On the Reduction of Stress Concentration Factor Around a Notch Using a Functionally Graded Layer

W1923126270 · 수집 2026-10-04 18:12:17

T10219 Composite Structure Analysis and Optimization

CoT 라벨링

라벨링 전

PDF 파일

자동 다운로드한 PDF PDF 보기

기본 정보

DOI 제목 발행년도 저널 URL 페이지 열기 PDF URL PDF 열기 발행기관 피인용수 라이선스

추가 정보 초록 본문

DOI: http://dx.doi.org/10.1590/1516-1439.025115

On the Reduction of Stress Concentration Factor Around a Notch

Using a Functionally Graded Layer

_Sadek Gouasmi_ _a,b_ _, Abdelkader Megueni_ _a,b_ _*, Aboubakar Seddik Bouchikhi_ _a,b_ _,_

_Kamel Zouggar_ _a,b_ _, Abderahmane Sahli_ _a_

_aFaculty of Technology, University of Sidi Bel Abbès, Sidi Bel Abbes, BP 89,_

_Cité Ben M’hidi, 22000, Algeria_
_bLaboratory of Structures and Solids Mechanics – LMSS, Faculty of Technology, University of Sidi Bel_

_Abbès, Sidi Bel Abbes, BP 89, Cité Ben M’hidi, 22000, Algeria_

Received: May 5, 2015; Revised: August 30, 2015

The aim of this paper is to study the effect of functionally graded material (FGM) layer around
a notch in a plate in three dimensions. Using exponential law of variation of Young’s modulus and a
coefficient of Poisson constant, the stress concentration factor (SCF) depends on the gradation direction
of the constituent materials. The finite element method is used to study the performance of FGM layer
around a notch in a ceramic plate under in tensile load. A parametric study is performed for several
geometric and mechanical parameters such that width of the FGM layer and the ratio of FGM layer
components. The effect of notch radius is also studied.

Keywords: _FGM layer, notch, plate, exponential law, SCF, finite element method_

1. Introduction

Since the beginning of the century, the use of composite
materials in the form of plate and beam has expanded
considerably to this day whether in the automotive, construction,
and more recently in aeronautic. The composite materials
have significant advantages over traditional materials.
In conventional multilayer structures, layered composite
materials are used to improve the performance (mechanical,
thermal, acoustic...) of the structure (plate). The disadvantage
of this type is to create stress concentrations at the interfaces
due to the change of mechanical and thermal properties.
In the late 80s, a team of Japanese researchers has proposed
to overcome these difficulties by developing new materials
known as functionally graded materials (FGM). Functionally
graded materials are composites in which the material
properties vary continuously as a known function of the
spatial position; these materials are usually associated with
particulate composite where the volume fraction of particles
varies in one or several directions. The initial development of
FGMs is designed to serve as a thermal barrier 1 . Today, there
have been more and more numerous modern engineering
applications of FGM, like the spacecraft, rocking engine
casings and packaging materials in the microelectronics
industry, biomaterials (dental implants) and others 2,3 .

The metallurgical field has also been the subject of recent
research work dealing with the behavior of functionally
graded steels 4,5 . The effect of a band in FGM around a hole
in a homogeneous plate was analyzed for the reduction of the
SCF under biaxial loading 6 . Many researchers were interested
in calculating the stress concentration factor (SCF) in FGM
plates for two types of holes (circular, elliptical) 7-12 . In the

present work, we used the finite element method to calculate
the stress concentration factor at the edge of a notch in a
homogeneous ceramic plate and TiB FGM plate for different
combinations ceramic-metal. The calculation of FGM’s
stress concentration SC involves three directions (x, y) and
the direction of the radius of the notch. To reduce the SCF,
we used a FGM layer around a notch in a homogeneous
ceramic under uniaxial load. The graded finite elements
are implemented in the FE software Abaqus 13 to verify the
UMAT used subroutine (Appendix).

2. Problem Formulation

A Functionally graded material plate with a semicircular
lateral notch is subjected to uniaxial tensile load equal to
σ0 =100 MPa (Figure 1). Geometric characteristics of the
plate are:

Width W = 20mm; height H = 2W and thickness denoted
t=2 mm. The lateral notch has a radius r = a.

The origin of the coordinates coincides with the center
of the notch lateral. For reasons of symmetry of loading
and geometry, half of the plate has been studied (Figure 1).
The boundary conditions of the model were imposed by
constraining the y-displacements (U2 = 0) at y = 0 and
rotations around the axes ox and oz (UR3 = 0 and UR2= 0)
at y = 0 in the Oxz plane of symmetry of the full model.
The properties of the different constituents of the FGM
(metal, ceramic) used are indicated in Table 1.

The Figure 2 shows the law graph for different combinations
of metal-ceramic. The FGM material is expressed by the
following exponential law:

E(ξ) E βξ (1)

With:

β= _1 log_  _E1_ 
_L_  _E2_ 

(2)

Figure 1. Geometric model of the plate with a lateral notch circular,
(a) the entire plate, (b) half of the plate.

Table 1. Material properties of the FGM used in calculations.

Nature Young’s modulus (GPa) Poisson ratio

Al 72 0.3
Ti 110 0.3
Cu 124 0.3
Ni 215 0.3
TiB 375 0.3

Figure 2. Exponential variation of Young’s modulus for different
combinations of the constituents.

Where E2 = E(0) is the Young’s modulus of the metal and
E1 the Young’s modulus of ceramic.

Different functionally graded material configurations
can be obtained by varying the gradation direction of the
constituent materials as:

ξ= x and L=W for x-FGM (Figure 3a).
ξ= y and L=W for y-FGM (Figure 3b).

ξ= _x2_ + _y2_ and L=W _2_ for r-FGM (Figure 3c).

3. Finite Element Modeling

For modeling the Young’s modulus in a desired direction,
the subroutine UMAT Abaqus was used 14 . The routine is written
in FORTRAN language and runs in parallel with the Abaqus
solver. It allows us to establish an algorithm to calculate the
variables used by the solver Abaqus. The routine was coded
so that the material stiffness matrices are established with
appropriate material properties, namely, the Young’s modulus.
In this study, the Poisson’s ratio is assumed constant, since it
has been shown that variations in the Poisson ratio are much
less important than the Young’s modulus 15 . Calculating the
stiffness matrix requires the use of the Gaussian quadrature 16 .

A half symmetric model of a 20 × 20 mm² plate and
a thickness t with 2 mm radius center notch was used for
verification as shown in Figure 4. The mesh was refined
around the center notch and was graded in the direction
moving away from the notch toward the outer boundaries.
The initial mesh consisted of 13810 C3D20R quadratic
brick, full integration stress elements with 276200 nodes.
The base Young’s modulus is TiB with E1 =375 GPa and E2
is the metal Young’s modulus. The Poisson’s ratio was held
constant equal to 0.3.

4. Results and Analyses

The finite element method has allowed us to calculate the
normalized stress concentration of a notched plate noted Kt
for different directions of FGM and a constant normalized
thickness (t/W=0.1). The stress concentration allows increasing

Figure 4. Model verification mesh with 13810 C3D20R elements.

Figure 5. Variation of normalized normal stresses vs normalized x
for different gradation direction of Ti-TiB FGM.

Figure 6. Strain max contours for different gradation direction distribution of FGM (a) x-FGM variation, (b) y-FGM variation, (c) r-FGM

the stress at the notch root. The factor Kt is defined by the
ratio of the maximum stress to the applied stress σ0:

_Kt_ = σσ _max0_

_t_ = σσ _max_

= σ _max_ (3)
σ _0_

σ22max. The value of σ22max is taken in the middle of the thickness
where it is maximum (Figure 6). The maximum values are
naturally obtained at notch edge (Figure 6). We deduce that
the direction of FGM along r gives the minimum value.
We note that FGM distribution management for the rest of
the study. The gain of normalized stress concentration (Kt)
of an r-FGM plate at the edge of the notch is 44% compared
to a homogeneous ceramic plate.

The field of deformation shape of the plate is similar to the
graduated FGM modulus for the three directions (Figure 7).
The maximum deformation is lower in the case of r-FGM.

In the case of a homogeneous and isotropic plate, Kt is
called stress concentration factor. The Figure 5 shows the
normalized distribution of σ22 stress versus x from the edge
of the notch for the tree direction of FGM and the normalized
distribution of σ22 stress of ceramic.

In our case, the stress σ _max_ represents the normal stress
along the y axis at the edge of the notch at y = 0 and noted

_4.1. Parametric study_

The profile of the stress distribution σ22 normalized
along the x-axis is the same for the various combinations of
metal‑ceramic. The decrease of the stress concentration factor

Figure 8. Variation of normalized normal stresses vs normalized x
for different combinations of the FGM.

Figure 9. Variation of the stress concentration factor Kt based on
the normalized x for different ratios E1 / E2.

Figure 10. Variation of stress concentration factor as a function of

at the edge of the notch is proportional to the decrease in
the E1/E2 report. For a normalized depth graduation of FGM
between 0.4 and 0.7, the normalized stress tend towards the
value 1, for a standardized depth graduation of FGM superior
to 0.7, the stresses decrease slightly (Figure 8). The greatest
reduction of the stress concentration factor at the edge of the
notch is obtained for the combination AL-TiB and is about
60% compared to the TiB ceramics (Figure 9).
Figure 10 shows the evolution of Kt as a function of
the normalized notch radius for different combinations of
the FGM. Kt increases with increasing radius of the notch
for all combinations. E2 is the Young modulus of ceramic.

_4.2. Effect of the FGM layer_

To analyze the effect of the FGM layer, a ceramic plate
in TiB contains a r-FGM layer in Ti-TiB with a width h
(h=b-a) around a circular lateral notch of radius a (Figure 11).

We observe that an increase of the width h corresponds
to a decrease of the Kt value at the rim of the notch and,
at the same time, we have an increase of the interface Kt
(Figure 12). The Kt at the rim of the notch is noted Ka
and the other Kb. The use of a standard FGM layer Ti-TiB

Figure 11. A geometric model FGM layer in a ceramic plate.

Figure 12. Variation of normalized normal stresses vs normalized

Figure 13. Variation of Ka and Kb depending on the normalized
width of FGM layer.

Table 2. The gain of SCF Ka and Kb for different normalized width
of layer.

Normalized Gain (%)

h/W ΔKa/Kt ΔKb/Kt

0.1 25.54 46.50
0.2 32.26 58.60
0.3 36.83 60.75
0.4 41 64

Normalized

h/W

References

1. Hirano T, Teraki J and Yamada T. On the design of functionally
gradient materials _._ In: Yamanouchi M, Koizumi M, Harai T
and Shiota I, editors. _Proceedings 1st International Symposium_
_on Functionally Gradient Materials_ ; 1990; Sendai. Sendai:
Functionally Gradient Materials Forum; 1990. p. 5-10.

2. Watari F, Yokoyama A, Saso F, Uo M, Ohkawa S and Kawasaki
T. EPMA elemental mapping of functionally graded dental
implant in biocompatibility test. In: Shiota I and Miyamoto
Y, editors. _Proceedings 4th International Symposium on_
_Functionally Graded Materials_ ; 1996; Tsukuba. Heidelberg:
Elsevier; 1996. p. 749-754.

3. Oonishi H, Noda T, Ito S, Kohda A, Yamamoto H and Tsuji E.
Effect of hydroxyapatite coating on bone growth into porous
Titanium alloy implants under loaded conditions. _Journal of_
_Applied Biomaterials_ [. 1994; 5(1):23-27. http://dx.doi.org/10.1002/](http://dx.doi.org/10.1002/jab.770050105)
[jab.770050105.](http://dx.doi.org/10.1002/jab.770050105)

4. Abolghasemzadeh M, Pour HSS, Berto F and Alizadeh Y.
Modeling of flow stress of bainitic and martensitic functionally
graded steels under hot compression. _Materials Science and_
_Engineering A_ [. 2012; 534:329-338. http://dx.doi.org/10.1016/j.](http://dx.doi.org/10.1016/j.msea.2011.11.077)
[msea.2011.11.077.](http://dx.doi.org/10.1016/j.msea.2011.11.077)

5. Jam JE, Abolghasemzadeh M, Salavati H and Alizadeh Y. The
effect of notch tip position on the charpy impact energy for
bainitic and martensitic functionally graded steels. _Strength_
_of Materials_ [. 2014; 46(5):700-716. http://dx.doi.org/10.1007/](http://dx.doi.org/10.1007/s11223-014-9604-0)
[s11223-014-9604-0.](http://dx.doi.org/10.1007/s11223-014-9604-0)

6. Sburlati R, Atashipour SR and Atashipour SA. Reduction of
the stress concentration factor in a homogeneous panel with
hole by using a functionally graded layer. _Composites. Part_
_B, Engineering_ [. 2014; 61:99-109. http://dx.doi.org/10.1016/j.](http://dx.doi.org/10.1016/j.compositesb.2014.01.036)
[compositesb.2014.01.036.](http://dx.doi.org/10.1016/j.compositesb.2014.01.036)

7. Sburlati R. Stress concentration factor due to a functionally

(h/W=0.4) leads to a significant decrease of Ka with a gain of
41% and a gain of 64% for the Kb with respect to a ceramic
plate. We observe that the value of Kt = 2.02 of a complete
r-FGM is close to that of Ka (Ka = 2.2) of a plate with the
same size of the layer.

Figure 13 shows the evolution of Ka and Kb depending
on the width of the layer. The values of Ka and Kb tend
respectively to 2.2 and 1.32. The gains of Ka and Kb
are summarized in Table 2 which are respectively 41%
and 64%.

5. Conclusion

The use of FG-layer is another way to increase the
performance of the notched structures.

The use of a r-FGM or r-FGM layer in a ceramic notched
plate allowed to draw the following conclusions:

- The variation of the FGM properties in the direction of

notch radius (r-FGM) offers the best favorable stress
concentration factor compared to other directions.

- The smallest ratio E1 / E2 is one that gives the best

guarantee.

- The use of FGM layer around the notch of a standardized

width less than 0.5 gives the best possible performance.

_Journal of Solids and Structures_ . 2013; 50(22-23):3649-3658.
[http://dx.doi.org/10.1016/j.ijsolstr.2013.07.007.](http://dx.doi.org/10.1016/j.ijsolstr.2013.07.007)

8. Makwana AB, Panchal KC and Gandhi AH. Stress analysis of
functionally graded material plate with cut-out. _International_
_Journal of Advanced Mechanical Engineering_ . 2014; 4(5):495500.

9. Enab TA. Stress concentration analysis in functionally graded
plates with elliptic holes under biaxial loadings. _Ain Shams_
_Engineering Journal_ [. 2014; 5(3):839-850. http://dx.doi.](http://dx.doi.org/10.1016/j.asej.2014.03.002)
[org/10.1016/j.asej.2014.03.002.](http://dx.doi.org/10.1016/j.asej.2014.03.002)

10. Kubair DV and Bhanu-Chandar B. Stress concentration factor
due to a circular hole in functionally graded panels under uniaxial
tension. _International Journal of Mechanical Sciences_ . 2008;
[50(4):732-742. http://dx.doi.org/10.1016/j.ijmecsci.2007.11.009.](http://dx.doi.org/10.1016/j.ijmecsci.2007.11.009)

11. Yang QQ, Gao CF and Chen WT. _Stress concentration in a_
_finite functionally graded material plate_, Physics. _Mechanics &_
_Astronomy_ [. 2012; 55(7):1263-1271. http://dx.doi.org/10.1007/](http://dx.doi.org/10.1007/s11433-012-4774-x)
[s11433-012-4774-x.](http://dx.doi.org/10.1007/s11433-012-4774-x)

12. Saini PK and Kushwaha M. Stress variation around a circular
hole in functionally graded plate under bending. _International_
_Journal of Mechanical, Aerospace, Industrial and Mechatronics_
_Engineering_ . 2014; 8(3):536-540.

13. Dassault Systèmes. _Abaqus software manual: version 6.10_ .
Providence: Dassault Systèmes Simulia Corp; 2011.

14. Buttlar WG, Paulino GH and Song SH. Application of graded
finite elements for asphalt pavements. _Journal of Engineering_
_Mechanics_ [. 2006; 132(3):240-249. http://dx.doi.org/10.1061/](http://dx.doi.org/10.1061/%0x28ASCE%0x290733-9399%0x282006%0x29132:3%0x28240%0x29)
[(ASCE)0733-9399(2006)132:3(240).](http://dx.doi.org/10.1061/%0x28ASCE%0x290733-9399%0x282006%0x29132:3%0x28240%0x29)

15. Sadd MH. _Elasticity, theory, applications & numerics_ . Boston:
Elsevier Academic Press; 2009.

16. Manneth V. _Numerical studies on stress concentration in_
_functionally graded materials_ . [Dissertation]. Kingston:

Appendix. User subroutine UMAT x-FGM.

c User subroutine for implementation of a continuous variation
c of the material elastic properties between integration points.
c
c ABAQUS 6.11 - user subroutine UMAT for functionally graded materials
c where E(X)
SUBROUTINE UMAT (STRESS, STATEV, DDSDDE, SSE, SPD, SCD,
1 RPL, DDSDDT, DRPLDE, DRPLDT,
2 STRAN, DSTRAN, TIME, DTIME, TEMP, DTEMP, PREDEF, DPRED, CMNAME,
3 NDI, NSHR, NTENS, NSTATV, PROPS, NPROPS, COORDS, DROT, PNEWDT,
4 CELENT, DFGRD0, DFGRD1, NOEL, NPT, LAYER, KSPT, KSTEP, INC)
C
INCLUDE ‘ABA_PARAM.INC’
C
CHARACTER*80 CMNAME
DIMENSION STRESS (NTENS), STATEV (NSTATV),
1 DDSDDE (NTENS, NTENS), DDSDDT (NTENS), DRPLDE (NTENS),
2 STRAN (NTENS), DSTRAN (NTENS),TIME(2),PREDEF(1),DPRED(1),
3 PROPS(NPROPS),COORDS(3),DROT(3,3),DFGRD0(3,3),DFGRD1(3,3)
c
E1=props (1)
E2=props (2)
ANU=props (3)
X= COORDS (1)
Y= COORDS (2)
W= 20
c
c W is Width of plate and a is radus of lateral notch
c Determine L
L=W
c Determine material properties based on global coordinates of gauss points.
c COORDS(1) is X-coordinates of gauss points.
c PROPS is defined by users.
c The function can be also defined by users.
E=E2*exp (log (E1/E2)/L)*X
c COMPUTE JACOBIAN
c
amu=E/2.0d0/(1.0d0+ANU)
alambda=E*ANU/(1.0d0+ANU)/(1.0d0-2.0d0*ANU)
c
do i=1, ndi
do j=1, ndi
if (i.eq.j) then
ddsdde(i,i)=alambda+2.0d0*amu
else
ddsdde(i,j)=alambda
endif
enddo
enddo
do i=ndi+1, ntens
ddsdde(i,i)=amu
enddo
c
c STRESSES AND STRAINS AT END OF TIME STEP
c
do i=1, ntens
do j=1, ntens
stress(i)=stress(i)+ddsdde(i,j)*dstran(j)
enddo
enddo
c
return

본문 출처 PDF: PDF 보기 · 다른 PDF는 위 [PDF 업로드]로 올릴 수 있습니다.

활용가능성

이력 0건

아직 편집 이력이 없습니다. 저장할 때마다 이전 내용이 여기에 백업됩니다.