Indentation of an orthotropic layer by a semi

~
lnt. J. Engng Sci. Õïé. 25, Íï. 11/12, ññ. 1531-1534, 1987
ÑÞnted ßç Great ÂÞtain. ÁI1 Þghts reserved
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Copyright @ 1987 Pergamon Journals Ltd
INDENTATION OF ÁÍ ORTHOTROPIC LAYER
ÂÕ Á SEMI-INFINITE PUNCH
Ç. G. GEORGIAD1St
33-35 G, PapandreouSt., 16231Athens, Greece
Abstract-In thispaper,a p!anee!asticity
prob!emtreatedinío!íingpunchindentation
of an
orthotropic stÞñ resting ïç a ÞgßdfÞctßïn!essfoundation. Á semi-infinitepunch and an
infinite!y !ong !ayer were considered,so that the prob!emwasamenab!eto the Wiener-Hopf
technique. App!ying FïuÞer transforms to the goíeming differentia! equations and the
imposedboundaryconditions,we succeeded
ßçformu!atinga Wiener-Hopf re!ationshipwhich
then yie!dedthe asymptoticestimationof the stressand disp!acement
fie!d ßçthe immediate
íicinity of the edgeof stamp.
1. INTRODUCTION
Contact problems ïß a similar type concerning isotropic layers were reported ßç [1]. Éç
particular, Alblas and Kuipers ßç a series ïß papers [2-4] solíed approximately
problems ïß indentation ïß an isotropic compressible and incompressible layer by a
stamp ïß a finite base. ÂÕ using exponential and cosine Fourier transforms, they
reduced the problems ßç integral equations solvable by the Wiener-Hopf method.
Aleksandrov and Voroíich [5] haíe also worked earlier ïç similarproblems within the
same analytical context. lt is noted that our problem is simpler than those previously
cited since it iníolíes a semi-infinite punch. However, our method ïß solution differs
essentially from the procedures employed there. Éç addition, we have considered an
orthotropic material for the layer and this is a novel subject .ßçliterature.
2. BASIC PRELIMINARIES
The material we considered was orthotropic with two mutually orthogonal axes ïß
elastic symmetry ßç the plane. With respect to the principal material-axes ïß an
orthotropic plate, the elastic constitutive expression relating the in-plane stressesand
strains is [6]
ó÷ ] = Cl2
Cl1
Cl2
óÕ
C22
ÏÏ
åÕ
å÷
(2.1)
[
][
[
Ô×Õ
Ï
Ï
C66
]
¾×Õ
Consider now a displacement-potential formulation, analogous to that utilized
frequently ßç elastodynamics
a
Ux=-a:;(ö + ø),
a
UÕ=ay(áö+bØ),
uz=O
(2.2)
where ö and Ø are the displacement potentials and the new constants are giíen by
á = cllPl
-
C66
Cl2 + C66
= (Cl2 + C66)Pl,
C22 - C66Pl
b = CllP2 Cl2
C66
+ C66
= (Cl2 + C66)P2
C22 - C66P2
(2.3)
Éç (2.3) Ñé and Ñ2 are the roots ïß the characteristic equation
CllC66P2+ (Ct2+ 2Cl2C66
- CllC22)P
+ C22C66 =Ï
(2.4)
Substitution(2.2) into the equationsïß equilibrium leads to Laplace type equations
satisfiedby the displacementpotentials
ó2ö
ó2ö
~+Playz=O,
t Presentaddress:ÌÅÁÌ,
ó:éø
ó2ø
~+P2ayz=O
The Uniíersity of Michigan,Áçç Arbor, ÌÉ 48109-2125,
U.S.A.
1531
(2.5)
1532
Ç. G. GEORGIADIS
Finally, stressesare given ßç temlS of the displacement potentials by
ó÷
=(1 + á)C66~
(1 + b)C66~
âé
ay2 +
â2
ay2
óÕ =
-(1
+
á)C66
ï2ö
-Æ
-
(2.6.1)
ï2ø
(1 + b )C66 -;-2
Ï×
(2.6.2)
dX
ï2ö + (1+ b) a;-ay
ï2ø ]
Ô×Õ
=C66[ (1+ á) a;-ay
(2.6.3)
3. STATEMENT ÁÍÏ
SOLUTION ÏÑ ÔÇÅ PROBLEM
As is shown ßç Fig. 1, we consider an elastic orthotropic body ßç the form of an
infinitely long stÞñ with height h. This layer of mateÞaÉis ßçfrictionless contact with a
rigid foundation. Á semi-infinite Þgßd punch indents the layer producing a uniform
displacement Uo. It is noted that this vertical displacement may be an arbitrary function
u(x) without any change ßç the procedure followed. Then the boundary conditions can
be wÞtteç as
uy(x,h)=-uo
for
-ïï<÷<Ï
(3.1.1)
uy(x,h)=m(x)
for
Ï<÷<ïï
(3.1.2)
óÕ(÷,h)=n(÷)
for
-ïï<÷<Ï
óÕ(×'h) = Ï
for Ï<÷ < 00
(3.1.3)
(3.1.4)
Ô×Õ(÷,h)=Ï
fo~ -ïï<÷<ïï
(3.1.5)
Uy(x,Ï) = Ï
for -00 < ÷ < 00
(3.1.6)
Ô×Õ(×'
Ï) = Ï
for
(3.1.7)
-ïï<÷
<00
Obviously, the determination of the as yet unknown functions m(x) and n(x)
completes our aim, íßÆ.the detemlination of the vertical stressesand displacements ïç
the surface of the layer. The existence of infinite and semi-infinite domains ßç the
boundary conditions ßç addition of eqns (2.5) implies the suitability of the WienerHopf method [7].
Applying the exponential FïuÞer transform to eqns (2.5) gives
~
ö*(ù, Õ) = Ï
(3.2.1)
a2
-ù2ø*(ù, Õ) + â2--ú ø*(ù, Õ)= Ï,
(3.2.2)
-ù2ö*(ù,
Õ) + âé
ay
Õt
[
i
é
ïß
"
---
rigid foundation
÷
Fig. 1. Punchindentationof an orthotropicelasticstÞñrestingïç a Þgid foundation.
Indentation
ïß anorthotropic
layer
1533
with the following solutions
ø*(ù, Õ) = Áé(ù)eVéùÕ+Á2(ù)e-véùÕ
(3.3.1)
ø*(ù, Õ) = Âé(ù)eV2ùÕ
+ Â2(ù)e-V2ùÕ,
(3.3.2)
where ù = ó + ßôis the complexvariableßçthe Fourier-transformplane and Vj = âéÉ/2
(j = 1, 2).
Further, due to evennessïß the displacement potentials ßç Õ the above expressions
take the following simplified form
ø*(ù, Õ) =Á(ù)cïsh(véùÕ)
ø*(ù, Õ) = Â(ù)cïsh(V2ùÕ)
(3.4.1)
(3.4.2)
Designate now as m~(ù) and n~(ù) the half Fourier transforms ïß the functions ïß
interest over positive and negative x-axis, respectively. The (+) and (-) subscripts
denote that the functions are analytic above or below a certain line ßn the complex
ù-ÑÉane.
Application ïß the Fourier transform to the boundary conditions and elimination ïß
Á(ù) and Â(ù) from the resulting system gives the following Wiener-Hopf equation
m~(ù)
=~
(-.!!
)
~
C66 1 + b
1+ á
[ 1
ù --cot
h(
õé
n~(ù)
h)
1
õéù +-cot
V2
h(
õ2ù
h)]
+-:--~~
é(2ð) ù
(3.5)
The kemel ßç the above equation seems to be too complicated for a closed form
solution. But ßßit suffices to find only asymptotic results near the point (Ï, h) ßn the
physical plane we may consider Nilsson's [8] procedure. Following the latter we find
hm
. m(x) = -uo + uo õé + V2 É/2
)
(
h
÷-Ï+
xlrl.
(3.6)
ðõéõ2
and
1.
( )
lmnx=÷-Ï-
(õé (-.!! iC66UO
~ )
+
õ2)(õé
õ2)É/2
(ðh)É/2(VÉV2)3/2
1+b
-.1/2
÷
.
(3 7)
1+á
Relations (3.6) and (3.7) are the asymptotic expressionsïß the Uy(x, h )-displacement
and the óÕ(×'h )-stress for small values ïß ×. It is noted that for ÷ = Ï eqn (3.6) gives
Uy(x, h) = -uo as is expected. Equation (3.7) also gives real values ïß the pressure
under the stamp for negative ×, since it contains the imaginary õçßé.The stresspresents
a square root singularity tending to infinity as the position ïß the edge ïß the punch
(comer) is approached. The latter result is common to indentations by punches having
sharp edges.
Since ßçliterature such expressionsas (3.6) and (3.7) were çïÉ given explicitly for the
isotropic case, we confine ourselves Éï compare our result with that given by Alblas
and Kuipers [2] for a finite-base punch. Indeed, their relations (2.21) and (2.24) for the
asymptotic values ïß the normal stress and displacement near the edge ïß the punch
have the same functional form as relations (3.6) and (3.7) ßç the present work.
Relations ßç both papers include the (ðh )-l/2-term, whereas stress behaves as the
inverse square root ïß distance and displacement as the square root ïß distance.
When the material becomes isotropic the orthotropic material constants take the
value õé = V2 = 1. Éç this case eqn (3.6) gives for the displacement profile
Uy(x,h)=-uo+u°;h
( 2 ) ÷É/2 for
É/2
÷-ï+
(3.8)
However, isotropic results for the stressescan not be extracted by setting õé = V2= 1
1534
Ç. G. GEORGIADIS
becauseboth the numerator and denominatorvanish ßç (3.7). This is a well-known
result also appearingßç other physical contexts (see for instanceßç [9] for a crack
problem). The limitingisotropic casemay be deducednumeÞcaÉÉÕ
by assignvaluesto
õé, V2near unity.
4. CONCLUSIONS
Á plane contactproblem was solvedßçthis paper within the context of linear and
orthotropic elastostatics.The solution was obtained by direct application of the
Fourier transform and the Wiener-Hopf technique.The mathematicalanalysiswas
greatly simplified by the choice of a semi-infinite punch. However, our problem
possesses
a 10tof practicalinterestßçthe caseof smallheightsof the layer ascompared
to the length of the baseof a punch, viz. for thin layers.
REFERENCES
[1] G. Ì. L. GLADWELL, ContactProb/emsin the C/assica/Theoryï[ E/asticity.Sijthoff and Noordhoff,
Alphen aanden Rijn (1980).
[2] J. Â. ALBLAS and Ì. KUIPERS,Acta Mech.8, 133(1969).
[3] J. Â. ALBLAS and Ì. KUIPERS,Acta Mech.9, 1 (1970).
[4] J. Â. ALBLAS and Ì. KUIPERS,Acta Mech.9,292 (1970).
[5] V. Ì. ALEKSANDROV and É.É.VOROVICH, ÑÌÌ 24, 462(1960).
[6] S. G. LEKHNITSKII, Theoryï[ E/asticityï[ an AnisotropicE/asticBody. Holden-Day,SanFrancisco
(1963).
[7] Â. NOBLE, MethodI'Basedon the Wiener-Hop[ Technique.PergamonPress,New York (1958).
[8] F. NILSSON,lnt. J. Fracture8,403 (1972).
[9] Ñ. Ê. SATARATHY and Ç. PARHI, lnt. J. EngngSci. 16,147 (1978).
,;--",.I,j~ ti }.; j-f, (Revisedversionreceived6 July 1987) -É'-', "É'
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