Journal ï/ Elasticity 44: 229-254, 1996. 229@ 1996 Kluwer Acádemic Publishers. Printed in the Netherlands.

Dynamic Thermoelastic Effects for Half - Planes and

Half-Spaces with Nearly-Planar Surfaces

~l L.M. BROCK1, Ì. RODGERS1 and H.G. GEORGIADIS2É é Engineering Mechanics, University ï/ Kentucky, Lexington, Kentucky 40506, U.S.A.J 2 Mechanics Division, Nátionál Technicál University ï/ Athens, Zográphou 15773 Greeqe

Received 14 June 1996

Abstract. The effects of non-planarity ïç the dynamic surface temperature changes induced forplane-strain and 3D problems ïç the nearly-planar surfaces of, respectively, coupled thermoelastichalf -planes and half -spaces by surface heat fluxes are treated. The nearly-planar nature of the surfacesallows the problem solutions Éï be wÞtteç, following a standard perturbation scheme, as seÞes expan-sions ßç a dimensionless surface contour amplitude parameter. The first, or zero-order, terms representthe ideal (planar) surface so.lutions, while the second, or first-order, terms represent couections for

non-planarity.Because the characteÞstic thermoelastic time is of Ï(lï-7)ìs, large-time asymptotic founs of

the exact integral transform solutions can be used. These can be inverted exactly and used ßç Green'sfunction operations Éï yield analytic, or integrals of analytic, expressions. Two types of thermalloading for the half-plane and yet a third type of thermalloading for the ha1f-space are considered.CïmÑaÞsïç of the zero- and first-order surface temperature changes for each case indicate that çïç-ÑÉanaÞtÕ gives Þse for large times Éï changes ßç surface regions beyond those predicted by an idealsurface ana1ysis. Moreover, the magnitudes of these changes can be more significant than the idea1surface results.

Key words: thermoelasticity, surface, nearly-planar, dynamic, temperature

1. Introduction

Analyses of the response Éï dynarnically-applied mechanica1 surface loads [1-4] are important ßç contact mechanics, tÞbïlïgÕ and geotechnics. The relatedcase of thermal surface loading [5, 6] a1so has application to those fields when

* " prorninent temperature changes are expected. Thermalloading studies often treatÉ '~ un- ~r partially-coupled thermoelastic ~olids. Moreo~er, they -l~e their mechanic.a1" '. loadmg counterparts - generally conslder the matenal Éï be an ldea1 half-plane m

~! plane-strain cases or an ideal half-space ßç full 3D problems. That is, the surfaceis planar. However, thermal studies for the idea1 surface [7] indicate that couplingeffects can influence dynarnic response, and non-thermal studies [8-10] show thateven weak surface non-planarity can affect response.

This article, therefore, considers the dynamic response due to therma11oadsapplied to the surfaces of isotropic, homogeneous coupled thermoelastic solidsmodeled for plane-strain and 3D problems as, respectively, nearly-planar, i.e. weak-ly non-planar, half-planes and ha1f-spaces. While the mathematica1 framework for

230 L.M. BROCK ÅÔ AL.

the complete dynamic response is proíided, the compmsons of non-planar andideal surfaces are based ïç the surface temperature changes engendered by íar-ious loadings. Because the surfaces are nearly planar, standard [11] perturbationmethods are used, cf. [8~10]. Moreoíer, existence of a small, i.e. Ï(10-8)ìs,thermoelastic characteÞstßc time al1ows the use of robust [12] large-time solutionsgiíen completely ßç terms of analytic, or integra1s of analytic, functions.

Specifically, the plane-strain examples treated are a half-plane loaded ïç itssurface either by a heat flux applied at a point (line load ßç the out-of-plane direction)or by a heat flux imposed oíer a uniformly-expanding smp. The 3D example is ahalf-space subjected to a heat flux ïç a fixed finite area of its surface. ÂÕ nearly-planar, we mean that the difference between surface length (area) and its planar '

projection is negligible, and that functions eíaluated ïç the surface can be wÞtteças Taylor seÞes expansions about the projection plane. The analysis begins ßç thenext section with the ha1f-plane subjected to both thermal and mechanicalloading.Integral transform and Green's function methods proíide large-time expressionsfor the surface temperature change due to purely therma11oading. The ha1f-spaceproblem is then considered. Among the results obtained is that eíen weakly çïç-planar surfaces exhibit temperature changes ßç regions where the idea1 (planar)surface does çïÉ Moreoíer, the magnitudes of these additional change:s can bemore significant than those induced ïç the ideal surfaces.

2. Plane-Strain Half-Plane Problem

Formulation: Consider the ha1f-plane ßç Figure 1. In terms ofthe Cartesian coor-dinates (×, Æ), it is defined as Æ > é\é(÷), where the surface amplitude é\ > Ï isdimensionless and the surface contour function j(x) is single-ía1ued, lies ßç C2,and has a magnitude that nowhere exceeds one length õçßé For time t ~ Ï thehalf-plane is at rest at a uniform temperature Ôï(Ê). That is, for t ~ Ï, Æ > é\é(÷)the initia1 conditions are

(u,è) = Ï, (1)

where u(x, Æ, t) = (ux, uz) is the displacement íector and è(÷, Æ, t) is the change

ßç temperature from ÔÏ. For t > Ï a normal (n) heat flux and normal and shear (s)tractions are imposed õñïç the surface. The (n, s)-directions, indicated ßç Figure "

1, can be found from their unit íectors ç = (nx, nz) and s = (sx, sz), where

é\1' -1nx = ""J1~(ëf~' nz = ºl=:;(ëé~' Sx = nz, 8Æ = -nx (2)

and (') denotes differentiation with respect to the argument. Thus, for t > Ï, Æ =é\! (÷) the boundary conditions are

fJèóns = Ôï, ón = óï, & = Ñ, (3)

.

DYNAMIC THERMOELASTIC EFFECTS 231

Æ=ëf(÷)

ß

Figure 1. Schematic of nearly-planar half-plane.

where the imposed tractions (ÔÏ, óï) and heat flux F are functions of (x,t) andfinite for finite t > Ï for all ×. The perhaps more natura1 dependence of thesequantities ïç (8, é), where 8 is the distance measured along the surface, could beobtained by using the formula

8 = ~x Ji+(ëð du, (4)

but it is more coníenient to work with (×, z)-based quantities. Indeed, we nowrewÞte (3) ßç terms of these quantities as

n. ó = (nzTo + n÷óï, -nxTO + nÆóï), n. Vè = Ñ, (5.1,3)

where ó(÷, Æ, t) is the 2Ï stress tensor.Fort > Ï,Æ > ëÉ(÷) thegoíemingequationsare [13, 14]

[}2u\72u+ (m2 -1)VÄ + êVè - b28t2 = ï, ê = êï(4 - 3m2) < Ï, (6.1)

h 2 [} ( m2å )-\7è+- -Ä-è =0, (6.2)

á [}t êé

1-ó = [(m2 - 2)Ä + êè]É + Vu + uV, (6.3)ì

where Ä(÷, Æ, t) is the dilatation and (\72, V) are the Laplacian and gradientoperators ßç the ×-Æ plane. Éç (6) the thermoelastic constants

h- ~ å- ~ (~ )2- ìmCh' - Ch m ' (7)

Vd 1 1m = -, b = -, á = -Vr Vr Vd

232 L.M. BROCKET AL.

are used, where the constants (êï,k,Ch,ì) are, respectively, the coefficient ofthemlal expansion, conductivity, specific heat and shear modulus, while Vd is theclassical (non-themlal) dilatational wave speed, Vr is the rotational wave speed, and(á, b) are the cïðeSÑïçdßçg slownesses. The symbol h is a themloelastic character-istic length with values typically [12] of 0(10-4)ìm, while å is the dimensionlessthemloelastic coupling constant with values typically [13] of 0(10-2). Éç addition,we require that (ï, è) be continuous ßç (x,z,t) and finite for all (×,Æ > ëÉ(÷))for finite t > ï. '

Dropping the è-temlS from (6.1,3) and the Ä-temlS from (6.2) reduces thefomler to the standard [15] thermal diffusion equation. It should be noted thatthe fully linear coupled fomls (6.1,3) are achieved by replacing the instantaneoustemperature by its initial value Ôï. As Chadwick [13] has noted, this approximationis valid when lè /Tol « 1, and is a useful first step ßç studyingcoupled themloelasticeffects ßç any case.

Ôï solve this problem, we first rewrite (5) ßç view of (2) as

ë!'ó÷ - ó×Æ = -Ôï + ë!'óï, (8.1)

ë!'ó×Æ - óÆ = -ë!,ôï - óï, (8.2)

ë!,~ - ~ = FJl+W~, (8.3)è÷ 8Æ

and then apply a standard [11] perturbation scheme based ïç the assumption thatthe half-plane surface is nearly planar. That is, the amplitude ë is small enoughthat (ï, è) evaluated along Æ = ëÉ(÷) can be written as a Taylor series expansionabout Æ = Ï, e.g.

o(ëI)=o(O)+ëI~+~~~+"" (9)8Æ 2 8Æ2

and that the difference between surface length (4) and its projection onto the x-axisis negligible, i.e.

ÉëÉ,l « 1. (10)

This suggests that the square-root radical ßç (8.3) be expanded as a power series ßçë!' and that (ï, è) be written as the series

(ï, è) = (ïï,èï) + ë(ïé,èé),+ë2(02,èv +"'. (11)

Use of (9)-(11) and the è-cïuçterparts ßç (1), (6), (8) and the associated bounded-ness/continuity conditions, and requiring that the resulting equations be satisfiedindependently of ë gives an equation system for each function set (Oi, èi), (ß =

.

OYNAMIC THERMOELASTIC EFFECTS 233

1,2,... ,). Specifica11y, each set (Oi, Oi) must satisfy for Æ> Ï the initial and gov-erning equations, (1) and (6), and the boundedness/continuity conditions, while fort > Ï, Æ = Ï the first two sets (00,00) and (01,01) satisfy the boundary conditions

800ó×ÆÏ = Ôï, óÆÏ = óï, & = -F (12)

and

É'( ) É8ó×ÆÏ , 8óÆÏó×ÆÉ = ó÷ï-óï -~' óÆÉ =2jTo-j8'

2 Æ Æ (13)

~=J'~-j~8Æ 8÷ 8Æ2 '

respectively. At this point, we assume that the first two sets - which are ßç viewof (11) the zero- and first-order cïçtÞbutßïçs - sufficiently represent the problem

solutions. Ôï obtain these two sets, a general problem is defined ßç the next sub-section and solved exactly ßç a multiple-transform space. Asymptotic forms are ßçlater sub-sections inverted analytically and used as the basis of Green's functionsfor the two sets for a particular choice of (ôï, óï, Ñ).

General half-plane problem: transform solotions: Consider an ideal half-planeÆ > Ï which for t ~ Ï is at rest under the same uniform temperature ÔÏ and fort > Ï is subjected to the surface loading

80ó×Æ = Ô÷, óÆ = ÔÆ, 8:; = Q. (14)

Here (ô ÷, Ô Æ, Q) are largely arbitrary functions of (÷, t). Equations (1) and (6) andthe boundedness/continuity conditions hold for Æ > Ï, and this system is attackedby means of the Laplace unilatera1 [16] and bilateral [17] transforms

GJ = [ÏÏù(t)e-stdt,ù* = 1 00 GJ(x)e-SPXdx (15.1,2)

Jo -00

, over (÷, t). Here s is real and positive, andp is ßç general complex. Application of(15.1,2) to (6) ßç view of (1) gives the complete multiple transform set

[ 0* ] [ Ù+ Ù- Ï } [ Á+ e-sa+z]- su; = -ñ -ñ â Á- e-sa_z , (16.1)

su: á+ á- Ñ !Âe-sâÆ

[ ó* ] [ 2ñá 2ñá Ô ] [ Á e-sa+z ]ó;Æ = ì Ô+ + Ô- - 2ñâ Á: e-sa-z (16.2)

ó: -Ô -Ô -2ñâ ! e-sâÆ

234 L.M. BROCK ÅÔ AL.

for Æ> Ï, where (A:J::, Â) are arbitrary functions of (s,p) and

á:J:: = fbi-="P'2, â = Jb'2~, (17.1)

Ô = b2 - 2ñ2, T:J:: = 2á1- b2,

êÙ:J:: = b2 M:J::, M:J:: = m1- 1,å (17.2)

Ì+Ì- = --, b:J:: = ám:J::,

ã

2m:J::=W+~~::I::{~-=~R, ã = áhs. (17.3)

Boundedness as Æ ~ 00 requires that Re( á:J::, â) ~ Ï ßn the p-planes cut alongIm(p) = Ï, Re(p) > (b:J::, b). Because s is real and positiíe, ßé can be shown that

m+ > 1 > m_(b+ > á > b_),

Ì+ > Ï > Ì_(Ù+ > Ï > Ù_),ã > Ï, (18.1)

m2(1 + å) - 1m+ > m(b+ > b), Ï < ã < m2(1 + å). (18.2)

Application of (15) Éï (14) ßn íiew of (1) and use of (16) and (17) then giíes theequations necessary Éï obtain (A:J::, Â) as

[ Á+ ] [ -2ñÙ_á_â ÔÙ_á- -R- ] [ bT; ]~- = ~ 2ñÙ+á+â -ÔÙ+á+ R+ bT;. (19)

â -ÔÌ -2ñÍ 2ñÔ(á- - á+) ~Q*

,Here

Ì = Ù+á+ - Ù_á_, Í = á+á_(Ù+ - Ù-), (20.1)

R = 4ñ2âÍ + ô2 Ì = Ù+á+R- - Ù_á_R+,(20.2)

R:J:: = 4Ñ2á:J::â + ô2,

and the similarity ßn form of R:J:: with the classical Rayleigh function [18] is noted.Substitution of (19) ßnÉï (16) completes the transform solution. Iníersion can

be carried ïõÉ without recourse Éï numeÞcal schemes [19] by making use of a

.

DYNAMIC THERMOELASTIC EFFECTS 235

large-time approximation. Arguments based ïç the ÔaubeÞan theorems [16] showthat this approximation is obtained by iníerting asymptotic forms íalid near s = Ï.Á glance at (16.3) shows that s often appears as a factor of the dimensionlessíaÞabÉe ã. Therefore, we base the small-s approximation ïç expansions of (16)near ã = Ï. Howeíer, the thermoelastic characteÞstßc length h of Ï(10-4)ìm andthe dilatational slowness á, generally [18] of O(10-4)sjm, giíe a characteÞstßctime há of Ï(10-8)ìs that is also a factor ïfã. Thus, a smaÉÉ-ã expansion ofthetransforms (16) does not seíerely restrict s itself, and the resulting iníersions maybe robust. We set Æ = Ï, therefore, ßç (16) and deíelop the transforms

è* = -~ + ~-Õé-Ó, Ó = ~(TT* - 2ñâô*), (21.1)b+s ê 1 + å ìR- Æ ÷

(8è)* d2è* áb2 å s2- = sñè*, - = -b+sQ* - ---Ó, (21.2)8÷ dz2 Ê 1 + å m+

!ó* = -~-Ñ* - !ô* + 2(b2 - b=-)Ó, (21.3)ì ÷ m2b+s ì Æ

1 dó* 2ê s-~ = ~ b pQ* - -âô; + 2spa_(a- - â)Ó, (21.4)

ì dz m + ì

!~ = -~~ + ~Tâ(á- - â)ô; - ~á_T;. (21.5)ì dz m+ R- ìR- ìR-

Éç (21) the expressions (17.1,2) still hold, but m:l: ßç (17.3) are replaced by theasymptotic forms

{ø+å 1m+ = -, m- = Ð-;-::' (22)ã ÕÉ+å

In the next two sub-sections, exact iníersions of (21) are giíen for useful choicesof (Ô×,ÔÆ, Q).

Green's function and zero-order surface temperature change: Consider thecase

(Ô÷, ÔÆ, Q) = (Ô÷, ÔÆ, ñ)ä(÷ - î)ä(t - ô), (23)

where (î, ô) are real and Ô> Ï, the symbols (Ô÷, ÔÆ, ñ) are now constants and ä()is the Dirac function. For(23), iníersion ïfè* ßç (21.1) proíides aGreen's functionthat for a specific choice of 10ading functions (ôï, óï, F) can be used to generatethe surface temperature changes induced for both the zero- and first-order solutioncïçtÞbutßïçs.

236 L.M. BROCK ÅÔ AL.

ÔÏ begin the process, operation ïç (24) by (15.1,2) and substimtion into (21.1)ßç view of (17.1,2) and (22) gives for Æ = Ï

è* = _Qr=~ =e-sñî-SÔy~ö

b2 å 1 -+- - -(ÔÔÆ - 2Ñâô÷)e-SÑî-sô. (24)'

ìê 1 + å R-

For the first term ßç (24), the inverse operation of (15.1,2) can be performed byinspection and use of standard transform tables [20] to yield

Ãh- ä(÷ - î)-Ñy~( -ß-+å) J7r(t=-T)' t > Ô. (25)

Tuming now to the other terms ßç (24), the inverse operation of (15.2) is [17]

ù = ~ ! ù* espx dp, (26)

where integration is along the Bromwich contour. Substimtion of the second termßç (24) into (26) gives, then, the formal operation

se-sr ! Ô---:- - eSÑ(÷-î) dp (27)2ðé R- '

where the analyticity of Ô / R- allows the Bromwich contour to be taken as theIm(p )-axis. The integrand of (27) decays exponentially as Ipl -t 00 for Re(p) >Ï,î > ÷ and Re(p) < Ï,× > î, and exhibits the branch cuts Im(p) = O,b- <É Re(p) É < b. Smdy of (20.2) ßç light of (22) indicates that R- has the zeroesÑ = ::É::Ñï, bo > b ßç this cut p-plane. The value bo is the effective thermoelasticRayleigh slowness for long times. Ôï obtain bo, the work of [14] is followed, andthe product-splitting operations ofNoble [21] used to wÞte R- ßç the form

R- = 2(b:' - b2)k+k_(p2 - bä), (28.1)

1 l b du -1 4u2áâÉç k:!:(p) = -- - tan, (28.2)

ð b- u::l:: Ñ ô2

á = ~2-~, â = J~~"ß;2-ß, (28.3) ,where k:!: are analytic everywhere except, respectively, the regions Im(p) = Ï, b- <Re(p) < b and Im(p) = Ï, -b < Re(p) < -b_, and the new definitions for (á, â)are introduced. The value of bo follows explicitly from (28) as

1 b2 1bo = - = - (29)

VR J2(~~ k+(O)'

DYNAMIC THERMOELASTIC EFFECTS 237

where VR denotes the effectiíe therrnoelastic Rayleigh speed. With (28) and (29)ßç hand, Cauchy residue theory can be used to rewrite (27) ßç the more tractableforrn

s l b- g(p) e-SÔ-SÑl÷-îÉ dp - spo e-SÔ-Sbol÷-îÉ, (30.1)ð b-

É : 4ñ2áâô2 T(bo)g(P) = 16ñ4á2â2 + ô4' Ñï = 4bo(b~ - b2)k+(bo)k_(bo)' (30.2)

The iníerse of(30.1) follows from (15.1) by inspection as

8 [ g(ò) ]- Ç(ò - b_)H(b - ò) - ñïä(ò - bo) ,8ô ðl÷ - îÉ (31)

É-ôò = j;-=-î-ú'

where Ç() is the Heaíiside function and É > ô. The iníersion of the remainingterrn ßç (24) follows along the same lines so that, ßç summary, for Æ = Ï, É > ô

è = _Qr=~~~ = ä(÷ - î) + ~ ~ ~ ~í~( -ß-+å) Õð(É - ô) ê 1 + å ì 8É

[ g(ò) ]÷ ;j;-=-î-úÇ(ò - b_)H(b - ò) -ñïä(ò - bo)

-~ ~ ~ ~ [ -!!.~H(ò - b_)ê 1 + å ì 8É ðl÷ - îÉ

--.!!!:.~H(b - ò)] , (32.1)

ðl÷-î!

4ñ3áâ2 ñâô2, gd(P) = 16ñ4á2â2+Ô4' gr(P) = 16ñ4á2â2+Ô4' (32.2)

As indicated aboíe, (32.1) can serve as a Green's function for both the zero- and, first-order contributions. We now introduce the first loading case listed at the outset

for the ha1f -plane, that of a constant heat flux applied to a surface point. Specifically,atx = Ï,Æ = ë!(÷) fort > Owehaíe

ÔÏ(×, É) = óï(÷, É) = Ï, F(x, É) = Ñä(÷), (33.1,3)

where F is now a constant. Substitution of (î, ô) for (÷, É) ßç (33.1,3) and thenreplacßçg (Ô÷, ÔÆ, ñ) with (33.1,3), respectiíely, ßç (32.1) and integrating the result

238 L.M. BROCK ÅÔ AL.

with respectto î andT overtheranges (-00,00) and (Ï, t) thengivesthezero-ordersurface temperature change

2F ~t00 = 1_{1 É_') -ä(÷)('4 = Ï, t > Ï). (34)

Õð(1+å) á

ÂÕ a similar process, inversions of (21.2,5) give the cïçtÞbutßïçs required for (13): ~

8200 á 800 1 2ê-8 2 = -h (1 +å)- 8t ' -ó÷Ï = 200,Æ ì m (35.1)

1 8ó×ÆÏ - 2ê 800ì a;:- - -~ &'

! ~ = ~hêF [ ~H(ò - b_)H(b - ò)ì 8Æ 1 + å ðl÷l

-ñïä(ò - bo)], ò = &ú . (35.2)

Here (Æ = Ï, t > Ï) and 800É8÷ follows directly from (34).The Æerï-ïrdercïçtÞbutßïç is the solution forthe purely thermalloading (33) ïç

an ideal half-plane. Equation (34), ßç particular, indicates that for large times ïçlÕthe point ofthermalloading (÷ = Ï) e×ÑeÞeçces a temperature change. The effectsof a non-planar surface ïç this change will appear ßç the first-order cïçtÞbutßïç,which is now developed.

First-order surface temperature change: Ôï obtain ÏÉ ïç the surface, (33.1,2),(34) and (35) are substituted into (13), and the vaÞabÉes (×, t) replaced by thedummy vaÞabÉes (î, Ô). The Þght-hand sides of (13) replace, respectively, theterms (Ô÷, ÔÆ, ñ) ßç (32.1), and (32.1) is then integrated with respect to î andÔ over the ranges (-00,00) and (O,t). This Green's function operation gives acombination of multiple integrations that can be reduced by integration variablechanges to a somewhat more explicit result. Specifically, when Ï < t < b-lxl,

ÏÉ =.jiiF [É(×)ä(×) - ~f'(×)ä'(×)] . (36.1)'

For × > Ï, b_x < t < bx the term

h {;;t;;;á 3F Ï 8 l t/x U9ddu åb4áh F-2åvh ÷- +-ð(1 + å) É() 8t b- Jt-=-uX (1 + å)2 ð

.

DYNAMIC THERMOELASTIC EFFECTS 239

8 l t/x [ 1 (t - bOX) 1 (t + bOX)]÷ -Ñï- -É +-1 gdu

8t b- u - bo u - bo u + bo u + bo

Ñï /L2 , L2 8 ft/x [ 1 (UX - t

)+LY~+b58tJb- ~+;rl ~1 (t - UX

)]+ É 9 du(bO-u)2 bo-u

1 8 l t/x l b [ 1 (t - õ÷)-- - gdu -É

, ð 8t b- é/÷ t - õ÷ u - õ

1 É(t-UX )] gdv+ ~ -;;-::--;; ~

1 8 l b l t/x [ 1 (õ÷ - t)-- - gdu -É

ð at b- b- õ÷ - t u + õ

+~ É(~ )] ~ (36.2)

t+ux u+v u+v

must be added to (36.1), while for × > Ï, bx < t < box this term takes the form

-2åJh/~~ =3FI(0)x~ ( ft/x ugdduV ð(1 + å) 8t Jb- Jt-=-uX"

- ft/x ugr dU )+ -~~ ~Jb Jt-=-uX" (1 + å)2 ð

l b [ 1 (t - bOX)×-ñÙ !'b- (u-bO)2 u-bo

1 É,(t + bOX)] d+ 9 U(u+bO)2 u+bo

, Ñï ~b2 fb [ É!, (t - UX )+LÕu=ï + uoJb- (bo - u)3 ~1 !' (UX - t

)] d -(bO+u)3 ~ 9 u

l 1b l b 8 [ 1 (t - õ÷)-- gdu --Éð b- b- 8t t - õ÷ u - õ

240 Gr:)Er'r;1:; L.M. BROCK ÅÔ AL.

.1 1(t - UX )] gdv+~ ~ ~

1l b l b Ï [ 1 (vx - t)-- gdu --1ð b- b- ot õ÷ - t u + v

+~ 1(~ )] ~. (36.3)t+ux u+v u+v -:

For ÷ > Ï, t > box, ßç addition to (36.3), the term

åb4áh 2 /L2 é L2[ ( ) ( )]' ;.- (1 )2Fb_ÑïÕb=ï +Ïß)u+1 u+ +u-1 u- ,+ å (36.4)

box :i: tU:i: = -~

must be added to (36.1). For ÷ < Ï the terms (36.2,4) arise again, with ÷ replaced byIxl. Equations (36) indicate that the first-order cïçtÞbutßïç to surface temperaturechange has waíe-like behaíior for large times. For example, the gd- and gr-termsßç (36.2,3) represent, respectiíely, effectiíe dilatational waíes of speed Vd~and rotational waíes, while (36.4) represents thermoelastic Rayleigh waíes. Thisbehaíior is ßç contrast to the zero-order cïçtÞbutßïç (34).

More importantly, (36) indicates that the non-planar surface exhibits a large-time temperature change eíerywhere, not just at the disturbance (loading) ñïßçéThis change depends ïç the contour profile function 1 and its slope 1'. For insightinto this non-planareffectthat is independentof (>., Ñ), we compare íalues attainedby èé for ÷ > Ï with íalues attained by èï at ÷ = Ï by plotting from (34) and (36)the dimensionless ratio

× = É èé (×, Ï, t)É (37)

èï(Ï, Ï, t)

vs t for two íalues of × > Ï ßç Figure 2. The thermoelastic region is taken to besimilar to a stainless steel [12] with properties

å = 0.00712, h = 1.67(10-4)ìm,(38.1)

ì = 73.1 GPa, ÔÏ = 294 Ê,"

Vd = 5735 mjs, Vr = 3065 mjs,(38.2)

VR = 2824mjs, há = 2.91(10-8)ìs,

while the surface is giíen the ÑeÞïdßc contour

Ð×1(÷) = cos ym, L = 1 mm. (39)

.

OYNAMIC THERMOELASTIC EFFECTS 241

(106)

120

100 x=O.lmm

.é /80 f (÷) = cos -ôï (m)

, ÷ L=lmm60

40

20

Ï 0.1 0.2 0.3 0.4 0.5

t(ìS)Figure 2. Á ratio of first-order to zero-order surface temperature change VÂ time.

Figure 2 indicates that, depending ïç the specific (small) va1ue of the surfaceamplitude ë, the magnitudes of the surface temperature change due to non-planaritymay actually be larger than that produced ïç the ideal half-plane. This effect does,however, appear to decrease with time.

It should also be noted that, while small, the time sca1e shown ßç Figure 2is orders of magnitude larger than the thermoelastic characteÞstßc time há givenßç (38.2). Moreover, extensive additional calculations indicate that the first term. ßç (36.1) and the 9d, 9r-terms ßç (36.2,3) dominate 01. This may becatlse the

remaining terms are of O(h) and Ï( Vh) sma11er, respectively. This observationwill be employed as the problem of a purely thermalload over an expanding regionofthe half-plane surface is now treated.

3. Plane-Strain Half-Plane Problem: Surface Load over Expanding Strip

Consider again the situation depicted ßç Figure 1, except that, ßç place of (33.1,3),a purely thermalload acts uniform1y over a stÞñ that expands uniform1y from the

242 L.M.BROCKETAL.

point ÷ = ï, Æ = .ëé(÷):

ÔÏ(×, t) = óï(÷, t) = Ï, F(x, t) = F Ç (t - ~)Ç(÷),V (40.1,3)

V <VR.

Here (Ñ, õ) areconstants, wherethe subcÞtßcaÉSÑeedvßsthesurface stÞÑe×Ñansßïç ~

rate. Áð aSÕmmetÞcaÉ expansion is chosen, to offer another contrast with the firstcase treated. In íiew of (4) and (10) the difference between the length of thestÞñ and its projection onto the x-axis is negligible. Substitution of (î, ô) for i(×, t) ßç (40) and then replacing (Ô÷, ÔÆ, Q), respectiíely, with (40.1,3) ßç (32) andperforming the same Green's function integrations used before giíes ßç place of(34) the zero-order surface temperature change

2Ñ ~ 180 = 1-/1 é ~\) -(é - cx)(z = Ï,÷ > O,t > cx), c = -. (41)vð\l+å} á V

Similarly, the components required for (13) can be obtained from differentiationof (41), the formal operations (35) and the result

! ~ = ~hêF [ 4g(ò) Ç(ò-b_)Ç(b-ò)ì 8Æ 1 + å ðl÷l (c - ~)

ñïä(ò - bo) T(c) ~ ] t

-c-bïsgç(÷)+jR~äti-cx)Ç(×) , (=j;i' (42)

With (35), (42) and (42) aíailable, the same Green's function process based ïç (32)and (13) can again be used to giíe an expression for ÏÉ ïç the surface ofthe region.In light of the behaíior of (36), ïçlÕ dominant terms are presented here. Thus, for÷ > ï, Ï < t < b_x

. ~ 3 á 8 Ud+ b_x + tÏÉ =4fVh ;ð+-;) FmLd- j(u)Iddu, Ud:i: = c:!:b_' (43)

where

l wc w2gd dw l wC w2gr dwId = , Ir = ,

b yt - cu - wlx - ul b yt - cu - wlx - Ul- (44)t- cuWc = 1 É,x-u

.

DYNAMIC THERMOELASTIC EFFECTS 243

while for ÷ > Ï, b_x < t < bx~ 3 á 8 Ud+ÏÉ = 4EVh ;Ôú+å) Ñ8ú [~ j(u)Iddu

(Ur+ ] bx ::É:: t:. - JUr- F(u)Ir du, Ur:f: = --;;Ð' (45.1)

and for ÷ > Ï, bx < t < cx

. ~ 3 á 8 Ud+ Ur+

ÏÉ = 4EVh ;ð+å) Ñ8ú [~ j(U)Iddu - ~ j(u)Ir dU] (45.2)

For ÷ > Ï, t > cx the term

Fj(x) (45.3)

must be added to (45.2) and finally, for ÷ < Ï, t > Ï ïçlÕ (45.2) goíems.Equations (41) and (43)-(45) show that the zero-order cïçtÞbutßïç for large

times alters the surface temperature ïçlÕ ïç the expanding 10ading stÞñ, while thefirst-order cïçtÞbutßïç that represents çïçÑlanaÞtÕ alters the surface temperatureelsewhere as well. Moreoíer, (45.3) and the Ï( Jh)-behaíior of (41) indicates that,eíen ïç the 10ading stÞñ, the effect of nonplanmty may be pronounced, dependingof course ïç the íalue of >.. and the time t.

Ôï complete this study, the related 3D half-space problem is now examined.

4. 3D Half-Space Problem

Formulation: Considerthe half-space shown ßç Figure 3. Éç terms ofthe Cartesiancoordinates ÷ = (×, Õ, Æ), it is defined as Æ > >..j (×, Õ), where the surface amplitude>.. > Ï is dimensionless and the surface contour function j(x, Õ) is single-ía1ued,lies in C2 and has a magnitude that nowhere exceeds one unit of length. For time

.. t ~ Ï the half-space is at rest under a uniform temperature Ôï(Ê). That is, fort ~ Ï, Æ > >"j(x, Õ) condition (1) again holds, except that now u = u(x, é) =(ux,Uy,uz) and 0= O(x,t). For t > Ï a normal (n) heat flux is imposed ïç the

. half-space surface. As shown in Figure 3, the normal direction has the unit íector

1 ( 8! 8! )n= /1 é I\T'7l'I2 >"-8 ,>"-8 ,-1 . (46)yl+I>..VjI2 ÷ Õ

Thus, for t > Ï, Æ = >..j (×, Õ) the boundary conditions are

n. ó = Ï, n. VO = Ñ, (47)

244 L.M. BROCK ÅÔ AL.

Æ '

Figure 3. Schematic of nearly-planar half-space.

where now (ó, V) are the 3Ï stress tensor and gradient operator, and the heatftux F is a continuous function of (×, Õ, t) and remains finite for finite t > Ï forall (×, Õ). Éç view of (46) the boundary conditions (47) can be wÞtteç ßç terms ofx-based quantities as

8! 8!.ëa;ó÷ + .ëèÕó×Õ - ó×Æ = Ï, (48.1)

8! 8!.ëa;ó×Õ + .ëèÕóÕ - óÕÆ = Ï, (48.2)

8! 8!.ëa;ó×Æ + .ëèÕóÕÆ - óÆ = Ï, (48.3)

.ë~ ~ + ~ ~ - ~ = FJú+ßëV!r. (48.4)8÷ 8÷ 8Õ 8Õ 8Æ

For t > Ï, Æ > .ëé(÷, Õ) the governing equations are again (6), where now thedilatation Ä = Ä(÷, t) and \72 is the full 3Ï Laplacian operator. Additionalrequirements that (u, è) be continuous ßç (÷, t) and finite forall (×, Õ, Æ > .ëé(÷, Õ)) "

for finite t > Ï are imposed.Ôï solve the system, the standard [11] perturbation scheme used previously is

applied under the assumption that .ë is small enough that (u, è) evaluated alongÆ = .ëé(÷, Õ) can be wÞtteç as a Taylor seÞes expansion about Æ = Ï ofthe form

(9) and that

I.ëVjl«1. (49)

.

DYNAMIC THERMOELASTIC EFFECTS 245

Equation (49) guarantees, cf.( 4), that the difference between half -space surface areaand its projection onto the ×-Õ plane is negligible. Then, the seÞes (11) for (õ, è)are introduced, the square-root radical is expanded as a power seÞes ßç É ë V 1 É, anda problem statement for each set (õß, èß)(i = 1,2,... ,) arises. The statement foreach set includes (1), (6) and the boundedness/continuity requirements, while forÆ = Ï, t > Ï the boundary conditions for the first two sets are

~ :1. óèïó×ÆÏ = óÕÆÏ = óÆï = ï, & = -F (50)

;)and

óé óé óó ×ÆÏó×ÆÉ = a;ó÷ï + ÂÕó:É:ÕÏ - laz'

óé óé óóÕÆÏóÕÆÉ = ÂÕóÕÏ + a;ó:É:ÕÏ - 1 az'óó ÆÏ

óÆÉ = - 1a;-'

~ = ~~+~~-1~, (51óÆ ó÷ ó÷ óÕ óÕ óÆ2 )

respectiíely. Again we assume that the first two-sets - the zero- and first-ordercïntÞbutßïns - are sufficient to represent the solution. As before, a Green's functionapproach is used to find the two cïnéÞbutßïns, and, as the first step, a general half-space problem is now defined and solíed by integral transform methods.

General half-space problem: transform solution: Consider an ideal half-spaceÆ > Ï which for t ~ Ï is at rest under the same uniform temperature ÔÏ and for

t > Ï is subjected to the surface loading

~ óèó:É:Æ = Ô÷, óÕÆ = ÔÕ' óÆ = ÔÆ, & = Q, (52)

"where (Ô ÷, ÔÕ' ÔÆ, Q) are largely arbitrary functions of (×, Õ, é). Equations (1) and(6) and the boundedness/continuity conditions hold for t > Ï, Æ > Ï, and thissystem is treated by the unilateral Laplace transform (15.1) and the multiple bilateral

Laplace transform

ù* = f é: ù(÷, Õ) e-s(px+qy) dx dy. (53)

246 L.M. BROCK ÅÔ AL.

Here S is real and positiíe and both (Ñ, q) are generally complex. Application of(15.1) and (53) ßç light of (1) and (6) giíes the complete multiple transform set

è* [ Ù+ Ù- Ï Ï ] Á+ e-sá+Æ *1 Ï Á -sá-ÆSUx -ñ -ñ - e

su~ = -q -q Ï 1 !Â÷ e-sâÆ , (54.1) ~

suz á+ á- Ñ q ~By e-sâÆ

*ó×Õ -2pq -2pq qâ ñâ r

* 2 2 '71 Á -sá+Æó×Æ ñá+ ñá- -.Lq pq + e* 2 2 '71 Á -sá-ÆóÕÆ qa+ qa- pq -.Lp - e

- ì (542)ó* - ç1 ç1 2 â Ï ~B e-sâÆ .÷ .Lq+ .Lq- Ñ â ÷

ó; Ôñ+ Ôñ- Ï 2qâ ~By e-sâÆ

ó; -Ô -Ô -2ñâ -2qâ

for Æ > Ï, where (A:i:, Â÷, ÂÕ) are arbitrary functions of (Ñ, q, s) and (17.2,3) and(18) still hold. Equation (17.1), howeíer, must modified to giíe

a:i: = Jbi - ñ2 - q2, â = Vb2 - ñ2 - q2, (55.1)

Ô = b2 - 2(Ñ2 + q2), T:i: = 2ai - b2,

Ôñ=Ô+ñ2, Tq=T+q2, Tp:i:=T:i:+2p2,2 (55.2)

Tq:i: = T:i: + 2q .

Application of(15.1) and (53) to (52) ßç light of(I), (17.1,2), (18), (54) and (55)then giíe

Á+ 1 -2ÑÙ_á_â -2qÙ_á_â Á- 2ñÙ+á+â 2qÙ+á+â ~~

sâÂ÷ - -4q2âÍ - ÔÔñÌ Ñq(4âÍ - ÔÌ)

sâÂÕ Ñq(4âÍ - ÔÌ) -4ñ2âÍ - TTqM .1.;

ÔÙ R ] 17*-á- - - ì ÷

-ÔÙ+á+ R+ t7;2 2 (56)-2ñâ Í 2ñÔâ (á- - á+) 17*

ì Æ

-2qâ2Í 2qTâ2(á- - á+) ~Q*

.

DYNAMIC THERMOELASTIC EFFECTS 247

Here (20.1,3) still hold, but (20.4) is replaced by

R:I: = 4(Ñ2 + q2)á:l:â + ô2. (57)

With the transform problem thus solíed, numeÞcaÉ iníersion is again aíoidedby making use of the same robust large-time approximation employed earlier.Therefore, by analogy with (21) the following asymptotic results for Æ = Ï when

It '¾ is small are presented:

0* Q* b2 å Ó Ó é ( * â * â *)= --b + - -1 ' = - R ÔÔÆ - 2ñ Ô÷ - 2q ÔÕ , (58.1)~ +s ê + å ì -

(80 80)* d20* áb2 å s2-8 ' _ï = (p,q)sO*, -d 2 = -b+sQ* - - -

1 -Ó, (58.2)÷ Õ Æ Ê + å m+

!ó* = --~Ñ* - !Ô* + 2(b2 - b2)Eì ÷ m2b+s ì Æ -

2pq2 (ô ) * 2q [ 2 ( Ô)]+ ì:Ç=- â - 4á- Ô ÷ + ì":Ç-=- âÔ + ñ 4á- - â

* 4q2 â *÷ÔÕ + - R á- ÔÆ' (58.3)ì -

!ó* = --~Ñ* - !ô* + 2(b2 - b2)Eì Õ m2b+s ì Æ -

2ñ [ 2 ( Ô)] * 2qp2 (ô )+ì":Ç-=- âÔ + q 4á- - â Ô÷ + ì":Ç-=- â - 4á-

* 4ñ2 â * 5÷ÔÕ + - R á- ÔÆ' ( 8.4)ì -

. 2 21 * 4á ê Q* b [(2 2 b2) * ( 2 b2) *]-ó×Õ = -- b2 pqs + âR q - qTx + 2ñ - ÑÔÕì + ì-

É " * +k(T - 2á_â)[Ñqô; + (Ñ2 - q2)(PT; - qT;)], (58.5)

1 dó×Æ 2ê Q* Sâ * 2 ( â)Ó 586- - d = Æ- b Ñ - - Ô÷ + spa- á- -, ( .)ì Æ m + ì

1 dó;Æ 2ê * s * )- -d = Æ- b qQ - -âÔÕ + 2sqa_(a- - â Ó, (58.7)ì Æ m + ì

248 L.M. BROCKET AL.

1 dó* b2,.. TQ* 2Ââ_-!. = ---+-Ô(á_-â)(Ñô*+qô*)ì dz m~ R- ìR- ÷ Õ

b4s-- R á-ô;, (58.8)ì -

where again the asymptotic forms (23) govern. The exact transform inversion of(58) required to complete the solution process are now presented. ~

Green's function, zero- and first-order surface temperature changes: TheGreen ' s function surface loading ~

(Tx,Ty,Tz,Q) = (Ô×'ÔÕ'ÔÆ,Q)ä(÷-î)ä(Õ-ç)ä(t-ô) (59)

is introduced, where (Tx,Ty,Tz,Q) are now constants and (î,ç,ô) are real, withô > Ï. Operating ïç (59) with (15.1) and (53) and substitution into (53.1) gives forÆ = Ï the surface temperature change transform

è* = -QC~ = e-SÔ-S(Ñî+qç) + ~ :.-- -.!.-y~ø ìê l+å R-

÷ (Ôô; - 2ñâô; - 2qâô;) e-SÔ-S(Ñî+qç). (60)

The use of [20] allows the first term ßç (60) to be inverted by inspection, cf. (25):

j-h- ä(÷ - î)ä(Õ - ç)-Ñy~( ú+-å) V1r(t-=-T) , t> Ô. (61)

Turning now to the other terms ßç (60), the inverse operation of (53) is [17]

ù= (~)21Iù*es(Ñ÷+qÕ)dÑdq, (62)

where integration ßç the complex p-q plane occurs along Bromwich contours. ':)

Substitution of the second term ßç (60) into (62) gives the formal result

(~ )2 e-STÉÉ..É-- eSÑ(÷-î)+sq(Õ-ç) dpdq. (63) {.

2ð-é R-

The rotation with Jacobian ofunity

ññ -+ -ñ(÷ - î) - q(y - ç), qp -+ -Ñ(Õ - ç) + q(x - î),

!é- ,,\2 é é.. ~\2 (64)Ñ = V (÷ - î)2 + (Õ- ç)2

.

DYNAMIC THERMOELASTIC EFFECTS 249

allows (63) to be wÞtteç as

(~ )2J J ..!-.- e-SPPdpdq, (65)

2ð? R-

where the property ñ2 + q2 -7 ñ2 + q2 means that (á_,â,Ô,R_) preserve theforms given ßç (55.1) and (57) ßç terms of the new set (Ñ, q) defined by (64). The

.(

Bromwich contours can be taken along the Im(p)- and Im( q)-axes. Éç particular, theintegrand of (65) is an even function of q, so that introducing the dummy variablechange q = ßõ, where v is real, allows (65) to be rewÞtteç as

;J

82 100 1 J ô-e-ST dv---: -e-sPPdp. (66)ð ï 2ð? R-

The p-integrand decays exponentially as Ipl -7 00, Re(p) > Ï, and exhibits thebranch cuts Im(p) = Ï, Â- < IRe(p) É < Â, where

Â- = #-~, Â = Jb2+-;;2. (67)

Study of (57) ßç view of (64) and (22) shows that R- has the zeroes Ñ = :É::Âï, Âï >Â ßç this cut p-plane. Despite the appearance of the real vaÞabÉe v, a processanalogous to that employed for (27) can again be used, with the result that, ßç placeof (28) and (29), one can wÞte

R- = 2(b=- - b2)K+K_(p2 - Âä), (68.1)

l 1b u du -14u2áïâïlnK:i:(P) = -- 12 '-2 1-2' --2 tan '712' (68.2)

ð b- v u- + v- v u- + v- :É:: Ñ .L Ï

áï = ~Æ-=~, âï = J~=-ú;2ú, ôÏ = b2 - 2u2 (68.3)

.. and

- Âï = É Ro( 2 ~ 2 )~(o) , Ro = (b2 + 2v2f - 4v2 Â-Â ~ ï. (69)

.é V 2(b2 - b:') +

Here K:i: are analytic everywhere except, respectively, the regions Im(p) =Ï, Â- < Re(p) < Â and Im(p) = Ï, -Â < Re(p) < -Â-. Cauchy theorycan now be used to rewÞte the p-integration ßç (66) ßç the more tractable form

! {Â G(p)e-SPPdp-Poe-sBop, (70.1)ð JB-

250 L.M. BROCK ÅÔ AL.

4(Ñ2 - õ2)áâô2G(p) = 16(Ñ2 - õ2)2á2â2 + ô4'

(70.2)Ñï = Ô(Âï) ,

4Bo(~ - b2)K+(Bo)K_(Bo)

where now

á = VP2 - v2 - b:', â = Vlp2 - v2 - b21, (71):

Ô = b2 - 2(Ñ2 - v2).The inversion of (66) follows by inspection from (15.1) as ~

1 82 100[ G(ò) ]- 2 dv -Ç(ò - Â_)Ç(Â - ò) - Ñïä(ò - Âï) ,

ð 8t ï ðñ (72)É-ôò--- ,

Ñ

where t > ô. Inversion of the remaining terms ßç (60) follows the same line, sothat ßç summary, cf. (32)

è = - I~=¸== ä(÷ - î)ä(Õ - ç) + ~ ~ ~ ~ÑY~å) Õð(t-ô) ê 1+å ð 8t2

(ÏÏ [ G(ò) ]÷ 10 dv -;ñÇ(ò - Â_)Ç( - ò) - Ñïä(ò - Âï)

+2 [(~) ~ + (7)~] koo dv

÷ [ ~H(ò - Â-) - ~H(ò - Â)] , (73.1)

ðñ ðñ

4ñ(Ñ2 - õ2)áâ2Gd(P) = 16(Ñ2 - õ2)2á2â2 + ô4' .,

(73.2) .

ñâô2Gr(P) = 16(Ñ2 - õ2)2á2â2 + ô4~

for Æ = Ï, where t > ô.ÂÕ analogy with (32.1), (73.1) can serve as the Green's function for both the

zero- and first-order cïçtÞbutßïçs for the present half -space problem. As a specificexample of purely thermalloading, we treat a uniform heat flux applied for t > Ïoverthe fixed surface area Æ = >"!(×,Õ), Rw:%:; Ôï:

F(x, Õ, t) = Ñ, Vx2-+;2 :%:; ôï, t > Ï. (74)

.

DYNAMIC THERMOELASTIC EFFECTS 251

Here (F, ôï) are constants and, ßç view of (49), the difference between the areaand its true circle projection onto the plane Æ = Ï is negligible. Substitution of(î,ç,ô) for (x,y,t) ßç (74), setting Q = F and (Ô×'ÔÕ,ÔÆ) = Ï ßç (73.1), andintegrating the result with respect to (î, ç) and ô over the ranges (-00,00) and(Ï, é), respectively, gives the zero-order surface temperature change, cf. (34):

2F ~t.' èï = 1-(1 é ~\) -H(ro - ô),

ãð\l+å) á (75)

r = Vx2-+-;2 (Æ = Ï, t > Ï).~

ÂÕ a process similar to this, inversions of (58) for the case (74) give the termsnecessary for the right-hand sides of (51):

ó2èï á óèï-a:;2" = h(1 + å)7it' (76.1)

1 1 8 á ê{;i{;;; .,fat 3

-ó÷ï = -óÕÏ = -3b 2 F (1 ) -~H(To - ô), (76.2)ì ì ð + å V r5 - r2

ó ) (óó÷ï óóÕÏ)a;(ó×Æï,óÕÆÏ = - &' 8Õ ' (76.3)

1 8áhêF 100 100 u2 + 2v2-ó÷Ï= dvduì Õ ð2(1 + å) ï ï u2yu2 + v2

÷ [gdH(u - b_) - grH(u - b)]

ôí u2 + v2 - t÷

vu2r5 - (t - rvuÆ-+-û2r

÷H(uro - t + T~2+-;;2), (76.4)

~ .!. ~ = ~~~ ~ (ÏÏ dv

ì óÆ ð(1 + å) at Jo

~ 1 {b du 9 ry u2 + v2 - t

÷-; Jb- VU2+V2~ vu2r5 - (t - rvuÆ-+-û2)2

-1 ÷H(uro - t + T~2+-;;2)1ç Ñï Bor - t

-2 H(boro - Bor), (76.5)bo vbBr5 - (t - Bor)2

252 L.M. BROCKET AL.

where (8èï/ 8÷, 8èï/ 8Õ) follow directly from (75), (g, gd, gr) are defined by (30.2)and (32.2), and bo is the thermoelasticRayleigh slowness defined by (29).

Ôï obtain èé ïç the surface, (75) and (76) are substituted into (51) and thevariables (÷, Õ, t) replaced by the dummy vaÞabÉes (î, ç, ô). The Þght-hand sidesof (51.1,4) replace. respectively, the terms (rx, ry, rz, ñ) ßç (73.1), and the result isthen integratedwithrespectto (î, ç) and rovertheranges (-00,00) and (Ï, é). ThisGreen's function operation can be, with convenient integration variable changes,put ßç a more tractable form. Again drawing ïç the e×ÑeÞeçce gained with (36), :-

ïçlÕ dominant terms are kept, so that for Æ = Ï, cf. (45),

4Ñ hJt; dj(x, Õ) ""èé = (1 ) - d ä(rï-r)+Fj(×,Õ)Ç(ôï-ô)

ð +å á ô

-~Vhr=~ =3F !! j(u,v)dudvð V ð(1 + å) íé-ñ3 vôä - u2 - v2

{é/ ñ dD (é/ ñ dD÷ Jb- wgdd; dwH(t -b_p) - Jb wgrd; dwH(t - bp), (77)

where (u, v)-integration is over the circle yIU2 + v2 < ro,

t wF(f) ~- pwD = ~~+P1VE(f) - /.J -é- ~..' Ã = , (78)

wp yt+pw t+pw

and (Ñ, Å) are complete elliptic integrals ofthe first and second kinds of modulus

Ã.The first term ßç (77) is, like èï ßç (75), established instantaneously over the

surface loading area âW ~ Ôï. The second term ßç (77) represents wave-likedisturbancesthatextendbeyondthis area, and is of Ï( Vh). Therefore, the effectofnon-planarity is, as ßç the previous examples, the generation of surface temperaturechanges not seen for large times ïç the ideal (plane) surface and, depending ïçthe surface contour amplitude ë, the production of - at least temporarily - surface

temperature changes of larger magnitude than those generated ßç the loading area )by the zero-order (ideal) solution.

5. Some Observations ..

This analysis considered the effect of nonplanarity ïç the dynamic surface tem-perature changes induced for plane-strain and 3D problems ïç the nearly-planarsurfaces of, respectively, coupled thermoelastic half-planes and half-spaces by sur-face heat fluxes. The analysis used the nearly-planar nature of the surfaces to presentthe surface temperature change as an expansion ßç a dimensionless surface contouramplitude parameter. The first term ßç the expansion was the temperature change

.

DYNAMIC THERMOELASTIC EFFECTS 253

that would aÞse ïç an ideal (planar) surface, while the second term represented thefirst-order cïçéÞbutßïç due Éï çïçÑlanaÞtÕ. The two terms were compared for ahalf-plane 10aded either by a ñïßçé source (line source ßç the out-of-plane direction)heat flux or a heat flux imposed over an aSÕmmetÞcaÉÉÕ-e×Ñandßçg surface stÞñ,and for a half-space 10aded by a heat flux imposed uniform1y over a fixed finitearea. This compmson showed that non-planarity gives Þse Éï surface temperaturechanges ßç areas beyond those predicted for large times by an ideal surface analysis,

..; and that the changes associated with nonplanmty may ßç some instances be more

significant than those for the ideal surface..- The solution forms were either analytic, or integrals of analytic, expressions.

,) Despite their large-time nature, these forms were robust because the characteÞstßc

thermoelastic time used for scaling was of Ï(10-8)ìs. The contour functions forthe surfaces were general - they had ïçlÕ Éï lie ßç C2 - and have amplitudes that

were small enough Éï render the differences between surface lengths/areas andtheir planar projections negligible, and Éï allow surface quantities Éï be wÞtteç asTaylor series expansions about the projection plane.

The results obtained under these restÞctßïçs suggest that, for thermalloads atleast, the role of even weakly non-planar surface contours may be important ßç thedynamic surface temperature fields. Future work along these lines will considersurfaces that are subjected Éï mechanicalloads.

References

1. J. Cole and J. Huth, Stresses produced ßç a half-plane by moving loads. ASME }ournal ÏÉ AppliedMechanics 25 (1958) 433-436.

2. G. Eason, The stresses produced ßç a semi-infinite solid by a moving surface force. /nternátionál}ournal ÏÉ Engineering Science 2 (1965) 581-609.

3. D.L. Lansing, The displacements ßç an elastic ha1f-space due Éï a moving concentrated normalload. NASA TR R-238, Langley Research Center (1966).

4. Õ.Á. ChuÞÉ0V, óç the effect of a normalload moving at a constant velocity along the boundaryof an elastic half-space. Prikladnayá Mátemátika ß Mekhanika 41 (1977) 125-132.

5. J.R. Barber, Thermoelastic displacements and stresses due Éï a heat source moving over thesurface of a half-plane. ASME }ournal ÏÉ Applied Mechanics 51 (1984) 636-640.

6. M.D. Bryant, Thermoelastic solutions for thermal dßstÞbutiïçs moving over half space surfacesand application Éï the moving heat source. ASME }ournal ÏÉ Applied Mechanics 55 (1988) 87-92.

~ 7. L.M. Brock and H.G. Georgiadis, Steady-state motion of a line mechanicaVheat source over

a half-space, University of Kentucky College of ÅçgßçeeÞçg. August 1995, Technica1 Report(1995).

'; 8. Lord Rayleigh, óç the dynarnical theory of gratings. Proceedings olthe Royál Society ÏÉ London.. Á79 (1907) 399-416.

9. J.W. Dunkin and A.C. ÅÞçgeç, The reflection of elastic waves from a wavy boundary of ahalf-space, ßç Proceedings 4th U.S. Nátional Congress on Applied Mechanics. UniversityofCalifomia, Berkeley (1962).

10. L.M. Brock, Wave equation Green's function approximations for nearly-planar half-planes andha1f-spaces. /nternational }ournal ÏÉ Engineering Science 21 (1983) 649-662.

11. G.F. CÌÞer and C.E. Pearson, Pártiál Differentiál Equations. Academic Press, New York (1988).12. L.M. Brock, Transient thermal effects ßç edge dislocation generation near a crack edge. /nterna-

tional }ournal ÏÉ Solids áÌ Structures 29 (1992) 2217-2234.

254 L.M. BROCKET AL.

13. Ñ. Chadwick, Thermoelasticity: the dynamical theory. Éç É.Í. Sneddon and R. ÇßÉÉ (eds), Progressin Solid Mechanics, íïé. 1. North-Holland, Amsterdam (1960).

14. L.M. Brock, Transient three-dimensiona1 Rayleigh and Stoneley signal effects ßç thermoelasticsolids. 1ntemationál Joumál of Solids ánd Structures (Éï appear).

15. H.S. Carslaw and J.C. Jaeger, Conduction ï! Heát in Solids. Oxford University Press, London(1959).

16. É.Í. Sneddon, The Use of 1ntegrál Tránsforms. McGraw-Hill, New York (1972).17. Â. van der Ñïl and Ç. Bremmer, Operátionál Cálculus Based on the Two-Sided Lapláce /ntegrál.

Cambridge University Press, Cambridge (1950).18. J.O. Achenbach, Wáve Propágátion in Elastic Solids. North-Holland/American Elsevier, Ams- ~

terdam (1973). -:<:l::19. O.G. Ouffy, Ïç thenumerica1 inversion ofLaplace transforms: comparison ofthreenew methods '

ïç characteristic problems from applications. ACM Tránsáctions on Máthemáticál So.ftwáre 19 \1(1993) 333-359. ~

20. Ì. Abramowitz and É.Á. Stegun, Handbook ofMáthemáticál Functions. Oover, New York (1970).21. Â. Noble, Methods Básed on the Wiener-HopfTechnique. Pergamon, New York (1958).

,()

""~

1

.