STRESS CONCENTRATION EFFECTS IN HIGHLY LOCALIZED ...

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


3.2.3 Results……………………………………………………………………20

3.3 Infinite Plate with Stress Free Elliptical Hole – Biaxial Loading……………....22

3.3.1 Introduction………………………………………………………………22


iii

iv

3.3.3 Results……………………………………………………………………27

3.4 Infinite Plate with Stress Free Elliptical Hole – Uniaxial Loading……………..31

3.4.1 Introduction………………………………………………………………31


3.4.3 Results……………………………………………………………………32

3.5 Contact in Half Space - Uniform Distributed Loading………………………….35

3.5.1 Introduction………………………………………………………………35


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.


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.


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.


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.


-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.


-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.


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.

STRESS CONCENTRATION EFFECTS IN HIGHLY LOCALIZED ...

Documents