Camp"f." '" 5''''<11I'''. Vol. I. pp. 603-621. Pergamon Press 1911. Printed in Great Britain

ANALYSIS OF NONLINEAR, DYNAMIC COUPLEDTHERMOVISCOELASTICITY PROBLEMS BY THE

FINITE ELEMENT METHOD

J. T. OOENt and W. H. ARMSTRONG+

The University of Alabama, Huntsville, Alabama, U.S.A.

Abstract-This investigation deals with the numerical solution of a class of nonlinear problem in transient,coupled, thermoviscoclasticity. Equations of motion and heat conduction are derived for finite clementsof thermomechanically simple materials and these are adapted to special classes of thermorheologicallysimple materials. The analysis involves the solution of large systems of nonlinear integrodilTercntial equa-tions in the nodal displacements and temperatures and their histories. As a representative example, thegeneral equations are applied to the problem oftransiem response of a thick-walled hollow cylinder subjectedto limc-varying inlernal and external pressures, temperatures, and heat fluxes. Thc integration schemeused to solve the nonlinear equations employs a linear acceleration assumption, rcpresenlation of nonlinearintegral terms by Simpson's rule, and the iterative solution of large systems of nonlinear algebraic equationsat each reduecd time step by the Newton-Raphson method. Various numerical results are given and arecompared with the linearized, isothermal. and quasi-static solutions.

1. INTRODUCTIONTm interconvcrtibility of mechanical work and heat was recognized long ago byThompson (Count Rumford, 1797) and their equivalence was experimentally verified byJoule and Kelvin in the nineteenth century. Indeed, it is common experience that mechanicalworking of a solid body generates heat and that, conversely, heating a body producesmechanical work. It is also common experience that the thermomechanical properties ofmany materials may vary with time, temperature, and the rate at which they arc deformed.Plastics, polymers. biological tissues, and most metals. for example. may continue todeform under constant applied load. and the rate of continued deformation may be highlysensitive to changes in temperature. The response of such materials is dependent upon thehistory of the local motion as well as the history of the temperature; i.e .. the materialpossesses a 'memory' of its past response.

While such thermomechanical phenomena have been widely observed for centuries.general continuum theories aimed at describing them have emerged only recently in theworks of Coleman [I] and others. Owing to the complexity of the general theories. theyhave provided largely qualitative information on the thermomechanical response ofmaterials with memory. In fact, the use of such theories in the experimental characterizationof actual materials constitutes, in itself, a difficult problem requiring much additionalstudy; and applications of these theories to the quantitative solution of nonlinear boundary-and initial-value problems of practical importance do not appear to be available.

This paper is concerned with the formulation and numerical solution of a class ofnonlinear problems in dynamic. coupled. thermoviscoelasticity. General finite-element

t Professor and Chairman, Enginecring Mechanics, UAH.t Engineer, Bocing Huntsville. and Graduate Student, UAH.

603

604 J. T. ODEN and W. H. ARMSTRONG

modcls are derived for the equations of motion and heat condition of typical elements ofdissipative materials. These include thcrmomechanical coupling and. for thc classes ofmaterials considered. assume the form of systems of nonlinear integrodifferentiaI equationsin nodal displacements and temperatures and their histories. In the case of the coupledheat conduction equation for an element, it is shown that forms for the discrete modeldiffer significantly from that- of the classical theory and cannot be obtained by simplyadding a dissipation integral as is often done.

Applications of the finite-element method to problems in linear. isothermal, visco-elasticity have been presented [2-4] and solutions to linear heat conduction problemswere given [5, 6]. Refercnces [7-10] considered applications of the method to problemsin coupled thermoelasticity, and the formulation of gencral discrete models of thermo-mechanical behavior was discussed [II]. More rccently. Taylor et al. {l2] described afinite-element analysis of a class of thermorheologically simple materials in which a non-linear temperature term was included. In their analysis, solutions of the uncoupled, linearheat conduction equation were used as input to the equations dcscribing quasi-staticmotion of a finite elcment. Cost [13] considered thcrmomechanical coupling phenomenain thermorheologically simple materials and prcsented numerical solutions to representativelinear non-isothermal problems.

In the present study, the general equations governing the finite element model areused to obtain special forms of the equations for a class of thermorheologically simplematerials. These are applied to the problem of transient response of a thick-walled cylindersubjected to time-varying internal pressure and temperature. A description of the numericalprocedurcs used in the analysis is also given. Briefly, the analysis employs a 'semi-explicit'integration scheme based on a quadratic differencc approximation of displacements andtemperaturcs (i.e., a linear acceleration scheme) and a technique in which nonlinear integralterms are cxpanded according to Simpson's rule and the displacement and temperaturchistories at only one timc interval in the past need be retained. In this latter respect, theprocedure is similar to that used in finite-element studies of linear thermoviscoelasticityproblems [12], However, in the present nonlinear analysis, a system of nonlinear algebraicequations in unknown nodal displacements and temperatures are encountered at eachtime incremcnt, making the method not a true explicit scheme. At each step the systemof nonlincar equations are solved itcratively by the Newton-Raphson method. Variousnumerical rcsults are givcn and compared with thc linearized. isothermal, and quasi-staticsolutions. 1t is found that including the effects of inertia, mechanically-dependent internaldissipation, and material nonlinearities can lead to significant departures from the resultsof the linear, quasi-static analysis.

2. THERMOMECHANICAL PRELIMINARIESConsider a continuous body at a uniform temperature To occupying a fixed homo-

geneous reference configuration Co. Particles X in Co are locatcd by an intrinsic systemof curvilinear coordinates (i=(i(x), x=(x!. x2 • .\'3) being a fixed, cartesian. spatial frameof referencc. We trace the behavior of the body from Co to other configurations C byidentifying the displacement vector l1(~, t) and the temperaturc change T=T(~, t) at theparticle ~ at time 1. The deformation and absolute temperaturc () at a particle ~ at taredetermined by

8(~,t)=To+T(~. t) (2.1)

Analysis of Nonlinear, Dynamic Coupled Thermoviscoelasticity Problems 605

where Ylj are the components of the Grcen-Saint-Venant strain tensor with respect to thebasis gl = ox/a(1 in the undeformcd body, III and III are the covariant and contravariantcomponents of u, and the semi-colon denotes covariant differentiation with respect to thematerial coordinates (I. In (2.1) it is understood that T and II I: j are functions of ~ and t.The functions

er(s)=e(~, t-s) (2.2)

where O~s< ro are referred to as the total histories of the strain and temperature at ~ attime I.

We are conccrned with the behavior of a class of thermomechanically simple materials;that is, materials whose response at a particle ~ at time t depends upon the histories of thedeformation and tcmperature at ~ [I]. Such materials may be characterized by a collectionof constitutive equations for the free energy fP, the stress tensor (Yij, the specific entropy t/,and the heat flux vector ql of the form

'Xl

cp= <P [y~,O~; 1,0, g].=0

co

alj = ~ij [1~,O~; 1,0, g].=0

'"1/= 9t[1~,I.I~; 1,e,g].=0

co1_ QI [ I fJl. 0]q - 1"" 1, , g

.=0

Hereco co<I> [-], , , ., QI [ ]

5=0 .=0

(2.3)

are functionals of the past histories 1~,O~, wherc 1~,O~denote the restrictions of y'(s), orCs)to the open interval S E (0, ro), and they are functions of the current values 1=1'(0),0 = 0'(0),and g=gradB; alj and qi denote contravariant components of the stress and heat fluxmeasured per unit deformed surface area and referred to the convccted coordinate lines(I in C.

The constitutive functionals must be form invariant under transformations of theobserver reference frame and under transformations of the material frame of referenceswhich belong to the isotropy group of the material (e.g., [14, 15]). In addition, the func-tionals (2.3) must be consistent with the basic physical laws of conservation of mass,balance of linear and angular momentum, conservation of energy, and the Clausuis-Duheminequality. In particular, we require that (2.3) satisfies the local energy equation

(2.4)

and the dissipation inequality

(2.5)

606 J. T. ODEN and W. H. ARMSTRONG

where (1 is the internal dissipation,

(1=pO/i-pll-q';i (2.6)

Altcrnately, (2.4) can be used as a definition of a. In these equations, p is the mass densityin C and II is the heat supplied from internal sources per unit mass.

By introducing (2.3) and (2.4) into (2.5) and observing the arbitrariness of the rates"Ii)' t), and g for fixed total histories of "Ii} and (), Coleman [I] arrived at the followingfundamental properties of thermo mechanically simple materials:

«>

1. The free energy functional <I> [-] is independent of the temperature gradient g:.=0

'"Og <I> [-]=0.=0

(2.7)

2. A thermomechanically simple material is characterized by only two constitutivefunctionals. one for the free energy and one for the heat flux. The constitutive functionalsfor stress and entropy are determined from the free energy functional by

«> '"ail = ty)i ["(~, ()~; "(, 0] :DY'J <I> ["(~, O~; "(,0]

.=0 .=0

'" '"/1= 91 ["(~, O~; "(,0]: -Do <I> [1~,O~; 1,0].=0 .=0

(2.8)

(2.9)

where Dy/! and Do are partial differential operators and, in accordance with (2.7), we haveeliminatcd the dependcnce on g in the arguments of the constitutive functionals for q>.aI], and 'I.

3. The internal dissipation (2.6) in thermomechanically simple matcrials is dcterminedfrom the free energy functional by

where bY'J' bo are Frechet differential operators and the vertical stroke indicates that thefunctionals arc linear in arguments that follow: i.e.,

'"~ [-Iy~,~~].=0

is lincar in y~and O~,where

and o<s<oo.

y~= -:/'(S) 0'= -~O'(s)r ds (2.11 )

Analysis of Nonlinear, Dynamic Coupled Thermoviscoelasticity Problems

The operators DYI}' Do, DYt}' and 00 in (2.8), (2.9), and (2.10) are defined by

00 0 00

Dy,}<1> [1~, 8~; 1,0]= lim- <1> [1~,O~; 1+lXr,O].=0 «~oaas=o

00 d 00

Do <1> [1~,O~; 1,0]= lim- <1> [1~.8~; 1,O+exfJs=O a-+Ooexs=o

0') ~ 00

b11} <1> [1~.8~;1,0ly~]= Iim~ <1> [i,+lXr" O~; 1,0]s=O «-+ooas=o

00 iJ 00

00 <1> [i" (1,.; 1.0lt}~]= lim- <1> [1~,O,+j,; 1,0].=0 «-+oa~s=o

607

(2.12)

where 0: is a scalar and r'(s), P(s) are total histories of arbitrary tensor-valued and scalar-valued functions rand f. Due to the decomposition of total histories into past historiesand current values, DYt} and Do amount to ordinary partial differentiations with respectto Y/j and O.

In summary, we require for a specific thermomechanically simple material, functionals

00

<l> [1~,O~; 1,0].•=0

and00

Q' [1~,O~; 1,0, g].•=0

for the free energy and the heat flux; the stress, entropy, and internal dissipation func-tJOnals are then determined from (2.8), (2.9), and (2.10). In the following section weexamine specific forms of these functionals for an important subclass of simple materials.

3. THERMORHEOLOGICALLY SIMPLE MATERIALSExperimental evidence obtained from tests on a large class of viscoelastic materials

has led to the identification of an important subclass of materials with memory. commonlyreferred to as thermorheologically simple materials. This classification arose from theobservation that, among certain amorphous high polymers which approximately obeyestablished linear and nonlinear viscoelastic laws at uniform temperature. are a groupwhich exhibit a simple property with a change of temperature; namely. a translationalshift of various material properties when plotted against the logarithm of time at differentuniform temperatures. The shift phenomenon is the basic characteristic of all thermorheo-logically simple materials and makes it possible to establish an equivalent relation betweentemperature and III t.

To fix ideas, consider the case of isothermal deformations of an isotropic, linearlyviscoelastic solid where stress is defined by a constitutive law of the form:

(3.1)

608 .I. T. ODEN and W. H. ARMSTRONG

where Ylj and Y arc thc dcviatoric and dilitational components of Y/j and J(I- s) andK(1-s) arc relaxation mouuli. Now in the case of a thermorheoiogically simplc matcrial.it is possible to write thc relaxation moduli at uniform temperature IJ, e.g. J{t-s) as afunction of /11 I, dcnotcd EO<'" I). Then the shift property is apparent if

(3.2)

where f(O) is a shift function relative to eo such that dfld/>O and f(lJo)=O. With thisproperty, the relaxation modulus curve will shift towered shorter times with an increasein e. By introducing a shift factor,

then

b(O)=expf(O) (3.3)

(3.4)

where ~ is a reduced time given by

~= rb(O) (3.5)

Ifit is possible to invert (3.5) so as to obtain t=g(~), then a constitutive equation for stresscan be obtaincd of the form

(3.6)

wherein we have used the transformations

(3.7)

That is, when transformed to a reduced time. the constitutive equations for a thermorheo-logically simple material assume the same form as the constitutive law for the isothermalcase.

While the above development applies to the special case of linear rcsponse of a bodyat various uniform temperatures, the basic ideas can be extended to finite deformations ofbodies subjected to transient, non-homogeneous temperatures [15]. For example, if it isassumed that the relaxation properties at a particle ~ depend only on the currcnt temperatureO(~, t) at ~. it may be postulated that an increment ~e in reduced time is related to anincrement ~I in real time according to ~e=b[O(~, t)]M. Then, instead of (3.5), we have

(3.8)

Clearly, (3.8) reduces to (3.5) in the case of uniform, constant temperatures.

Analysis of Nonlinear, Dynamic Coupled ThcrmoviscoeJasticity Problems 609

As a representative example of a constitutive functional for the free energy of a thermo-rheologically simplc material, considcr the functional proposed by Cost [13]:

Here all quantities have been transformed to reduced time; i.e., Yij('. e)=Yij(', g(e)), ...etc. (t=g(~»). Also e=f(t). as given by (3.8), e'=f(t') and e"=f(t") are dummy variablesin reduced time. In (3.9), CPo is a constant. Dij( ), I( ). K( .), G(,). and m(, ) arematerial kernals. and C( is the coefficient of thermal expansion. Integrating (3.9) by partsand introducing the resulting equation into (2.8). (2.9), and (2.10) yields the folIowingconstitutive equations for stress. entropy, and internal dissipation:

(3.10)

(3.11)

(3.12)

610 J. T. ODEN and W. H. ARfl.fSTRONG

In these equations 3K(e')= 3K(0, e'), m(O = 11/(0, 0=3a2'3K(O, 0. Thc materialkernals 3K( ) and 2G( ) are relaxation bulk and shear moduli.

To conclude this scction, an assumption is made concerning the heat constitutiveequation for the thcrmorheologically simple material. Among a variety of forms thatmight be selected for the functional is the linear law proposed by Christenscn and Naghdi[16].

(3.13)

where K(~-n is a thermal conductivity kernel and 9, j=o9/oej. The fact that (3.13) isnot completely reconcilable with (2.5) is the subject of a future note by the authors.

4. FINITE ELEMENTS OF DISSIPATIVE MEDIAThe construction of general finite-element models of nonlinear thermomechanical

behavior has been discusscd elsewhere [II, 17, 18]; conscquently, only thc main rcsultsneed be stated here. Following the usual finite-element procedure, we decompose a con-tinuum into a collection of elements connected smoothly together at nodal points. Weisolate a typical element e containing Ne nodal points and represent the local displacementfield and temperature field by approximations of the form

(4.1)

where the repeated nodal indcx is sununed from I to Ne. Here uf and TN are thc values of thecovariant displacemcnt components and the temperature increment T(~, t) == O(~, t) - To atnode N and t/JN(~) and qJN(~) are local interpolation functions defined so that t/JN(~M)=qJN(~M)=Ot:, at nodc ~M, and t/J~)(~)=qJ~)(~)=O if ~ does not belong to element e. Byforming energy balances for the element (or, equivalently, by applying Galerkin's methodto local forms of the laws of conservation of linear momentum and energy), it can beshown [II, 17, 18] that the gcneral equations of motion and heat conduction for a finiteelement are

/11 jiM +f q1joY'jdV=PNM k ~ N Nkv. Ollk

(4.2)

(4.3)

Here ve is the undeformed volume of element e, 11/NM is the mass matrix, Pt.'k are the com-ponents of generalized force at node N, qN is the generalized normal heat flux at node N.and UN is the generalized dissipation at node N:

(4.4)

Analysis of Nonlinear, Dynamic Coupled Thermoviscoelasticity Problems 61 I

where Fk and Sk are components of body forces and surface tractions respectively and niis a unit vector normal to A e' Note also that for the finite element

(4.5)

wherein rj~ are the Christoffel symbols of the second kind, g"lf is the contravariant metrictensor in Co. and

(4.6)

To specialize (4.2) and (4.3) so that they apply to materials of the type defined by(2.8). (2.9), and (2.10), note that for the finite element

2y,}(s)=g;' ". j(s) + gj . ".:(o5)+II.:(S)' ".j(s)

II ~(s)=g'(liJ bl_.I• r.1)IIN(I)(s)• I .,. N. I, 'I' N , j

ot(s) = To +CPN(~)TN(I\S)

g= VO=CPN.ir"'gi (4.7)

where g, and gi are the biorthogonal base vectors in the initial configuration and 1I~(tI(S)

and TN(t)(s) are the total histories of the nodal displacement components and temperatures.Thus, the arguments in the functionals of (2.8), (2.9), (2.10). and (2. I I) can be recast interms of the histories of oN and TN:

a:. f.C.

DUj <I> [-]= ~ Ij[o~(t), T~(l): u,'/, TN]3=0 _,=0

<Yo aoD(J <I> [-]= ~r [u~'(r), T~(t); UN, TN]

5"'0 .•=0

'" '"~ [-I..,~,O~]= ~ [u~('), T~(I): UN, TNlp~(r), t~(')]s=O saO

IT 00

QI [_, g]= 0 i[II~(t), T~(t); U,'/, TN]s=o .•=0

(4.8)

It follows that the equations of motion and heat conduction for finite elements of materialsof this class are

(4.9)

(4.10)

where, for simplicity, we denote by [-] the arguments listed in (4.8).

61:! J. T. ODEN and W. H. ARMSTRONG

Thermorheologicalfy simple materialsFurther spccialization of (4.9) and (4.10) can be obtained by restricting our attention

to thermorheologically simple matcrials of the type described in the previous section.We shall now consider a class of such materials for which it is possible to transform allfield quantities to corresponding quantities in reduced time: i.e. if f(~, 1) is a function ofreal time (, a function g(~) = t exists such that we can write

(4.11)

Then it is easily shown that

(4.12)

where j = dJ/de and b[ ] is the shift factor at ~ at reduced time ~ corresponding to f:

b[i)(~, e)]=b[O(~, t)) (4.13)

Notc that d~/df=O, since ~ denotes a material point (i.e .. (i are intrinsic coordinates). Itfollows that

li;\I = b(O)Wiii\( = Jj2(D)W + Jj(8)6(iMt'

0= b(O)O (4.14)

etc., wherein dependence of the quantities (r.'f, 8 etc. on ~ is understood. Similar relationshold for Ii. qi, and (1.

To cite a specific example. consider the material defined by (3.10)-(3. I3): suppose2Yij=u,; }+II}; /. and p(hi~p1'oli· Then, introducing (3.1O)-{3.13) into (4.9) and(4.10) after making the transformations (4.8). and transforming the resulting equationsinto reduced time with the aid of (4.11), (4.12), and (4.13), we obtain the following localequations of motion and heat conduction of a thermorheologically simple element inreduced time:

",~\rti~+m~~~uf+a~"\{kf:2G(e- O:~!d~' +b~Mkf: [3K(e-O-2G(~-n]~i. de'

-CMNkf: 3K(~-naa~~ de' =fMk (4.15)

(4.16)

Analysis of Nonlinear, Dynamic Coupled Thcrmoviscoclasticity Problems 613

The multidimensional arrays appearing in these equations are defined in Table I, Thequantities r)k appearing in Table 1 are the ChristoOel symbols of the second kind for thelocal element coordinates (i.

We observe that the behavior of a collection of such finite elements is described by alarge system of nonlinear integrodifferential equations. We discuss numerical solutions ofspecial cases in the following sections.

TABLEI. Multidimensional arrays

a~.I" =fpc tb[ "','I, J('" M, jO" + "'",,10 J, - 2"',\lgkrrj~) - "'Nr~jfIlJ.,]dv

b~\fN'= f...1!J[f" ••,"'N. i -"'.'1("'''', nO.,qj-tfr",ok,r~nr~j)]dv

kMN =f <P.If,,<PN .• dI1

",

h~t~ =3ClToK(O/ CPM("'N, j-"'Nr'j)dvJ t'.

dH~{ =*{ <P.I/WJ"'N.•"'l .. n - 2"'.'1, .I/I,.rln - 21/1,\·I/IL.• rj:+2I/1NI/ILr.~r",~ + I/IL. il/lN. jdvc•

...M. N. L= 1,2, .... Nc

5. NUMERICAL ANALYSISTo obtain quantitative solutions of (4.15) and (4.16), we shall assume that the

kernels appearing in the integrals in these equations pertain to matcrials with fadingmemory; i.e. the current response is not appreciably ctTected by strains and temperaturesat times distant in the past. This property is often assured by considering each kernel to

614 J. T. OlJEN and W. H. ARMSTR01'OG

be representable in the form of a decaying Prony series; for example, a typical matcrialkernel K(~) is defined by a series of the form

n

K(~)= L K1exp( -elvj)

.=1(5.1)

where Kj and VI are experimentally-determined constants, ~~O. and vj>O. Kernels of thctype G(e;. n are assumed to have the property

(5.2)

so that, in accordance with (5. I), we can write

n

G(~-C ~-~")= L Glexp[(-2~+f+e")I).I]1=1

(5.4 )

etc., )'1 being a constant> O.Now consider a typical integral appearing in one of the constitutive equations cited

earlier; e.g. define

(5.5)

We obtain an alternate representation of (5.5) which is easily adapted to numerical approxi-mation by dividing the reduced time interval [0, e] into,. subintervals [~m' 'm+I]'11/= I. 2, ... , r+ I. Then. using the representation (5,1) for the kernel in (5.5), we have

(5.6)

If hm=~m+ t -em is the time step, we may use Simpson's rule to approximate the integralsin (5.6):

(5.7)

To proceed further, the rates iJi/Jlae lUust be replaced by diflcrence approximations.In the present study, we assume a quadratic variation of (j~ between nodes Ill-I, Ill, III+1

Analysis of Nonlinear, Dynamic Coupled Thcrmoviscoelasticity Problems 615

which Icads to approximations of 0(112) of the rates. Introducing these into (5.7) andrearranging terms, we arrive at the following form of (5.5):

wherein

and

IGNj= L GiJl(u, A.)i

(5.8)

(5.9a)

,-IHG(CiJ, Ai) = L h-~{Ci7(m+ 11- Ci7<III-1»exp(~mj)'i) + i«(j~'m+ t) - Ci7cm»)exp[(~m + IIm/2)/A.,]

m=l

(5.9b)

(5.9c)

In (5.9a) we have separated the histories HG(u7, ;'i) from the current values CG(l1jv,Ai);that is, the value of the integral at the current time e,+ II is obtained by adding to theaccumulated sum ut e, the contribution between e, und e,+ h. By the notation ii7Im)' (t71111-t)

etc., we mean ft7(em), 117(em - II), etc.Using the notation of (5.9), the equations of motion und heat conduction for a thermo-

rheologically simple finite element assume the forms

1\ _ (1)".\/+ 121 'M (. i bi ')'G JM( ') bl "K JM( ,)f','u-m .....\/Uj 1tI ...f/'iUj + a.\H'ij- .\lNj,- k ik II, I. + MNjt... k il U,I• k

..

- C.\/N} LK".Jt1(T, 1')l

2/lllij "KkJM( ,)JL( ')+2(d(tlij dI2I,j)"GkJM( ')JL( ))- (N.I/Lt... - il< II,) jk 11,1 NML - NML t...- il< II, A Jl Ii, .k V" " ..1.1<

(5.IOa)

(5.tOb)

Thus, upon connecting elements together, we obtain a system of highly nonlinear differentialequations in the nodal values Cif, 1'N and their histories.

616 J. T. ODEN and W. H. ARMSTRONG

6. INFINITE CYLINDER

For illustration purposes, we now consider applications of (5.10) to the problem of aninfinite, thick-walled, circular cylindrical tube subjected to prescribed mechanical andthermal conditions at its inner and outer boundaries. For the locall1nite-element approxi-mations of the displacement components and the temperature wc use the simplex approxi-mations ifJN(r)=aN+bNr, N= 1. 2, 0N and bN being constants dependent only on the lengthof a radial element (bt=-b2=-I/(r2-rl); 01=r2/('2-'I)' a2=-rt/('2-'t) for anelement bctween nodes at radii 'I and '2)' By introducing this form of the interpolationfunctions into the arrays in Table I we obtain all properties of the equations of the discretemodel that are independent of specific material properties.

To proceed further, we must identify the form of thc shift function blO). Williams el al.[19] proposed the following empirical expression for b(O) for a wide variety of polymers,polymer solutions, organic and inorganic glasses:

10glob(0) = :x(0- To)/(fJ +0- To) (6.1 )

Here IX and fJ are constants. Experimental data [15] show that for 10 - Tol ~ 50°C, IX ~ 9and fJ~ 100, so that if To=O°C,

b(O)= I09T/lloo+n (6.2)

where, for the finite element, T= ifJN(r)TN =(aN +"Nr)TN. Transforming to reduced time,we find that

(6.3a, b)

wherein

(6.3c)

and fN is represented by the quadratic difference scheme described earlier.In the numerical integration scheme, we treat the current value of b(O) at time e as a

constant and allow 6 and f to lag the current values by the amount for the current stepsize h,. The values of Jj and of are recomputed at the end of each time step and are carriedforward into the next time interval to determine nodal displacements and temperatures.This procedure makes it possible to express the mass matrices m~},~and m\7~ in theparticularly simple forms

(6.4a, b)

where

(6.4c)

and iJiN(')~aN+bN(rl +r2)/2. Similar procedures are used to evaluate the remainingintegrals in the equations of motion and heat conduction.

Analysis of Nonlinear, Dynamic Coupled Thermoviscoelasticity Problem, 6t7

Neglecting body forces, and intcrnal heat sources, the gencralized forces and heatfluxes of (4.4) become

(6.5a, b)

(no sum on N), where SN(e) and QN(e) are prescribed functions of reduccd time. Alter-nately, we can also prescribe arbitrary time-dependent nodal displacements and tem-peratures. For convective heat transfer problems, we set

(6.6)

..

where IN is the film coefficient at node N and To * is the temperature of a medium in contactwith the cylinder at a boundary.

To complete the statement of the problem. we consider a hypothetical material forwhich po=1O-3.lb-sec2jin.4, lX=10-5 in.jin.°C, the dilitational relaxation kernelK(~)= 10 x 105exp( -e/l0-s)+5 x 105exp( -'/oo),lb/in.2, the shear relaxation kernelG(e)= I x IOsexp( -ejlO- 5)+0'1 x 105exp( -~joo).lbjin.2, and the thermal conductivitykernel of thc heredity-type heat flux law(3.l3)of K(e)=O'IOcxp( - e/lO- 5)+0·01 exp( - elm)in.Ib/in.oC sec.

Numerical procedureWe now apply the above equations to the analysis of a cylinder with an inner radius

of 10·0 in. and an outer radius of 11·0 in. for which the 1·0 in. wall is divided into twenty0·05 in. clcments. The system of nonlinear differential equations governing this model areintegrated using the explicit quadratic scheme mentioned earlier; howevcr, owing to thenonlinear heredity terms, we encounter at each time step a system of nonlinear algebraicequations in the current values of the nodal displacements and temperaturcs. In theexamples to be cited below, these equations arc solved by Newton-Raphson iteration,Starting values for the Newton-R,~phson process are obtained from the initial conditions(e,g .. if tiN(eo)=k"(eo)= 1'N(eo)= 1'N(eo)=O. thcn central difference approximations of u,liN and fN lead directly to expressions for liN, i"". tN. and TN at time eo +h in terms offiN and 1'11' at eo +Jr. 1nitial values so computed are introduced. directly into (5.10) toobtain displacements and tcmperaturcs at eo + /z and the value of l'[see (6.3c)] to be usedin the second time step. For the second timc step, wc again use the quadratic approximationIi", + t :::; (3u",+ I -4ulII +um _ 1 )j2/z for various rates. This results in adequate data to initiatethe Newton-Raphson iteration process. Once convergence of a specified degree is obtained.another reduced time increment is added and the process is continued.

Element stresscs are computed at the cnd of each time stcp by introducing nodaldisplacements and temperatures into the stress constitutive equation (3.10). AlI quantities(nodal displacemcnts, temperatures, and clement stresses) are computed as functions ofreduced time and are then transformed to real time by using the rclation

M= dejb[O(~. e)] (6.7)

Here de=/z is the reduced time increment and b(O) is given by (6.2). The real time at stepr+ I is then tr+/zlb[O(~. UJ. Note, however, that different values of real time should beconsidered for tiN and 1'N than for clement stresses: more specifically. for the nodal valuesaN, TN thc shift factor is associated with a temperature change at a single node whereas

618 J. T. ODEN and W. H. ARMSTRONG

the stress tcnsor is associated with a shift factor that involves the entire element-temperaturefield and. consequcntly, depends upon the temperature changes at every node of theelement. In the former case we simply set the real time increment litr associated withnodal quantities ttl', fr equal to II/hI' where hI' is the (global) value of h(lJ) at node r. Inthe latter case (transformation of stresses), we set the real time increment in element e,/ilk)' equal to II/ble) wherein b(e) is an averaged shift factor for element e.

Numerical resullsWe shall consider as an example the dynamic response of the cylinder due to purely

mechanical pressure, step loading p = 101I(t) psi (h(/) being the unit step function) appliedat the inner boundary, while the thermal boundary conditions correspond to convectiveheat transfer. Film coefficients of 0·1 and 1·0 in.lb/in.2 °C sec. respectively are assumedfor the inner and outer boundaries, The heat supply from internal sources is assumed tobe zero and the initial temperature of the cylinder and the surrounding media are takento be 300oK. Heat is thus generated through thermomechanical coupling. Since energyis dissipated in the system, the internal temperature increases during the transient part ofthe motion while heat is cxpelled at different rates at each boundary, For the materialparametcrs and the loading considercd, a steady-state condition is reached at approximately10-3 sec. We can also obtain, for purposes of comparison, the solution of the isothermalcasc by ignoring all thermal effects. setting fj = I. 6 =0. and by treating the integrationvariable as real time.

Nondimensional plots of the computed radial and circumferential stress waves atvarious real timcs are shown for the isothermal and non-isothermal cases in Figs. I and 2;the displacement profiles are shown in Fig. 3. While stress and displacement profiles areobtained at different times for the isothermal and non-isothermal cases (owing to thenecessity of transforming all computed quantities to real time in the non-isothermal case).the results do indicate a lag in the non-isothermal stress waves and a noticeable decreasein amplitude. Circumferential stresses are initially compressive but become tensile almosteverywhere in the cylinder at t= 10-'\ sec. The circumferential stress wave requires approxi-mately 2·5 x 10- 5 sec to traverse the 1·0 in, thickncss of the cylinder in the isothcrmal

-0,8

-0-2

0,0100

------ISOTHERMAL-.-- NONISOTHERMAL

/1'0

RADIUS (inches)

FIG. I. Stress propagation in a thermorheologically simple cylinder-nondimensional radialstress versus radial distance for various real times.

Analysis of Nonlinear, Dynamic Coupled Thermoviscoelasticity Problems 619

-06

-0'6

------ISOTHERMAL--- - NONISOTHERMAL

Time in Seconds

FIG. 2. Stress propagation in a thermorheologically simple cylinder-nondimensional circumferentialstress versus the radial distance for various real times.

,()"4

- ISOTHERMAL---- flONISOTHERMAL

Time In Seconds

..'"!I-Zl<J:l:UJuc(

..Ja..lfl6..JSoc(Cl:

10-910'0 10'2 10'6 10'6 11'0

RADIUS (Inches)

Fro. 3. Radial displacement profiles for various real times.

620 J. T. ODENand W. H. ARMSTRONG

case and approximately 4'0 x 10- 5 sec in the non-isothermal case. Displacement profilesare qualitatively the same for the isothermal and non-isothermal cases. The non-isothermaldisplacements are higher than those predicted by the isothermal analysis at the outerboundary, but they are lower on the inner boundary. This is due chiefly to the relativemagnitudes of the film coefficients at these boundaries. We remark that the results indicatedwere computed using a reduced time increment of 10-6 sec; rather rapid convergence ofthe Newton-Raphson scheme was observed at each time step.

Figure 4 indicates the computed temperature change over the thickness of the cylinderat various values of real time. Note that the mechanically loaded inner boundary is thefirst to experience an increase in temperature. The maximum temperature is reached atapproximately t = 4 x 10- 5 sec, and the temperature appears to decrease thereafter.

FIG. 4. Nondimensionalized temperature distributions at various real times.

Ackllowfedgemellf-Support of this work by the U.S. Air Force Office of Scientific Research under ContractF44620-69-C-0124 is gratefully acknowledged.

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[5] E. L. WILSONand R. E. NICKELL, Application of finite element method to heat conduction analysis.Nuclear Engng. Design 4, 276-286 (I966).

Analysis of Nonlinear, Dynamic Coupled Thermoviscoclasticity Problems 621

[6] E. B. BECKERand C. H. PARR,Application of the finite clement method to heat conduction in solids.Technical Report S-117, Rohm and Haas Redstone Research Laboratories, Huntsville, Alabama (1967).

[7] R. E. NICKELLand J. J. SACKMAN,Approximate solutions in linear, coupled thermoclasticity. J. Appl.Mech. 35, 255-266 (1968).

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[14] C. TRUESDfLl.and W. NOLL, The nonlinear field theories of mechanics. Encyclopedia of Physics Ill/I(edited by S. FLUGGE),Springer-Verlag (1965).

[15] A. C. ERIl"GEN,MechanicsofContinlla. John Wiley, New York (1967).[16] G. L1ANIS,Nonlinear thcrmorheologically simple materials. Fifth IlIIernational COl/gresson Rheology,

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in nonlinear visco-elasticity. Developments in Theoretical and Applied Mechanics. Proceedings, 4thSECT AM, Pergamon Press, London.

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(Receil'ed 25 March 1971)