Journal of Elasticity, Vol. 6, No. 2, April 1976 Noordhoff International Publishing - Leyden Printed in The Netherlands
Two contact problems in anisotropic thermoelasticity D. L. CLEMENTS and G. D. TOY Department of Applied Mathematics, University of Adelaide, Australia (Received July 8, 1974; revised November 19, 1974)
ABSTRACT Some basic equations recently derived by Clements are used to consider two contact problems in anisotropic thermoelasticity. The first problem concerns the determination of the thermal stress in an anisotropic half-space due to a heated load over a section of the boundary, The second problem concerns the indentation of a half-space by a heated rigid punch. In particular, indentation by a cylindrical punch is considered and some numerical results obtained. ZUSAMMENFASSUNG Einige Grundgleichungen, die Clements neulich abgeleitet hat, dienen als Grundlage, um zwei Kontaktprobleme in der anisotropiscben Thermoelastizit~it zu betrachten. Im ersten Problem geht es datum, die W~irmespannung in einem anisotropischen Halbraum zu bestimmen, die von einer erhitzten Last fiber einem Teil der Grenzlinie hervorgebracht wird. Im zweiten Problem geht es darum, einen Halbraum durch eine erhitzte starre Punze auszuzacken. Insbesondere wird das Auszacken durch eine kreisartize Punze betrachtet, und es ergeben sich numerische Angaben.
1. Introduction
Thermoelastic contact problems for isotropic materials have been discussed by a number of authors. For example, George and Sneddon [1], Keer and Fu [2], Fu [3] and Barber [4] considered various thermoelastic punch problems involving isotropic plates and half-spaces while Korovchinski [5] examined a thermoelastic problem connected with the contact between two elastic cylinders. In contrast, thermoelastic contact problems for anisotropic materials have attracted little attention. In the present paper some basic equations recently derived by Clements [6] are used to consider two thermoelastic contact problems for an anisotropic half-space. The first problem concerns the determination of the stress in an anisotropic half-space when the boundary of the half-space is subjected to a heated load over a strip of constant finite width and infinite length. Some numerical results are given for a specific transversely isotropic material. These serve to determine the effect of an increase in the Journal ~?fElasticity 6 (1976) 137 147
D. L. Clements and G. D. Toy
138
temperature of the load on the hormal displacement under the loaded strip. The results indicate that, for the particular material under consideration, an increase in temperature causes a decrease in the indentation of the surface of the half-space by the load. The second problem is the mixed boundary-value problem of a heated punch indenting an anisotropic half-space. The specific case of a cylindrical punch is examined and an expression obtained for the width of the region of contact. Some numerical results are considered for a particular anisotropic material and these show that an increase in the temperature of the punch causes a decrease in the width of the region of contact.
2. Fundamental equations Take Cartesian coordinates x~, x2, x3, with x 2 vertical and suppose the half-space x2>0 is filled with a homogeneous anisotropic elastic material in which the stress and temperature fields are independent of the Cartesian coordinate x3. The temperature distribution T(xj, x2) in the half-space satisfies the heat conduction equation O2T
2ij c~xic~x2
O,
(2.1)
where 2~j= 2j~ are the coefficients of heat conduction and the repeated suffix convention (summing from 1 to 3) has been used. This summation convention will be used subsequently for Latin suffices. If the temperature
r(xt, 0 ) = f ( x l ) ,
(2.2)
is prescribed on the boundary x2 = 0 of the half-space then the temperature in xz > 0 may be written in the form (see Clements [6]) {A(p) exp (ipz') + A(p) exp( - ip~')} dp, =
(2.3)
0
where the bar denotes the complex conjugate and
A(p) =
f({) exp(-ip{)d{, --o0
A(p) =
f
f ( { ) exp(ip{)d{.
(2.4)
--oO
Also, in (2.3), z'=x 1+zx2 where ~ is the root with positive imaginary part of the quadratic equation )'11 +2)'12"C+222 "c2= 0.
(2.5)
The displacement and stress in the half-space x2>0 may be written in the form (see Clements [6])
Uk =~, A~tfi~(z~) + Z A~lp~(z~) + Ck~(Z ) + Ck~(Z.'), O~(z~) + Nij~)'( z') + Nij~'( ~') - fiij T, T,
ct
-
(2.6) (2.7)
Two contact problems in anisotropic thermoelasticity
139
where e takes the values 1, 2, 3, primes denote derivatives and
o A(p)p-1 exp (ipz')dp.
+(z') = ~
O~(z) = U~ o E~(p) exp(ipz)dp,
(2.8) (2.9)
with the E~(p) being arbitrary functions which will be chosen to satisfy particular boundary conditions on the boundary x2 = O. In (2.6), the Ck are defined by the equations
Dik Ck = 7i,
(2.10)
where ~/~ = -i[fiil+zfii2],
(2.11)
Dik = Cilkl ~-'~Cilk2 @ T,gi2kl ~-'g2 Ciik2 ,
(2.12)
i is the square root of minus one, the Cokl are the elastic constants and the fiij are the stress temperature coefficients which may be obtained from the coefficients of thermal expansion CtkZby the relations flij ~ CijklO~kl •
Also, the Ak~ are the solutions to the equations
(Ciikl + p~cilkz + P~Cizk* + P~2 Cizkz)Ak~ = O,
(2.13)
where the p~ are the roots (with positive imaginary part) of the sextic [Cilkl + pCilkz + pCizkl + pZ Cizk2 [ =
O.
(2.14)
In (2.7), the Lij~ and Nij are defined by
Lij~ = (Cokl + p~ Cok2) Ak~ ,
(2.15)
Nij = (¢ijkl + ZCijk2)Ck.
(2.16)
Finally, in this section, we use (2.6)-(2.9) and (2.3) to obtain expressions for the displacement and stress in the form 1
Uk = ~
~o
f O {~ Ak~E'(p) exp(ipz~)+CkA(p)p-1 exp(ipz')}dp,
~i~ = ~
L~j~E~(p)ip exp(ipz~)+(iNo-fiij)A(p ) exp(ipz') dp,
(2.17) (2.18)
0
where ~ denotes the real part of a complex number. On x2 = 0 these equations may be written in the form
uk =
Tk(p) exp(ipxl)dp, 7C
aij = 1
(2.19)
0
i °° Ui~(p)exp(ipxl)dp,
(2.20)
D. L. Clements and G. D. Toy
140 where
Tk(p) = Z Ak~,E~,(P)+ CkA(p)P- ' ,
(2.21)
ot
Uij(p ) = i p~ Lij ~E~(p) + (i Nij- flij) A (p) .
(2.22)
o~
3. Strip loading on xz = 0
We suppose a heated strip comes into contact with the half-space over the region [x~[
T(xl,0)
= { f ( o 1) for for
22(xl,0) :
for for
0
Ixlla,
(3.1)
f la,
(3.2)
at2(x~, O) = a23(xt, O)--O for all x~. In order to demonstrate the procedure to be followed for such a problem it will be sufficient to consider the simple case f ( x l ) = To and g(x~)= ao where To and ao are constants. Then, from (2.4),
A(p) = 2Top -~ sin pa.
(3.3)
Substituting (3.3) into (2.3) and integrating, we obtain
T : To/zc{tan-l(x'+a+z-'x2~\
T2X2
tan-l(x'-a+z'x2~t,
/
'~2X2
(3.4)
J)
where z='c 1 +iz2 (zl, z2 real). Also, from (2.20) and (3.2)
U12=0~
U22= -Zaop-l sinpa,
U32=0.
(3.5)-(3.7)
Hence, using (2.22)
E~(p) = 2M~ [( - Nj2 - i flj2) To + i ao c5j2]p - 2 sin pa,
(3.8)
where
L~2,M~j = 6~j.
(3.9)
Substituting (3.8) into (2.17) and (2.18) it follows that
~uk _ 2 ~ ~X 1
7g
i
Ak~M~j[(-N~2-iflj2) To+iao6~2] exp(ipz~)
0
+ To(C k exp(ipz')} p-~ sin padp, air '= ~z2 J(~o {~ L~j~M~k[(-- iNk2 + flk2) T°--a°c3k2) exp(ipz~)
(3.10)
Two contact problems in anisotropic thermoelasticity
141
+ (i Ni~- flij) To exp (ipz')} p-1 sin pa dp.
(3.11)
In particular, on x2 = 0 Ou~ _ 2 ~X 1
{iBkj[(-Nj2-ifij2)To+iao6j2]+iToCk}
7C
foop-~ sinpaexp(ipxx)dp, 0
(3.12) aij = - ~ 7C
Lij~M,k[(--iNke+fika)To--ao6ke]
where Bkj = 2 Ai~M~j.
(3.14)
At this point it may be noted that some simplification occurs in (3.12) and (3.14) if one or more of the Xl=0 planes is a plane of elastic symmetry. For example, if the xl = 0 plane is a plane of elastic symmetry then it may be shown (see Clements [6]) that C1, Nll, N22, N33 and N23 have zero real part while C2, C3, N12 and N13 have zero imaginary part. Also it may be shown (see Clements [7]) that, in this case, B12 , B13 , B21 and B31 have zero imaginary part and all other Bkj have zero real part. Similar results for the Ak~ and Lij~ are available in Clements [7]. In addition, it may be readily shown that, when the Xl=0 plane is a plane of elastic and/331 are all zero. Hence symmetry, the stress-temperature coefficients/312, fi13, it is apparent that, when such symmetry exists, then, for example, the factor
f121
2 {B2j [ ( , Xj 2 "iflj2 ) To + i ao 6j2] + To C2},
7"C
occurring in (3.12)(with k= 2) has zero imaginary part so that (3.12) may be rewritten in the form 0u2 Oxl
1 {B2j[(-Nj2-ifijz)To+iaoc~jz]+ ToC2} log xlXl+a-a , rc
(3.15)
where the result oo
x ! _}_ a
fo p- 1 sin pa sin pxldp = ½ log
xl-a
'
has been employed to obtain (3.15). Now denoting by A the difference between Ou2/~?xl at temperature To= T' and OUz/OXl at temperature To=0 it follows, from (3.15), that A = •uz ~X1 where
To=T '
_ Ou2 = KT'log ~Xl To=0
K = 1 {Bzj(Nj2 + ifij2)- C2}. 7~
xl+a xl--a
(3.16) ' (3.17)
142
D. L. Clements and G. D. Toy
We now apply the results of this section to a particular transversely isotropic material. If the x~ and x2-axes lie in the transverse plane so that the x3-axis is normal to the transverse plane then the non-zero c~jk~,c~j and 2ij are Cllli
:
¢3333,
C2222 ~
C l 1 3 3 ~ C2233 , 1
C1313 ~ C2323 ,
C1212=2(C1111--C1122),
~11 = ~ 2 2 ,
C1122~ ~33,
"~11 = '~22, "~33"
If the xa-axis is kept fixed and the x2 and x3-axes are rotated through an angle 0 the constants referred to the rotated frame are given by Cijkl = aimajnakpalqCmnpq,
fi'ij = ai,,aj, fim, ,
2~j = ai,,ay~,,,,
where [a~j] =
cos 0
sin
- sin 0
cos
.
Under this rotation it is apparent that, regardless of the value of 0, the x~ = 0 plane is a symmetry plane so that the formula (3.15) is valid for all 0. We consider the specific transversely isotropic material which, referred to symmetry axes with the x~-axis normal to the transverse plane, has constants CIlll : 16.5, cl122 = 3.1, c1133 = 5, c3333 = 6.2 c1313 = 3.92, 1 0 6 ~ 1 t = 60.8, 1 0 6 ~ 3 3 -~ 14.3, 211/)t33 = 1.17. If the elastic constants are multiplied by 10 j~ then the units for these constants are dynes/cm a while the coefficients of linear thermal expansion are for a temperature increase of one degree centigrade. These are the values of the material constants for a crystal of zinc although they are chosen here merely as an example, The values of K (occurring in (3.16)) for various values of 0 are listed in Table 1. The values, when taken in conjunction with (3.16), show that the effect of increasing the temperature Table 1 0
15 °
30 °
45 °
60 °
75 °
90 °
l0 s K
2.00
1.87
1.73
1.71
1.87
1.96
is to decrease the displacement gradient OlA2/~X1 for x 1 < 0 and increase the gradient for xa >0. Also the difference between the displacement gradients at To=0 and T O= T' varies appreciably with 0 and has a minimum in the vicinity of 0 = 60 °. This shows that the effect of increasing the temperature is to decrease the indentation of the surface of the half-space by the load. This will be the case regardless of the value of 0 although the decrease in indentation varies with 0.
4. Indentation by a rigid punch
We suppose the half-space xz>O is indented by a heated rigid punch in the region
Two contact problems in anisotropic thermoelasticity
143
Ix1]
T(xl, O) = { f ( o 1) for ]xaIa, azz(Xl,0)=0
for
IXxl>a,
~2(x1, 0) = ~23(xl, 0) = 0
Uz(X~,O)=g(Xl)
for
(4.1)
--a
for all
a22(Xl, O ) d x ~ = - P ,
(4.2),(4.3)
x~,
Ixl[
(4.4) (4.5)
where f(x~) and g(xl) are even functions of Xl and P is the total force exerted on the half-space by the punch. Use of (2.20) and (4.4) shows that, in this case, (4.6)
U12 = U23 = 0 .
If we suppose the punch is such that the normal stress is everywhere finite on x2 = 0, then it will be sufficient for the present analysis to choose U22 in the form
i o r(t) Jo(pt)dt,
Uz2(P) = .
(4.7)
0
where r(0 is a real function to be determined and, in the usual notation, Jo is a Bessel function. It may be readily verified that, with this choice of/522, the boundary condition (4.2) is satisfied. Use of (2.22) together with (4.6) and (4.7) now yields 1
E~(p) = - P {M~j [(Nj2 + i fij2) A(p) + i6j2 Uzz] },
(4.8)
where M~j is defined by (3.9). Substituting (4.8) into (2.21) and (2.19) we obtain
[( Nj2 + i/~j2) A (p) + i ~j2 U~ 2] + Ck A (p)},
r~(p) = p - 1 { _ Bkj
1 0
(~U2
(co
Oxl - ~ Oxl ~ Jo {-B2j[(Nj2+ifij2)A(p)+iaj2U22] + CzA(p) } p-* exp (ipxt)dp.
(4.9)
Hence B22 U22 exp(ipxl)dp = q(x~) for Ixll < a ,
1 7C
(4.10)
0
where
q(xl) = 9'(xl) + 1 ~z
i °~ {iB2i(Nj2+ifij2)_iC2}A(p ) exp(ipxl)dp.
(4.11)
.1o
It can be shown that B22 has zero real part (see Clements [8]) and hence, on employing (4.7), equation (4.10) yields ~-o
sin pxl dp
o
r(t) Jo(pt)dt = q(xl)/(iB22 ) for lxll < a .
(4.12)
The left hand side of (4.12) is an odd function of x 1 and hence the right hand side must be also. Hence the results of this section are only applicable for materials for
144
D. L. Clements and G. D. Toy
which q(xl) is an odd function. That q(xl) is certainly odd for many important classes of materials may be shown, from (4.11), by employing results due to Clements [6]. Here, as an illustration of the procedure, we shall show that q(xl) is an odd function for the class of materials for which the Xl=0 plane is a plane of elastic symmetry (the important transversely isotropic and orthotropic materials are included in this class). We first note that, since g(x 0 and f(xl) are even functions of Xl, the function g'(xl) is an odd function of Xl and A(p) has zero imaginary part. Now, as has been indicated in the previous section, if the x~---0 plane is a plane of elastic symmetry then B2~, C2 and N~2 have zero imaginary part while B22, B23, N22 and Na3 have zero real part. Hence it follows that the second term on the right hand side of (4.11) is an odd function of xl and therefore, since g'(x 0 is odd, q(xl) is also an odd function of xl. Now
i
0
jo(pt)sinpx~dp =
0
(x
-t2)
for 0 < x l < t for t
(4.13) ,
and hence, returning to (4.12) and interchanging the order of integration, we obtain 1~ xl r(t)dt _ q(xl) J0 ( x ~ - t2)~ iB22
for 0 < x l < a .
(4.14)
Equation (4.14) is an Abel's integral equation for r(t) with solution r(t) -
2
d i t sq(s)ds for 0 < t < a
iB2: ~
(4.15)
o(t2-s2) ~
Once r(t) has been determined, (4.7) and (4.8) yield E~(p) and then the stress and displacement throughout the half-space can be calculated by (2.19)-(2.22), It remains to satisfy the condition (4.3). Now, from (2.20) and (4.7) it follows that
f
1 "0 r(t)dt 0"22(X1, O)~-- ~1( ~
f
0 cos PXl Jo(pt)dt
r(t)dt
=n)~,(tZ-x~),
for 0 < x l < a .
Hence
i
2 a '~ r(t)dt -a cr22(xl' O)dxl = ~- 0 dxl x, ( t 2 - x ,2) -~ --
f ;
;
o
r(t)dt
and using (4.15) and (4.3) 2 a sq(s)ds - iB~2fo (a2_s2)6
P.
(4.16)
Equation (4.16) may be used to determine a so that the width of indentation 2a may be evaluated. If the rigid punch has the form of a circular block then the appropriate form for g(xl) is
g(x 0 = 7-fix 2
(4.17)
145
Two contact problems in anisotropic thermoelasticity
where 7 is a constant and f = l / 2 R where R is the large radius of the punch. Hence, in this case we may write the condition (4.16) in the form 2 iB22
rca2f+ K
where K is
a=(
sds A ( p ) sin psdp 0 (a2-$2) ½ 0
defined by (3.19).
= P,
(4.18)
ff(xl)=0 then (4.18) reduces to
~iPB22/¢ nfl ) '
(4.19)
in agreement with a result previously obtained by Clements [3] using an alternative method. If the circular block is held at a constant temperature To then A(p) is given by (3.3) and, use of integration by parts in (4.18) yields arc { a f + 2 T o K } = p iB22
(4.20)
and hence T°K
a-
fi-
+
{(~K~)2 + ( i P B 2 2 ~ + \ nf /J '
(4.21)
If ToK/fl is sufficiently small then, with error of O[(ToK/f)2], (iPB22~ ~
a=\ rcf / -
r°K
(4.22)
f
Hence the difference between a at To = T' and To = 0 is given by alto: r ' - air0= o = - Z ' K / f t .
(4.23)
It is apparent from (4.23) and Table 1 that, in the case of the transversely isotropic material zinc the effect of increasing the temperature of the circular punch is to cause the width of the region of contact to decrease.
Acknowledgment The authors are grateful to Dr. J. R. Barber of the University of Newcastle-upon-Tyne for a helpful criticism of an earlier draft of this paper. REFERENCES [1] George, D. L. and Sneddon, I. N., The axisymmetricBoussinesq problem for a heated punch, J. Math. Mechs., 11 (1962) 665 689. [2] Keer, L. M. and Fu, W. S., Some stress distributions in an elastic plate due to rigid heated punches, Int. J. Engng. Sci., 5 (1967) 555-570. [3] Fu, W. S., Indentation of an elastic half-space due to two coplanar heated punches, Int. J. Engng. Sci., 8 (1970) 337-349. [4] Barber, J. R., Indentation of the semi-infinite elastic solid by a hot sphere, Int. J. Mech. Sci., 15 (1973) 813 819.
D. L. Clements and G. D. Toy
146
[5] Korovchinski, Plane contact problem of thermoelasticity during quasi-stationary heat generation on the contact surfaces, J. Basic Engng., 87 (1965) 811-817. [6] Clements, D. L., Thermal stress in an anisotropic elastic half-space, S I A M J. appl. Math., 24 (1973) 332-337. [7] Clements, D. L., The response of an anisotropic elastic half-space to a rolling cylinder, Proc. Camb. Phil. Soc., 70 (1971) 467-484. [8] Clements, D. L., The motion of a heavy cylinder over the surface of an anisotropic elastic solid, J. Inst. Maths. Applics., 7 (1971) 198~206.
Appendix Dr. J. R. Barber has pointed out to the authors that, for the problems considered in this paper, there exists a useful relationship between the temperature profile and the load distribution on x2--0. The discussion which follows is based directly on some analysis made available to the authors by Dr. Barber. Equation (3.17) may be used in (4.9) to yield
c3xt
7r c?xl ~ )o [TzKA(p)+iB22Uz2(p)]p -1 exp(ipxl)d p
(1.1)
and hence if iB22 U22(P) = - ~ K A ( p )
(1.2)
then the surface x2=0 remains plane. Now, from (2.3)
T(xt, O) = 1
A(p) exp(ipxl)d p o
= _ 1.~ f oo iB22U22(p) exp(ipxi)d p 0
rcK
(A.3) '
where (1.2) has been used to substitute for A(p). If the x t = 0 plane is a plane of elastic symmetry then K is real and, since iB22 is also real, it follows that FiB22~ 1
Coo
T(xl, 0 ) = - L ~e_j~_ ~ Jo U22(p)exp(ipx1)dp = _ iB2za22(Xl, 0) rcK
(A.4)
Hence, a stress distribution --rcK
a22(xl, 0) ffl2 = G32 = 0
iB22 T(xl, 0),
(A.5) (A.6)
Two contact problems in anisotropic thermoelasticity
147
on x2 = 0 will give rise to zero normal displacement on this plane. Therefore, if we have the boundary conditions on x2 = 0 for for
0.22(X1,0) = { p(xi)
0
ab,
(1.7)
and
T(x> 0) = -{q(;1)
for for
a < x l b ,
(1.8)
then the stress (r22 over the contact region will differ from the corresponding isothermal value by 60_22 ~- -- rcKq(xl, O)
iB22
(A.9)
It follows that, in the problem of section 4, the effect of a constant temperature profile T(xx, 0) = To over Ix1]< a is to increase the stress over the contact region by -~zKTo/iB22 and increase the total load by 2TcKToa/iB2> This result may be used to deduce (4.20) directly from the isothermal result (4.19). We put
p = px + 2zcKToa iB22
(A.10)
where P denotes the total load, P* the load in the corresponding isothermal problem and the final term the increase in load due to the increase in temperature. Now, from (4.19) • I
a=
tiP'B'2t
t
(A.11)
rc/~ J
and using (A.10) to substitute for px it follows that a 2 _ iPB22
2KToa ,
in agreement with the result (4.20) obtained in section 4.
(A.12)
