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Journal of Thermal Stresses

ISSN: 0149-5739 (Print) 1521-074X (Online) Journal homepage: http://www.tandfonline.com/loi/uths20

EFFECT OF THERMAL CAPACITY ONTHERMOELASTIC INSTABILITY DURING THESOLIDIFICATION OF PURE METALS

Faruk Yigit , Nai-Yi Li & James R. Barber

To cite this article: Faruk Yigit , Nai-Yi Li & James R. Barber (1993) EFFECT OF THERMALCAPACITY ON THERMOELASTIC INSTABILITY DURING THE SOLIDIFICATION OF PUREMETALS, Journal of Thermal Stresses, 16:3, 285-309, DOI: 10.1080/01495739308946231

To link to this article: https://doi.org/10.1080/01495739308946231

Published online: 25 Apr 2007.

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EFFECT OF THERMAL CAPACITY ON THERMOELASTIC INSTABILITY

DURING THE SOLIDIFICATION OF PURE METALS

Faruk Yigit Department of Mechanical Engineering and Applied Mechanics

The University of Michigan Ann Arbor, Michigan 48109

Nai-Yi Li Alcoa Technical Center

Process Design and Reliability Division Alcoa Center, Pennsylvania 15069

James R. Barber The Uniuersity of Michigan, Ann Arbor

If unidirectional solidification of pure metals at constant cooling rate is perturbed by a small spatially sinusoidal heat flux at the casting/mold inletface, corresponding responses with sinusoidalperturbation are obtained for the evolution of the casting/meit interface, for the temperature and stress fields in the solidified casting, and for the contact pressure at the mold surface. The results show that the perturbation in contact pressure tends asymptotically to a maximum ualue at larger values of time. The magnitude of the contact pressure perturbation is decreased by the inclusion of thermal capacity, and this effect is enhanced at higher mean cooling rates. This indicates that materials with higher specific heat (or lower thermal diffusivities) might be less susceptible to thermoelastic instabilities associated with the contact pressure and its dependent thermal contact resistance at the casting/mold interface.

INTRODUCTION

Any spatial perturbation in mold temperature during unidirectional solidification will cause a corresponding perturbation in the temperature and residual stress in the casting. This in turn will cause nonuniformity in the contact pressure at the casting/mold interface and in extreme cases may lead to air-gap formation (see [Ill.

Richmond et al. [2] developed an idealized analysis of this problem, in which a nominally uniform heat flux at the casting/mold interface is given a small, spatially sinusoidal perturbation. Their analysis of the thermomechanical stresses was based on a nonuniform beam approximation and is appropriate only for small times, i.e. when the thickness of the solidified layer is small in comparison with the wave- length of the perturbation. More recently, a stress function formulation of this

The authors wish to express their gratitude to Dr. Owen Richmond of Alcoa Technical Center for suggesting this research topic and for his continued support and guidance during all phases of the work.

Journal of Thermal Stresses, 16:285-309,1993 Copyright O 1993 Taylor & Francis

0149-5739/93 910.00 + .OO

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NOMENCLATURE

coefficients associated with stress function [see Eq. (4911 dimensionless coefficient [see Eq. (69)l specific heat Young's modulus displacement potential function [see Eq. (4411 dimensionless displacement potential function [see Eq. (72)] residual stress function dimensionless residual stress function liquid head of the molten metal thermal diffusivity thermal conductivity latent heat of fusion wave number first-order perturbed contact pressure [see Eq. (5511 dimensionless first-order perturbed contact pressure [see Eq. (74)l heat flux at mold surface position of the melt front dimensionless position of the melt front (see eq. 17) dimensionless position of the melt front (see eq. A.1) time temperature in casting temperature in casting subject to a sinusoidal disturbance dimensionless temperature [see Eq. (1911

dimensionless temperature [see Eq. (A.311 melt temperature dimensionless melt temperature [see Eq. (23)l difference between melt and mold temperature lateral spatial variable spatial variable through the casting thickness dimensionless parameter [see Eq. (17)] dimensionless parameter [see Eq. (A.l)] error function coefficient of thermal expansion spatial step size Stefan number [see Eq. (I.)] dimensionless parameter associated with thermal diffusivity [see Eq. (2211 dimensionless parameter associated with the mean melt front [see Eq. (7111 dimensionless time [see Eq. (IS)] constant [see Eq. (3311 dimensionless time when the mean melt front reaches the location Y Poisson's ratio

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EFFECT OF THERMAL CAPACITY ON THERMOELASTIC INSTABILITY 287

density stress components dimensionless time displacement potential stress function [see Eq. (4911 tanh( 5 Y ) tanh(?) zeroth-order part first-order part particular solution homogeneous solution

problem was developed by L1 and Barber [3], who also considered the effect of coupling between the heat conduction and stress problems through the occurrence of a pressure-dependent thermal contact resistance at the casting/mold interface.

Both of these treatments make the simplifying assumption that the Stefan number

is very much less than unity, where c, L are the specific heat and latent heat of solidification respectively, and AT is the difference between the solidification and mold temperatures. This is equivalent to the assumption that the thermal capacity of the solidified material has a negligible effect on the temperature field. It is reasonable for the freezing of ice but not for the solidification of metals in contact with molds near to room temperature, for which Stefan numbers are typically in the range 1 < E < 3 [4]. In Richmond's problem 1:2], where the heat flux at the mold is specified, the temperature drop across the solidified layer and hence also the effective Stefan number increase with time from zero at the onset of solidification. Thus, the idealized solution should function as an asymptotic limit to the true solution at small times but will be increasingly in error as time progresses.

In this paper, the effect of this approximation is assessed by developing a new stress function solution of Richmond's problem for materials of nonzero but constant specific heat; in other words, the value of thermal diffusivity is not infinitely large. This generalization necessitates a numerical solution of the prob- lem, but the linear perturbation method permits the unidirectional unperturbed process and the first-order perturbed solution to be uncoupled, leading to two one-dimensional ordinary differential equations, which are solved by finite differ- ence method.

FORMULATION OF THE PROBLEM

We consider the two-dimensional problem of a liquid initially at its melting temperature T,, in the half space y > 0. It is assumed that the liquid is in contact

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F. YIGIT ET AL.

Liquid

Solid

Mold

Fig. 1 Geometry of nonuniform directional solidification.

with a plane mold at y = 0, on which the heat flux is a prescribed function of one space coordinate and time t . After a time t , the liquid solidified near the mold forms a solidified shell of thickness s ( x , t ) . Thus, s ( x , t ) defines the moving interface between the solid and liquid phases as shown in Fig. 1. The prescribed heat flux is assumed to be nearly uniform in x so that the perturbation from unidirectional solidification, s =s, ( t ) , is small. We assume that the thermal diffu- sivity k, conductivity K, and density p of the solid phase are constant and independent of temperature.

The Heat Conduction Problem

The temperature field of the solid T ( x , y , t ) must satisfy the heat conduction equation:

with the boundary conditions

T ( x , s, t ) = T,

where y = s(x , t ) defines the instantaneous position of the solid/melt boundary. Notice that the smallness of the perturbation from unidirectional solidification implies that d s / d x << 1 and hence permits us to write the boundary condition ( 4 )

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EFFECI' OF THERMAL CAPACITY ON THERMOELASTIC INSTABILITY 289

in terms of x, y rather than in a slightly rotated coordinate system normal to the local melt front. Following Richmond 1:2], we consider the case in which the prescribed heat flux is of the form

where Q, and Q, are assumed to be time-independent constants. The second part of Eq. (6) defines a small sinusoidal perturbation on the constant heat flux Q, at the casting/mold interface. Since Q, < Q,, we anticipate a corresponding small sinusoidal perturbation in the temperature field and the thickness of the solidified layer, which can be expressed by the equations

We assume that the amplitude of this perturbation is small in comparison with its wavelength (i.e., ms,(t) =c l), in which case the slope of the moving front, ds/dx, is very much less than unity.

Substituting Eq. (7) into Eq. (2) and separating periodic and uniform terms, we obtain

Since the perturbation is small, we can expand the temperature field in the vicinity of the mean solid/melt interface position, y = so(t), in the form of a Taylor series. The boundary condition (3) can then be written by separating periodic and uniform terms and dropping second- and higher-order and product terms in the small quantities, TI and s,, giving the two following equations

A similar procedure applied to the boundary condition (4) also yields the two equations

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Finally, the solid/mold boundary condition (51, with Eq. (6), gives

Dimensionless Formulation

Before proceeding to the solution, it is convenient to introduce the following dimensionless variables

The governing equations (9) and (10) for T,(Y, 8 ) and T,(Y, 0) then become

where

The boundary conditions (ll), (13), (151, corresponding to the zeroth-order temperature field T,,(Y, 01, become

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EFFECT OF THERMAL CAPACITY ON THERMOELASTIC INSTABILITY 291

and the boundary conditions (12), (14), (16), corresponding to the first-order temperature field T 1 ( y , O), can be written as

Thus, the heat conduction problem is reduced to the determination of two pairs of functions T0(Y, 01, S,(O) and T,(Y, O), S l ( 0 ) in Eqs. (20) and (211, which satisfy the boundary conditions (23)-(25) and (26)-(28) respectively.

Approximate Analytical Solutions

Equations (201, (21) with boundary conditions in (231425) and (26)-(28) cannot be solved in closed form, but an asymptotic solution for small values of 6 can be obtained by assuming a power series in time for the thickness of the solidified layer and a double power series for the temperature distribution [4, 51, with the results

Like other asymptotic solutions, the series solutions in Eqs. (29), (30) diverge beyond a certain point depending on the value of 8 . Figure 2 shows the partial sum of S,(O) with one, two, three, and four terms respectively and indicates clearly that the series solution in Eq. (29) is not practically useful beyond about 0 = 0.1. Since the first term-equivalent to Li's idealization of zero Stefan number-is good enough for most practical purposes in 0 < 0 < 0.1, we conclude that the asymptotic solution is of little value for our present purpose'. However, we shall see below that the series is useful in providing a starting condition for a numerical solution of the problem.

Several other approximate methods of solution [6-81 have been proposed for solving the problem of this section. Goodman 191 presented solutions to several problems using a "heat balance integral" technique similar to the momentum integral method in fluid flow problems. His results compared favorably with the exact solution for small values of Stefan number. Lardner [ l o ] and Prasad and Agrawal [ I l l obtained an approximate solution for small values of time by using

' A considerable improvement in the convergence of the series in Eq. (29) can be achieved using Shanks transform [IS].

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292 F. YIGlT ET AL.

Fig. 2 Numerical solution (solid line), approximate analyiical solutions and asymptotic with one, two, three, and four terms respectively for the solidification front.

series solutions

Biot's variational principle. El-Genk and Cronenberg 1121 presented an analytical solution to this problem and claimed that it was the first exact solution. However, Cho and Sunderland [I31 showed i t was not exact since the suggested solutions for the solidification moving front do not satisfy the heat conduction equation (20). In their analysis, the temperature distribution was assumed to have a similar profile to that without phase change, i.e.,

where 6, is a constant. This expression satisfies the heat conduction equation (20) and the boundary condition (25), but it cannot be made to satisfy boundaly conditions (23), (24) simultaneously and hence does not constitute an exact solution.

Imposing boundary condition (23) alone, they obtained the condition

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EFFECT OF THERMAL CAPACITY ON THERMOELASTIC INSTABILITY 293

for S,(O), whereas imposing (24) alone gives

with 0 , = 1/27r. Equations (31), (32) define an approximate solution of the problem where the energy balance at the solidification front is relaxed, while Eqs. (31), (33) define an approximate solution of the problem where temperature continuity at the solidification front is relaxed.

Cho and Sunderland 1131 surmized that the exact solution would lie between these two approximate solutions. They also expressed the opinion that the condi- tion (33) would lead to a better approximation than (32). Both these hypotheses are supported by the numerical results presented in the next section. In particular, we found that the maximum error in S,(O) resulting from the use of Eq. (31) with condition (33) is 3%. The results of both approximations are also shown in comparison with the asymptotic solution (29) and the numerical solution in Fig. 2.

Cho and Sunderland's method can also be used to obtain an approximate solution to the first-order problem defined by Eq. (21) with boundary conditions (26)-(28) and initial temperature zero. The governing equation (21) is first solved for the half-space without phase change, using the mold boundary condition (28) with the result

Following Cho and Sunderland, the first-order solidification front S,(O) can be approximated by imposing one only of the remaining conditions (26) or (27). The resulting approximations are compared with the numerical solution in Fig. 3 for the case of c = 0.5. As before, the correct solution lies between the two approximate solutions, but the approximations are significantly less good than that in the zeroth-order case.

Numerical Solution of the Heat Conduction Problem

Equations (20) and (21) with boundary conditions in (23)-(25) and (26)-(28) cannot be solved in closed form, and therefore we must resort to a numerical solution. Two alternative methods may be used for this purpose.

If fixed grid points are used, the moving solid/melt boundary requires special treatment, and discontinuities tend to arise during time increments for which this boundary passes through one of the grid points [141. An alternative approach is to use a Lagrangian method in which there is a fixed number of grid points in the solid phase, so that the space step size, 6, increases with the thickness of the solidified layer as it grows in time.

In the present work, the Lagrangian scheme has been implemented, such that the solid/melt interface is exactly identified with the last grid point of the mesh at

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Fig. 3 Numerical solution (solid line) and approximate analytical solutions for the first-order solidifica- tion front in the casc of 0.5.

all times. However, since all the space step elements are moving with time, convective tcrms have to be included in the updating algorithm for temperature.

The instantaneous temperature field in the solidified shell is represented by the temperatures at N + 1 equally spaced points including the end points. The growth of the solidifying layer during the next time increment can be determined from the finite difference form of Eq. (24). The temperature field in the layer is then updated using the heat conduction equation.

In order to obtain a solution in a reasonable number of steps, it is necessary to choose the time increment as large as possible subject to the constraint that the scheme is kept stable. In general, it is found that the maximum time step for stability is proportional to S ' / K where 6 is the smallest dimension in the discretization. At the beginning of the process, the thickness of the solidified layer and 6 are very small, and therefore we would need an extremely small time step. However, the first term of the series solution becomes progressively more accurate at small values of time, and we can therefore start the process with a small but finite thickness, using the first term of the series solution to define the initial values for the temperature field. The evolution of the mean layer thickness, S,(O), over an extended range of 0 is shown in Fig. 4.

Once T,(Y, O),S,(O) have been determined, a similar finite difference algo- rithm can be developed to determine T,(Y, O), S,(f l) through Eqs. (21), (261428).

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EFFECT O F THERMAL CAPACITY ON THERMOELASTIC INSTABILITY 295

Fig. 4 Numerical solution for the solidification front at large 8.

THE DEVELOPMENT OF THE STRESS FIELD

To determine the stress field, we follow the procedure developed by Li and Barber [3] , noting, however, that in his case the simplification E = 0 permitted the temperature field to be obtained in closed form, whereas in the present problem it is determined by a numerical algorithm and, therefore, is defined in discretized form. Since the analysis is closely related to that in [3], only the essential steps are presented in the following derivations, readers being referred to [3] for more details.

The Zeroth-Order Solution

We first consider the zeroth-order stress field, corresponding to the temperature field T o ( y , t ) defined by Eq. (91, which is determined independently from the solution of the first-order problem. Since To(y, t ) is a function of y and t only, the first equation in (3 ) of [3] becomes

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where the displacement potential, 4,, satisfies the condition

and the superscript p indicates that 4, defines a particular solution of the thermoelastic equations. Also Eq. (2) of [3] will take the form

Using Eqs. (35), (371, uxP,, can be determined in the form

where To(y, t ) is known in discrete form. Since the displacement potential 4, is a function only of y and t , the other two stress components will be zero, i.e.,

To complete the solution, we must superpose a homogeneous stress field and assign the arbitrary constants to satisfy the boundary conditions (20)-(22) of [3]. It is easily verified that this is achieved by the uniform biaxial field

so that the complete zeroth-order solution is

Note that p g ( H - s o ) is the hydrostatic pressure at the freezing front due to the liquid head of molten metal, H. It follows from Eq. (43) that the zeroth-order contact pressure at the casting/mold interface is p g ( H -so).

The First-Order Solution

We next consider the thermoelastic problem corresponding to the first-order temperature field T,(y , t)cos(m). A suitable particular solution can be obtained by

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EFFECT OF THERMAL CAPACITY ON THERMOELASTIC INSTABILITY 297

assuming a potential in the form

and using Eq. ( 2 ) of [3] to determine the function f ( y , t )

Note that this equation must be satisfied for all t , and hence it is essentially an ordinary differential equation for the function f ( y , t ) in which t appears only as a parameter. We can write the particular solution for the first-order stresses using Eqs. (3) of [3] and Eq. (44) as follows

where ('1 denotes differentiation with respect to y . We represent the homogeneous solution of the first-order problem in terms of

an Airy stress function @. In view of Eqs. (5), (6) of [3], the time derivative of @ must be biharmonic. It can be verified by substitution that the appropriate form is

where the time-dependent coefficients a , ( t ) , a , ( t ) , a , ( t ) , a , ( t ) and time-indepen- dent function g ( y ) are to be determined from the mechanical boundary conditions corresponding to the first-order problem [3]. Using Eqs. (1) and (4 ) of 131 and Eqs. (46)-(49), we can construct the complete solution of the first-order problem in the form

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Also, using the elastic constitutive relation for plane strain condition to determine the strain components and hence solve for the displacements, we obtain

where (.) denotes differentiation with respect to time t . We now consider the boundary con"dtions corresponding to the first-order

problem. Since the perturbation on the stress field is small, we can expand the stress field in the vicinity of the mean solid/melt interface position, y = s , ( t ) in the form of a Taylor series. Then the first boundary condition in Eq. (20) of [3] can be written, dropping second, higher, and product terms in small quantities, a,, s,, as follows

Separating periodic and uniform terms and using Eq. (421, we obtain the boundary condition for ax,, at y = s,( t) , i.e.,

Applying the same procedure to the remaining boundary conditions in Eq. (20) of

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EFFECT OF THERMAL CAPACITY ON THERMOELASTIC INSTABILITY 299

[3], we obtain

u,,,(x,s,,t) = 0; ayyl(x,so, t ) = 0 (56)

Also from Eqs. (21), (22) of 131, we have

Applying boundary conditions (57), using Eqs. (51), (531, we can determine the time derivatives of coefficients, a2(t) and a&), i.e.,

We can therefore write

without loss of generality, since constants of integration will lead to time-indepen- dent terms in @ that can be subsumed under g(y).

Substituting for the stress components from Eqs. (50)-(52) into the remaining boundary conditions (55), (56), we obtain the equations

where we have used Eqs. (59) to eliminate a2(t) and a&). These three equations

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300 F. YIGIT ET AL.

must be satisfied for all values of t, and hence we can use them to eliminate a,(t) and a,(t). Let us define w = tanh(ms,(t)); therefore, we obtain

which serves to determine the unknown residual stress function g(y). Once g(y) is known, we can recover a,([) and a&) by solving Eqs. (601, (62), with the result

Finally, we determine the perturbation in contact pressure p,(t)cos(mr) at the casting/mold interface, from Eqs. (25) of [3] and (52), (59) as

p l ( t ) = m2[a,(t) -f(O,t)l (66)

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EFFECT OF THERMAL CAPACITY ON THERMOELASTIC INSTABILITY 301

where we have imposed the arbitrary condition g(0) = 0 , since the free constants in the solution of Eq. (63) can be assigned to satisfy this condition.

Solution Procedure

The stability of the unidirectional solidification process depends on the relation between the perturbation Ql in the imposed heat flux and the resulting perturba- tion in the contact pressure, p,, at the casting/rnold interface, given by Eq. (66). In order to evaluate this pressure perturbation, we have to determine the coefficient a , ( t ) from Eq. (641, which in turn requires the evaluation of the functions, f ( y , t ) , g ( y ) from the differential equations (451, (63) respectively.

Now consider equations (451, (631, (64), (66) written in the dimensionless form as follows:

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where

F. YlGlT ET AL.

In Eqs. (681, (69), ('1 denotes differentiations with respect to the dimensionless variable Y .

It is convenient to recast Eq. (68) in the Y domain by defining the function Oy(Y) such that O= O,(Y) is the solution of the equation Y = So(0), i.e,. O = Oy(S,,(0)) or Y=S,(O,,(Y)). In effect, Oy(Y) is the time at which the mean melt front reaches the location Y . Using this notation, Eq. (68) takes the form

where w = tanh( 4' Y ) = tanh(So(Oy)).

The Limiting Solution for Zero Stefan Number

The principal purpose of this paper is to compare its predictions with the previous solutions due to Li and Barber [3] and Richmond et al. [2], which were restricted to thc case where the Stefan number is negligible, in order to estimate the influence of Stefan number on the development of the perturbed field. In the present case, the limit of zero Stefan number corresponds to the range 0 << 1 when 5~ 1, and it is convenient to define the new timelike variable 7 = (O( = mQ,t /pL) in order to approach the limit in a regular manner.

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However, in [3, 21, it is assumed that the solidified layer is thin as well, so that the temperature field can be assumed to be linear with y. This is equivalent to the assumption that the first term in Eq. (10) is much larger than the remaining terms and is only reasonable as long as T < 1. We demonstrate this in the appendix by developing the solution for [+ 03 as a limiting case of the present more general theory. In particular, we find that the dimensionless amplitude of the perturbation in the solidification front is given by

from Eq. (A.lO),whereas the approximate theory of Li and Richmond leads to the linear relation S1(7) = Q,T/Q,,, which is a good approximation only as long as T << 1. In physical terms, this means that the growth of the perturbation will deviate significantly from linearity when the thickness of the solidified layer so is of the same order as the wavelength of the perturbation 27r/m.

Numerical Implementation

For finite g, both the zeroth- and first-order temperature fields are known only in discrete form as a result of the finite difference algorithm derived during the section on the numerical solution of the heat conduction problem. The solution for the amplitude of the contact pressure perturbation must therefore be found by a corresponding finite difference solution of Eq. (67) for F(Y, 0) followed by Eq. (75) for the residual stress function G(Y) . In this procedure, it is convenient to use Eq. (67) to substitute for the second derivative F", to reduce the inaccuracies inherent in numerical differentiation.

We use the same above-mentioned algorithm, good convergence being achieved with 100 spatial grid points in the solid phase. Also, the use of the Lagrangian method necessitates that the procedure be initiated at a nonzero value of time, but Li's solution provides a rigorous asymptotic solution to the problem when T = (0 1, and hence it can be used to determine suitable initial conditions.

In the limiting case where 5- w, the temperature fields and the function F are determined in closed forms [see Eqs. (A.41, (A.111, (A.1311, but the finite difference algorithm was used2 to solve the differential equation (A.15).

RESULTS

We first illustrate the approximation inherent in the Li and Richmond solutions by comparing results for the limiting case t + w. Figure 5 shows the solution obtained

It is in fact possible to obtain an explicit solution to this equation, but the resulting expression involves integrals that have to be evaluated numerically. It was found that there was no saving of computer time in using this solution as compared with a direct discretization of the original differential equation.

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Fig. 5 Richmond's, Li's, and limiting solution (see Appendix) for the perturbation in contact pressure as a function of time.

from the analysis given in the appendix in comparison with results from (i) Richmond's nonuniform beam theory and (ii) Li's stress function solution. Note that the perturbed contact pressure is expressed in the T-domain, not in the Bdomain. We also note that Li's solution reduces to that of Richmond if the stress function is approximated by a four-term truncated power series in Y. Both approximations give acceptable accuracy if T < 0.5, but divergence beyond that point is rapid. The exact solution predicts that the contact pressure perturbation grows monotonically, approaching a limiting value of 0.63 at large 7 . The perturba- tion is essentially constant after about T = 4. A physical explanation of this behavior is provided by the consideration that when T > 27r, the thickness of the solidified layer is greater than the wavelength of the perturbation, in which case variations in conditions at the solidification front will have relatively little influence on those at the mold boundary. The contact pressure perturbation .is therefore frozen at a value determined by the history of the transient process.

For a given zeroth-order heat flux, Q,, the parameter 5 can be considered as describing the effect of finite thermal diffusivity for the material or, equivalently, of the thermal capacity of the material being nonnegligible. Figure 6 shows the development of the perturbation in contact pressure as a function of 7 , for various values of 5 , these results being obtained using the finite difference algorithm and the Eqs. (671, (691, (701, (75). The limiting solution for 5 + m developed in the appendix is shown for comparison and provides some check on the numerical procedures used.

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EFFECT OF THERMAL CAPACITY ON THERMOELASTIC INSTABILITY 305

Fig. 6 Perturbation in contact pressure as a function of time, for various values of 5.

Figure 6 shows that in each case the perturbation tends monotonically to a limiting value but that the limit is approached more slowly at lower values of 4'. The argument advanced above for the case 4' + m tempts us to expect stabilization of the perturbation when the zeroth-order solid layer thickness exceeds the wavelength-i.e, when S J T ) > 2 ~ . This is indeed found to be the case for [> 20. However, for lower values of 5- corresponding to significant thermal capacity in the material-the transient first-order temperature field continues to vary signifi- cantly as long as the dimensionless parameter kt /a2 , where a is the appropriate length parameter that here might be taken as equal to the wavelength 27r/m, is small. We therefore should anticipate variation in the thermal stress field and, in particular, in the contact pressure perturbation as long as this inequality is satisfied, which in our dimensionless notation translates into the condition [T < 8 r 2 . The numerical results confirm that, for [ < 20, the perturbation approaches its limiting value on a time scale proportional to [T . The asymptotic value at large 7 is shown as a function of 4' in Fig. 7.

CONCLUSIONS

The above results demonstrate that the inclusion of thermal capacity effects reduces the perturbation in contact pressure associated with a given perturbation in heat flux at the mold surface and hence might be expected to reduce the tendency to instability in a fully coupled problem such as that of Li and Barber 131.

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306 F. YlGlT E T AL.

10-1 100 101 102 103 104 105

5 Fig. 7 Maximum perturbation in contact pressure as a function of 5.

The magnitude of this effect depends on the parameter [ that contains the mean heat flux Q,, [see Eq. (22)], so the effects of thermal capacity will become more important a t large cooling rates.

The perturbation in contact pressure approaches monotonically to a maximum value as solidification proceeds. The magnitude of this maximum is relatively inscnsitive to [, varying through a factor of about 4 in the range 0.1 < 3 <a.

This paper also demonstrates how the linear perturbation method of (Li and Barber [3] and Richmond et al. [2]) can be adapted to governing equations that require numerical solution, without requiring a full two-dimensional numerical solution. This technique is, in principle, extendable to more realistic material constitutive laws, including those with temperature-dependent coefficients.

APPENDIX

To obtain the solution in the limit as [+ m, corresponding to the case of zero Stefan number, it is convenient to define the new dimensionless parameters3 as

' Wc note that it would be eossible to solve the entire problem in terms of these parameters. I-lowcver, the tempcratyre field T is then generally a function of the parameter 5 as well as the dimcnsionlcss position Y and time T . By contrast, To depends upon Y, 0 only [see Eqs. (20). (23)-(2511, y d hcnce this presentation is to be preferred when discussing the temperature field. Notice also that S,, is identical with 5 of Eq. (71).

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EFFECT OF THERMAL CAPACITY ON THERMOELASTIC INSTABILITY 307

The Temperature Field

Substituting these relations into the asymptotic solution (29), (30) of the zeroth- order solution for temperature, we find that

Thus, the zeroth-order temperature field is always linear in the limit 5 - m, and the melt front advances linearly with time, as assumed by Li and Barber [3] and Richmond et al. [2]. However, the governing equation for the first-order term in the temperature (21) becomes

and, in the limit 5 - r m, it reduces to

with solution

which is not linear in 3, except in a limiting sense when ?< 1. To complete the solution, we express the boundary conditions (26)-(28) in the new notation and take the limit as 5 -t w, with the result

d 1 0 Q, =-

dl ; Q"

where we have used the limiting forms of Eq. (A.4) to substitute for the zeroth-order functions go and f,,.

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308 F. YIGIT ET AL.

Using Eqs. (A.7), (A.8) and eliminating the two arbitrary functions A(T), B(T), we obtain the ordinary differential equation

for the amplitude of the perturbation in the solidification front, with solution

The functions A(7), B(7) can then be found by back substitution, the resulting first-order temperature field being

The Stress Field

Equation (67) can be written in terms of the new variables [see Eqs. (A.l)-(A.311 in the form

and, after substituting for f l from Eq. (A.10, a suitable solution is found as

Finally, we write Eqs. (701, (75) in terms of the new variables and allow C -, m, with the result

= (1 - A2)[hZ?- 2A- f ] f c o s h ( ? )

where A = tanh(.$(~,(f))) = tanh(9).

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EFFECT O F THERMAL CAPAClTY ON THERMOELASTIC INSTABILITY 309

REFERENCES

1. 0. Richmond and N. C. Huang, Interface Instability During Unidirectional Solidification of Pure Metal, Pmc. 6rh Canadian Congress of Applied Mechanics, Vancouver, pp. 453-454, 1977.

2. 0 . Richmond, L. G. Hector Jr., and J. M. Fridy, Growth Instability During Non-Uniform Direc- tional Solidification of Pure Metals, ASME J. Appl. Mech., vol. 57, pp. 529-536, 1990.

3. N.-Y. Li and J. R. Barber, Thermoelastic Instability in Planar Solidification, Inr. J. Mech. Sci., vol. 33, pp. 945-959, 1991.

4. H. S. Carslaw and J. C. Jaeger, Conduction of Hear in Solids, Oxford University Press, London, England, 1956.

5. G. W. Evans 11, E. Isaacson, and J. K. L. MacDonald, Stefan-Like Problems, Quart. Appl. Math., voL 8, pp. 312-319.1950.

6. G. E. Bell, A Refinement of the Heat Balance Integral Method Applied to a Melting Problem, Inr. J. Heat Mass Transfer, vol. 21, pp. 1357-1361, 1978.

7. M. El-Genk and A. W. Cronenberg, Some Improvements to the Solution of Stefan-Like Problems, Int. J. Hear Mass Transfer, vol. 22, pp. 167-170, 1978.

8. D. Langford, The Heat Balance Integral Method, Int. J. Hear Mass Transfer, vol. 16, pp. 2424-2428, 1973.

9. T. R. Goodman, The Heat Balance Integral and its Application to Problems Involving Change of Phase, ASME J . Hear Tmnsfer, vol. 80, pp. 335-341, 1958.

10. T. J . Lardner, Biot's Variational Principle in Heat Conduction, A M 4 I . , vol. I , pp. 196-206, 1963. 11. A. Prasad and H. C. Agrawal, Biot's Variational Principle for a Stefan Problem, AIAA I.. vol. 10,

pp. 325-327, 1971. 12. M. El-Genk and A. W. Cronenberg, Solidification in a Semi-Infinite Region with Boundary

Conditions of the Second Kind: An Exact Solution, Letters in Hear Mass Transfer, vol. 6, pp. 321-327, 1979.

13. S. H. Cho and I. E. Sunderland, Approximate Temperature Distribution for Phase Change of a Semi-Infinite Body, ASME J. Hear Transfer, vol. 103, pp. 401-403, 1981.

14. N.-Y. Li and 1. R. Barber, Residual Stresses in Castings with Axisymmetric Solidification, ASME Tmnspori Phenomena in Manufacturing, FED vol. 90, Pfund et al., ed., pp. 27-34, 1989.

15. D. Shanks, Non-Linear Transformations of Divergent and Slowly Convergent Sequences, J. Math. Phys., vol. 34, pp. 1-42, 1955.

Receiued October 2. 1992

Address correspondence to Nai-Yi Li.

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