Development of Strong Surfaces

8/3/2019 Development of Strong Surfaces

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Th. Zisis1

e-mail: [email protected]

A. Kordolemis

A. E. Giannakopoulos

Department of Civil Engineering,

Laboratory for Strength of Materials and

Micromechanics,

University of Thessaly,

Volos 38336, Greece

Development of Strong SurfacesUsing Functionally GradedComposites Inspired by NaturalTeeth—Finite Element andExperimental VerificationFunctionally graded materials (FGMs) are composite materials that exhibit a microstruc-ture that varies locally in order to achieve a specific type of local material propertiesdistribution. In recent years, FGMs appear to be more interesting in engineering appli-cation since they present an enhanced performance against deformation, fracture, and

fatigue. The purpose of the present work is to present evidence of the excellent strength properties of a new graded composite that is inspired by the human teeth. The outer surface of the teeth exhibits high surface strength while it is brittle and wear resistant,whereas the inner part is softer and flexible. The specific variation in Young’s modulusalong the thickness of the presented composite is of particular interest in our case. The

present work presents a finite element analysis and an experimental verification of anactual composite with elastic modulus that follows approximately the theoretical distri-

bution observed in the teeth. DOI: 10.1115/1.3184038Keywords: functionally graded composites, surface strength, finite element analysis,FEA, experiments

1 Introduction

Functionally graded materials FGMs are characterized by a

specific distribution of certain properties. The property distribu-

tion is the result of a compositional and structural variation over

volume. Various approaches based on particulate processing, pre-

form processing, layer processing, and melt processing are used to

fabricate the FGMs, resulting in layered materials designed for

specific applications. In recent years, FGMs have become the

point of interest in engineering due to their performance against

deformation, fracture, and fatigue 1, while several FGMs are

manufactured by two phases of materials with different properties.

Accordingly, since the volume fraction of each phase gradually

varies in the gradation direction, the effective properties of FGMs

change along this direction. An important benefit of graded mate-

rials is the suppression of crack propagation through redistribution

of stresses at surfaces or interfaces subjected to mechanical and

tribological loading 2.

The present paper builds upon a previous work by Giannako-

poulos et al. 3, where the problem of the application of a normal

or tangential line load upon a semi-infinite layered elastic sub-

strate was presented. It was assumed that the layered substrate

presented a variation in Young’s modulus that follows a distribu-

tion of the type E = E 0 y0/

y. This type of Young’s modulus distri-bution emanates from the human teeth, which is characterized as a

hard and tough functionally graded composite 3. The outer sur-

face of the teeth exhibits high surface strength while it is brittle

and wear resistant, whereas the inner part is softer and flexible. In

their work, the analytical stress and distribution fields were pre-

sented and compared with the analytical expression given from

Flamant’s solution. It was shown that the normal loading leads tocompressive layer at the surface that can effectively shield thepropagation of possible pre-existing cracks. Here, the same elasticdistribution is assumed, and the analytical results are comparedwith numerical solutions. Furthermore, experimental evidencesthat support the theoretical conclusions are presented.

2 Analytical Formulas for the Loading of a Semi-

Infinite Substrate

The problem of a line load acting on the surface of an elastichomogeneous semi-infinite substrate Fig. 1 has been examinedpreviously, and is referred to as the Flamant problem 3. Thestresses in the Cartesian coordinates are

xx = rr sin2 ˆ = −2P

x2 y

x2 + y22−

2Q

x3

x2 + y22

yy = rr cos2 ˆ = −2P

y3

x2 + y22−

2Q

xy2

x2 + y222.1

xy = rr sin ˆ cos ˆ = −2P

xy2

x2 + y22−

2Q

x2 y

x2 + y22

Accordingly, the Cartesian displacements u x , u y are given by

u x =P1 +

E

xy

x2 + y2−

P1 −

E arctan

x

y

2.2

u y = −2P

E v + 1

2

x2

x2 + y2+ ln

x2 + y2

y

where y is the thickness of the substrate.Giannakopoulos et al. 3 showed that the stress field in a func-

tionally graded composite for the normal load in the Cartesiancoordinates Fig. 1 takes the following forms:

1Corresponding author.

Contributed by the Materials Division of ASME for publication in the J OURNAL OF

ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received January 14, 2009;final manuscript received May 13, 2009; published online November 5, 2009. Re-

view conducted by Hussein Zbib.

Journal of Engineering Materials and Technology JANUARY 2010, Vol. 132 / 011010-1

Copyright © 2010 by ASME

Downloaded 30 Nov 2011 to 203.153.32.220. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm



xx = rr sin2 ˆ =− P

r sin sin2 ˆ = −

P

y

x2

x2 + y2

yy = rr cos2 ˆ = −P

y

y2

x2 + y22.3

xy = −P

ysin ˆ cos ˆ = −

P

y

xy

x2 + y2

Furthermore, the Cartesian displacements u x , u y are

u x =P

E 0 y0− x + 1 + y arctan

x

y

2.4

u y =P

E 0 y0− y − 1 + x arctan

x

y +

Py

E 0 y0

where y0 is a reference depth at which E = E 0 and y is the thick-ness of the substrate.

Note that for plane strain, the following replacements must betaken into consideration:

E → E

1 − 2

2.5

→

1 −

3 Finite Element Simulations

In the current section, we perform a set of finite element simu-lations. The extracted stress and displacement fields are presentedand compared with the previously presented analytical formulas.The possibility of the propagation of preexisting cracks in thesemi-infinite composite substrate is explored.

3.1 Finite Element Formulation. In order to approach theproblem of a substrate loaded numerically by a line force, weconstruct a two-dimensional substrate under plane strain condi-tions. The commercial finite element program, ABAQUS Standard4, was employed, and a typical mesh used for the simulations isshown in Fig. 2. Full integration scheme was used. No specialtypes of elements were used, either close to the line force so-called “singular elements” or at infinity so-called “infinite ele-ments”. It is recognized that at distances very close to the force,the solution is not accurate, and only in between the point of application of the force and the outer boundary is sufficient accu-racy expected. The mesh was gradually refined to yield results instresses that converged with available analytical results at dis-

tances, which are more than three elements around the point of application of the force. A mesh sensitivity study revealed thatadequate accuracy is achieved by a finite element FE mesh thatcomprises about 169,000 nodes and 56,000 8-noded quadrilateralplain strain elements CPE8 in ABAQUS notation. The displace-

ment boundary conditions were: u y =0 along the bottom of the

mesh, u x =0 for the middle node of the bottom surface of the

mesh, and a traction free side boundary. The substrate was loadedat the middle of the top surface by a line force of 1 N/m, actingeither normally or tangentially. The minimum element size around

the region of the line force was he = 0.12 m. Results are presented

for mesh dimensions equal to 6500he243he. Furthermore, alllengths were in meters m and stresses in Pascal Pa.

3.2 Material Parameters. We performed two sets of calcula-tions. The first set comprised of three calculations for normal, andthree for tangential line load, according to which we consideredthe substrate to be homogeneous isotropic with Young’s modulus

E = 0.666 Pa, 1 Pa, and 5 Pa, and Poisson ratio =0.3. Then, in asecond set of calculations, the substrate was assumed to comprise

a stack of layers of equal thickness l. Each layer was treated ashomogeneous isotropic with a linear elastic response with Young’s

modulus E . In the case of the layered substrate, the Poisson ratioinsignificantly affects the attained stress field 5, so we assumedit constant and equal to 0.3 for all the layers. The Young’s modu-

lus varied with respect to depth y according to E = A / y, where A

= E 0 y0. Accordingly, the introduced model described a substratethat comprised of a top surface, which was very stiff, while thebottom of the substrate was very compliant. Simulations were

carried out for E 0, which is equal to 0.666 Pa, 1 Pa, and 5 Pa, and

for layer thickness l = 1 m, 0.5 m, and 0.25 m, while y0 =1 m for

all the cases considered here, unless otherwise stated. Then, yvaried with the thickness as described earlier Fig. 3. The zero

value of y corresponds to the upper surface of the substrate. Pre-liminary FE simulations, verified that six layers are adequate in

Fig. 1 The applied vertical P and horizontal Q line loads actingat the surface of a semi-infinite substrate

Fig. 2 „a … Region of the mesh used for the finite element simu-lations with the appropriate boundary conditions. „b … Blown upview of the mesh around the center line. The mesh comprises95517 nodes and 31600 8-node biquadratic plane strain quad-rilateral elements „CPE8 in ABAQUS notation…. Minimum elementlength: h e =0.12 m.

011010-2 / Vol. 132, JANUARY 2010 Transactions of the ASME




order for the solution with discrete layers to approach the continu-ous variation in the elastic modulus solution. Furthermore, accord-

ing to equation E = E 0 y0 / y, an infinite Young’s modulus corre-sponds to the top layer, but for numerical purposes, a value of

Young’s modulus equal to E 0 y0 / l at y = 0 was assigned for all the

cases considered here. Thus, as shown in Fig. 3, for l =0.25 m

and E 0 = 1 Pa, for example, Young’s modulus at the first layer willbe 4 Pa.

3.3 Finite Element Results. First, we present the results forthe homogeneous substrate that is loaded normally and tangen-

tially. From the analysis, it is clear that Young’s modulus E servesas a simple amplification factor for the strains and displacements,but does not affect either the distribution or the magnitude of the

stress fields, so results for the case of E =1 Pa are shown, unlessotherwise stated. Figure 4 presents the stress distribution along a

path on the x-axis at depth equal to y =−1.95 from the surface of the substrate. Analytical results according to Eq. 2.1 are shown,combined with the FE solution for normal and tangential lineload. An excellent agreement is observed for all the presentedstress components for both types of loading verifying the validity

of the FE model. Accordingly, results for the case of a graded

substrate with l =0.25 m and E 0 =1 Pa, are shown in Fig. 5.

Again, the stress distribution along a path on the x-axis at depth

equal to y =−1.95 from the surface of the substrate is presented,and analytical results from Eq. 2.3 are compared with the FEsolution for the case of normal line load. A very good agreementis obtained and is concluded that Eq. 2.3 can accurately capturethe response of the layered substrate upon normal line loading.Figure 6 presents the normalized vertical and normal displace-

ments u x and u y at y / y0 =−0.65 y =−1.95 m and y0 = 3 m for thecase of a homogeneous and a FGM substrate. Analytically, Eqs.2.2 and 2.4, and numerical results for both cases, are pre-sented. For the case of the FGM substrate, we considered a layer

thickness l = 3 m, a Young’s modulus E 0 at y0 =3 m equal to 1000

Pa, and a substrate thickness y= 243 m. We remind the readerthat the increase in Young’s modulus by a factor of 1000 does notaffect the attained stress field but only the magnitude of the at-tained displacement field. It is observed that for the homogeneouscase, a good agreement is attained between analytical and FE

Fig. 3 Variation in the continuous Young’s modulusE , used inthe FE model, as a function of the depth y for layer thickness„a … l = 1 m , „b … l =0.5 m, and „c … l =0.25 m „E 0 =1 Pa, y 0 = 1 m…

Fig. 4 Stress distribution along a path on the x -axis at depthequal to y / y 0 =−1.95 from the surface of the substrate. Analyti-cal and FE results for „a … normal line load and „b … tangential lineload for homogeneous substrate „l =0.25 m, E 0 =1 Pa, y 0= 1 m….





results. For the case of the FGM substrate, the analytical and FEresults present a good qualitative agreement. Quantitatively, theydiverge by a factor of about 2. This is due to the compliance of thegraded substrate when compared with the homogeneous one, dueto second order factors that have not been taken into account intothe analytical solution, and finally due to the fact that the displace-ment field in a two-dimensional problem has multiple solutionsdepending on the way that the far field displacements areconstrained.

Figure 7 presents contours of equivalent, in-plane normal andshear stresses, at a domain close to the region of the appliednormal load. Contours correspond to the mesh at the undeformedconfiguration, and results are shown for homogeneous and layered

substrate with l =1, 0.5, and 0.25 m. It is concluded that the stressfields shrink normally to the region of the applied load while theyexpand laterally for reduced layer thickness Fig. 7a and 7b.

Contours of in-plane yy and xy stresses show that the stress fieldis shrinking close to the surface without laterally expanding as the

layer thickness reduces, but the effect is minor. Furthermore, we

note that the xx, and yy stress fields are compressive so a mi-

crocrack advance along the x- or y-axis is judged unlikely due toshielding.

Figure 8 presents contours of undeformed shape of equivalent,

in-plane normal and shear stresses for a layered substrate with l

=1, 0.5, and 0.25 m. Here, we examine the case of tangentiallyapplied load. The homogeneous case is added for reference. As

expected, contours of in-plane xx and yy show a compressivestress field advancing in front of the point of application of thetangential force, while a tensile stress field forms behind. Again,the attained stress fields shrink normally to the applied load while

they expand laterally. Only the xy stress field seems to shrink inboth directions with reduced layer thickness. The tensile stressfield behind the load of application supports the possibility of theadvance of a preexisting crack for the case of a homogeneoussubstrate. Nevertheless, due to the concentration of the attainedstress field that is very close to the surface, with reducing layer

Fig. 5 Stress distribution along a path on the x -axis at depthequal to y / y 0 =−1.95 from the surface of the substrate. Analyti-cal and FE results for „a … normal line load and „b … tangential lineload for FGM substrate „l =0.25 m, E 0 =1 Pa, y 0 = 1 m….

Fig. 6 Displacement distribution along a path on the x -axis atdepth equal to y / y 0 =−0.65 from the surface of the substrate.Analytical and FE results are shown for „a … vertical displace-ments u x and „b … normal displacements u y for homogeneousand FGM substrate „y =−1.95 m, l = 3 m , E 0 =1000 Pa, y 0 = 3 m ,y =243 m….





Fig. 7 „a … Equivalent stress field for „i… homogeneous „ii… graded with layer thickness l = 1 m , „iii… graded with l =0.5 m, and „iv…graded with l =0.25 m substrate, with E 0 =1 Pa and y 0 =1 m, normally loaded with line force of 1 N/m; „b … in-plane xx stress fieldfor „i… homogeneous „ii… graded with layer thickness l = 1 m , „iii… graded with l =0.5 m, and „iv… graded with l =0.25 m substrate,with E 0 =1 Pa and y 0 =1 m, normally loaded with line force of 1 N/m; „c … in-plane xy stress field for „i… homogeneous „ii… gradedwith layer thickness l = 1 m , „iii… graded with l =0.5 m, and „iv… graded with l =0.25 m substrate, with E 0 =1 P and y 0 = 1 m ,normally loaded with line force of 1 N/m; „d … in-plane yy stress field for „i… homogeneous „ii… graded with layer thickness l = 1 m , „iii… graded with l =0.5 m, and „iv… graded with l =0.25 m substrate, with E 0 =1 Pa and y 0 =1 m, normally loaded with lineforce of 1 N/m





Fig. 8 „a … Equivalent stress field for „i… homogeneous „ii… graded with layer thickness l = 1 m , „iii… graded with l =0.5 m, and „iv…graded with l =0.25 m substrate, with E 0 =1 Pa and y 0 =1 m, tangentially loaded with line force of 1 N/m; „b … in-plane xx stressfield for „i… homogeneous „ii… graded with layer thickness l = 1 m , „iii… graded with l =0.5 m, and „iv… graded with l =0.25 m sub-strate, with E 0 =1 Pa and y 0 =1 m, tangentially loaded with line force of 1 N/m; „c … in-plane xy stress field for „i… homogeneous „ii…graded with layer thickness l = 1 m , „iii… graded with l =0.5 m, and „iv… graded with l =0.25 m substrate, with E 0 =1 Pa and y 0=1 m, tangentially loaded with line force of 1 N/m; and „d … in-plane yy stress field for „i… homogeneous „ii… graded with layerthickness l = 1 m , „iii… graded with l =0.5 m, and „iv… graded with l =0.25 m substrate, with E 0 =1 Pa and y 0 =1 m, tangentiallyloaded with line force of 1 N/m





thickness, it is concluded that the tangential loading does not de-velop detrimental tensile stresses, and the possibility of the ad-vance of a pre-existing crack minimizes, as expected from thetheoretical limit.

4 Materializing the Stiffness Distribution as Volume

Fraction of Particles

Hashin and Shtrikman 6 and later, Wallpole 7,8, providedvery accurate upper and lower bounds of the elastic properties of composites, based on the second-phase volume fractions and theelastic properties of the constituents 9. Paul 10 derived somesimple and direct approximate formulae for the elastic compositemodulus, and Ravichandram 11 gave more refined approxima-tions. More recent approximate average properties for the com-posites are based on the Mori and Tanaka approach 12. All theabove formulae pertain to composites that are homogeneous in anaverage sense. Long range inhomogeneous composites can bemodeled by assuming spatial variability of the composition 2.Next, we will present some examples of such approach.

Budianski 13 developed an elegant self-consistent compositetheory. In the limit case of a composite made of rigid sphericalinclusions and an incompressible matrix, the effective elasticproperties give an incompressible material with effective shearmodulus

=m

1 − 52

c

, 0 ci 2/5 4.1

where m is the shear modulus of the matrix, and ci is the volumeconcentration of the rigid inclusions assumed perfectly bonded

with the matrix. Interestingly, for ci =0.4, the shear modulus isinfinite. Using Eq. 4.1, the elastic modulus distribution of the

type E = E 0 y0 / y can be reduced to a concentration composition

ci y =2

51 −

E m y

E 0 y0

4.2

where E m = 3 m is the elastic modulus of the incompressible ma-

trix, and E y = 3 y. Note that the composition decreases lin-

early from ci =0.4 at y =0, down to ci =0 at y = E 0 y0 / E m. If H is thetotal layer thickness, we must have

HE m

E 0 y0

1 4.3

The stiffening of an incompressible matrix by rigid particles hasbeen addressed by Suquet 14.

S = 1 + c p21 − c p

5−

21c p1 − c p2/32

101 − c p7/3

−1 4.4

where c p is the volume fraction of the particles.Enamel is made of needlelike mineral crystals, 15–20 nm thick,

and 1000 nm long with a ratio of mineral to collagen rich proteinmatrix on the order of 1:2. Gao et al. 15 gave an estimate of the

composite modulus E of such microstructures.

1

E =

121 − cm

E pcm2 k 2+

1

cm E c4.5

In Eq. 4.5, cm is the volume concentration of the mineral, k is the

aspect ratio of the mineral plates 40 in the case of enamel, E p isthe elastic modulus of the mineral 114 GPa in the case of hy-

droxyapatite, and E c is the elastic modulus of the protein 4.3GPa for keratine. It is clear that if any of the above parametersbecomes a function of position, so does the composite modulus.

5 Experiments

5.1 Material Preparation. Deshpande and Cebon 16 per-formed uniaxial compression experiments on various idealized as-

phalt mixes, with volume fractions of aggregate ranging from40% to 85%. The aggregate stiffens the bitumen and it was foundthat the experimental results compare with a modified version of Hashin’s composite sphere model. For high values of the bitumenstiffness 1 GPa, the bitumen behaves as an elastic solid. At largestrains, the response of the bitumen is nonlinear. For the 50 pen

bitumen at 20°C and at stresses above 0.1 MPa, the bitumen canbe approximated as a power law material 17.

A mold was manufactured in order to cast and compact thespecimens. An oil lubricant mixture of glycerine and valvolinewas applied to the inner surface of the mold made from tin foilsto prevent sticking of the bituminous mixture. The mixture con-sisting of the correct amounts of bitumen and aggregate clean,round stones of 10 mm diameter was transferred to the mold in

layers, as follows: First, the bitumen was heated to 190°C in anoven for 90 min, and then the aggregates were poured in with thebitumen melt following right after. The ambient temperature was

9–13°C. The mechanical properties of the matrix bitumen andthe particles aggregates are shown in Table 1. A wooden hum-mer was used as the dropping weight of 50 N, falling from aheight of 1m and using 25 drops per layer. Air voids were of the

order of 4%. Each specimen had dimensions of 1500480

300 mm3. The specimens were constructed, each with 6 layersof constant thickness of 80 mm. Each layer was made by localmixing of the aggregates and the bitumen in two sublayers. Thetotal time for the casting was around 8 h for the homogeneousspecimen, and about 12 h for the graded specimen. In order tokeep the surface strength characteristic similar to the two speci-mens, the surface layer was made the same.

The first specimen was constructed to have constant elastic

modulus E → E 0 by keeping a constant percentage of the aggre-gate within each layer. The second specimen was constructed to

have a variable elastic modulus according to E = E 0 y0 / y. This wasachieved by the gradual change in the percentage of aggregateswithin each layer. In order to achieve the variation in the modulus,we used a rule of mixture that is pertinent to the materials we used16.

E

E bit

= 1 +2.5

p

ca

1 − ca

p

5.1

where E is the composite modulus, E bit is the elastic modulus of

the bitumen, ca is the volume percentage of the aggregates, and pis a parameter given by

p = 0.83 log 4 10

10

bitPa 5.2

The amount of material used for each specimen is shown in Tables2 and 3, respectively.

The thermal expansion of the bitumen is 210−4 K−1. As the

bitumen is cooled down from 190°C to 20°C, a 12% volumeshrinkage could lead to tensile stresses at the bitumen-aggregateinterface. However, creep strains develop, and given sufficienttime, they will negate the thermal strains. The bitumen followsDorn’s law of creep 18, and the creep strains are given by

c = sst + 0.0721 − exp− 26sst 5.3

Table 1 The mechanical properties of the materials used forthe composite specimens †6,7‡

Properties Bitumen 40/50 Aggregates calsius rock

Elastic modulus GPa 1 40–75Compressive strength MPa 6 80–120Tensile strength MPa 1.3

Density kg/m3 1041 2700





ss = 3.73 10−4s−1

106 Pa2.4

exp−228 103 J mol−1

8.314 J mol−1 K−1 T

5.4

where c is the creep strain, ss is the rate of thermal strain, isthe tensile stress that develops at the aggregate-bitumen interface

about 2 kPa, t is the time in seconds, and T is the average

temperature 378 K. The minimum time needed for the creep

strain to negate the thermal strain can be estimated from

c =12

100

1

3

= 4 10−2

This would require t = 28 h. Given that the specimens were testedthirty days after casting, we can safely assume that the thermalstrains were completely relaxed, and no residual stresses remainedinto the specimens.

5.2 Specimen Loading and Observations. We designed aself-stressing truss that could apply small normal and high tangen-tial load to the specimen. The truss is shown in Fig. 9. The generaltwo-dimensional theory of Filon 19 assures the global equilib-rium and stability of the specimen. The “nodal” application of theloads were done by using steel T-beams that were inserted to adepth of 4 cm, as shown in Figs. 9 and 10. The surface load wasapplied at three different points at each specimen at distances of 50 cm from each other, as shown in Fig. 9. The truss geometrywas modified accordingly, to keep the loading the same.

For the homogeneous specimen, due to the dead weight of thebars of the truss, a concentrated load of 10 kp at the end of the

truss was transferred to a distributed load of P =333 N /m and

Q =2333 N /m. Upon application of an additional concentratedload of 50 kp, the specimen failed by developing a deep verticalcrack, normal to the tangential load direction Fig. 11. Obviously,the large tensile stresses of the tangential load are responsible forthe expansion of the 4 cm initial slit, due to the insertion of theT-beam.

We performed the same loading sequence to the graded speci-men. The only damage that we observed was a local plastic twist

around the surface point of the load application Fig. 12. Notethat the surface layer of the grade specimen was prepared in thesame way as the homogeneous specimen.

Furthermore, we assured that the loading was fast enough toavoid the so-called thermal softening effect, i.e., development of heat due to fast local plastic straining. In the bitumen used, ther-mal softening can be avoided provided the strain rate is smaller

than 0.03 s−1 17. The largest linear stress that can develop in thebitumen is 2 kPa. This means that the elastic stresses must be

smaller than 60 Pa/s. If the change in the tangential load is Q, thenwithin the first layer, the typical change in the stress is

Table 2 Quantities of material used for the homogeneousspecimen „E =E 0…

Layer from surfaceAggregates

kgBitumen

kg

1 162 8322 162 8323 162 8324 162 8325 162 8326 162 832Total 972 4992

Table 3 Quantities of material used for the graded specimen„E =E 0y 0 / y …

Layer from surfaceAggregates

kgBitumen

kg

1 26 10182 2423 11373 2327 12224 2247 1285 218 13196 21126 1367Total 1389 7343

Fig. 9 Side view of the specimens during loading. The self-tension mechanism with P =F / b , Q = 7F / b , and b =30 cm.

Fig. 10 Overall view of the specimens, loading and layers

Fig. 11 Failure of the homogeneous specimen





2Q

y0

60 Pa/s ⇒2

y0

Q

t 60 Pa/s 5.5

In our case, the change in tangential load is Q =4200 N and

y0 =0.08 m. This would result in a time change of t 1.02 s.This time scale was less than the time scale of the load applica-tion, and so no thermal softening was experienced by the bitumenin the present tests.

6 Conclusions

In the present work, the analytical approach of the line loadingof a semi-infinite layered substrate was presented. The substrate

was modeled with a varying Young’s modulus with respect to itsthickness of the type E = A / y, where y is the depth from the sur-face. Accordingly, a finite element model was constructed andsimulations for the line loading of a graded substrate were carriedout. We found that FE and analytical results of the stress field fora functionally graded material are in good agreement. It wasshown that a composite substrate that approximates an elastic

modulus distribution of the type E = A / y leads to almost zero ten-sile stresses due to the application of tangential surface loads. The

normal loading creates a highly compressive surface layer that canshield the substrate from the opening of existing microcracks. It isconcluded that the tangential loading does not develop detrimentaltensile stresses, making the substrate particularly strong againstfrictional forces. The above findings are supported by the damagemode observed from a series of experiments performed using steelT-shaped beams inserted into the surface of bitumen specimensunder different types of loading. The present results can be veryuseful in the development of armor plates, special bearings, andtransportation pavements.

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Fig. 12 Failure of the graded specimen


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Development of Strong Surfaces