Upload
manoj-manoharan
View
68
Download
0
Tags:
Embed Size (px)
DESCRIPTION
Stress Concentration Effects
Citation preview
STRESS CONCENTRATION EFFECTS IN HIGHLY LOCALIZED FUNCTIONALLY GRADED MATERIALS
BY
STEVEN P. BERLO
A SPECIAL PROBLEMS PAPER SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
MASTERS OF SCIENCE
IN
MECHANICAL ENGINEERING AND APPLIED MECHANICS
UNIVERSITY OF RHODE ISLAND
May 2009
Abstract
In this study localized nonhomogeneity in material property is evaluated for its reducing
effect on stress concentration. A number of traditional plane elasticity problems were
solved using the finite element method to evaluate highly localized variation in Young’s
modulus. Two-dimensional infinite plane theory with both biaxial and uniaxial far field
loading was applied to problems with remote stress free holes, both circular and elliptical.
Young’s modulus was varied in both intensity and gradation depth in the in plane x and y-
coordinate directions with gradation originating on, and following the shape of the stress
free hole. In addition, a brief study for a uniformly loaded half-space contact problem is
presented with localized modulus gradation originating on the loading surface and graded
in the loading direction only.
ii
Table of Contents
1. Introduction……………………………………………………………………….…1
1.1. FGM Background………………………………………………………………..1
1.2. Focus of Current Study…………………………………………………………..2
2. Modeling……………………………………………………………………………...3
2.1 Finite Element Analysis Pre and Post-Processing……………………...……...…3
2.2 Modeling of FGM………………………..……………………………………….4
2.2.1 Element Type and FGM Modeling………………………………………..5
2.2.2 User Subroutine…………………………………………………………...6
2.3 FEA FGM Model Verification………………………………………………...…7
3 Analysis and Results……………………………………………………..………...12
3.1 Plate with Stress Free Circular Hole – Biaxial Loading……………………..…..12
3.1.1 Introduction……………………………………………………………….12
3.1.2 FEA Model and Boundary Conditions…………………………………..12
3.1.3 Results……………………………………………………………………14
3.2 Infinite Plate with Stress Free Circular Hole – Uniaxial Loading………………18
3.2.1 Introduction………………………………………………………………18
3.2.2 FEA Model and Boundary Conditions…………………………………..18
3.2.3 Results……………………………………………………………………20
3.3 Infinite Plate with Stress Free Elliptical Hole – Biaxial Loading……………....22
3.3.1 Introduction………………………………………………………………22
3.3.2 FEA Model and Boundary Conditions…………………………………..23
iii
iv
3.3.3 Results……………………………………………………………………27
3.4 Infinite Plate with Stress Free Elliptical Hole – Uniaxial Loading……………..31
3.4.1 Introduction………………………………………………………………31
3.4.2 FEA Model and Boundary Conditions…………………………………..31
3.4.3 Results……………………………………………………………………32
3.5 Contact in Half Space - Uniform Distributed Loading………………………….35
3.5.1 Introduction………………………………………………………………35
3.5.2 FEA Model and Boundary Conditions…………………………………..36
3.5.3 Results……………………………………………………………………42
3.5.4 Discussion………………………………………………………………..47
4 Conclusions…………………………………………………………………………52
4.1 Conclusions……………………………………………………………....52
Appendix A. Sample UMAT User Subroutine…………………………………………55
Appendix B. Additional Results for Contact in Half-Space Problem………………..…57
Bibliography…………………………………………………………………………….59
1 Introduction
1.1 Functionally Graded Materials – Background
Functionally graded materials, FGMs, are a classification of materials that posses
material properties that vary gradually as a function of position. A prime example is the
earth’s crust, where soil becomes more compacted with depth. FGM properties are
purposely varied to achieve smooth variations such as to avoid abrupt property changes
in material and may exhibit either isotropic or anisotropic properties (Kim and Paulino,
2002). FGMs differ from conventional composite materials in that there exists no
delamination of layered material as a result of stress concentrations at the layer interfaces,
which typically create material discontinuity.
Deliberately grading the properties of a material is aimed at optimizing the response
under mechanical loading, including both structural and thermal behaviors. Graded
materials have been increasingly promoted over the past 2 decades, particularly as a type
of barrier coating in the aerospace industry for resolving thermal problems as
encountered in high performance aircraft, such as gas turbine engines and rocket nozzles
(Batra and Rousseau, 2007). In terms of the structural applications, FGMs have been
seen in history in such applications the hardening of swords, as in the Samurai where the
outer material was hardened leaving a more ductile, tougher material at the core.
However, most theory and application of FGMs have occurred relatively recently for use
in engineering materials. Typical structural application of FGMs can include use with
gears and bearings for wear resistance, where standard homogeneous material would not
support both high wear resistance and high toughness.
1
1.2 Focus of Current Study
To date, typical evaluation of FGMs has been for the most part limited to linear,
exponential or power law gradation through the thickness of a material. There have been
numerous studies on the effects of fracture due to this through thickness gradation
(Erdogan, Wu, 1997), as well as for stress concentrations due to geometrical
discontinuity (Venkataraman, 2003, Matsunaga, 2008, Manneth, 2009). The general
focus of these studies involved modifying the materials Young’s modulus properties
using linear, exponential or power law functions, which were applied across or spanned
the depth of the material boundaries.
The focus of this study is on the stress concentration effects due to a more localized
Young’s modulus gradation. The traditional infinite plate problem with stress free center
holes, both circular and elliptical, were evaluated with local gradation in Young’s
modulus around the hole. This includes both radial gradation for the case of the circular
hole, and a shape dependent gradation in the case of the elliptical hole, where the spatial
gradation follows the elliptical geometry. In addition, a brief study was conducted on
localized gradation for a simple half-space contact problem where local gradation occurs
near the loaded surface and is limited to the depth direction.
2
2 Modeling
2.1 Finite Element Analysis Pre and Post-Processing
The method used in this study to model and evaluate the local gradation effects focused
on the use of Abaqus finite element software to model the stress concentration problems.
The gradation was modeled at the element level using an Abaqus user subroutine that
mapped the variation in elastic modulus as a function of spatial coordinates at Gauss
points within each element. This method results in smooth and continuous variation
across the element.
Once the user subroutine was established it was initially evaluated against known closed
form solutions from Sadd (2009) for both linear and radial gradation. After the
associated models where verified, the appropriate gradation functions were established
and evaluated using MATLAB before being coded into the user subroutine.
With each of the studies conducted, convergence of the FEA solution was accomplished
to ensure that accuracy was optimized. In addition, the FEA model was evaluated against
the closed form solution where available. In all cases the closed form solution did exist
for the homogeneous case so all models were initially evaluated against the homogeneous
solutions. Once the modeling accuracy was established, they were run with the
appropriate gradation functions coded into the user subroutine.
For the evaluation of results, a path of nodal points was established on the discretized
model in the direction of interest, for example, along the radial symmetric boundary
3
running perpendicular to the load direction for the case of the circular hole with uniaxial
far filed loading. The nodal stresses were then extracted from the FEA model, as a
function of coordinate, and read directly into MATLAB for further evaluation and
plotting.
2.2 Modeling of FGM
There are generally two approaches to modeling the gradation of material properties
using finite elements. Homogeneous elements can be used in such a fashion that the
elements are assembled in rectangular rows that are aligned with the gradation direction.
Each row of homogeneous elements is then assigned the varied material property for the
midline of the row. This produces a step-wise approximation where the stiffness matrix
for a specific element is assumed constant and has the property assigned at the centroid of
the element (Santare and Lambros, 2000).
Although using the homogeneous element can provide reasonable results, it does not lend
itself to capturing geometry that is not rectangular in shape. Furthermore, due to the high
stress gradients that are inherent to stress concentration problems, a more accurate
method of capturing the gradients, without using an extremely large number of elements,
is preferred. The more accurate method of modeling material property variation is with
the isoparametric element where the spatial variation in the property can be assigned at
each Gauss point within the element. Normally the components of the stiffness matrix,
Ke, contain constant material properties for an element. By assigning spatially dependant
properties at each Gauss point, the stiffness matrix provides variation across the element,
4
resulting in a full, smooth transition across each element. The user subroutine is used to
map the modulus gradation over the boundary area of the model.
2.2.1 Element Type and FGM Modeling
The element type selected for this study is the plane stress CPS8, bi-quadratic, full
integration isoparametric element. This element supports the use of the user defined
material parameters UMAT subroutine. The general formulation of the element, and the
application of the material gradation to the element, is described by Bathe and Wilson
(1976), as well as Santare and Lambros (2000). To formulate the stiffness matrix, a set
of shape functions is established resulting in the following matrix of displacements
components:
i
n
ii UxNxu )()(
1∑
=
= (2.1)
where u(x) is the matrix of displacement components within the element, Ni(x) is the
matrix of shape functions and Ui are the nodal displacements for each of the nodes, n.
Taking the derivative of the shape functions results in the infinitesimal strain components
of equation (2.2), where Bi(x) is populated with the Ni(x) derivatives.
∑=
=n
iii UxBx
1)()(ε (2.2)
At each point the stress components are calculated from the strain and material property
matrix C(x)
)()()( xxCx εσ = (2.3)
5
For the nonhomogeneous case, the material matrix, C(x), consists of a set of properties
that are spatially dependant. The element stiffness matrix, Ke, is then defined as the
linear function that maps the nodal displacements to the nodal forces, fi,
ie
i UKf = (2.4)
Per the principle of virtual work, the work done by the nodal forces must equal the work
of deformation within the element. Equating these quantities, the element stiffness
matrix is derived as
∫=eV
Te dVxBxCxBK )()()( (2.5)
where the integral is taken over the volume of the element.
2.2.2 User Subroutine
To model the spatially dependent Young’s modulus in this study, the Abaqus UMAT user
subroutine was used. The subroutine is written in FORTRAN language and runs in
parallel to the Abaqus solver. It allows the user to establish an algorithm to calculate user
variables that will be passed into the Abaqus solver. For this study the subroutine was
coded such that the material and stiffness matrices were established with the appropriate
spatially dependent material properties, i.e., Young’s modulus. Poisson’s ratio was
assumed to be constant since it has been shown that variations in Poisson’s ratio have
much less significance than Young’s modulus (Sadd, 2009). The method required for
establishing the stiffness matrix requires equation (2.5) to be integrated numerically using
6
Gauss quadrature (Manneth. 2009). Using Gauss quadrature, equation (2.5) is evaluated
at specific Gauss points (xi, yi) within the element through the following relation
jiji
N
i
N
jiijiji
Te wwyxJyxByxCyxBK ,),(),(),(),(1 1
∑∑= =
= (2.6)
where i and j correspond to the element integration points, J is the determinant of the
Jacobian matrix, and wi and wj are the weights of each Gauss point. The UMAT code
that was used in this study was a modified version of that established by Manneth (2009)
and is listed in Appendix A.
2.3 Finite Element FGM Model Verification
To verify the UMAT user subroutine, a finite element model was evaluated against a
known closed form solution. The goal was to assess the performance of the subroutine
and its link to the Abaqus solver, as well as to assess the performance within a high stress
gradient stress field. Since the closed form solution for an infinite plate with a circular
hole has been established for nonhomogeneous elastic modulus, it was used for the
subroutine verification. The solution is developed in Sadd (2009) and is derived from the
hollow cylindrical domain under uniform internal and external pressure loading, where
the modulus is graded radially with a power-law variation of Young’s modulus.
n
arErE ⎟
⎠⎞
⎜⎝⎛= 0)( (2.7)
7
This can be applied to the rectangular plate with stress free center hole model by defining
a large boundary to hole radius ratio. The stress field for radial and tangential stress is as
follows
[ 2/)2(2/)2(2/)2(
0 nkknkkk
nk
r rarab
bP +−−++−−+
−−
−=σ ] (2.8)
⎥⎦⎤
⎢⎣⎡
−+−−
++−−+
−−= +−−++−
−+2/)2(2/)2(
2/)2(0
22
22 nkknk
kk
nk
r rank
nkrnk
nkab
bPννν
νννσ (2.9)
where νnnk 442 −+= , P0 = -T (biaxial far field boundary traction), b is the radius of
the outer boundary, or in the case of the rectangular plate the half width, and a is the
center hole radius.
A quarter symmetric model of a 40 x 40 unit thin plate with 1unit radius center hole was
used for verification. This was modeled as a square 20 x 20 unit plate with a 1 unit radius
hole in the corner (a/b = 20), as shown in figure 2-1. The mesh was refined around the
center hole and was graded in the direction moving away from the hole toward the outer
boundaries. The initial mesh consisted of 471 CPS8 bi-quadratic, full integration plane
stress elements with 1581 nodes.
8
T
0= Tu x
Figure 2-1. Verification mesh with 471 CPS8 elements.
The biaxial loading boundary conditions consisted of edge pressures on the outer
boundaries as well as symmetry along the x and y-axes. The base Young’s modulus, E0,
was 200 GPa with a Poisson’s ratio of 0.25, which was held constant. The initial
evaluation was to the homogenous solution, where n in equation (2.7) was set to zero.
The closed form solution results in a maximum normalized tangential stress, σθ/T = 2.0 at
the edge of the hole. The FEA model returned a result of 1.994, which is an accuracy of
0.3%.
Next the model was evaluated with the gradation parameter n = 0.2. The FEA solution
returned a normalized maximum tangential stress σθ/T = 1.534, where the analytical
solution is 1.550, or about 1.0% accuracy. The solutions were plotted for comparison
along the radial line from the edge of the hole extending to the boundary. It was noticed
that although the gradient in the vicinity of the hole was adequately captured when
a b
u = 0y
9
compared to the closed form solution, it appeared to slightly diverge approaching the
outer traction boundary, which is likely due to the rectangular boundary which is not
present in the analytical solution. The mesh was then refined for both convergence study
as well as to assess the behavior near the boundary. Meshes of 1980 and 2350 elements
were generated, both of which had refinement near the edge of the hole. Although the
normalized stress converged to 1.997 for both, the near boundary behavior was
unchanged. Since this far field loading problem is focused on localized stress at the hole,
this was deemed acceptable.
The mesh consisting of 1980 elements, with 6141 nodes, was used to further verify the
UMAT user subroutine for radial gradation with values n = -0.2, 0 and 0.2. The results
are plotted in figure 2-2, along with the 1980 element mesh. The FEA solution is
generally very good throughout the domain as compared to the analytical curves. For n =
0.2, the 1980 element FEA solution was 1.535 compared to 1.550 for the theoretical
(1.0%). The y-direction stress contours are shown in figure 2-3, where the tangential
stress is represented along the x-axis. For n = -0.2 the FEA solution returned 2.546 where
the closed form is 2.550 (0.2%).
10
Figure 2-2. 1980 element mesh and resulting UMAT verification curves.
y
x
Figure 2-3. Y-direction stress contours for radial gradation for n = 0.2.
11
3 Analysis and Results
3.1 Infinite Plate with Stress Free Circular Hole – Biaxial Loading
3.1.1 Introduction
The following studies the effect of local Young’s modulus gradation around a stress free
circular hole in an infinite plate subject to far field biaxial loading. The modulus was
graded in the radial direction emanating from the edge of the hole into the section toward
the outer traction boundaries. As in the verification problem for user subroutine in
Section 2.3, Young’s modulus was varied starting at the edge of the hole, but now was
limited in depth such that gradation did not extend to the outer boundaries (except for the
initial run in to establish a baseline). The intent was to model a Young’s modulus ratio,
E/E0, at the edge of the hole, for both increase and decrease, to grade the variation within
general vicinity of the hole. The intensity of the modulus increase as well as the
gradation depth were controlled and evaluated. The goal was to create a highly localized
gradation to reduce the magnitude of stress concentration.
To quantify the depth of gradation, it was assumed that when the graded modulus E
reached 99.9% of E0, the gradation was considered complete and this location was
considered the “depth of gradation”. This distance is referred to as the percentage depth
of the overall boundary length.
3.1.2 FEA Model and Boundary Conditions
The finite element model used for this study was leveraged from the verification of the
user subroutine, Section 2.3. Again, the evaluation criteria and convergence confirmation
12
for this problem were completed where the homogeneous solution, equations (2.8) and
(2.9), were used to determine accuracy. The localized radial gradation function used to
control the modulus ratio at the edge of the hole and depth of gradation was
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡+=
⎥⎦
⎤⎢⎣
⎡−⎟
⎠⎞
⎜⎝⎛−
2
1
0 1 arn
keEE (3.1)
where E is the variable modulus, E0 is the base modulus, k is the modulus modifying
parameter, n is the gradation depth modifier, and a is the radius of the center hole, which
for this model was equal to 1. Throughout this study the far field modulus,
E0 = 1 GPa with Poisson’s ratio ν = 0.3 and held constant.
For this study, the initial k and n parameters were varied as k = 5.0, 2.5, 1.0 and n = 0.1
and .015. Using these parameters, the gradation is plotted in figure 3-1 as a function of
normalized section depth, r/a. The positive k value reflects surface stiffening, or material
stiffening at the edge of the hole.
Figure 3-1. Gradation for n = 0.015 (full gradation) and n = 0.10 (50% gradation depth).
13
Negative k values were also evaluated, which reflects surface softening at the edge of the
hole. The negative parameters varied as k = -0.8, -0.7, -0.6, -0.4 which were each
evaluated at n = 0.1, 0.2, 0.3 and 2.0. The gradation depths for these n parameters are
shown in figures 3-2 thru 3-5 with their respective results.
3.1.3 Results
The initial models run for the surface stiffening case, positive k values, showed an
increase in the normalized stress at the edge of the hole for both the 50% and full
gradation depth (n = 0.1 and 0.015 respectively), figures 3-2 and 3-3. As the maximum
tangential stress for the homogeneous case is 2.0, the increase in stress concentration
exceeded this in all cases with k>0 and is more severe for the shallow gradation depth of
50%.
Figure 3-2. Surface stiffening, k>0, for n = 0.10 (50% depth).
14
Figure 3-3. Surface Stiffening, k>0, for n = 0.015 (full depth).
The resulting maximum normalized tangential stress occurred at the edge of the hole, as
expected, and are listed in Table 3-1. The increase in stress at the hole appears to be
directly related to modulus ratio where it is more pronounced as the modulus ratio is
increased. The intensity of the stress for each modulus ratio appears to be related to the
depth of gradation, where the more shallow gradient results increased stress level.
Table 3-1. Resulting stress concentrations for surface stiffening, k>0.
k E/E 0 max n Depth σ θ /T at hole0.10 50% 2.5670.02 100% 2.4180.10 50% 3.0040.02 100% 2.7110.10 50% 3.3860.02 100% 2.9435 6.0
1 2.0
2.5 3.5
The results for the surface softening case, k<0, where more favorable in terms of reducing
the stress concentration at the hole, where in all of these cases the tangential stress at the
hole was reduced. Typical behavior is shown in figures 3-4 and 3-5. It is interesting to
15
note that the typical behavior exhibits an increase in tangential stress at a depth that
appears to coincide with gradation depth, and is more pronounced as the modulus ratio is
reduced. The case of k = -0.8 and n = 2.0 represents the most dramatic decrease in the
stress concentration at the hole, at 0.562, but shows an additional stress peak within the
depth of the plane equal to 1.400, again coincident with, or slightly inside, the gradation
depth.
Figure 3-4. Surface Softening, k<0, for n = 0.1 (45% grad depth), biaxial loading.
Figure 3-5. Surface softening, k<0, for n = 2.0 (14% grad depth), biaxial loading.
16
Plots of edge modulus ratio are shown in figure 3-6 for the varying gradation depths. The
general trend can clearly be seen where for each modulus ratio the depth of gradation
appears to have a more dramatic effect on the maximum stress within the section, where
as the modulus ratio has more of an effect on the magnitude of stress concentration at the
edge of the hole. Furthermore, it appears that as gradation depth is increased, the stress
gradient from the hole to the interior peak becomes smoother with its peak slightly inside,
or at a slightly more shallow depth, than the gradation depth. This is clearly the case for
n = 0.1. The results are tabulated in Table 3-2 for all cases evaluated.
Figure 3-6. Effects of gradation depth for Young’s modulus ratio, biaxial loading.
17
Table 3-2. Stress concentration for surface softening, k<0, biaxial loading.
k E/E 0 max n Depth σ θ /T max σ θ /T at hole0.1 45% 1.184 0.8390.2 35% 1.209 0.7750.3 28% 1.230 0.7382.0 14% 1.400 0.5620.1 45% 1.153 1.0850.2 35% 1.177 1.0250.3 28% 1.200 0.9882.0 14% 1.360 0.8150.1 45% 1.282 1.2820.2 35% 1.230 1.2300.3 28% 1.197 1.1972.0 14% 1.337 1.0340.1 45% 1.590 1.5900.2 35% 1.552 1.5520.3 28% 1.529 1.5292.0 14% 1.410 1.410
-0.6 0.4
-0.4 0.6
-0.8 0.2
-0.7 0.3
3.2 Infinite Plate with Stress Free Circular Hole – Uniaxial Loading
3.2.1 Introduction
The case of the stress free center circular hole in an infinite plate with far field uniaxial
loading was investigated next. Gradation was again in the radial direction emanating
from the edge of the hole toward the outer boundaries. Here the modulus ratio was
focused on surface softening, as the previous biaxial study indicated that the stress
concentration reduction was dependant on softening and not stiffening. As in the biaxial
case, the intensity of the modulus decrease as well as the gradation depth was controlled
and evaluated.
3.2.2 FEA Model and Boundary Conditions
As for the biaxial case, the FEA model was leveraged from the initial user subroutine
verification. The boundary conditions were unchanged, as shown in figure 2-1, with the
18
exception of the boundary loads where the y-direction load was dropped. The x-direction
load remained at 1 MPa. The radial gradation was controlled by equation (3.1) with the
parameters k = -0.8, -0.7, -0.6, -0.4 which were each evaluated at n = 0.1, 0.3 and 2.0.
The evaluation criteria was again the homogeneous closed form solution which is derived
in Sadd (2009) and results in a maximum tangential, or hoop, stress around the edge of
the hole given by
)2cos21(),( θθσ θ −= Ta (3.2)
where θ and a are defined in figure 3-7.
x
y
a θ T T
Figure 3-7. Stress free hole for infinite plate with uniaxial loading.
The maximum tangential stress is located on the edge of the hole at θ = 90o, or on the
axis perpendicular to the direction of loading. For the homogeneous case the stress
concentration is equal to 3.0.
19
3.2.3 Results
The behavior due to uniaxial loading was found to be quite similar to that of the biaxial
case. While the stress concentration at the hole was reduced for each case evaluated,
there did exist an additional stress peak that coincided with the gradation depth. As in the
biaxial case, the greatest reduction in stress concentration at the hole is found at k = -0.8
and n = 2.0, where the normalized tangential stress is 0.920. Again, the peak stress shifts
from the edge of the hole to a depth of approximately 14% where it has a normalized
value of 1.532, a 67% increase from the edge stress. This is shown in figure 3-8 where
the gradation depth n = 2.0 is plotted for the various k values. The resulting stress
concentrations and peak stresses are tabulated in Table 3-3.
Figure 3-8. Surface softening, k<0, for n = 2.0 (14% grad depth), uniaxial loading.
20
Table 3-3. Stress concentration for surface softening, k<0, uniaxial loading.
k E/E 0 max n Depth σ θ /T max σ θ /T at hole0.1 45% 1.248 1.2430.3 28% 1.299 1.0962.0 14% 1.532 0.9200.1 45% 1.621 1.6210.3 28% 1.470 1.4702.0 14% 1.482 1.2790.1 45% 1.927 1.9270.3 28% 1.785 1.7852.0 14% 1.598 1.5980.1 45% 2.399 2.3990.3 28% 2.296 2.2962.0 14% 2.151 2.151
-0.8 0.2
-0.7 0.3
-0.6 0.4
-0.4 0.6
Figure 3-9 plots each of the modulus ratio values to the varying gradation depths. The
behavior is again quite similar to the biaxial case, where the stress concentration at the
hole appears to be a function of the Young’s modulus ratio, and the gradation depth
appears to effect the transition and formation of an additional stress peak at depth.
21
Figure 3-9. Effects of gradation depth for Young’s modulus ratio, uniaxial loading.
3.3 Infinite Plate with Stress Free Elliptical Hole – Biaxial Loading
3.3.1 Introduction
In this study the localized Young’s modulus gradation is applied to the elliptical shaped
hole with far field biaxial tensile loading. The same methods are applied to vary both the
modulus ratio at the edge of the hole and the depth of gradation. However, here the
spatial gradation is a function of the elliptical geometry and not a function of the radial
distance from the edge of the hole, as for the circular shape. The gradation follows the
shape of the elliptical hole as it emanates from the edge toward the outer boundaries.
22
For this study, two elliptical cases subjected to biaxial loading are evaluated. The first
has a major to minor axis ratio (a/b) of 2:1, followed by the case of 5:1.
3.3.2 FEA Model and Boundary Conditions
The general model for consisted of a 40 x 40 unit square plate with a centered ellipse
with its major axis in the horizontal, or x-direction. The FEA model took advantage of
symmetry on both the vertical, y-axis and horizontal x-axis and consisted of a 20 x 20
unit square boundary with a quadrant of the ellipse located at the midpoint of symmetry.
The remaining boundary conditions consisted of both x and y-direction tensile loads of 1
MPa on the outer boundary edges.
To evaluate model accuracy, convergence to the closed form analytical solution for the
homogeneous case was evaluated. The homogeneous solution for biaxial loading is
derived from superposition of the two uniaxial cases for horizontal loading (1) and
vertical loading (2) as shown in figure 3-10.
23
yy S=σ∞
Figure 3-10. Superposition of uniaxial far field loading (courtesy of Sadd 2009).
From Sadd (2009), the solution to each individual problem for the boundary
circumferential stress component is given by
⎟⎟⎠
⎞⎜⎜⎝
⎛+π−ϕ−−π−ϕ−+
=ϕσ
⎟⎟⎠
⎞⎜⎜⎝
⎛+ϕ−−ϕ−+
=ϕσ
ϕ
ϕ
1)2/(2cos2)2/(2cos212)(
12cos22cos212)(
222
222)2(
121
211)1(
mmmmS
mmmmS
y
x
(3.3)
where 121 , mababm
babam −=
+−
=+−
= , and φ is the angle measure counterclockwise from
the x-axis.
(1) Horizontal Uniaxial
x
y
xx S=σ∞
a b
x
y
xx S=σ∞
ab =
yy S=σ∞
y
a+ b x
(2) Vertical Uniaxial
24
Using superposition and letting m = m1 = - m2
⎟⎟⎠
⎞⎜⎜⎝
⎛+ϕ−−ϕ++−
+⎟⎟⎠
⎞⎜⎜⎝
⎛+ϕ−−ϕ−+
=
ϕσ+ϕσ=ϕσ ϕϕϕ
12cos22cos212
12cos22cos212
)()()(
2
2
2
2
)2()1(
mmmmS
mmmmS yx
(3.4)
For the case of equal biaxial tensile loadings where SSS yx == , boundary tangential stress becomes
⎟⎟⎠
⎞⎜⎜⎝
⎛+ϕ−
−=ϕσϕ 12cos2
)1(2)( 2
2
mmmS (3.5)
For the case with b > a, the maximum value of this stress is found at φ = π/2
abS
mmS
mmmS 2
1)1(2
12)1(2)2/( 2
2
max =⎟⎠⎞
⎜⎝⎛
+−
−=⎟⎟⎠
⎞⎜⎜⎝
⎛++
−=πσ=σ ϕ (3.6)
With b/a = 2, σmax = 4S, while for b/a = 5, σmax = 10S
The initial FEA model was constructed for a/b = 2 and consisted of 1976 elements and
6531 nodes. Note that the FEA geometry has rotated the ellipse 90o from the illustration
shown in figure 3-10. As before, the isoparametric CPS8, bi-quadratic, full integration
element was used with the UMAT user subroutine defining spatial gradation. The
function used to establish the elliptical dependent spatial gradation is given by the similar
form
[ ][ 21)(0 1 −−+= rnkeEE ] (3.7)
where
2
2
2
2
by
axr += (3.8)
25
The modulus ratios studied where k = -0.8, -0.6, -0.4 each evaluated at gradation depth
parameters of n = 0.1, 0.3 and 2.0.
The convergence study, listed in Table 3-4, resulted in an accuracy to the closed form
solution to 1.7%.
Table 3-4, convergence study for biaxial elliptical hole.
Elements Nodes Maximum Normalized Stress Accuracy
1976 6135 4.093 2.3%4949 15164 4.081 2.0%5499 16834 4.069 1.7%6524 19939 4.068 1.7%
The refined mesh consisting of 5499 elements was used for the study, shown in figure 3-
11, as it represented 1.7% accuracy to the closed form homogeneous solution and
provided a mesh that was consistent in element aspect ratio, while the 6524 element mesh
resulted in skewed elements. The mesh was refined on the edge of the hole with a graded
mesh that became less dense toward the outer boundary.
y
uy = 0
u x =
0
T
T
x
a
b
Figure 3-11. FEA mesh and boundary conditions, elliptical hole with biaxial loading.
26
For the a/b = 5 case, the boundary was increased to 50 x 50 units such that the ratio of the
overall width to the major axis, a, remained at 10:1 for the infinite model to be valid.
Similar mesh refinement and convergence study resulted in a mesh consisting of only
1863 elements (5798 nodes) with a maximum normalized stress of 10.098 at the edge of
the hole on the major axis, which is an accuracy to 1%. In this case mesh refinement was
held close to the high stress gradient area near the hole and relaxed toward the outer
boundaries.
3.3.3 Results
The behavior for the elliptical case is very similar to that of the circular hole. A reduction
in the normalized tangential stress at the edge of the hole was indicated in all cases
evaluated with k<0. There exists a similar behavior of the stress on the major axis
direction (x-axis) in that it tends to lose its monotonic decay moving away from the edge
of the hole. This is more pronounced for the shallow depth gradation, n = 2.0, 28%, and
low modulus ratio, k = -0.8, as seen in figure 3-12, along with typical gradation curves,
for a/b = 2.
Figure 3-12. Surface softening, k<0, for n = 2.0 and a/b = 2, biaxial loading.
27
The reduction in stress concentration at the edge of the hole again appears to the related
more to the modulus ratio than gradation depth, as shown in Figure 3-13. A second stress
peak is again observed as the gradation depth in reduced.
Figure 3-13. Effects of gradation depth for a/b = 2, biaxial loading.
The maximum tangential stress is located on the edge of the hole for all cases except for k
= -0.8 and n = 2.0, in which case there is a slight increased peak at approximately 18%
depth. Results for the a/b = 2 case are tabulated in Table 3-5.
28
Table 3-5. Stress concentration for surface softening, k<0, for a/b = 2, biaxial loading.
k E/E 0 max n Depth(x) σ θ /T max σ θ /T at hole0.1 90% 1.988 1.9880.2 70% 1.819 1.8190.3 57% 1.735 1.7352.0 28% 1.492 1.4340.1 90% 2.853 2.8530.2 70% 2.709 2.7090.3 57% 2.633 2.6332.0 28% 2.345 2.3450.1 90% 3.397 3.3970.2 70% 3.200 3.2000.3 57% 3.246 3.2462.0 28% 3.037 3.037
-0.4 0.6
-0.8 0.2
-0.6 0.4
For the case of a/b = 5, the maximum tangential stress appears on the edge of the hole in
all cases evaluated. The stress along the major axis symmetry boundary appears much
smoother, however, when compared to the a/b = 2 case. Figure 3-14 shows the typical
gradation depths, along with the resulting tangential stresses in the major axis direction
through 1/5 of the total section depth (width). As can be noticed, the stress just inside of
the edge of the hole has a sharp gradient similar to the a/b = 2 case, as well as for the
circular hole, and exhibits monotonic decay as it extends toward the center of the width
of the plate. This difference in behavior may be attributed to the gradation shape that
extends around the ellipse. On the minor axis, y-direction, the gradation depth is more
shallow than for the major axis direction due to the elongation of the elliptical shape in
the x-direction. As the gradation emanates from the edge of the hole it reaches the base
modulus, E0, much quicker along the y-axis direction than the x-direction. This causes a
more non-uniform modulus across the diagonal width, say at 450, which may provide
stiffer material in this section of the plate that helps reduce the effects of the y-axis
29
loading. Regardless, the maximum stress occurs on the x-axis (major axis). The results
for the a/b = 5 case are tabulated in Table 3-6.
Figure 3-14 Effects of gradation depth for a/b = 5, biaxial loading
Table 3-6. Stress concentration for surface softening, k<0, for a/b = 5, biaxial loading.
k E/E 0 max n Depth(x) σ θ /T max σ θ /T at hole0.1 90% 5.034 5.0340.3 47% 4.610 4.6102.0 28% 4.105 4.1050.1 90% 7.037 7.0370.3 47% 6.683 6.6832.0 28% 6.528 6.5280.1 90% 8.345 8.3450.3 47% 8.103 8.1032.0 28% 7.814 7.814
-0.4 0.6
-0.8 0.2
-0.6 0.4
30
3.4 Infinite Plate with Stress Free Elliptical Hole – Uniaxial Loading
3.4.1 Introduction
In this section the plate with center elliptical hole is subject to uniaxial loading in the
direction perpendicular to the major axis. The modulus gradation was identical to that for
the biaxial case, where the spatial gradation follows the shape of the elliptical hole given
by equations (3.7) and (3.8). The modulus ratio was evaluated for hole edge surface
softening, where k < 0. The two elliptical cases evaluated for biaxial case, a/b = 2 and
a/b = 5, were again evaluated here for uniaxial case.
3.4.2 FEA Model and Boundary Conditions
The FEA models were leveraged from the biaxial study. The boundary conditions were
identical with the exception of the loading, where the x-direction load was removed. The
resulting load boundary condition was a uniform tensile load of 1 MPa on the outer edge
of the plate in the direction perpendicular to the major axis, as in figure 3-15.
T
uy = 0
u x =
0
Figure 3-15. Mesh and boundary conditions for elliptical model for a/b = 2, uniaxial load.
31
For both a/b cases, the modulus ratios where k = -.08, -.06, -.04 and where evaluated at
gradation depths n = 0.1, 0.3 and 2.0.
The closed form analytical solution for the uniaxial case was established as part of the
biaxial solution and is given by the first of equations (3.3) for b>a. Equation (3.3) is
rotated 90o relative to the FEA model as shown in figure 3-15 and has the major axis in
the b-direction. The maximum tangential stress solution for the homogeneous case of
a/b = 2 is 5.0, and for a/b = 5 is 11.0. The FEA model validation solutions were 5.105
(2%) and 11.099 (0.9%) respectively.
3.4.3 Results
The results for the a/b = 2 case are very similar to the biaxial case in that the stress
concentration on the edge of the hole is reduced with reduction in modulus ratio. Again,
as the depth of gradation becomes more shallow, the tangential stress along the x-
symmetry boundary becomes non-monotonic, which is most pronounced at the shallow
depth of n = 2.0, and in particular for the modulus ratio of k = -0.8. Figure 3-16 plots the
resulting tangential stresses along the major, x-axis direction along with the typical
gradation curves. For all cases evaluated the stress is reduced at the edge of the hole and
is maximum at this point. These results are shown in Table 3.7.
32
Figure 3-16. Effects of gradation depth for a/b = 2, uniaxial loading.
Table 3-7. Stress concentration for surface softening, k<0, for a/b = 2, uniaxial loading.
k E/E 0 max n Depth(x) σ θ /T max σ θ /T at hole0.1 90% 2.510 2.5100.3 47% 2.173 2.1732.0 28% 1.825 1.8250.1 90% 3.602 3.6020.3 47% 3.296 3.2962.0 28% 2.962 2.9620.1 90% 4.281 4.2810.3 47% 4.068 4.0682.0 28% 3.822 3.822
-0.4 0.6
-0.8 0.2
-0.6 0.4
33
The a/b = 5 case also shows similar behavior to the biaxial case. The tangential stress
along the major axis direction is smooth and is reduced to a greater degree as the modulus
ratio is reduced. Figure 3-17 shows the resulting stresses along the major, x-axis
direction along with the typical gradation curves. The results are listed in Table 3-8.
Once again, the k = -0.8 modulus ratio (20%) and shallow gradation depth of n = 2.0
(28%), appear to provide the most significant reduction in stress concentration at the edge
of the hole, with a reduction from 11.099 to 4.512, or 59%.
Figure 3-17. Effects of gradation depth for a/b = 5, uniaxial loading.
34
Table 3-8. Stress concentration for surface softening, k<0, for a/b = 5, uniaxial loading.
k E/E 0 max n Depth(x) σ θ /T max σ θ /T at hole0.1 90% 5.532 5.5320.3 47% 5.051 5.0512.0 28% 4.512 4.5120.1 90% 7.488 7.4880.3 47% 7.334 7.3342.0 28% 6.880 6.8800.1 90% 9.180 9.1800.3 47% 8.900 8.9002.0 28% 8.591 8.591
-0.4 0.6
-0.8 0.2
-0.6 0.4
3.5 Contact Loading – Uniform Normal Distributed Loading in Half Space
Following is a brief study that evaluates the stress field and stress concentration effects of
a simulated contact mechanics problem using localized Young’s modulus gradation.
3.5.1 Introduction
A typical analytical approach to modeling contact mechanics problems is with a
distributed load on an elastic half space. Since contact stresses within an elastic body are
highly concentrated close to the contact region and decrease rapidly away from the area
of contact, stresses can be calculated to a good approximation by considering a body as a
semi-infinite elastic solid bounded by a plane surface, or an elastic half-space. Providing
the boundary dimensions are large enough as compared to the contact area, the stresses
near the contact region are not dependent on boundary shape or constraint far from the
contact area. In addition, body curvature can be neglected, as contact elements are
typically rounded or spherical in shape, and the stress field for this non-uniform contact
can be approximated with a uniform loading (Johnson, 1985).
35
The stress field below the surface directly under the contact area reaches a peak
maximum shear stress slightly below the surface. For the homogeneous case this occurs
at a depth of y/a = 1.0 below the surface, with y being the axis parallel to the applied load
and a being the half length of a symmetrical uniform load centered on y (figure 3-18).
For ductile materials it is theorized that this maximum shear stress is responsible for the
fatigue failure of contacting elastic elements, in which a crack originates below the
surface at the peak maximum shear stress point and propagates to the surface under
repeated loading (Shigley and Mischke, 1989). Thus for the evaluation of stress
concentration effects in this current study, the maximum shear stress is the criteria to
which the local gradation effects were evaluated.
The local gradation was constructed such that the modulus had both increasing and
decreasing behaviors measured from the contact loading surface. The transition from the
surface modulus back to that of the base material value, or depth of gradation, was also
varied.
3.5.2 FEA Model and Boundary Conditions
The model used in this study consisted of a small boundary section with a uniform load
applied to the free surface with Young’s modulus varying from the free surface into the
depth of the section. As shown in figure 3-18, the uniform surface load was applied over
a small section of the surface and was centered on the y-axis, which is positive in the
direction into the depth of the section. The boundary size that was modeled was 20 x 20
units with a loading surface of 1/10 of the overall boundary, or 2 units. The modulus
36
gradation was localized toward the loaded surface and varied into the depth of the section
in the vertical direction only.
a
y
x
Figure 3-18. General contact model for vertical modulus gradation.
The finite element model consisted of a 20 x 20 unit 2-D plane with a uniform pressure
load of 1 MPa over 2 units acting on the free surface centered on the y-axis. The mesh
consisted of 2496 CPS8 full integration quadrilateral elements with 7689 nodes and was
refined under the uniform load and graded from the surface in to the depth of the section.
The section was constrained at the lower free edge using zero displacement boundary
conditions for both the x and y-directions. Figure 3-19 shows the mesh and boundary
conditions used for this contact problem study.
37
Figure 3-19. Finite element mesh and boundary conditions for contact problem.
Local Young’s modulus gradation was modeled using an exponential function such that
the modulus was varied starting at the loaded surface, y = 0, and graded to various depths
into the section. The modulus was graded in the y-direction only using equation (3.9).
( )2
10nykeEE −+= (3.9)
The parameters k and n were varied such that the modulus at the surface, as well as the
depth of transition, or gradient, could be modified. The k parameter modifies the
modulus to either increase or decrease, and the n parameter produces a gradient depth
modifier. To obtain preliminary results the parameters initially selected for the study
Y
X
20
ux = uy = 0
P
20
38
were: k = -0.8, -0.4, 0.4, 0.8 with n = 0.02, 0.10, 1.0. Once the general behavior of the
model was understood, the additional parameters of k = 2.0 and k = 3.0 were added for
additional evaluation. The gradation depth parameters, n, are shown graphically for k =
0.8 and -0.8 in figure 3-20. The depth and gradient of the curves are typical for all k
values evaluated.
Figure 3-20. Typical gradation depth shapes for various n values.
Prior to evaluating modulus gradation, the homogeneous case was run and compared to
the analytical solution. The analytical solution for the stress field for half-space under
uniform loading is derived from superposition of the single concentrated normal force
solution, resulting in the following stress field (Sadd, 2009):
( ) ([ 1212 2sin2sin22
θθθθπ
σ −+−−=P
x )] (3.10)
( ) ( )][ 1212 2sin2sin22
θθθθπ
σ −−−−=P
y (3.11)
39
[ 12 2cos2cos2
θθπ
τ −=P
xy ] (3.12)
where θ1 and θ2 are defined in figure 3-4.
P
θ1 θ2 a a
y
x
Figure 3-4. Half-space under uniform loading over –a>x>a (Sadd, 2009).
The homogeneous model was evaluated with a Young’s modulus of 1GPa and Poisson’s
ratio of 0.3. Since the area of interest is directly under the load, the resulting stress
distribution was obtained from the nodal values along the y-axis and compared to the
analytical solution in terms of maximum shear stress. The theoretical maximum shear
stress along the y-axis can be calculated using only equations (3.10) and (3.11) since the
in plane shear stress is zero along the y-axis due to symmetry. Thus, with the x and y-
axes being principal, the maximum shear stress is determined using equation (3.13).
yx σστ −=21
max (3.13)
The normal stress values along the y-axis were extracted from the finite element model
and the resulting maximum shear stress was calculated using equation (3.13). The model
40
did return in-plane shear stress values, which may have resulted from slight asymmetry
within the mesh, but these were insignificant in magnitude and thus ignored. The
resulting comparison between the FEA model and the closed form solution is shown in
figure 3-21, where it can be seen that there exists some deviation in peak value.
Figure 3-21. Maximum shear stress for theoretical and FEA solutions.
The peak normalized maximum shear stress for the closed form solution is 0.3183. The
FEA solution returned a value 8.8% higher at 0.3465. Increasing element count and
mesh refinement showed insignificant improvement but did confirm convergence of the
initial FEA model. It was determined that since the model was capturing the behavior as
compared to the theoretical solution, i.e., the maximum shear stress curves followed the
same shape along the y-axis with the peak occurring at y/a = 1.0, it would be adequate to
evaluate the general, relative behavior of the FGM contact model.
41
3.5.3 Results
Evaluation of the gradation effects were assessed relative to the FEA solution of the
maximum shear stress values for the homogeneous model. In general, the behavior of the
graded models appeared to follow that of the closed form solution in that the peak
maximum shear stress occurred at a depth of y/a = 1.0. For the case of surface softening,
k = -0.8 and -0.4, the peak maximum shear stress showed an increase for the deeper
gradation parameters n = 0.02 and 0.10 of up to 7.3%, while the more shallow case, n =
1.0, showed slight decrease of up to 4.1%. The increased peak maximum shear stress is
shown in figure 3-22, which is for the case of n = 0.10 for both k = -0.8 and -0.4.
In addition to the maximum shear stress, the σy stress was evaluated on the y-axis and
compared to the FEA homogeneous model. For the case of surface softening, all
gradation depth variations showed a slight increase in the compressive stress, with the
typical behavior shown in figure 3-23. The σy stress value at the critical y/a = 1.0 depth
was extracted from the model and recorded. The results for peak maximum shear stress
and σy stress at y/a = 1.0 for surface softening are shown in Table 3.9.
42
Figure 3-22. Maximum shear stress for surface softening for n = 0.10.
Figure 3-23. Normal stress, σy, along y-axis for k = -0.8, surface softening.
43
Table 3.9. FEA results for contact problem, surface softening case.
k E/E 0 at surface nNormalized Peak Max
Shear Stress along y-axis
τmax
Normalized Max Stress along y-axis @ y/a=1.0
σy
0(homogeneous) 1 - 0.3465 -0.799
0.02 0.3717 -0.8240.10 0.3699 -0.8541.00 0.3328 -0.8690.02 0.3557 -0.8040.10 0.3561 -0.8161.00 0.3395 -0.827
-0.8 0.2
-0.4 0.6
For the case of surface stiffening, where k values are positive, the peak maximum shear
stress follows an opposite trend to that of surface softening, k<0. In all of the cases
evaluated, the FEA model predicts a decrease in peak maximum shear stress when the
gradation depth is increased, for n = 0.10 and 0.02, whereas for the more shallow case of
n = 1.0 the peak increases. In addition, the σy stress along the y-axis exhibits an opposite
trend as well where the compressive stresses are reduced slightly at depth. Figure 3-24
shows the typical behavior of maximum shear stress for the deeper gradation parameters,
n = 0.02 and n = 0.10, where n = 0.10 is plotted for modulus parameters k = 0.4 and 0.8.
Figure 3-25 shows the vertical normal stress, σy along the y-axis, where the trend shows a
reduction in compressive stress at depth, which is observed in all of the cases evaluated
for the positive k parameter.
44
Figure 3-24. Maximum shear stress for surface stiffening at gradation depth n = 0.10.
Figure 3-25. Normal stress, σy, along the y-axis for n = 0.10, surface stiffening.
The results for the surface stiffening study are given in Table 3.10. As indicated, the
most dramatic decrease in peak maximum shear stress occurs at k = 3.0 with a gradation
depth of n = 0.10, where the peak is reduced by 8.3%.
45
Table 3.10. FEA results for surface stiffening case.
k E/E 0 at surface nNormalized Peak Max
Shear Stress along y-axis
τmax
Normalized Max Stress along y-axis @ y/a=1.0
σy
0(homogeneous) 1.0 - 0.3465 -0.799
0.02 0.3351 -0.7940.10 0.3339 -0.7821.00 0.3597 -0.7600.02 0.3400 -0.7960.10 0.3392 -0.7891.00 0.3533 -0.7780.02 0.3255 -0.7910.10 0.3233 -0.7701.00 0.3777 -0.7230.02 0.3202 -0.7900.10 0.3178 -0.7641.00 0.3917 -0.700
2.0 3.0
3.0 4.0
0.8 1.2
0.4 1.4
However, for the case of shallow stiffening where n = 1.0, again with surface stiffening at
k = 3.0, there tends to be an increase in the peak maximum shear stress within the depth
of y/a = 1.0 approaching the loading surface. Conversely, at greater gradation depth the
peak maximum shear stress is reduced. It was also observed (Table 3.10) that the peak
decreased from a gradation depth n = 1.0 to n = 0.1, but showed a slight increase from the
n = 0.10 level as the depth was increase to n = 0.02. Figure 3-26 shows maximum shear
stress for gradation depth at k = 3.0 along the y-axis. It is noted that for n = 1.0 and to a
lesser degree n = 0.10, there is an increase in maximum shear stress at the loaded surface
which differs from the typical behavior and curve shape exhibited by lower k values.
Furthermore, the peak maximum shear stress appears to occur at a slightly deeper y/a
point for the n = 0.10 case as opposed to the n = 1.0 case, where it remains at y/a = 1.0.
46
Figure 3-26. Maximum shear stress along the y-axis for k = 3.0.
3.5.4 Discussion Although this contact study suggests that there may be the ability to reduce the stress
concentration effects with localized gradation of Young’s modulus, the FEA model
exhibits behavior that appears to be suspect at elevated k values. For the case of k = 3.0,
E/E0 = 4, and n = 1.0, and n = 0.10, the maximum shear stress is elevated at the surface
as compared to zero at the surface for the theoretical homogeneous case. Investigating
the normal stress components along the y-axis, where there appeared to be little change in
σy stress for all cases, there appears to be a drastically increased horizontal, σx, stress at
and directly below the surface to a depth of y/a = 0.22. Increasing the total element count
from 2496 to 9986 with mesh refinement in the area directly below the surface to a depth
that exceeds y/a = 1.0 had no effect on the resulting stress field in the model. In addition,
the model was reconstructed such that the load was applied over 1/20 the overall section,
as opposed to 1/10, with no change to this behavior. Thus, the behavior appears not to be
related to mesh density or discretization error.
47
The σy stress along the y-axis for k = 3.0 from the surface to a depth of y/a = 5.0 is shown
in figure 3-27. The stress appears to vary little for n = 0.02 and 0.10 within the depth
from the surface to y/a = 1.0. For n = 1.0 there is clearly a decrease in compressive σy
stress approaching y/a = 1.0. A reduction in σy stress would suggest that the maximum
shear stress be reduced, which is clearly not the case. This suggests that the horizontal
principal stress, which would normally be compressive, should show reduction as well.
Figure 3-27. Normal stress, σy, along the y-axis from surface to y/a = 5.0 for k = 3.0.
The horizontal, σx stresses for k = 3.0 are shown in figure 3-28. From inspection of
figure 3-28 it is clear that the horizontal stress for n = 1.0 is the cause of the spike in
maximum shear stress at the surface, as well as for the n = 0.10 case. For n = 1.0 the
graded region remains within the y/a = 0 to y/a = 2.0 (ref to figure 3-20). This is more
than likely due to the significant relative increase in stiffness in the horizontal direction as
48
compared to the highly graded vertical direction, where the material outside, or deeper
than the graded region is more compliant.
Figure 3-28. σx stress along the y-axis for k = 3.0.
The shallow gradation depth along with increased E/E0 creates a very high modulus
gradient close to the surface that appears to act as a material discontinuity (but is clearly
not a physical discontinuity).
The displacement in the vertical direction for the k = 3.0 case is shown in figure 3-29. As
expected the displacement is reduced as the depth of gradation increases, where the
region of stiffness in the vertical direction emanating from the surface is increased.
49
Figure 3-29. Vertical displacement for along y-axis for k = 3.0.
Since the horizontal stress is the culprit of the increase in maximum shear stress at the
surface for this model, the displacement and stress distributions were plotted along
horizontal paths at the surface and at depths of y/a = 0.2 and y/a = 1.0. A plot of the σx
stress for these various depths is shown in figure 3-30. It is interesting to note that the
compressive stress values are reduced with depth directly under the load and actually
become tensile at the critical y/a = 1.0 depth. In addition, the stress outside of the loaded
area becomes tensile at and slightly below the surface. The remainder of the plots are
listed in Appendix B for further review.
It appears that the highly localized, high E/E0 ratio case results in a material discontinuity
effect due to the extreme nature of the highly localized modulus gradient for this contact
problem.
50
Figure 3-30. σx stress along horizontal at various depths.
51
4 Conclusions
Localized material gradation was evaluated for its effect on and ability to reduce stress
concentrations. Two-dimensional infinite plane theory with far field loading was applied
to problems with remote stress free holes, both circular and elliptical, as well as a for a
simple uniform loaded contact problem. Young’s modulus was varied in both intensity
and gradation depth in both the x and y-coordinate directions with gradation originating
on, and following the shape of the stress free hole. For a uniform loaded contact
problem, the modulus was varied in the direction of loading only, originating on the
loaded surface.
It is concluded that for the remotely located stress free hole problem, reducing the
modulus ratio E/E0<1 at the hole (local softening effect) has a reducing effect on the
resulting stress concentration at the edge of the hole. This effect is more prominent with
shallow gradation depths but introduces additional stress increases at locations within the
graded region of the section.
Increasing the modulus ratio E/E0>1 on the edge of a circular hole (local stiffening effect)
and grading it to the base modulus within a defined depth resulted in an increase in the
stress concentration on the edge of the hole. The magnitude of the stress increase was
greater where the gradation depth becomes shallow, or closer to the hole.
Reduction of the modulus ratio at the hole, E/E0<1, had a reducing effect on the stress
concentration at the hole in all cases evaluated, for both circular and elliptical shapes. It
52
was determined that the tangential stress level at the edge of the hole was related more to
the modulus ratio, where greater reductions in stress concentration were observed as E/E0
ratios were reduced. This behavior was typical for both the circular and elliptical shaped
holes, in cases of both uniaxial and biaxial loading.
The depth of gradation was observed to affect the decay in tangential stress moving away
from the hole into depth. As the modulus ratio is reduced the transition in stress moving
from the edge into the far field loses its monotonic decay. As the gradation depth
becomes shallow, a second stress peak begins to form with greater intensity for the lower
E/E0 ratios. This is more prominent for the circular hole and elliptical hole with a/b = 2,
for both uniaxial and biaxial far field loading. This second tangential stress peak was
observed to exceed the stress at the edge of the hole in cases of circular shape with low
E/E0 ratio, accentually moving the stress concentration off the edge and into the depth of
the section. For the a/b = 5 elliptical hole, the gradation depth had little effect on the
stress levels away from the edge of the hole.
It is also concluded that for the uniform loaded contact problem, increasing E/E0>1
(surface stiffening effect) in conjunction with a deeper gradation depth reduced the
maximum shear stress concentration under the load, although only slightly. With
increased modulus ratio and shallow gradation depth the maximum shear stress
developed and increases dramatically on and just under the loaded surface.
53
A modulus ratio E/E0 <1 (surface softening effect) resulted in the peak maximum shear
stress to increase for deeper gradation cases, with a slight reduction for a more shallow
gradation depth. For E/E0 >1, the opposite behavior was observed. The peak maximum
shear stress shows a slight decrease for deeper gradation depth and an increase for
shallow depth. As the modulus ratio was increased to E/E0= 3 and 4, the maximum shear
stress at the loaded surface showed a dramatic increase for the more shallow depths. This
was found to be attributed to an increase in the horizontal compressive stresses at the
loaded surface at decreasing gradation depth.
It is further concluded that for shallow gradation depth along with very high or very low
E/E0 creates an extremely high modulus gradient that has a similar effect of a material
discontinuity, thus losing the effects of smooth functional grading.
54
Appendix A
UMAT User Subroutine for Radial Gradation, k = -0.8 and n = 0.3.
C C ABAQUS 6.5 - user subroutine UMAT for functionally graded materials C where E(x,y) C C ********************************************************************* C SUBROUTINE UMAT(STRESS,STATEV,DDSDDE,SSE,SPD,SCD, 1 RPL,DDSDDT,DRPLDE,DRPLDT,STRAN,DSTRAN, 2 TIME,DTIME,TEMP,DTEMP,PREDEF,DPRED,MATERL,NDI,NSHR,NTENS, 3 NSTATV,PROPS,NPROPS,COORDS,DROT,PNEWDT,CELENT, 4 DFGRD0,DFGRD1,NOEL,NPT,KSLAY,KSPT,KSTEP,KINC) C implicit real*8(a-h,o-z) parameter (nprecd=2) C CHARACTER*8 MATERL DIMENSION STRESS(NTENS),STATEV(NSTATV), 1DDSDDE(NTENS,NTENS),DDSDDT(NTENS),DRPLDE(NTENS), 2STRAN(NTENS),DSTRAN(NTENS),DFGRD0(3,3),DFGRD1(3,3), 3TIME(2),PREDEF(1),DPRED(1),PROPS(NPROPS),COORDS(3),DROT(3,3), 4STRAIN(3),S(3),PS(3),AN(3,3),D(4) C C X=COORDS(1) Y=COORDS(2) C E0=1.E9 ANU=0.3 C a=2 E=E0*(1-.8*(exp(-.3*(((X**2+Y**2)**0.5-1)**a)))) C C1=E/(1.-ANU**2) C2=E/(2.*(1.+ANU)) C C COMPUTE JACOBIAN C D11=C1
55
D12=ANU*C1 D22=C1 D33=C2 C DDSDDE(1,1)=D11 DDSDDE(2,1)=D12 DDSDDE(3,1)=0.0 DDSDDE(1,2)=D12 DDSDDE(2,2)=D22 DDSDDE(3,2)=0.0 DDSDDE(1,3)=0.0 DDSDDE(2,3)=0.0 DDSDDE(3,3)=D33 C C STRESSES AND STRAINS AT END OF TIME STEP: C S1=STRAN(1)+DSTRAN(1) S2=STRAN(2)+DSTRAN(2) S3=STRAN(3)+DSTRAN(3) C STRESS(1)=D11*S1+D12*S2 STRESS(2)=D12*S1+D22*S2 STRESS(3)=D33*S3 C RETURN END
56
Appendix B Horizontal displacements and corresponding normal stress, σx, at various depths for
uniform loaded contact problem at k = 3.0, n = 1.0.
Figure B-1. Horizontal displacement and corresponding σx stress the surface for half-space contact problem.
Figure B-2. Horizontal displacement and corresponding σx stress at y/a = 0.2 for half-space contact problem.
57
Figure B-3. Horizontal displacement and corresponding σx stress at y/a = 1.0 for half-space contact problem.
58
59
Bibliography Abaqus, Analysis User’s Manual, Version 6.5, H.K.S, Pawtucket, RI, 2004. Batra, V., Rousseau, C-E, Behavior of Nonhomgeneous Materials Subject to Bearing Load, J. Eng. Materials and Tech., April 2007, pp. 248-254. Birman, V., Byrd, L.W., Modeling and Analysis of Functionally Graded Materials and Structures, Applied Mech’s Review, 2007, pp.195-216. Erdogan, F., Wu, B. H., The Surface Crack Problem for a Plate With Functionally Graded Properties, J of Applied Mechanics, 1997, Vol 64, pp. 449-456. Johnson, K. L., Contact Mechanics, Cambridge University Press, London, 1985. Kim, J-H., Paulino, G. H., Isoparametric Graded Finite Elements for Nonhomogeneous Isotropic and Orthotropic Materials, J. Applied Mechs, July 2002, pp. 502-514. Manneth, V., Numerical Studies on Stress Concentration in Functionally Graded Materials, Master’s Thesis, University of Rhode Island, 2009. Matsunaga, H., Stress Analysis of Functionally Graded Plates Subjected to Thermal and Mechanical Loadings, Composite Structures, 87, 2009, pp. 344-357. Sadd, M. H., Elasticity, Theory, Applications & Numerics, Elsevier Academic Press, Boston, 2009. Santare, M. H., Lambros, J., Use of Graded Finite Elements to Model the Behavior of Nonhomogeneous Materials, J. of Applied Mechs, 2000, pp.819-822. Shigley, J. E., Mischke, C. R., Mechanical Engineering Design, McGraw-Hill, Inc., New York, 1989. Venkataraman, S., Sankar, B. V., Elasticity Analysis and Optimization of a Functionally Graded Plate with Hole, American Inst. of Aerodynamics and Astronautics, 2003-1466, pp.1-13.