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The correspondence principle of linear viscoelasticity for problems that involve time-dependent regions
IMJ.
E~~RX Sci.. 1973, Vol. t 1, pp. 1%
140.
Pergamon Press.
Printed in Great Britain
THE CORRESPONDENCE PRINCIPLE OF LINEAR VISCOELASTICITY FOR PROBLEMS THAT INVOLVE TIME-DEPENDENT REGIONS G. A. C. GRAHAM
Department of Mathematics, Simon Fraser University, Burnaby 2, British Columbia, Canada
G. C. W. SABIN Department of Mathematics,
Memorial University, St. John’s, Newfoundland,
Canada
Abstract-A version of the correspondence principle of linear viscoelasticity theory that is based on an alternative form of the Laplace transform inversion theorem and is valid for bodies that OCCUPY time-dependent ablating regions of space is presented and used to derive solutions to certain thermoviscoelastic boundary value problems for hollow ablating right circular cylinders. The application of this version of the correspondence principle to bodies occupying fixed regions of space but for which the boundary regions, over which different types of boundary conditions are specified, vary with time is investigated and exploited to study the thermoviscoelastic problem of an infinite body containing an extending penny shaped crack on whose surface various thermal conditions may prevail.
1. INTRODUCTION THE STANDARD
way to solve boundary value problems in the linear quasi-static theory of viscoelasticity is to apply the Laplace transform, with respect to time, to the timedependent field equations and boundary conditions. Then, if a solution to the transformed problem can be found, the solution to the original problem is reduced to transform inversion. This method of solving viscoelastic stress analysis problems is referred to as the Correspondence Principle. In its classical form (see Lee[l] and Sternberg[2]) the correspondence principle is inapplicable unless the shape of the body under consideration and the type of condition specified at points of its boundary remain invariant with time. However, certain variations of the principle, introduced by one of the present authors[3,4] and by Ting [S] remain valid, for bodies occupying fixed regions of space, even if the type of condition specificed at points of the boundary changes with time. The purpose of this paper is to present a version of the principle that is applicable to the solution of viscoelastic boundary value problems for bodies that occupy time-dependent ablating regions of space. The only other genera1 method of solution to such problems appears to be the extension of the Papkovich-~euber stress-function solution of homogeneous and isotropic elasticity that was introduced and applied in Graham]6]. In section 2 of the present paper, following a statement of the fundamental field equations of linear thermo-viscoelasticity theory and the classical correspondence principle, we give another version of that principle which is based on an alternative form of the Laplace transform inversion theorem (see appendix) and is valid for bodies that occupy time-dependent regions of space. The application of this version of the correspondence principle to bodies occupying fixed regions of space but for which the boundary regions, over which different types of boundary conditions are specified, vary with time is investigated in some detail and the results of [S] and [4] are recovered. 123
G. A. C. GRAHAM
124
and G. C. W. SABIN
In section 3 the new correspondence principle is used to derive solutions to certain thermoviscoelastic boundary value problems for hollow ablating right circular cylinders. In section 4 the method is used to study the distribution of thermal stress in an infinite viscoelastic solid containing an extending penny shaped crack on whose surface various thermal conditions may prevail. 2. TIME-DEPENDENT
REGIONS.
THE
CORRESPONDENCE
PRINCIPLE
Suppose a time-dependent region respresented by R(r), with boundary 8R (t), 0 < t < ~0, is occupied by a homogeneous and isotropic linear viscoelastic solid. For future convenience it is appropriate to denote by I? the set of all (x, t) such that x is the position vector of a point of R (t) and 0 < t < COwhile an entirely similar notational agreement will apply to any time-dependent region of space. We will assume that as time progresses the region ablates in the sense that R ( tz) is contained in R (r,) whenever tz 2 tI. Thus, material may actually be removed from the body as time progresses. The case when R (t) remains constant for all time will therefore be a special example of an ablating region. Let uI(x, t), eij(x, r) and oij(x, t), each of which is a function defined onR, denote the Cartesian components of displacement, strain and stress respectively. Using indicial notation the strain-displacement relations, the stress equations of motion appropriate to the quasi-static theory and the constitutive equations of the material appear as follows (e.g. see Sternbergr21): k,j(X* qj,j(X, sij(X,
t)
=
{Gf
where eti and sil denote defined through
t) =
t) +Fi(x, hj}
Ui,j(Xy
t) =
(X3 t) 9
(Tu(X,
0, uklek(x,f)
the deviatoric
eU(x, t) = eij(x, t)
t) + Uj,i(X,
-&jekk(x, t):
=
t)
t),
(2.1)
= Cjji(X,
t),
(2.2)
{G~*~[EM-~~TI)(X,~),(2.3)
components
Sij(X, 1) =
of strain and stress which are
uij(X,
t)
f),
-&~kk(X,
(2.4)
in which ati is Kronecker’s delta. Here G, and Gz which are functions of time defined for 0 s t < m, refer to the relaxation functions of the material in shear and isotropic compression respectively. The components of body force density Fi (x, t) and the temperature T(x, t) are presumed to be given on i? and (Ystands for the constant coefficient of thermal expansiont. Here we have adopted the notation for ‘Sheltjes Convolutions’ used by Gut-tin and Stemberg [7] and Stemberg [21 so that if fand g are two functions of position and time then [f* dg] stands for the function defined by [f*
dg] (x, t) =f(x,
t)g(x, 0) +
t f(x,
J0
t--7)
w
dr,
whenever the right hand side is meaningful. In addition to equations (2.1)-(2.4) which must hold on I? certain boundary conditions must be given in order to specify a probtWhen the coefficient of thermal expansion depends on temperature a ‘pseudotemperature’ duced in terms of which the constitutive equations reduce to the form (2.3). See Sternberg[Z].
may be intro-
Correspondence
125
principle of linear 6scoelasticity
If we prescribe the surface displacement and traction, respectively, on complimentary subsets cV?,(t), dR (t) - 8R1 (t) of the boundary 8R (t) then the boundary conditions take the form
lemt.
UI(X, t) = Ul(x, t)
on
CO?,
Q(X, t)nj(x, t) = Ti(x, t)
on
a&-al?,
(2.6) I
where Q(X, t) are the components of the outward unit normal to aR(t) and UI(x, t), Ti (x, t) are given functions. We will now briefly consider the case of an elastic solid. It is immediate from (2.3) and (2.5) that for the particular choice
Gt (t) = 2~7 where h and
K
GB(t)
are constants, the stress-strain sil(x, t) = 2peii(x, t),
Odt
= 3K,
(2.7)
equations (2.3) reduce to
CT&X, t) = 3K[Ekk- 3d](x,
t).
(2.8)
In view of (2.4), equations (2.8) are a statement of Hooke’s law, modified to account for thermal expansion, provided p and K respectively are the shear and bulk moduli of the elastic material. Equations (2. l), (2.2) and (2.8) constitute the field equations of thermoelasticity which is they hold ond and are supplemented by conditions of the form (2.6) specify a one-parameter (the parameter is t) family of static thermoelastic boundary value problems. With a view to discussing the correspondence principle we now introduce the notation
f(x, P>= L{f(x, t); t + p} =
few f(x, t)
eppt dt,
(2.9) f(x, t) = L-V(x,p);
p + t),
for the Laplace transform with respect to time t of a function~(x. t) and for its inverse Laplace transform respectively. We shall denote by [uf(x, t, p), E$(x, t, p), a;(x,t,p)], (x, t) on 8, a set of solutions to the system of equations obtained from (2. l), (2.2), (2.8) by replacing 21.~by pc, (p) and 3~ by pc, (p). This set of solutions are presumed to be valid in Rl(p) > c, where c is some (real) constant. If R does not vary with time it is easy to show, by applying the Laplace transform to equations (2. I), (2.2), (2.3) and using the appropriate convolution and differentiation theorems, that the displacements, strains and stresses given by u{(x,~) =L-‘{fr
uB(x,r.p)
eFmdr;
eti(x, t) = L-’ {jr
E$(x, 7,~)
e+
dr;
]
p+t), p +
t
(2.10)
cru(x, t) = L-’ {,,” u$(x, 7,~) eern d7, tit is inherent in the theory as presented that no ‘initial conditions’ need be specified.
here that all field quantities vanish for times prior to t = 0, so
126
G. A. C. GRAHAM
and G. C. W. SABIN
generate a solution valid in I? of the field equations (2.1)-(2.3). Further, if 8R, (and therefore 8R - dR,) does not vary with time and the state [up, E;, @J is chosen to meet the boundary conditions uf(x, t,p) = Ui(x, r)
on
ad, (2.11)
as(x, t,p)~(x)
= Ti(x, t)
on
eR -%?,, J
where tij(x) refer to the, now time-invariant, components of the outward unit normal to aR, then the state [ui, Eij,crU] generated by (2.10) wilt satisfy the boundary conditions (2.6). Equations (2. IO) and (2.11) constitute a statement of the correspondence Principle.? (see Sternberg[2]). For time-dependent regions R (t)the integrals involved in (2.10) are unmeaningful unless for every I 3 0, rrp(x, r,p), $(x,~,p) and uFjj(x, 7,~) are defined for all x belonging to R(t)and 0 G Q-< CT;.In view of the assumption that the region is ablating this condition is equivalent to the requirement that [up+~g, cr:l be defined on the Cartesian product R(0) X [0, w) X {plRP(p) > c}.In order to remove this restriction so that [uf, ES, a~] need only be defined on i? X {pi Rl (p) > c} consider the relations
(2.12) x,~T,p) e-lHd7:
p -+ 0.1
Strictly speaking, these relations define [ui, Eij, aG] only on the set of all (x, t) such that x is a point of R(r) and 0 < t < 00. In order to alleviate this we shall assume that the functions defined through (2.12) have been continuously extended so as to provide values for [ui, eij, gij] on w. Then, to see that (2.12) provides US with a viscoelastic solution valid in I? we need only verify that the field equations (2.11, (2.2), (2.3) are satisfied by the state [LQ,E~~,P~~] generated through (2.12). Equations (2.1) and 12.2) are found to be satisfied by exploiting the fact that [uf, E&,CT&]satisfies them and using the results embodied in equations (A&), (A.13). By using the fact that [UC,~6, ~$1 satisfies the equations got from (2.8) by replacing 2~ and 3~ by pG(p) and pG(p) respectively and exploiting the result contained in equations (A.9), (A. 11). (A. 12) it is found that equations (2.3) are also satisfied. On substituting from (2.12) into (2.6) it is found that the mixed bounda~ value problem of thermoviscoelasticity is reduced to the determination of solutions [u;, E;, G.;] which satisfy the boundary conditions
It is useful to observe that if the components of displacement uf, 1 G i 6 3, (components of stress (+6, 1 6 i, j s 3) are independent of p throughout i? then the first tCertain modification of the Correspondence Principle to accommodate time-dependent 8R, and dR - aR, of a time-invariant region R have been given in [3], [4] and [S].
surface regions
Correspondence
127
principle of tinear viscoelasticity
(second) of equations (2.13) reduces to uf(x, t) = Ui(x, t)
on
ali,
(aa(x, r)nj(x, t) = Ti (x, t)
on
~@-a&).
(2.14) i
We will now briefly consider the particular circumstances that the region R does not vary with time. In this case it is instructive to consider a somewhat broader class of boundary conditions than is covered under (2.6). Thus, we will denote by u, and u, (o@ and a,) the vector components of displacement (traction) normal and tangential to aR respectively. Then u,, us, a,, a, are vector valued functions of (x, t) defined on aR x [0, w). Suppose that (a,, a,, bl, b,) are elements, taken in order, of any column of the matrix
Then, boundary conditions which prescribe the normal (tangential) components of the displacement and traction vectors on time-dependent complimentary subsets of aR may be written in the form ai(x, t) =Ai(x,
t)
On ai&,
bi(x, t) =&(x,
t)
on
1 s i g 2,
#-a&,
(i)
1 d i 6 2,
(ii)
(2.16) I
where Ai, Bi, 1 s i s 2, are prescribed vector valued functions. We wifl assume that aR&), 1 6 i s 2, are closed regions that are monotonically increasing with time.t Then for each point x on aR we define ti (x) , 1 G i s 2, through the relations ti(x) = Mfn (11x E aR,(t)}, Q(x) = m if
xEaR-dRi(t),
OSttm,
l~is2.
(2.17)
Thus, we have
x E
x E t3Ri(t)
if
t 3 ti(x)
aR--aRi(t)
if
t < ti(x),
1 d i 6 2.
(2.18)
Consider the family of elastic solutions [ue, ~5, @] that on (p 1RI (p) > c} satisfy the boundary conditions &(x, t,p) = a$“(x, t) bf(x, t,p) = Bi(X, t)
on on
a$, aE-a&,
1 G i G 2, 1 G i G 2,
(2.19)
tThat there is no essential loss in generality involved in this assumption may be seen by taking account of the fact that solutions to the field equations (2.1)-(X3) may be superimposed on one another and that (al, a,, b,, b,) may be chosen from any column of (2.15).
128
G. A. C. GRAHAM
and G. C. W. SABIN
where Bi, 1 G i s 2, are functions that have appeared in (2.16) and ui”, 1 4 i s 2, refer to vector valued functions that are as yet unspecified. Suppose that these solutions are such that af(x, t, p) = af*)(x, t, p)
on
a17- aRf,
bf(x,
on
al?,,
1 d i S
2, (2.20)
t,p)
= bi*‘(x,
t,p)
1C
i g
2.
I
Then for the viscoelastic solution generated by (2.12) we have that
By substituting
from (2.19), (2.20) in (2.21), taking note of (2.18) and using (A.6) it is found that (2.16) (ii) are identically satisfied while (2.16) (i) are satisfied provided a~“(x,t)+~~~L-l[~~~a~z~(x,~,p)e-”dr;
p-,5}=Ai(x,t), on
a&,
lGiG2,
(2.22)
where, for the purposes of integration, values for @‘(x, ti(x), p), 1 s i s 2, that are obtained from continuous extensions of the values of a$“)on JR -a&, 1 G i 6 2, may be assigned. Since ui2), 1 G i c 2, are related to all), 1 G i < 2, it is seen that (2.22) represent two integral equations for Uf(l) , 1 G i s 2. Similarly, by virtue of (2.19), (2.20), (2.21) it is found that 4
(X5t) = $I_ L-’
(j-1&‘(x,
7,
p)cPTdr;
on bi(x, t) = /iy I;-’
( j:,
p +
.$)
aR--a&,
lsiG2,
(i) (2.23)
) b:V,
7, PI cyr dc on
a&, lGiS2.
In certain important special circumstances fied. If Uf2’(x, t,p)
= p&(p)
di(X,
P --$t}
t)
(ii) I
the relations (2.22), (2.23) may be simplion
afi--a&, 1 6
i S
2,
(2.24)
where ki, UQzi,1 G i s 2, are functions only of the arguments indicated, then, by substituting into (2.22) and using (A.9), (A. 1 I), (A. 12) and (2.5) we obtain the relations &5(x,
t-9)
a%
Tdr=
w
Ai(x, t),
OII
p--j t},
1 G i C 2.
a&,
1 =Gi
6 2,
(2.25)
where xi(t)
= L-l{ki(p);
(2.26)
129
Correspondenceprinciple of linear viscoelasticity
At the same time (2.23) (i) become 4
(XTt) = diCx,
fIxi
t
di(X, t--7) vdr
0
If, in particular, k,(p) = l/p, 1 s i d 2, thenXi(f) (2.25), (2.27) reduce to @(x,
t) =Ai(X, 1),
ai(x,t)= respectively.
(2.27)
1 G iC2.
ai?-&,
on
I
Co)+
on
&*(x,1),
on
= 1, t 3 0, 1 d i s 2, and equations
1 6 i C 2,
a&, al?-&,
(2.28)
lsi42,
(2.29)
1 s i 9 2,
(2.30)
Simplifications also arise in (2.23) (ii) if bj*‘(X, t,p) =pmi(p)Bi(X,
t),
where mi, Bi, 1 6 i G 2, are functions after substitution, we obtain
h(x, f) = %(x9t) -40) + on
ai&,
on
a&,
only of the arguments
I
indicated,
for then,
r-r&4
gi(X, t-7)
0
Tdr, (2.3 1)
lSii2,
where Jfli(t) =L-‘{m*(p);
1 S i S 2.
(2.32)
a&, 1 S i S 2.
(2.33)
p + t},
which, if mi(p) = l/p, 1 s i s 2, reduces to bi(X9f)
=gi(Xyt),
on
If aR, (aR,) is independent of time then, by virtue of (2.17) we find that the first of (2.22) (second of (2.22)) reduces to @(x,
(&(x,
t) =Al(x,t) t) =A*(x,
t)
on
a&,
on
aA,.)
(2.34)
leaving one equation, which is generally an integral equation, to be solved for &)(a~‘)). A form of this result corresponding to the case when dR1 = aR and (ul, u2, bl, b2) is chosen from the first column of (2.15) has been presented by Ting[S], whose work generalized the results of an earlier study by one of the present authors [4], demonstrating in the process the superfluity of certain restrictive conditions introduced there. If aR1, aR2 are both independent of time then (2.12) supplemented with (2.19) and both of (2.34) provide us with an alternative version of the classical correspondence principle that is enshrined in equations (2. lo), (2.11). The classical correspondence principle admits straightforward generalization to ES. Vol. I1
No. 1 - 1
130
G. A. C. GRAHAM
and G. C. W. SABIN
anisotropic and non-homogeneous materials and to thermorheologically simple viscoelastic bodies provided the temperature field is either purely position-dependent or purely time-dependent (see [2]). It is not hard to verify that these generalized correspondence principles admit extensions analogous to those given in this section to the correspondence principle for homogeneous and isotropic bodies. 3. AN EXAMPLE
We will here consider the problem of determining the displacement, strain and stress fields prevailing in a hollow ablating right circular cylindrical region which is occupied by a homogeneous and isotropic linear viscoelastic material that is under the influence of prescribed distribution of temperature and body force. We assume that the mechanical response of the material is governed by equations (2. l), (2.2), (2.3). For our purpose it is convenient to introduce circular cylindrical co-ordinates (r, cp,z) which are related to rectangular Cartesian co-ordinates (x,, x2, x3) through x1 = r cos cp, x, = r sin cp, x, = 2, (3.1) O
0 =%$0 < 2r,
--03
If at any time t the cylinder occupies the region u(t) c r s b(t) where a (b) is a prescribed monotone increasing (decreasing) function of time then we write iT = {(r,‘p, The temperature to take the form
z, t) 1u(t)
c r s b(t),
0 c cp < 27r, --co < z < cc, 0 < t}.
(3.2)
and body force, which must be given at all points of R will be presumed
T=T(r,t),
F,=pdr,
Frp=FZ=O,
(3.3)
where p refers to the (constant) density of the viscoelastic material and w is a prescribed function of non-negative time.t At this stage it proves convenient to introduce the functions 0, @ which are defined on fi through the relations O(r, t) =
pT(p,
t) dp,
@(r, t) = ‘y-T(r.
t).
(3.4)
Further we shall recall from Gurtin and Sternberg[7] that f and its ‘Stieltjes inverse’ f-l, functions of time t, are defined to be related through [f*
df-‘1 (t) = [f-l*
df] (t) = 1,
t 2 0,
(3.5)
so that,
J(P).?%)
= P-2.
(3.6)
tThe body force given in (3.3) will account for the centrifugal force acting on the viscoelastic material if the cylinder is presumed to be rotating with angular velocity o about its axis of symmetry.
Correspondence
131
principle of linear viscoelasticity
Then, it may be shown that the system of equations obtained from (2.1), (2.2), (2.8) by replacing 2~ by JIG, (p) and 3~ by pG,(p) have solutions (see for example Timoshenko and Goodier[8]). u;(T, t,~) = 3r~(G,+2Gz)-‘(p)Cft)
+;PG;~(P)W)
+~pz~(p)(2G1+GJ1(~)O(~,t)-~p(2GI+Gp)-1(p)~e(f) ef;(r,t,p)
= 3~(G~+2G~)-‘(~)C(~) -3ap’c(p)
-sG;‘(p)D(t)
(2G1+ G,)-‘(p)@(r,
e&(r, f,p) = 3~(G~+2G~)-‘(~)C(~)
t) +p!2G,
+~G~‘(~)~(~~
+~p’~(p)(2G,+G,)-‘(p)Q(r, a& (r, t,j?) =C(t)
-%$
+G,)-“(p)o’(t),
3cY -~P~G,(~)G,~P)
f)-~p(2G,+GZ)-‘(p)wyt), (2G1+ Gz)-‘(p)@(r,
t)
(3.7)
--$p2(5G,+4G,)(p)(2G~+G,)-‘(p)02(t), @?J&, t,p) = C(t) +- “,i’) -t ~wJ~%P)%P) +$P’(GI &(r,
-4Gz)
(2G, + G,)-‘(p)@(r,
t)
(P) f2G1 +G,)-l(pW(i),
&P) = W(G,-GC,)(p)(G,+2G,)-‘(p)C(r) -~~P~~(P)~(P)
(2G1+ G,)-‘(p)T(r,
t)
+~~Z(G,-G,)(p)(2G,+G~)-‘(p)w’o,
where C and D are arbitrary functions of time. On substituting from (3.7) in (2.12) and using equations (A.@, (A. I 1) we obtain the following solution, that is valid on ii, to the viscoelastic field equations (2. I), (2.2), (2.3):-
&(T, I) = 3rC(G1+2Gz)-l*
+$
dC] (f) ++ [G;’ ;*dD] (t)
[G, * d(2G1 +Gz)-l
++de] (r, t) -+I(r),
132
e&r,
G. A. C. GRAHAM
t) =
3[(G,+2G,)-l -3cu[G,
ew(r,t)
* dC] (t) -;
s d(2GI+
[G,-” * dD] (t)
Gz)-l fi d@] (r, t) - 9p’zfl(t), 8
= 3[(G1+2Gz)-‘*dC](r)+$ +y
t) = C(t) -%$-$A(L
+q(r,
t) = C(t) + y+ +$
[G;l+dD](t)
[G, * d(2G1+ Gp)-l
r,,(r,
rz,(r,t)
and G. C. W. SABIN
t) -5
-3a[G1
(r, t)
a(t),
-y
[5G1+4Gz)
*dSZ](I),
3a[G1 + dG, * d(2G1+ G,)-’
[(Cl-4G,)
=2[(G,-G,)
++d@]
+ da] (Y, t)
*dQ](t),
*d(G1+2Gz)-l
*dC](r)
*dG, *d(2G,+G2)-1*dT](~,f)+~[(G,--G,)
*da](t),
where we have written A+, r) = [G, * dG, d d(2G, -I-G,)-’
*dOI (r, t), (3.9)
n(t)
= [ (2G1 + Gp)-l * d(W2)] (t).
We wil now consider condition
three specified
1
problems
each of which must satisfy the
%AQ(a)t t) = 0.
(3.10)
Thus, by virtue of (3.Q (3.10), we must have
DO> C(f) -a”Ct)
[(5Gl+4G2)
*iSI](
(3.11)
Case I: In this case we take the second boundary condition to be (3.12)
%,(W), tf = 0, which, by (3.8), implies that y[(5G,+4G2)
++da](t).
(3.13)
Correspondence
133
principle of linear viscoefasticity
Equations (3.1 l), (3.13) are simultaneously satisfied by 3a c(t) = [P(t) -aZ(t)]
IA(
+&r2(O+b2(Q] 8
1) -h(a(t),
t))
[(5G1+4Gz) *da](t), (3.14) ~ff2(~)A(~(~), f) -b2(t)A(a(t),
D(t) = [b’(t)ya2(t)] +~uz’f~bz’f)
[(5G, +4G2) * dQ] (t).
t)>
J
Equations (3.8), (3.14) then determine the complete solution to the problem in hand. Case 2 : Here we take the second boundary condition to be (3.15)
uT(b, t) = 0
where b is a constant. On substituting from (3.8) in (3.15) and using (3.9) we find that C(t)++[(G1+2G,)
~dG;‘*dD](t) -$(Gl+2Gz)
=$[(Gt+2G2)
SdG, *d(2G,+G,)-l
*da](t) *ddO](b,t).
(3.16)
Elimination of C between (3. I 1) and (3.16) then results in the integral equation
o(t)
+(GL+2Gz)
a2(t) +3b2
sdG;‘*dDJ(t)=F(r),
(3.17)
for D, where F(t) = ~ata(t),~)-~[(G~+ZG~)
+w*w 8[(5G1+4Gz)
*dGp*d(2GI+G2)-1*dO](b,t) *dnJ(t)+$[(G,+2G2)
*ddfl](t).
(3.18)
Equation (3.17) is amenable to nume~cai solution for D by the method outlined in Lee and Rogers[9]. Equation (3.11) then determines values for C and the complete solution to this specific problem will then be generated by (3.8). Case 3: Finally, we take the second boundary condition to be m&b, t) = -Bed& where b, B are positive constants:
t),
the former is prescribed
(3.19) while the latter may be
134
G. A. C. GRAHAM
and G. C. W. SABIN
chosen in such a manner that (3.19) expresses the assumption that the viscoelastic cylinder is bound by an elastic casing (e.g. see Mot-land and Lee[ IO]). Combining (3.8) with (3.19) we find that
C(f) =-B +$
9-g
Mb,
f)
-T
3[(G,+2Gz)-’
~~dC](t)++;1
[G:! *d(2G1+G2)+
*dO](b,
1
*dQZ](t)
[(5G1+4Gz)
+~dD](t) t,-+$I($)}
(3.20)
from which, with the aid of (3.11) C may be eliminated to yield D(t) {&+}+3B
[(G,+2G,)-’
+:d ($1
(f)+-$[G;’
~~dDl(t)= F(t),
(3.21)
where F is complicated, but prescribed, function of time, The solution to this case may now be determined by proceeding as in case 2. An approach to the problem considered in this section, that is valid if T = 0 and is to a certain extent similar to that presented here, has recently been presented by E. C. Ting[ 1 l] who specifically considered case 3. Edelstein[ 121 has solved the problem in that case for a viscoelastic material, with temperature-dependent mechanical response, in terms of Neumann series expansions. Other works on this problem, which are restricted to the case where either T = 0 or ablation is absent, are referred to in Christensen’s [ 131 book. 4. AN
In this and in the (2. I)-(2.31, viscoelastic body force
EXAMPLE
section, still working in terms of circular cylindrical coordinates (r,p,z) context of materials whose mechanical response is governed by equations we shall consider an axisymmetric (the axis of symmetry is r = 0) thermoproblem for a body occupying z L 0 that is under the influence of zero and boundary condition of the form G-,,(T, 0. t) = (r&r, 0, t) = 0, (Trr(r,O, 1) = -p(r.
t),
I(,(T, 0, t) = 0, where a is a prescribed monotone symmetric nature of the problem, being suppressed. The temperature throughout the body and is required
0 S r =z a(t),
(4.1)
r > a(r),
increasing function of time and, in view of the axithe dependence on p of all the field quantities is field will be presumed to satisfy Laplace’s equation to meet the boundary conditions
T(r,O,t) = Q(r,
z(r.O,I)
Y z= 0,
t), =O,
0
s r =za(t),
r > a(t),
(4.2)
Correspondence
135
principle of linear viscoelasticity
where Q is a prescribed function. In addition, all the components of stress and the temperature are required to vanish at infinity. Provided that p and Q are such as to ensure that uz(r, 0, t) > 0, 0 G r 6 a(t), we may consider this problem to be equivalent to that of a symmetrically loaded penny-shaped crack, with a prescribed temperature prevailing over its surface, that is situated in an infinite homogeneous and isotropic linear viscoelastic material with no body force acting. For future convenience we shall now record certain aspects of the solution to the thermoelastic problem governed by equations (2. I), (2.2), (2.8) and the boundary conditions (4. I), (4.2). In this context reference is made to a paper of Boroda~hev~l4~ from which we find that 9K --B(r, p+3K
(c’-F)“*
t),
0 s
r d a(t),
r > a(r)
(4.3)
(4.4)
where 9(c, t) =f
I’ PP(P,f) dp Iatt) o
(c2_pz)“2
’
0
s
c
(4.5)
zG a(t),
a9(c, t)/ac
o
am a(?-,
t)
=f I
dc
’
a(t),
(4.6)
0 s r s a(r).
(4.7)
(r2_c2)1/2
PQ(P, t) dp
o%qs=Fqo~’ I
dc,
r
Further, provided %Y(c, t)/ac, 0 s c G a(t) , exists and is continuous that
‘im {F-~(t))1’2~(f-la(t)* t)) =
r+a’(r)
’
we find from (4.6)
(2u(~))l,2~(u(~), t).
(4.8)
The problem in hand may be treated by the method outlined at the end of section 2 and in this regard it corresponds to the case when (aI, uz, 6,, b,) is chosen from the second column of (2.15) provided
(4.9H
The functions tl, t2 which are defined on 8R will be independent tI(r) = 0;
tz(r) = Mjn {t I r G a(t)).
of cpand satisfy (4.10)
tThe appearance of elastic constants in (4.4) precludes the application of the method described in [4] to the problem under consideration here.
136
G. A. C. GRAHAM
and G. C. W. SABIN
For the required viscoelastic solution to be generated through (2.12) (or (2.10)) we see from (2.19), with the aid of (2.34) that [uf, EL,$1 must be chosen in such a way that
u&(r,O, t,p) =-p(l)(r,
0 S r S u(t),
t).
(4.11)
where, by virtue of (2.22), we must have
uZ,(r, 0, 7, P) cpT ck
5)= (-)p(r,
P +
t),
0 S r C u(t).
(4.12)
However, by using the appropriate thermoelastic solution to the problem satisfying the boundary conditions (4.11) we find, with the aid of(3.6) and (4.4) that
t) dp
pQh
X
($(t)
-p2)l/2’
r
’
(4.13)
a(t),
where the values of g(l) are generated through (4.5), (4.6) by takingp = p”‘. On substituting from (4.13) in (4.12) and using arguments entirely similar to those used in deriving (2.25) and (2.28) we find that t
p”‘(r,
t) = p(r, t)
pQ(p, t-r) dp ($(t-7) -p2)“2’
K’(T) dr -3a! IT t-t2(rj (rZ-a2(t-~))1/2
I
0 C r S u(t)
(4.14)
= [G, +dG, sd(2G,+G,)-‘l(t).
(4.15)
where I
Then, by using the same arguments as those used to obtain (2.27), (2.29) we find that, for the required viscoelastic solution aw
t +
K’(T) dr (r2-u2(t-7))1’2
‘(‘-‘)
Q(
(~*&
while, with the aid of (2.3 1) and (4.3) it is found that
PQ(P, 1) dp (d(t) -p2)“2
, t-7) d _,,,L
’
r ’
a(t)*
(4.16)
Correspondence
principle of linear viscoelasticity
137
where J,(t) = [ (2G,+G,)
*d(Gr+2Gz)-’
*dGil](t), (4.18)
and values for V), 3(l) are generated through (4.3, (46) by substituting for p the quantity p”’ given by (4.14). The stress intensity factor which is defined by the relation N(t) = Jam_ ~+-~~~~~liz~zz~r,O,
0)
(4.19)
is, by virtue of(4.8) and (4.16) found to be given by
(4.20) where tmin= min (7 I a(r) = a(r) ). For the case of a material with similar behaviour in shear and dilatation we have that t-2v
G,(t) = - I+v
G,(t),
taOQ,
(4.21)
where v is a constant that plays the role of ‘Poissons ratio’, On substituting from (4.2 1) in (4,f4), (4.15), (4.16), (4.17), (4. IS), (4.2Qf we obtain a form of the solution, to the probtem in hand, that is appropriate to this mate&& An alternative form of this particular solution may be derived using the appropriate form of the method given in [IS]. The problem that arises when, instead of the temperature, the heat flux is prescribed across the surface of the crack has been treated for the case of an elastic body by Olesiak and Sneddonfl61. Their solution may be extended to viscoelasticity in the same manner as illustrated in this section. Acknowledgment-This investigation tute, University of Oxford, England. Research Council (G. A. C. 0.1, (G. C. W. S.) and grant No. A483 1of
was initiated while the authors were visiting the Mathematical InstiThe research was supported in part by grants from the British Science the Canron Limited-Sidney Hagg Memorial Graduate Scholarship the National Research Council of Canada.
REFERENCES [I] E. H. LEE, Proceedings ofthe first symposium on Naval Structural Mechanics (Edited by J. N. GOODIER and N. J. HOFF) p. 456. Pergamon (1960). [2] E. STERNBERG, Proceedings of the third symposium on Naval Structural Mechanics (Edited by A. M. FREUDENTHAL, B. A. BOLEY and H. LIEBOWITZ) p, 348 Pergamon (1964).
138
G. A. C. GRAHAM
and G. C. W. SABIN
Mech. Stosow. 5,772 (1967). [31 G. A. C. GRAHAM,Archwm. [41 G. A. C. GRAHAM,Q. appl. Math. 26, 167 (1968). [51 T. C. T. TING, Proceedings of the eleventh Midwestern Mechanics Conference (Edited by H. J. WEISS, D. F. YOUNG. W. F. RILEY andT. R. ROGGE) p. 591. Iowa University Press (1969). 161 G. A. C. GRAHAM, Prof. R. Sot. Edinb., Section A67, 1 (1965). r&on. Mech.Analysis 11,291 (1962). I73 M. E. GURTIN and E. STERNBERG,Arch. @I S. TIMOSHENKO and J. N. GOODIER, Theory ofElasticity, McGraw-Hill (195 1). [91 E. H. LEE and T. G. ROGERS,.//. appl. Mech. 30, 127 (1963). [lOI L. W. MORLAND and E. H. LEE. Trans. Sot. Rheol. 4.233 (1960). E. C.TING,AfAAJl.S, 18 (1970). Mech. 8, 174(1969). tt:; W. S. EDELSTEIN,Acta Theory o~~j~~oef~~~ici~y.Academic Press ( 197 1). 1131 R. M. CHRISTENSEN, 1141 N. M. BORODACHEV, Soviet oppf. Mech.. (translation of Prikladnaya Mekhanika~2.54 f 1966). [I51 G. A. C. GRAHAM, BcrN.math. Sot. Sci. math. R&pub. pop. ruum. 12 (60). 7 1 C1968). [I61 Z. OLESIAK and I. N. SNEDDON, Arch. rution. Mech. Analysis 4,238 (1960). (Received6
March 1972)
Appendix Suppose f(x, t,p), a function of (position) x, (time) t and (Laplace transform parameter) p is used to generate a function 9(x, t) of (position) x and (time) t through the equation 9 (x, t) = L-” { J0=f( x,T,p)
e-“‘dr:
p+
t},
t a 0,
p-5
),
t 3 0,
(A.1)
Then, if 9 is a continuous function oft, we can show that 9(x,
t) = litn_LT_{jif(x,r,~)
e-m;
(A.2)
where it is to be understood that the values of the right hand side of this equation on f B 0 are obtained by continuous extension from its values on t > 0. This result, which represents a slight interpretation of a result presented by Tingt[S], may be established in the following way. From (A. 1) we see that, for any non-negative “.$” we may write p-,5)
~(x.~)=L-‘I~~f(x,i,p)e-P’dr; +L-l{,Yf(
x.7.p)
e-“dr;
p+
[
1.
(A.3)
On introducing the change of variable T’ = r-t and exploiting the Faltung theorem for Laplace transforms it may be verified that the second term appearing on the right hand side of equation (A.31 vanishes for 0 < 5 =z 1. It follows that ~(x,~)=L-‘~~~/(x,~,p)e-~~dr:p~~},
Os(
(A.4)
from which we may obtain the result embodied in (A.2) for t > 0 and thence by continuous extension for f 5 0. Two particular corollaries of this result are worthy of mention. In the first place, if the functionf is independent ofp then, by virtue of the Laplace transform inversion theorem equation (A. 1)reduces to 9 (x, t) =f(x.
t),
(AS)
t z 0,
and equation (A.2) then becomes f(x, t) = Ii,: L,T’{II f(x, T) e-‘” dr; in the second instance we shall suppose that the functionfis .flX,T,P)
p -+ t},
t 2 0.
(A.6)
given by
=PY(P)wix,T,P).
(A.71
TThe form of the result given in [5] may be obtained from (A. I), (A.2) by interchanging the order of the limiting operation and Lacplace inversion in (A.2). That the formula obtained in this way is not always valid may be verified by considering the particular casef(x, 7, p) - 1.
Correspondence
139
principle of linear viscoelasticity
where y and w are given functions. Then, by substituting in equation (A. 1) and using the appropriate Faltung theorem we find that %(x,t)=[Y*dW](x,t)=[W*dY](x,t),
(AS)
r>O
where we have used the notation of (2.5) and where Y(t) = L-‘{y(p);
p + r},
W(x,t)=L-‘(~~w(x.7.p)e-“d+;
I > 0,
(A.9) P-W),
(A.lO)
r*O.
Thus, provided the variations with time of 9 and W are sufficiently smooth, we find, by virtue of (A.2) that
=[W*dY](x,i),
(A.ll)
r>O,
where Y is given by (A.9) and (A.12) In conclusion we observe that f(x,
7,
p) e-” dr:
p + 6
11 = IimL-1 t-+r-
p-‘t],
1 s i C 3,
(A.13)
provided the appropriate derivatives exist. R&um&Ceci est une version du principe de correspondance de la theorie de viscotlasticid lineaire, qui est baste sur une deuxieme forme alternative du theoreme de I’inversion de transformation de Laplace et est valable pour les corps occupant des regions de I’espace dont l’ablation est fonction du temps, et cette version est presentee et utilisee pour trouver des solutions a certains problemes de valeur de frontiere thermoviscoClastique pour des cylindres circulaires droits ablatifs creux. L’application de cette version du principe de correspondance B des corps occupant des regions dtfinies de l’espace,-mais pour lesquels les regions front&e, sur lesquelles dierents types de conditions de front&e sont specifies, varient avec le temps,-est soumise a la recherche et exploitee pour etudier le probleme de thermoviscoelasticite d’un corps infini contenant une crique en expansion en forme de piece de monnaie, SW la surface le laquelle differentes conditions thermiques peuvent prevaloir. Zu~~e~~u~-Eine Version des Ko~espondenzp~n~ps der linearen Viskoel~tizit~tstheo~e, die auf einer altemativen Form des Inversionstheorems eines Laplace-Transforms basiert und fiir K&per gliltig ist, die zeitabhangige Abbrennzonen des Raumes bewohnen, wird vorgelegt und dazu verwendet, Losungen fur gewisse thermoviskoelastische Grenzwertprobleme fur hohle gerade kreisformige Ablationszylinder abzuleiten. Die Anwendung dieser Version des Korrespondenzprinzips auf Korper, die feststehende Zonen des Raumes bewohnen, aber fiir die die Grenzzonen, ilber welche verschiedene Arten von Grenzbedingu~en festgelegt sind, sich mit der Zeit andem, wird untersucht und beniltzt, urn das the~ovisk~lastische Problem eines unendiichen Korpers zu untersuchen, der einen sich ausdehnenden miinzenformigen Riss enthPlt, und an dessen Obertliiche verschiedene Warmebedingungen herrschen. Sommario-Si presenta una versione de1 principio di corrispondenza della teoria di viscoelasticitl lineare basato su una forma altemativa de1 teorema d’inversione di trasfo~~ione di Laplace e valid0 per corpi occupanti regioni ablative di spazio dipendenti dal tempo, e la si impiega per ricavare soluzioni di certi problemi di valore limite tennoviscoelastici nei riguardi di cilindri circolari destrorsi ablativi e cavi. L’applicazione di questa versione de1 principio di corrispondenza ai corpi the occupano regioni fisse di spazio ma per i quali le regioni limite, sulle quali si precisano tipi differenti di condizioni limite, variano con il tempo b oggetto di studio ed e impiegata per studiare il problema di termoviscoelasticita di un corpo infinito contenete una incrinatura estesa a forma di penny sulla cui superficie possono prevalere varie condizioni termiche.
140
G. A. C. GRAHAM
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E~~RX Sci.. 1973, Vol. t 1, pp. 1%
140.
Pergamon Press.
Printed in Great Britain
THE CORRESPONDENCE PRINCIPLE OF LINEAR VISCOELASTICITY FOR PROBLEMS THAT INVOLVE TIME-DEPENDENT REGIONS G. A. C. GRAHAM
Department of Mathematics, Simon Fraser University, Burnaby 2, British Columbia, Canada
G. C. W. SABIN Department of Mathematics,
Memorial University, St. John’s, Newfoundland,
Canada
Abstract-A version of the correspondence principle of linear viscoelasticity theory that is based on an alternative form of the Laplace transform inversion theorem and is valid for bodies that OCCUPY time-dependent ablating regions of space is presented and used to derive solutions to certain thermoviscoelastic boundary value problems for hollow ablating right circular cylinders. The application of this version of the correspondence principle to bodies occupying fixed regions of space but for which the boundary regions, over which different types of boundary conditions are specified, vary with time is investigated and exploited to study the thermoviscoelastic problem of an infinite body containing an extending penny shaped crack on whose surface various thermal conditions may prevail.
1. INTRODUCTION THE STANDARD
way to solve boundary value problems in the linear quasi-static theory of viscoelasticity is to apply the Laplace transform, with respect to time, to the timedependent field equations and boundary conditions. Then, if a solution to the transformed problem can be found, the solution to the original problem is reduced to transform inversion. This method of solving viscoelastic stress analysis problems is referred to as the Correspondence Principle. In its classical form (see Lee[l] and Sternberg[2]) the correspondence principle is inapplicable unless the shape of the body under consideration and the type of condition specified at points of its boundary remain invariant with time. However, certain variations of the principle, introduced by one of the present authors[3,4] and by Ting [S] remain valid, for bodies occupying fixed regions of space, even if the type of condition specificed at points of the boundary changes with time. The purpose of this paper is to present a version of the principle that is applicable to the solution of viscoelastic boundary value problems for bodies that occupy time-dependent ablating regions of space. The only other genera1 method of solution to such problems appears to be the extension of the Papkovich-~euber stress-function solution of homogeneous and isotropic elasticity that was introduced and applied in Graham]6]. In section 2 of the present paper, following a statement of the fundamental field equations of linear thermo-viscoelasticity theory and the classical correspondence principle, we give another version of that principle which is based on an alternative form of the Laplace transform inversion theorem (see appendix) and is valid for bodies that occupy time-dependent regions of space. The application of this version of the correspondence principle to bodies occupying fixed regions of space but for which the boundary regions, over which different types of boundary conditions are specified, vary with time is investigated in some detail and the results of [S] and [4] are recovered. 123
G. A. C. GRAHAM
124
and G. C. W. SABIN
In section 3 the new correspondence principle is used to derive solutions to certain thermoviscoelastic boundary value problems for hollow ablating right circular cylinders. In section 4 the method is used to study the distribution of thermal stress in an infinite viscoelastic solid containing an extending penny shaped crack on whose surface various thermal conditions may prevail. 2. TIME-DEPENDENT
REGIONS.
THE
CORRESPONDENCE
PRINCIPLE
Suppose a time-dependent region respresented by R(r), with boundary 8R (t), 0 < t < ~0, is occupied by a homogeneous and isotropic linear viscoelastic solid. For future convenience it is appropriate to denote by I? the set of all (x, t) such that x is the position vector of a point of R (t) and 0 < t < COwhile an entirely similar notational agreement will apply to any time-dependent region of space. We will assume that as time progresses the region ablates in the sense that R ( tz) is contained in R (r,) whenever tz 2 tI. Thus, material may actually be removed from the body as time progresses. The case when R (t) remains constant for all time will therefore be a special example of an ablating region. Let uI(x, t), eij(x, r) and oij(x, t), each of which is a function defined onR, denote the Cartesian components of displacement, strain and stress respectively. Using indicial notation the strain-displacement relations, the stress equations of motion appropriate to the quasi-static theory and the constitutive equations of the material appear as follows (e.g. see Sternbergr21): k,j(X* qj,j(X, sij(X,
t)
=
{Gf
where eti and sil denote defined through
t) =
t) +Fi(x, hj}
Ui,j(Xy
t) =
(X3 t) 9
(Tu(X,
0, uklek(x,f)
the deviatoric
eU(x, t) = eij(x, t)
t) + Uj,i(X,
-&jekk(x, t):
=
t)
t),
(2.1)
= Cjji(X,
t),
(2.2)
{G~*~[EM-~~TI)(X,~),(2.3)
components
Sij(X, 1) =
of strain and stress which are
uij(X,
t)
f),
-&~kk(X,
(2.4)
in which ati is Kronecker’s delta. Here G, and Gz which are functions of time defined for 0 s t < m, refer to the relaxation functions of the material in shear and isotropic compression respectively. The components of body force density Fi (x, t) and the temperature T(x, t) are presumed to be given on i? and (Ystands for the constant coefficient of thermal expansiont. Here we have adopted the notation for ‘Sheltjes Convolutions’ used by Gut-tin and Stemberg [7] and Stemberg [21 so that if fand g are two functions of position and time then [f* dg] stands for the function defined by [f*
dg] (x, t) =f(x,
t)g(x, 0) +
t f(x,
J0
t--7)
w
dr,
whenever the right hand side is meaningful. In addition to equations (2.1)-(2.4) which must hold on I? certain boundary conditions must be given in order to specify a probtWhen the coefficient of thermal expansion depends on temperature a ‘pseudotemperature’ duced in terms of which the constitutive equations reduce to the form (2.3). See Sternberg[Z].
may be intro-
Correspondence
125
principle of linear 6scoelasticity
If we prescribe the surface displacement and traction, respectively, on complimentary subsets cV?,(t), dR (t) - 8R1 (t) of the boundary 8R (t) then the boundary conditions take the form
lemt.
UI(X, t) = Ul(x, t)
on
CO?,
Q(X, t)nj(x, t) = Ti(x, t)
on
a&-al?,
(2.6) I
where Q(X, t) are the components of the outward unit normal to aR(t) and UI(x, t), Ti (x, t) are given functions. We will now briefly consider the case of an elastic solid. It is immediate from (2.3) and (2.5) that for the particular choice
Gt (t) = 2~7 where h and
K
GB(t)
are constants, the stress-strain sil(x, t) = 2peii(x, t),
Odt
= 3K,
(2.7)
equations (2.3) reduce to
CT&X, t) = 3K[Ekk- 3d](x,
t).
(2.8)
In view of (2.4), equations (2.8) are a statement of Hooke’s law, modified to account for thermal expansion, provided p and K respectively are the shear and bulk moduli of the elastic material. Equations (2. l), (2.2) and (2.8) constitute the field equations of thermoelasticity which is they hold ond and are supplemented by conditions of the form (2.6) specify a one-parameter (the parameter is t) family of static thermoelastic boundary value problems. With a view to discussing the correspondence principle we now introduce the notation
f(x, P>= L{f(x, t); t + p} =
few f(x, t)
eppt dt,
(2.9) f(x, t) = L-V(x,p);
p + t),
for the Laplace transform with respect to time t of a function~(x. t) and for its inverse Laplace transform respectively. We shall denote by [uf(x, t, p), E$(x, t, p), a;(x,t,p)], (x, t) on 8, a set of solutions to the system of equations obtained from (2. l), (2.2), (2.8) by replacing 21.~by pc, (p) and 3~ by pc, (p). This set of solutions are presumed to be valid in Rl(p) > c, where c is some (real) constant. If R does not vary with time it is easy to show, by applying the Laplace transform to equations (2. I), (2.2), (2.3) and using the appropriate convolution and differentiation theorems, that the displacements, strains and stresses given by u{(x,~) =L-‘{fr
uB(x,r.p)
eFmdr;
eti(x, t) = L-’ {jr
E$(x, 7,~)
e+
dr;
]
p+t), p +
t
(2.10)
cru(x, t) = L-’ {,,” u$(x, 7,~) eern d7, tit is inherent in the theory as presented that no ‘initial conditions’ need be specified.
here that all field quantities vanish for times prior to t = 0, so
126
G. A. C. GRAHAM
and G. C. W. SABIN
generate a solution valid in I? of the field equations (2.1)-(2.3). Further, if 8R, (and therefore 8R - dR,) does not vary with time and the state [up, E;, @J is chosen to meet the boundary conditions uf(x, t,p) = Ui(x, r)
on
ad, (2.11)
as(x, t,p)~(x)
= Ti(x, t)
on
eR -%?,, J
where tij(x) refer to the, now time-invariant, components of the outward unit normal to aR, then the state [ui, Eij,crU] generated by (2.10) wilt satisfy the boundary conditions (2.6). Equations (2. IO) and (2.11) constitute a statement of the correspondence Principle.? (see Sternberg[2]). For time-dependent regions R (t)the integrals involved in (2.10) are unmeaningful unless for every I 3 0, rrp(x, r,p), $(x,~,p) and uFjj(x, 7,~) are defined for all x belonging to R(t)and 0 G Q-< CT;.In view of the assumption that the region is ablating this condition is equivalent to the requirement that [up+~g, cr:l be defined on the Cartesian product R(0) X [0, w) X {plRP(p) > c}.In order to remove this restriction so that [uf, ES, a~] need only be defined on i? X {pi Rl (p) > c} consider the relations
(2.12) x,~T,p) e-lHd7:
p -+ 0.1
Strictly speaking, these relations define [ui, Eij, aG] only on the set of all (x, t) such that x is a point of R(r) and 0 < t < 00. In order to alleviate this we shall assume that the functions defined through (2.12) have been continuously extended so as to provide values for [ui, eij, gij] on w. Then, to see that (2.12) provides US with a viscoelastic solution valid in I? we need only verify that the field equations (2.11, (2.2), (2.3) are satisfied by the state [LQ,E~~,P~~] generated through (2.12). Equations (2.1) and 12.2) are found to be satisfied by exploiting the fact that [uf, E&,CT&]satisfies them and using the results embodied in equations (A&), (A.13). By using the fact that [UC,~6, ~$1 satisfies the equations got from (2.8) by replacing 2~ and 3~ by pG(p) and pG(p) respectively and exploiting the result contained in equations (A.9), (A. 11). (A. 12) it is found that equations (2.3) are also satisfied. On substituting from (2.12) into (2.6) it is found that the mixed bounda~ value problem of thermoviscoelasticity is reduced to the determination of solutions [u;, E;, G.;] which satisfy the boundary conditions
It is useful to observe that if the components of displacement uf, 1 G i 6 3, (components of stress (+6, 1 6 i, j s 3) are independent of p throughout i? then the first tCertain modification of the Correspondence Principle to accommodate time-dependent 8R, and dR - aR, of a time-invariant region R have been given in [3], [4] and [S].
surface regions
Correspondence
127
principle of tinear viscoelasticity
(second) of equations (2.13) reduces to uf(x, t) = Ui(x, t)
on
ali,
(aa(x, r)nj(x, t) = Ti (x, t)
on
~@-a&).
(2.14) i
We will now briefly consider the particular circumstances that the region R does not vary with time. In this case it is instructive to consider a somewhat broader class of boundary conditions than is covered under (2.6). Thus, we will denote by u, and u, (o@ and a,) the vector components of displacement (traction) normal and tangential to aR respectively. Then u,, us, a,, a, are vector valued functions of (x, t) defined on aR x [0, w). Suppose that (a,, a,, bl, b,) are elements, taken in order, of any column of the matrix
Then, boundary conditions which prescribe the normal (tangential) components of the displacement and traction vectors on time-dependent complimentary subsets of aR may be written in the form ai(x, t) =Ai(x,
t)
On ai&,
bi(x, t) =&(x,
t)
on
1 s i g 2,
#-a&,
(i)
1 d i 6 2,
(ii)
(2.16) I
where Ai, Bi, 1 s i s 2, are prescribed vector valued functions. We wifl assume that aR&), 1 6 i s 2, are closed regions that are monotonically increasing with time.t Then for each point x on aR we define ti (x) , 1 G i s 2, through the relations ti(x) = Mfn (11x E aR,(t)}, Q(x) = m if
xEaR-dRi(t),
OSttm,
l~is2.
(2.17)
Thus, we have
x E
x E t3Ri(t)
if
t 3 ti(x)
aR--aRi(t)
if
t < ti(x),
1 d i 6 2.
(2.18)
Consider the family of elastic solutions [ue, ~5, @] that on (p 1RI (p) > c} satisfy the boundary conditions &(x, t,p) = a$“(x, t) bf(x, t,p) = Bi(X, t)
on on
a$, aE-a&,
1 G i G 2, 1 G i G 2,
(2.19)
tThat there is no essential loss in generality involved in this assumption may be seen by taking account of the fact that solutions to the field equations (2.1)-(X3) may be superimposed on one another and that (al, a,, b,, b,) may be chosen from any column of (2.15).
128
G. A. C. GRAHAM
and G. C. W. SABIN
where Bi, 1 G i s 2, are functions that have appeared in (2.16) and ui”, 1 4 i s 2, refer to vector valued functions that are as yet unspecified. Suppose that these solutions are such that af(x, t, p) = af*)(x, t, p)
on
a17- aRf,
bf(x,
on
al?,,
1 d i S
2, (2.20)
t,p)
= bi*‘(x,
t,p)
1C
i g
2.
I
Then for the viscoelastic solution generated by (2.12) we have that
By substituting
from (2.19), (2.20) in (2.21), taking note of (2.18) and using (A.6) it is found that (2.16) (ii) are identically satisfied while (2.16) (i) are satisfied provided a~“(x,t)+~~~L-l[~~~a~z~(x,~,p)e-”dr;
p-,5}=Ai(x,t), on
a&,
lGiG2,
(2.22)
where, for the purposes of integration, values for @‘(x, ti(x), p), 1 s i s 2, that are obtained from continuous extensions of the values of a$“)on JR -a&, 1 G i 6 2, may be assigned. Since ui2), 1 G i c 2, are related to all), 1 G i < 2, it is seen that (2.22) represent two integral equations for Uf(l) , 1 G i s 2. Similarly, by virtue of (2.19), (2.20), (2.21) it is found that 4
(X5t) = $I_ L-’
(j-1&‘(x,
7,
p)cPTdr;
on bi(x, t) = /iy I;-’
( j:,
p +
.$)
aR--a&,
lsiG2,
(i) (2.23)
) b:V,
7, PI cyr dc on
a&, lGiS2.
In certain important special circumstances fied. If Uf2’(x, t,p)
= p&(p)
di(X,
P --$t}
t)
(ii) I
the relations (2.22), (2.23) may be simplion
afi--a&, 1 6
i S
2,
(2.24)
where ki, UQzi,1 G i s 2, are functions only of the arguments indicated, then, by substituting into (2.22) and using (A.9), (A. 1 I), (A. 12) and (2.5) we obtain the relations &5(x,
t-9)
a%
Tdr=
w
Ai(x, t),
OII
p--j t},
1 G i C 2.
a&,
1 =Gi
6 2,
(2.25)
where xi(t)
= L-l{ki(p);
(2.26)
129
Correspondenceprinciple of linear viscoelasticity
At the same time (2.23) (i) become 4
(XTt) = diCx,
fIxi
t
di(X, t--7) vdr
0
If, in particular, k,(p) = l/p, 1 s i d 2, thenXi(f) (2.25), (2.27) reduce to @(x,
t) =Ai(X, 1),
ai(x,t)= respectively.
(2.27)
1 G iC2.
ai?-&,
on
I
Co)+
on
&*(x,1),
on
= 1, t 3 0, 1 d i s 2, and equations
1 6 i C 2,
a&, al?-&,
(2.28)
lsi42,
(2.29)
1 s i 9 2,
(2.30)
Simplifications also arise in (2.23) (ii) if bj*‘(X, t,p) =pmi(p)Bi(X,
t),
where mi, Bi, 1 6 i G 2, are functions after substitution, we obtain
h(x, f) = %(x9t) -40) + on
ai&,
on
a&,
only of the arguments
I
indicated,
for then,
r-r&4
gi(X, t-7)
0
Tdr, (2.3 1)
lSii2,
where Jfli(t) =L-‘{m*(p);
1 S i S 2.
(2.32)
a&, 1 S i S 2.
(2.33)
p + t},
which, if mi(p) = l/p, 1 s i s 2, reduces to bi(X9f)
=gi(Xyt),
on
If aR, (aR,) is independent of time then, by virtue of (2.17) we find that the first of (2.22) (second of (2.22)) reduces to @(x,
(&(x,
t) =Al(x,t) t) =A*(x,
t)
on
a&,
on
aA,.)
(2.34)
leaving one equation, which is generally an integral equation, to be solved for &)(a~‘)). A form of this result corresponding to the case when dR1 = aR and (ul, u2, bl, b2) is chosen from the first column of (2.15) has been presented by Ting[S], whose work generalized the results of an earlier study by one of the present authors [4], demonstrating in the process the superfluity of certain restrictive conditions introduced there. If aR1, aR2 are both independent of time then (2.12) supplemented with (2.19) and both of (2.34) provide us with an alternative version of the classical correspondence principle that is enshrined in equations (2. lo), (2.11). The classical correspondence principle admits straightforward generalization to ES. Vol. I1
No. 1 - 1
130
G. A. C. GRAHAM
and G. C. W. SABIN
anisotropic and non-homogeneous materials and to thermorheologically simple viscoelastic bodies provided the temperature field is either purely position-dependent or purely time-dependent (see [2]). It is not hard to verify that these generalized correspondence principles admit extensions analogous to those given in this section to the correspondence principle for homogeneous and isotropic bodies. 3. AN EXAMPLE
We will here consider the problem of determining the displacement, strain and stress fields prevailing in a hollow ablating right circular cylindrical region which is occupied by a homogeneous and isotropic linear viscoelastic material that is under the influence of prescribed distribution of temperature and body force. We assume that the mechanical response of the material is governed by equations (2. l), (2.2), (2.3). For our purpose it is convenient to introduce circular cylindrical co-ordinates (r, cp,z) which are related to rectangular Cartesian co-ordinates (x,, x2, x3) through x1 = r cos cp, x, = r sin cp, x, = 2, (3.1) O
0 =%$0 < 2r,
--03
If at any time t the cylinder occupies the region u(t) c r s b(t) where a (b) is a prescribed monotone increasing (decreasing) function of time then we write iT = {(r,‘p, The temperature to take the form
z, t) 1u(t)
c r s b(t),
0 c cp < 27r, --co < z < cc, 0 < t}.
(3.2)
and body force, which must be given at all points of R will be presumed
T=T(r,t),
F,=pdr,
Frp=FZ=O,
(3.3)
where p refers to the (constant) density of the viscoelastic material and w is a prescribed function of non-negative time.t At this stage it proves convenient to introduce the functions 0, @ which are defined on fi through the relations O(r, t) =
pT(p,
t) dp,
@(r, t) = ‘y-T(r.
t).
(3.4)
Further we shall recall from Gurtin and Sternberg[7] that f and its ‘Stieltjes inverse’ f-l, functions of time t, are defined to be related through [f*
df-‘1 (t) = [f-l*
df] (t) = 1,
t 2 0,
(3.5)
so that,
J(P).?%)
= P-2.
(3.6)
tThe body force given in (3.3) will account for the centrifugal force acting on the viscoelastic material if the cylinder is presumed to be rotating with angular velocity o about its axis of symmetry.
Correspondence
131
principle of linear viscoelasticity
Then, it may be shown that the system of equations obtained from (2.1), (2.2), (2.8) by replacing 2~ by JIG, (p) and 3~ by pG,(p) have solutions (see for example Timoshenko and Goodier[8]). u;(T, t,~) = 3r~(G,+2Gz)-‘(p)Cft)
+;PG;~(P)W)
+~pz~(p)(2G1+GJ1(~)O(~,t)-~p(2GI+Gp)-1(p)~e(f) ef;(r,t,p)
= 3~(G~+2G~)-‘(~)C(~) -3ap’c(p)
-sG;‘(p)D(t)
(2G1+ G,)-‘(p)@(r,
e&(r, f,p) = 3~(G~+2G~)-‘(~)C(~)
t) +p!2G,
+~G~‘(~)~(~~
+~p’~(p)(2G,+G,)-‘(p)Q(r, a& (r, t,j?) =C(t)
-%$
+G,)-“(p)o’(t),
3cY -~P~G,(~)G,~P)
f)-~p(2G,+GZ)-‘(p)wyt), (2G1+ Gz)-‘(p)@(r,
t)
(3.7)
--$p2(5G,+4G,)(p)(2G~+G,)-‘(p)02(t), @?J&, t,p) = C(t) +- “,i’) -t ~wJ~%P)%P) +$P’(GI &(r,
-4Gz)
(2G, + G,)-‘(p)@(r,
t)
(P) f2G1 +G,)-l(pW(i),
&P) = W(G,-GC,)(p)(G,+2G,)-‘(p)C(r) -~~P~~(P)~(P)
(2G1+ G,)-‘(p)T(r,
t)
+~~Z(G,-G,)(p)(2G,+G~)-‘(p)w’o,
where C and D are arbitrary functions of time. On substituting from (3.7) in (2.12) and using equations (A.@, (A. I 1) we obtain the following solution, that is valid on ii, to the viscoelastic field equations (2. I), (2.2), (2.3):-
&(T, I) = 3rC(G1+2Gz)-l*
+$
dC] (f) ++ [G;’ ;*dD] (t)
[G, * d(2G1 +Gz)-l
++de] (r, t) -+I(r),
132
e&r,
G. A. C. GRAHAM
t) =
3[(G,+2G,)-l -3cu[G,
ew(r,t)
* dC] (t) -;
s d(2GI+
[G,-” * dD] (t)
Gz)-l fi d@] (r, t) - 9p’zfl(t), 8
= 3[(G1+2Gz)-‘*dC](r)+$ +y
t) = C(t) -%$-$A(L
+q(r,
t) = C(t) + y+ +$
[G;l+dD](t)
[G, * d(2G1+ Gp)-l
r,,(r,
rz,(r,t)
and G. C. W. SABIN
t) -5
-3a[G1
(r, t)
a(t),
-y
[5G1+4Gz)
*dSZ](I),
3a[G1 + dG, * d(2G1+ G,)-’
[(Cl-4G,)
=2[(G,-G,)
++d@]
+ da] (Y, t)
*dQ](t),
*d(G1+2Gz)-l
*dC](r)
*dG, *d(2G,+G2)-1*dT](~,f)+~[(G,--G,)
*da](t),
where we have written A+, r) = [G, * dG, d d(2G, -I-G,)-’
*dOI (r, t), (3.9)
n(t)
= [ (2G1 + Gp)-l * d(W2)] (t).
We wil now consider condition
three specified
1
problems
each of which must satisfy the
%AQ(a)t t) = 0.
(3.10)
Thus, by virtue of (3.Q (3.10), we must have
DO> C(f) -a”Ct)
[(5Gl+4G2)
*iSI](
(3.11)
Case I: In this case we take the second boundary condition to be (3.12)
%,(W), tf = 0, which, by (3.8), implies that y[(5G,+4G2)
++da](t).
(3.13)
Correspondence
133
principle of linear viscoefasticity
Equations (3.1 l), (3.13) are simultaneously satisfied by 3a c(t) = [P(t) -aZ(t)]
IA(
+&r2(O+b2(Q] 8
1) -h(a(t),
t))
[(5G1+4Gz) *da](t), (3.14) ~ff2(~)A(~(~), f) -b2(t)A(a(t),
D(t) = [b’(t)ya2(t)] +~uz’f~bz’f)
[(5G, +4G2) * dQ] (t).
t)>
J
Equations (3.8), (3.14) then determine the complete solution to the problem in hand. Case 2 : Here we take the second boundary condition to be (3.15)
uT(b, t) = 0
where b is a constant. On substituting from (3.8) in (3.15) and using (3.9) we find that C(t)++[(G1+2G,)
~dG;‘*dD](t) -$(Gl+2Gz)
=$[(Gt+2G2)
SdG, *d(2G,+G,)-l
*da](t) *ddO](b,t).
(3.16)
Elimination of C between (3. I 1) and (3.16) then results in the integral equation
o(t)
+(GL+2Gz)
a2(t) +3b2
sdG;‘*dDJ(t)=F(r),
(3.17)
for D, where F(t) = ~ata(t),~)-~[(G~+ZG~)
+w*w 8[(5G1+4Gz)
*dGp*d(2GI+G2)-1*dO](b,t) *dnJ(t)+$[(G,+2G2)
*ddfl](t).
(3.18)
Equation (3.17) is amenable to nume~cai solution for D by the method outlined in Lee and Rogers[9]. Equation (3.11) then determines values for C and the complete solution to this specific problem will then be generated by (3.8). Case 3: Finally, we take the second boundary condition to be m&b, t) = -Bed& where b, B are positive constants:
t),
the former is prescribed
(3.19) while the latter may be
134
G. A. C. GRAHAM
and G. C. W. SABIN
chosen in such a manner that (3.19) expresses the assumption that the viscoelastic cylinder is bound by an elastic casing (e.g. see Mot-land and Lee[ IO]). Combining (3.8) with (3.19) we find that
C(f) =-B +$
9-g
Mb,
f)
-T
3[(G,+2Gz)-’
~~dC](t)++;1
[G:! *d(2G1+G2)+
*dO](b,
1
*dQZ](t)
[(5G1+4Gz)
+~dD](t) t,-+$I($)}
(3.20)
from which, with the aid of (3.11) C may be eliminated to yield D(t) {&+}+3B
[(G,+2G,)-’
+:d ($1
(f)+-$[G;’
~~dDl(t)= F(t),
(3.21)
where F is complicated, but prescribed, function of time, The solution to this case may now be determined by proceeding as in case 2. An approach to the problem considered in this section, that is valid if T = 0 and is to a certain extent similar to that presented here, has recently been presented by E. C. Ting[ 1 l] who specifically considered case 3. Edelstein[ 121 has solved the problem in that case for a viscoelastic material, with temperature-dependent mechanical response, in terms of Neumann series expansions. Other works on this problem, which are restricted to the case where either T = 0 or ablation is absent, are referred to in Christensen’s [ 131 book. 4. AN
In this and in the (2. I)-(2.31, viscoelastic body force
EXAMPLE
section, still working in terms of circular cylindrical coordinates (r,p,z) context of materials whose mechanical response is governed by equations we shall consider an axisymmetric (the axis of symmetry is r = 0) thermoproblem for a body occupying z L 0 that is under the influence of zero and boundary condition of the form G-,,(T, 0. t) = (r&r, 0, t) = 0, (Trr(r,O, 1) = -p(r.
t),
I(,(T, 0, t) = 0, where a is a prescribed monotone symmetric nature of the problem, being suppressed. The temperature throughout the body and is required
0 S r =z a(t),
(4.1)
r > a(r),
increasing function of time and, in view of the axithe dependence on p of all the field quantities is field will be presumed to satisfy Laplace’s equation to meet the boundary conditions
T(r,O,t) = Q(r,
z(r.O,I)
Y z= 0,
t), =O,
0
s r =za(t),
r > a(t),
(4.2)
Correspondence
135
principle of linear viscoelasticity
where Q is a prescribed function. In addition, all the components of stress and the temperature are required to vanish at infinity. Provided that p and Q are such as to ensure that uz(r, 0, t) > 0, 0 G r 6 a(t), we may consider this problem to be equivalent to that of a symmetrically loaded penny-shaped crack, with a prescribed temperature prevailing over its surface, that is situated in an infinite homogeneous and isotropic linear viscoelastic material with no body force acting. For future convenience we shall now record certain aspects of the solution to the thermoelastic problem governed by equations (2. I), (2.2), (2.8) and the boundary conditions (4. I), (4.2). In this context reference is made to a paper of Boroda~hev~l4~ from which we find that 9K --B(r, p+3K
(c’-F)“*
t),
0 s
r d a(t),
r > a(r)
(4.3)
(4.4)
where 9(c, t) =f
I’ PP(P,f) dp Iatt) o
(c2_pz)“2
’
0
s
c
(4.5)
zG a(t),
a9(c, t)/ac
o
am a(?-,
t)
=f I
dc
’
a(t),
(4.6)
0 s r s a(r).
(4.7)
(r2_c2)1/2
PQ(P, t) dp
o%qs=Fqo~’ I
dc,
r
Further, provided %Y(c, t)/ac, 0 s c G a(t) , exists and is continuous that
‘im {F-~(t))1’2~(f-la(t)* t)) =
r+a’(r)
’
we find from (4.6)
(2u(~))l,2~(u(~), t).
(4.8)
The problem in hand may be treated by the method outlined at the end of section 2 and in this regard it corresponds to the case when (aI, uz, 6,, b,) is chosen from the second column of (2.15) provided
(4.9H
The functions tl, t2 which are defined on 8R will be independent tI(r) = 0;
tz(r) = Mjn {t I r G a(t)).
of cpand satisfy (4.10)
tThe appearance of elastic constants in (4.4) precludes the application of the method described in [4] to the problem under consideration here.
136
G. A. C. GRAHAM
and G. C. W. SABIN
For the required viscoelastic solution to be generated through (2.12) (or (2.10)) we see from (2.19), with the aid of (2.34) that [uf, EL,$1 must be chosen in such a way that
u&(r,O, t,p) =-p(l)(r,
0 S r S u(t),
t).
(4.11)
where, by virtue of (2.22), we must have
uZ,(r, 0, 7, P) cpT ck
5)= (-)p(r,
P +
t),
0 S r C u(t).
(4.12)
However, by using the appropriate thermoelastic solution to the problem satisfying the boundary conditions (4.11) we find, with the aid of(3.6) and (4.4) that
t) dp
pQh
X
($(t)
-p2)l/2’
r
’
(4.13)
a(t),
where the values of g(l) are generated through (4.5), (4.6) by takingp = p”‘. On substituting from (4.13) in (4.12) and using arguments entirely similar to those used in deriving (2.25) and (2.28) we find that t
p”‘(r,
t) = p(r, t)
pQ(p, t-r) dp ($(t-7) -p2)“2’
K’(T) dr -3a! IT t-t2(rj (rZ-a2(t-~))1/2
I
0 C r S u(t)
(4.14)
= [G, +dG, sd(2G,+G,)-‘l(t).
(4.15)
where I
Then, by using the same arguments as those used to obtain (2.27), (2.29) we find that, for the required viscoelastic solution aw
t +
K’(T) dr (r2-u2(t-7))1’2
‘(‘-‘)
Q(
(~*&
while, with the aid of (2.3 1) and (4.3) it is found that
PQ(P, 1) dp (d(t) -p2)“2
, t-7) d _,,,L
’
r ’
a(t)*
(4.16)
Correspondence
principle of linear viscoelasticity
137
where J,(t) = [ (2G,+G,)
*d(Gr+2Gz)-’
*dGil](t), (4.18)
and values for V), 3(l) are generated through (4.3, (46) by substituting for p the quantity p”’ given by (4.14). The stress intensity factor which is defined by the relation N(t) = Jam_ ~+-~~~~~liz~zz~r,O,
0)
(4.19)
is, by virtue of(4.8) and (4.16) found to be given by
(4.20) where tmin= min (7 I a(r) = a(r) ). For the case of a material with similar behaviour in shear and dilatation we have that t-2v
G,(t) = - I+v
G,(t),
taOQ,
(4.21)
where v is a constant that plays the role of ‘Poissons ratio’, On substituting from (4.2 1) in (4,f4), (4.15), (4.16), (4.17), (4. IS), (4.2Qf we obtain a form of the solution, to the probtem in hand, that is appropriate to this mate&& An alternative form of this particular solution may be derived using the appropriate form of the method given in [IS]. The problem that arises when, instead of the temperature, the heat flux is prescribed across the surface of the crack has been treated for the case of an elastic body by Olesiak and Sneddonfl61. Their solution may be extended to viscoelasticity in the same manner as illustrated in this section. Acknowledgment-This investigation tute, University of Oxford, England. Research Council (G. A. C. 0.1, (G. C. W. S.) and grant No. A483 1of
was initiated while the authors were visiting the Mathematical InstiThe research was supported in part by grants from the British Science the Canron Limited-Sidney Hagg Memorial Graduate Scholarship the National Research Council of Canada.
REFERENCES [I] E. H. LEE, Proceedings ofthe first symposium on Naval Structural Mechanics (Edited by J. N. GOODIER and N. J. HOFF) p. 456. Pergamon (1960). [2] E. STERNBERG, Proceedings of the third symposium on Naval Structural Mechanics (Edited by A. M. FREUDENTHAL, B. A. BOLEY and H. LIEBOWITZ) p, 348 Pergamon (1964).
138
G. A. C. GRAHAM
and G. C. W. SABIN
Mech. Stosow. 5,772 (1967). [31 G. A. C. GRAHAM,Archwm. [41 G. A. C. GRAHAM,Q. appl. Math. 26, 167 (1968). [51 T. C. T. TING, Proceedings of the eleventh Midwestern Mechanics Conference (Edited by H. J. WEISS, D. F. YOUNG. W. F. RILEY andT. R. ROGGE) p. 591. Iowa University Press (1969). 161 G. A. C. GRAHAM, Prof. R. Sot. Edinb., Section A67, 1 (1965). r&on. Mech.Analysis 11,291 (1962). I73 M. E. GURTIN and E. STERNBERG,Arch. @I S. TIMOSHENKO and J. N. GOODIER, Theory ofElasticity, McGraw-Hill (195 1). [91 E. H. LEE and T. G. ROGERS,.//. appl. Mech. 30, 127 (1963). [lOI L. W. MORLAND and E. H. LEE. Trans. Sot. Rheol. 4.233 (1960). E. C.TING,AfAAJl.S, 18 (1970). Mech. 8, 174(1969). tt:; W. S. EDELSTEIN,Acta Theory o~~j~~oef~~~ici~y.Academic Press ( 197 1). 1131 R. M. CHRISTENSEN, 1141 N. M. BORODACHEV, Soviet oppf. Mech.. (translation of Prikladnaya Mekhanika~2.54 f 1966). [I51 G. A. C. GRAHAM, BcrN.math. Sot. Sci. math. R&pub. pop. ruum. 12 (60). 7 1 C1968). [I61 Z. OLESIAK and I. N. SNEDDON, Arch. rution. Mech. Analysis 4,238 (1960). (Received6
March 1972)
Appendix Suppose f(x, t,p), a function of (position) x, (time) t and (Laplace transform parameter) p is used to generate a function 9(x, t) of (position) x and (time) t through the equation 9 (x, t) = L-” { J0=f( x,T,p)
e-“‘dr:
p+
t},
t a 0,
p-5
),
t 3 0,
(A.1)
Then, if 9 is a continuous function oft, we can show that 9(x,
t) = litn_LT_{jif(x,r,~)
e-m;
(A.2)
where it is to be understood that the values of the right hand side of this equation on f B 0 are obtained by continuous extension from its values on t > 0. This result, which represents a slight interpretation of a result presented by Tingt[S], may be established in the following way. From (A. 1) we see that, for any non-negative “.$” we may write p-,5)
~(x.~)=L-‘I~~f(x,i,p)e-P’dr; +L-l{,Yf(
x.7.p)
e-“dr;
p+
[
1.
(A.3)
On introducing the change of variable T’ = r-t and exploiting the Faltung theorem for Laplace transforms it may be verified that the second term appearing on the right hand side of equation (A.31 vanishes for 0 < 5 =z 1. It follows that ~(x,~)=L-‘~~~/(x,~,p)e-~~dr:p~~},
Os(
(A.4)
from which we may obtain the result embodied in (A.2) for t > 0 and thence by continuous extension for f 5 0. Two particular corollaries of this result are worthy of mention. In the first place, if the functionf is independent ofp then, by virtue of the Laplace transform inversion theorem equation (A. 1)reduces to 9 (x, t) =f(x.
t),
(AS)
t z 0,
and equation (A.2) then becomes f(x, t) = Ii,: L,T’{II f(x, T) e-‘” dr; in the second instance we shall suppose that the functionfis .flX,T,P)
p -+ t},
t 2 0.
(A.6)
given by
=PY(P)wix,T,P).
(A.71
TThe form of the result given in [5] may be obtained from (A. I), (A.2) by interchanging the order of the limiting operation and Lacplace inversion in (A.2). That the formula obtained in this way is not always valid may be verified by considering the particular casef(x, 7, p) - 1.
Correspondence
139
principle of linear viscoelasticity
where y and w are given functions. Then, by substituting in equation (A. 1) and using the appropriate Faltung theorem we find that %(x,t)=[Y*dW](x,t)=[W*dY](x,t),
(AS)
r>O
where we have used the notation of (2.5) and where Y(t) = L-‘{y(p);
p + r},
W(x,t)=L-‘(~~w(x.7.p)e-“d+;
I > 0,
(A.9) P-W),
(A.lO)
r*O.
Thus, provided the variations with time of 9 and W are sufficiently smooth, we find, by virtue of (A.2) that
=[W*dY](x,i),
(A.ll)
r>O,
where Y is given by (A.9) and (A.12) In conclusion we observe that f(x,
7,
p) e-” dr:
p + 6
11 = IimL-1 t-+r-
p-‘t],
1 s i C 3,
(A.13)
provided the appropriate derivatives exist. R&um&Ceci est une version du principe de correspondance de la theorie de viscotlasticid lineaire, qui est baste sur une deuxieme forme alternative du theoreme de I’inversion de transformation de Laplace et est valable pour les corps occupant des regions de I’espace dont l’ablation est fonction du temps, et cette version est presentee et utilisee pour trouver des solutions a certains problemes de valeur de frontiere thermoviscoClastique pour des cylindres circulaires droits ablatifs creux. L’application de cette version du principe de correspondance B des corps occupant des regions dtfinies de l’espace,-mais pour lesquels les regions front&e, sur lesquelles dierents types de conditions de front&e sont specifies, varient avec le temps,-est soumise a la recherche et exploitee pour etudier le probleme de thermoviscoelasticite d’un corps infini contenant une crique en expansion en forme de piece de monnaie, SW la surface le laquelle differentes conditions thermiques peuvent prevaloir. Zu~~e~~u~-Eine Version des Ko~espondenzp~n~ps der linearen Viskoel~tizit~tstheo~e, die auf einer altemativen Form des Inversionstheorems eines Laplace-Transforms basiert und fiir K&per gliltig ist, die zeitabhangige Abbrennzonen des Raumes bewohnen, wird vorgelegt und dazu verwendet, Losungen fur gewisse thermoviskoelastische Grenzwertprobleme fur hohle gerade kreisformige Ablationszylinder abzuleiten. Die Anwendung dieser Version des Korrespondenzprinzips auf Korper, die feststehende Zonen des Raumes bewohnen, aber fiir die die Grenzzonen, ilber welche verschiedene Arten von Grenzbedingu~en festgelegt sind, sich mit der Zeit andem, wird untersucht und beniltzt, urn das the~ovisk~lastische Problem eines unendiichen Korpers zu untersuchen, der einen sich ausdehnenden miinzenformigen Riss enthPlt, und an dessen Obertliiche verschiedene Warmebedingungen herrschen. Sommario-Si presenta una versione de1 principio di corrispondenza della teoria di viscoelasticitl lineare basato su una forma altemativa de1 teorema d’inversione di trasfo~~ione di Laplace e valid0 per corpi occupanti regioni ablative di spazio dipendenti dal tempo, e la si impiega per ricavare soluzioni di certi problemi di valore limite tennoviscoelastici nei riguardi di cilindri circolari destrorsi ablativi e cavi. L’applicazione di questa versione de1 principio di corrispondenza ai corpi the occupano regioni fisse di spazio ma per i quali le regioni limite, sulle quali si precisano tipi differenti di condizioni limite, variano con il tempo b oggetto di studio ed e impiegata per studiare il problema di termoviscoelasticita di un corpo infinito contenete una incrinatura estesa a forma di penny sulla cui superficie possono prevalere varie condizioni termiche.
140
G. A. C. GRAHAM
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