical hole(t = 0.5) at different plate height to width ratios ( H /W ).

0

3

6

9

12

0.0 0.1 0.2 0.3 0.4 0.5 0.6



t = 1/4 t = 1/3 t = 1/2 t = 1

(a) x-FGM

0

3

6

9

12



t = 1/4 t = 1/3 t = 1/2 t = 1

(b) y-FGM

0

3

6

9

12



t = 1/4 t = 1/3 t = 1/2 t = 1

(c) r-FGM

0

3

6

9

12



t = 1/4 t = 1/3 t = 1/2 t = 1

(d) -FGM

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6

Figure 5 Inuence of elliptical hole shape factor ( t) on the SCFs for UDFGM plates at different FGM congurations. UDFGM with (a)horizontal distribution ( x-FGM ), (b) vertical distribution ( y-FGM ), (c) radial distribution ( r-FGM ) and (d) angular distribution ( h-FGM ).




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Comparison was made for the SCFs in a thin isotropic platehaving a central hole. Fig. 4 shows the FE results for SCF atdifferent normalized major radius ratios ( a/W ). For plate withcentral circular hole ( Fig. 4 a), i.e. shape factor t = 1, the re-sults of the developed FE models were compared with theequations presented by Heywood and Isida [36]. It is notedthat, plate height to width ratio ( H /W ) has an important inu-ence on the results of the stress concentration factors. Themaximum difference between the results of the developed FEmodels and equations results found for plate height to width

ratio H /W = 1 at normalized major radius ratio a/W = 0.5.By increasing H /W from 1 to 1.5 the maximum difference inSCF reduced from about 50% to 9% comparing to the Hey-woods equation results and from about 47% to 6% compar-ing to the Isidas equations results. While, for H /W greaterthan or equal 2 the maximum SCF was higher by about2.3% than the results of Heywoods equation and was lowerby about 0.2% than the Isidas equations results.

On the other hand, for elliptical hole with shape factort = 0.5 ( Fig. 4 b) the results were compared with the equations

6

7

8

9

10

11

12

13

14

0.0 0.1 0.2 0.3 0.4 0.5 0.6


S t r e s s c o n c e n t r a

t i o n

f a c t o r

( S C F ) (a) t = 1/4

x-FGM

-FGM r-FGM -FGM

Isotropic

4

5

6

7

8

9

10

11


S t r e s s c o n c e n t r a t i o n f a c t o r ( S C F ) (b) t = 1/3

x-FGM

-FGM r-FGM -FGM

Isotropic

3

4

5

6

7

8

9


S t r e s s c o n c e n t r a t i o n f a c t o r ( S C F ) (c) t = 1/2

x-FGM -FGM

r-FGM -FGM

Isotropic

2

3

4

5

6

7


S t r e s s c o n c e n t r a t i o n f a c t o r ( S C F ) (d) t = 1

x-FGM -FGM

r-FGM -FGM

Isotropic

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6

Figure 6 Inuence of UDFGM congurations (i.e. x -, y-, r -, or h-FGM ) on the SCFs for the different elliptical hole shape factor ( t).

50

100

150

200

250

300

350

400

0 0.2 0.4 0.6 0.8 1

Position ratio: (x / W) or (y / H) or (R / R0) or ( / 0)

M o d u l u s o f e l a s t i c i t y ( G P a )

m = 0.1 m = 0.5 m = 1.0 m = 5.0 m = 10.0

(a)

0.10

0.15

0.20

0.25

0.30

0.35

Position ratio: (x / W) or (y / H) or (R / R0) or ( / 0)

P o i s s o n s r a t i o

m = 0.1 m = 0.5 m = 1.0 m = 5.0 m = 10.0

(b)

0 0.2 0.4 0.6 0.8 1

Figure 7 Variations in the (a) Modulus of elasticity and (b) Poissons ratio through the UDFGM plate used for the differentcongurations (i.e. x -, y-, r -, or h-FGM ).

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presented by Isida [36]. Similarly, at normalized major radiusratio a /W = 0.5, increasing H /W from 1 to 1.5 the maximumdifference in SCF decreased from about 21% to 2% compar-ing to the Isidas equation results. While, for H /W greater than1.5 the maximum difference in SCF did not exceed 1.5%.

Therefore, based on the previous comparisons between theresults of the developed FE models and the literature equa-tions for the two cases (i.e. circular and elliptical holes), it

can be stated that high condence is established in the presentFE results. Moreover, it is recommended to use plate height towidth ratio ( H /W ) equal to or greater than 2 to neutralize theeffect of plate height on the SCF results. But in the presentinvestigation the implemented models used to perform a para-metric study to understand the effect of the material propertiesand the gradation direction on the SCF due to an ellipticalhole in functionally graded plates. Plate height has no effecton material gradation in x- and h-directions, while in the otherhand it will affect material gradation in y- and r-directions.Thus, the current investigation uses plate height to width ratioequal to one ( H /W = 1) as a special case which already hassome technological applications and also used in some casesin the literature such as in [5,9,12,13] .

4.2. Uniaxial loading

The numerical results of stress concentration factors (SCFs)are obtained for unidirectional functionally graded materials

(UDFGM) plates with central elliptic hole under uniaxialloading conditions. Different elliptical hole shape factors andUDFGM congurations are studied. Some of the resultsare obtained for isotropic plate also for sake of comparisonsince the SCFs of an isotropic plate under uniaxial tensionhave been studied extensively and good results have beenreported.

It has been noted that, the position at which the tensile

stress becomes maximum always found at the root of the major axis as it has the minimum radius of curvature ( q) Thisposition of the maximum tensile stress does not affected bythe gradation direction of the FGM. However, its magnitudeis affected by the gradation direction. Fig. 5 presents the rela-tionship between the normalized major radius of the ellipticalhole ( a/W ) and the maximum stress concentration due to thisopening hole. It shows the effect of elliptical hole shapefactor ( t = b/a) on the SCFs for different UDFGM congu-rations (i.e. for UDFGM with horizontal distribution(x-FGM ), vertical distribution ( y-FGM ), radial distribution(r-FGM ) and angular distribution ( h-FGM )). All of these con-gurations had linear composition variation through theplate ( m = 1, refer to Eq. (1)). The stress concentrationaround elliptical hole varies as the ratio of lengths of ellipticalhole minor axis (2 b) to its major axis (2 a) varies. As the ellip-tical hole shape factor decreased, the stress concentration atthe end of major axis of the ellipse increased for all UDFGMcongurations.

x-coordinate

y -

c o o r d

i n a t e

20 40 60 80 100

20

30

40

50

60

70

80

90

100

150

200

250

300

350

00

10

(a) m = 0.1

x-coordinate

y -

c o o r

d i n a t e

20 40 60 80 100

20

30

40

50

60

70

80

90

100

150

200

250

300

350

00

10

(b) m = 0.5

x-coordinate

y -

c o o r

d i n a t e

20 40 60 80 100

20

30

40

50

60

70

80

90

100

150

200

250

300

350

00

10

(c) m = 1

x-coordinate

y -

c o o r

d i n a t e

20 40 60 80 100

20

30

40

50

60

70

80

90

100

150

200

250

300

350

0

0

10

(d) m = 10

Figure 8 Elastic modulus 2D contour plots for horizontal distribution with m = 0.1, 0.5, 1 and 10.




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It should be noted that, by increasing the normalized majorradius of the elliptical hole ( a/W ) the stress concentration fac-tors always increased for UDFGM with horizontal distribu-tion ( x-FGM ) (Fig. 5 a) and for radial distribution ( r-FGM )(Fig. 5 c). On the other hand for UDFGM with vertical distri-bution ( y-FGM ); SCFs had a small decreasing trend when theratio a /W increased from 0.05 to 0.15 followed by an increas-ing in the SCFs when a/W ratio increased. As the elliptical hole

shape factor decreased this change in SCFs marked. Moreover,for UDFGM with angular distribution ( h-FGM ) (Fig. 5 d) theSCFs decreased by increasing a/W ratio to achieve its minimalvalue at a/W = 0.2. After that, SCFs return to increased byincreasing a/W ratio. This noticed obviously for all ellipticalhole shape factor ( t). This variation in trends may be inter-preted by the fact that as the a/W ratio varies there is nochange in material properties at the root of the major axisfor y- and h-FGM distributions. In contrast, material proper-ties at the root of the major axis vary with the variation ina/W ratio for x - and r-FGM distributions.

The effect of material gradation direction of UDFGM onthe SCFs for the different elliptical hole shape factors is shownin Fig. 6 . Results of SCFs for UDFGM distributed in x-, y-, r-,or h-directions are compared with each others and with theSCFs results of isotropic material plate. It can be noted that,for all elliptical hole shape factors of UDFGM with x-, y- and r-distributions the stress concentration factors at the

notch root of the plate are lower than the stress concentrationfactors of the isotropic material plate. This shows the benet of using FGM to reduce stress concentration in plates with open-ing holes. However, SCFs of UDFGM distributed in h-direc-tion are greater than SCFs of isotropic material plates forsmall a /W ratios. While, for a /W ratio of about 0.2 or greaterthe SCFs of UDFGM distributed in h-direction are reducedcompared to the SCFs of isotropic material plates.

Referring to Fig. 6 , regardless of the value of elliptical holeshape factor, x-FGM in which the material gradation is per-pendicular to the loading direction presents the lower SCFsup to a/W ratio of about 0.4. When a/W ratio is more than0.4 graded material with gradient parallel to the direction of loading (i.e. y-FGM ) shows the lower SCFs. In the case of the r -FGM distribution SCFs are lower than and very similarin trend to the SCFs of the isotropic material plates. This isobvious as a result of the parallelism between SCFs curvesfor r-FGM and isotropic material for all elliptical hole shapefactors. Moreover, as the elliptical hole shape factor decreasedthe difference between the different UDFGM congurationsincreased nearly for all normalized major radius ( a/W ).

The material gradation controlled by the composition var-iation parameter m as formulated in Eq. (1). The compositionsof constituent materials change as the position ratio x /W or y/H or R/R 0 or h/h0 in an exponential pattern by m. In otherwords, parameter m governs the volume fraction of the

2

3

4

5

6

7

0.0 0.1 0.2 0.3 0.4 0.5 0.6


S t r e s s c o n c e n t r a t

i o n

f a c t o r

( S C F )

m = 0.1 m = 0.5 m = 1.0 m = 5.0 m = 10.0

(a) x-FGM

2

3

4

5

6

7


S t r e s s c o n c e n t r a t

i o n

f a c t o r

( S C F )

m = 0.1 m = 0.5 m = 1.0 m = 5.0 m = 10.0

(b) y-FGM

2

3

4

5

6

7


S t r e s s c o n c e n

t r a t

i o n f a c

t o r

( S C F )

m = 0.1 m = 0.5 m = 1.0 m = 5.010

(c) r-FGM

2

3

4

5

6

7


S t r e s s c o n c e n

t r a t

i o n f a c t o r

( S C F )

m = 0.1 m = 0.5 m = 1.0 m = 5.0 m = 10.0

(d) -FGM

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6

Figure 9 Inuence of composition variation parameter ( m) on the SCFs for UDFGM plate has central circular hole ( t = 1) for differentUDFGM congurations.

8 T.A. Enab



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constituent materials, thereby determining the gradation of themechanical properties ( Fig. 7 ). To study the effect of composi-tion variation parameter ( m) on the SCFs ve m values wereused ( m = 0.1, 0.5, 1, 5, 10). Consequently, m = 10 has thelowest gradation of Youngs modulus and Poissons ratiostarting from the elliptical hole center (origin of the coordi-nates), whereas m = 0.1 has the highest gradation of the abovementioned properties as illustrated in ( Fig. 7 ). Moreover, to

illuminate the effect of composition variation parameter mon the FGM distribution through the plate, Fig. 8 shows elas-tic modulus 2D contour plots for horizontal distribution withm = 0.1, 0.5, 1 and 10. Thus, regulating m enables us to tailorproperty gradation, thereby achieving a better design of FGMplate.

Fig. 9 illustrates the inuence of composition variationparameter ( m) on the SCFs for UDFGM plate has central cir-cular hole (i.e. elliptical hole shape factor t = 1) for differentUDFGM congurations. It can be noted that changing thecomposition variation parameter ( m) does not necessarily im-plies reduction in the SCFs. For example, at m = 5 or 10 thestress concentration factors are increased for the differentUDFGM congurations compared to their values at m = 1.At m = 0.5 there is a minor change in the SCFs dependingon the UDFGM conguration and the value of normalizedmajor radius. This change shows an increasing in the SCFsfor x-FGM and r-FGM at a/W greater than 0.15 whereas itshows a decreasing in SCFs at a/W lower than 0.15.

Furthermore, for y-FGM and h-FGM using m = 0.5 resultsin decreasing of SCFs for a/W lower than 0.35 and nearlyno signicant change for a/W greater than 0.35 ( Fig. 9 b andd). Finally, decreasing the composition variation parameterto m = 0.1 results in growing at SCFs for all UDFGM cong-uration except for y-FGM at a /W lower than 0.15 and h-FGM at a /W lower than 0.2.

4.3. Biaxial loading

The stress concentration occurs near the hole when the platecontaining an elliptical hole in the center and subjects to biax-ial loads. The stress concentration grade depends on the biax-ial loading ratio ( k) (i.e. the ratio of loads applied in twodirections), and the elliptical hole shape factor ( t = a/b). Inthe developed FEM models, the arbitrary biaxial loading con-dition is introduced into the boundary conditions by consider-ing several cases of in-plane loads. This is achieved byintroducing the biaxial loading ratio ( k) into the boundaryconditions (refer Fig. 2 ii). Thus, by adopting any value of the k, other type of biaxial loading conditions can be obtained.Fig. 10 presents the effect of k on the SCFs at different normal-ized major radius of the elliptical hole ( a/W ) for differentUDFGM congurations (i.e. x-, y-, r-, or h-FGM ). As a simplecase, linear composition variation in the FGM through theplate ( m = 1) and unity elliptical hole shape factor ( t = 1)were considered.

1

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11



1.00 0.75 0.50 0.25 0.00-0.25-0.50-0.75-1.00

(a) x-FGM

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11



1.00 0.75 0.50 0.25 0.00-0.25-0.50-0.75-1.00

(b) y-FGM

1

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9

11

0.0 0.1 0.2 0.3 0.4 0.5 0.6



1.00 0.75 0.50 0.25 0.00-0.25-0.50-0.75-1.00

(c) r-FGM

1

3

5

7

9

11

13

0.0 0.1 0.2 0.3 0.4 0.5 0.6



1.00 0.75 0.50

0.25 0.00-0.25-0.50-0.75-1.00

(d) -FGM

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6

Figure 10 Inuence of biaxial loading ratio ( k) on stress concentration factors for the different UDFGM congurations using m = 1,t = 1.




10/12

Similar to the uniaxial case, it is worth to note that, the

SCFs increase gradually with the increasing of normalizedmajor radius of the elliptical hole ( a/W ) regardless of the valueof the biaxial loading ratio ( k) for UDFGM with horizontalor radial distribution ( x- or r -FGM ) (Fig. 10 a and c).Alternatively for vertical distribution UDFGM ( y-FGM );SCFs had a slight decreasing trend for a/W < 0.15 followedby an obvious increasing trend for a/W > 0.15 ( Fig. 10 b).Also, by increasing a/W ratio for angular distributionUDFGM ( h-FGM ) (Fig. 10 d) the SCFs decreased and attainedits minimal values at a /W of about 0.2. Then, SCFs increasedby increasing a/W ratio. Moreover, for all UDFGMcongurations the stress concentration is found higher whenuniaxial load ( k = 0) is applied compared to equi-biaxial load-ing (k = 1). While maximum SCFs are obtained at k = 1 i.e.at condition equivalent to pure shear oriented 45 to the ellipseaxes. In addition, it is found that, increasing positive biaxialloading ratios result in decreasing the SCFs (i.e. increase thestress attenuation rate). On the contrary, decreasing negativebiaxial loading ratios cause signicant increasing in stress con-centration factors. This is valid for all UDFGM congurationsand can be obviously noticed for x-FGM at small values of a /W and for h-FGM at higher values of a/W due to difference inFGM distribution.

Fig. 11 shows the inuence of material gradation directionof UDFGM on the SCFs for some biaxial loading ratios ( k).Results of SCFs for x -, y-, r -, and h-FGM are compared with

each others and with the SCFs results of isotropic material

plate. It is worth to note that, SCFs of UDFGM plates withx-, y- and r-distributions are lower than the SCFs of the isotro-pic material plate for all biaxial loading ratios. ComparingSCFs of UDFGM distributed in h-direction with SCFs of iso-tropic material plates show variable trend depending on thevalue of biaxial loading ratio ( k) and a /W ratio. For example,SCFs of h-FGM are greater than those of isotropic material fork = 1 (Fig. 11 a and d). While, SCFs of h-FGM compared toSCFs of isotropic material plates have the higher values for a /W < 0.2 and have the lower values for a/W > 0.2 for1 > k > 1. Also, for biaxial loading ratios of k = 1 theSCFs of x- and y-FGM are identical as expected since onecan obtain y-FGM distribution by rotating x-FGM distribu-tion 90 and vice versa. Noting that, this is valid only for cir-cular holes (i.e. elliptical shape factor of t = 1). For r-FGM distribution SCFs are lower than and have similar trends com-pared to SCFs of the isotropic material plates for all biaxialloading ratios. This is observable as a result of the nearly par-allelism between SCFs curves. Therefore, FGM with properdistribution can be used to increase the stress attenuation ratein plates with opening holes.

5. Conclusions

Stress concentration factor (SCF) due to elliptical hole in uni-directional functionally graded materials (UDFGM) plate sub-

1.5

1.7

1.9

2.1

2.3

2.5

2.7

2.9



x-FGM y-FGM r-FGM t-FGM Isotropic

(a) = 1.0

x-FGM -FGM

r-FGM -FGM

Isotropic1.5

2.0

2.5

3.0

3.5

4.0

4.5




x-FGM -FGM

r-FGM -FGM

Isotropic

(b) = 0 .5

2

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8

9




x-FGM -FGM

r-FGM -FGM

Isotropic

(c) = 0 .5

2

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6

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12

14

0.00 0.15 0.30 0.45 0.60 0.00 0.15 0.30 0.45 0.60

0.00 0.15 0.30 0.45 0.60 0.00 0.15 0.30 0.45 0.60




(d) = 1 .0

x-FGM -FGM

r-FGM -FGM

Isotropic

Figure 11 Inuence of UDFGM congurations on the stress concentration factors for some biaxial loading ratios ( k) (m = 1, t = 1).

10 T.A. Enab



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jected to uniaxial or biaxial loadings conditions is obtainedusing nite element method (FEM). The developed FEM mod-els are efcient and well veried. From the numerical resultsthe following points can be concluded:

1. For both isotropic and UDFGM plates, the SCFs dependon the normalized major radius of the elliptical hole(a/W ). Moreover, as the ratio of minor to major axis in

elliptical hole increases the stress concentration at the tipof major axis decreases. Also, the SCFs increase graduallywith the decreasing of biaxial loading ratio ( k).

2. SCFs are greatly affected by the gradation direction of functionally graded materials. SCFs for x-, y-, andr-FGM distributions are lower than the SCFs of the isotro-pic material plate for all biaxial loading ratios. While, com-paring SCFs of h-FGM with SCFs of isotropic materialplates show variable trend depending on the value of biaxialloading ratio ( k).

3. FGM composition variation parameter ( m) has a signi-cant effect on stress concentration and does not necessarilyimply reduction in the SCFs.

4. The solution presented here can be a tool for the designersto analyze the stress concentration problem under uniaxialor biaxial loading conditions.

5. Further study will be carried out to formulate a generalparametric equation for the maximum SCF in terms of the geometric parameters, loading conditions and FGMparameters.

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Tawakol A. Enab is an Assistant Professor atProduction Engineering and MechanicalDesign Department, Faculty of Engineering,Mansoura University, Mansoura, Egypt. Hehas received his Ph.D. from Savoie University,Chambery, France. His research interestscomprise: the mechanical behavior of advanced composite materials, piezoelectriccomposites, functionally graded materials andbiomaterials.

12 T.A. Enab
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Stress concentration analysis in functionally graded plates with elliptic holes under biaxial loadings