--transient thermal contact problem for axial crack in a hollow circular cylinder

8/12/2019 --Transient Thermal Contact Problem for Axial Crack in a Hollow Circular Cylinder

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Enginr rr ing Fr acture Mechanics Vol. 46, No. 6, pp. 931-943, 1993 0013-7944/93 6.00 + 0.00Printed in Great Britain 0 1993 Pcrgamon F ress td.

TRANSIENT THERMAL CONTACT PROBLEM FOR AXIALCRACK IN A HOLLOW CIRCULAR CYLINDER

CHAO-KUANG CHEN and BOR-LIH KU0Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C.

Abstract-The transient behavior of an axial-cracked hollow circular cylinder subjected to a suddenheating is investigated. It is shown that surface heating may induce compressive thermal stress near theinner surface of the cylinder which in turn may force the cracked surfaces to close together. Assumingthat the existence of the crack does not alter the temperature distribution, this problem can be dividedinto two parts and solved by the principle of superposition. First, the temperature and transient thermalstress distributions along the axisymmetric surface of the imaginary cylinder without a crack are obtainedby finite element/implicit time integration method. The calculated temperature and thermal stressdistributions are in good agreement with the values predicted by the analytical method. Secondly, theopposite senses of the stress distributions along the cracked surfaces, which are obtained previously,are treated as the traction boundary conditions, and the contact length and contact pressure of thereal cracked cylinder are obtained by a modified elimination finite element scheme. In this scheme, theconcepts of contact-node-pairs penetration, contact-double-forces and compliance matrix are introduced.The calculated results indicate that the contact length ratio becomes smaller when the crack length ratioincreases, and becomes larger as the radius ratio increases. Finally, the normal&d stress intensity factorfor the crack tip of the cylinder is obtained. It is shown that the larger the crack length ratio the higherthe stress intensity factor.

INTRODUCTIONFOR AN axial-cracked hollow cylinder subjected to thermal loading, two possible conditions mayoccur: the crack either opens wider or progressively closes together. When the inner surface of thecylinder is subjected to sudden cooling, it could result in very high tensile thermal stresses near theinner surface which tend to open the crack wider. Several authors [6-141 have solved crack openingproblems. Nied and Erdogan [6] analyzed the transient thermal stress problem for a circumferentialcracked hollow cylinder by the concept of superposition. Later on, Nied [A also solved the problemof thermal shock fracture in an edge-cracked plate by the same techniques. Noda and Sumi [8]obtained a transient solution for the problem of a Griffth crack in a thick plate by the finitedifference method. Delale and Kolluri [9] treated the thermal shock fracture problem for athick-walled cylinder with a radially embedded or edge crack. Noda et al. [ IO ] reated the thermalshock problem in a finite elastic body with an external crack. Kokini [l l] solved the thermal shockproblem in a cracked strip by the finite element method. Oliveira and Wu [12] calculated the stressintensity factors for internal and external axial cracks in hollow cylinders subjected to thermalshock by using a closed-form weight function formula. Chen and Weng [14] obtained coupledtransient thermoelastic response in an edge-cracked plate by the finite element/Laplace transformmethod. When the surface of the strip is subjected to heating, the surface experiences compressivethermal stresses which cause the opposing faces of the crack to come into contact. Nied [15]analyzed this kind of problem for an infinite edge-cracked strip by using a singular integralformulation.In the present study, the internal axisymmetric edge crack problem in a long hollow circularcylinder is depicted as shown in Fig. 1. The elastic hollow cylinder with an internal axial crackis initially at temperature T,,, and suddenly heated on the inner surface of the cylinder at theambient temperature T,. It is assumed that the existence of the crack does not alter the temperaturedistribution, and the problem can be separated into the following two parts and solved bythe principle of superposition, which has been widely used by many investigators. First, thetemperature and stress distributions in the imaginary body without a crack are calculated by finiteelement/implicit time integration method. Secondly, the opposite senses of the stress distributionsalong the crack surface, which are obtained previously, are treated as the traction boundaryconditions, and the contact length and stress intensity factor of the real cracked body are obtainedby a modified elimination finite element scheme.

931


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932 CHAO-KUANG CHEN and BOR-LIH KU0

Fig. 1. A cracked hollow cylinder initially at temperature T, heated by ambientinner surface. temperature T, on the

FORMULATION OF THE PROBLEM1. The thermal stress distri bution in the untracked hol low cyli nder

The basic equations of classical coupled thermoelasticity theory are summarized herein.Balance equations

(a) Mass conservation;=l+uk,&.

(b) Momentumor,, + Pb, = Pii 1 trii = aji .

(c) Energy (heat conduction)

wherePV = -4k.k + Q,

p. is the mass density in reference configurationp is the mass density in current configurationu, is the displacement vectoreij is the stress tensor6, is the body force vectorT is the absolute temperatureq is the entropy densityq, is the heat flux vectorQ is heat source.

(1)

(2)

(3)

A superposed dot () is used to denote material time differentiation and a comma (J) to denotepartial time differentiation with respect to coordinate xj (j = 1,2,3).Generalized constitutive equations

tt = rl0 + c.0 + f B jc 9 (4)


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933ransient thermal contact problem

whereC,,, is the tensor of elastic moduli

cij is the strain tensorKij is the termal conductivity tensorC, is the specific heatT,,, is the maximum temperature difference between the applied temperature on the plate and T,8 is the dimensionless temperature, 8 = (T - TJT,,,.Substituting eq. (4) into eq. (2) and eq. (3), we have the resulting field equations in terms ofdisplacements and temperature as follows:

P& + (B, Tm0 - C,,UU~,I,,= phi (5)&ed+ BijT4sjlTm (Kje,,), I = Q/T,, (6)

where flljTti,j/Tm is called the thermoelastic coupling term. The nonlinear energy balanceequation (6) can be linearized to the following form if the reference temperature To is chosen suchthat (T - TJT, 4 1:PC,~ + BilTo .llTm - (K,e.j), i= QlTm. (7)

In comparing eq. (6) with eq. (7), it is clear that the influence of the coupling term on solutionswill be always higher than that predicted by eq. (7). If the material is assumed to be homogeneousand isotropic, eqs (5) and (7) become~Uk,ki+CL(Ui,kk+Uk, )-a,T,(3~+2~)e,,-pPii i+pbi=O (8)

I@,,= (31 + 2p)a,T& /T,,, + pC,d - Q /T,,,, (9)where I and ~1 re Lames elastic constants [A = vE/(l + v)( 1 - 2v), p = E/2( 1 + v)], a, is the linearthermal expansion coefficient, E is the modulus of elasticity, and v is Poissons ratio.Finite element formation

Spatial discretization. In the finite element method, the variation within any element isdescribed in terms of the nodal values by means of the shape functions. Thus, we introduce twosets of shape functions of the displacement and temperature field as follows:

{u}= {:] = N]{u}I

n8 = c N;e, = w]{eq.i= IBy substituting into the integro-differential equations offrom eqs (8) and (9) with eqs (10) and (1 l), we obtain

i.e.

(10)(11)

variational form, which are derived

[KY)] ; [G)j+ . 0 i [@I

Pfl{& + [Cl@) + PWI = {FL (12)


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934 CHAO-KUANG CHEN and BOR-LIH KU0in which

[AC] = s p[N]qN] du[W] = su I $ iv]T[B2] dv + j ~[Nl~Nl{P}[~,Idom [P] = J pC [N]qN] du[WI = s P, IVW, 1do

n n[lv$-).j J K[M]qM] do + h[N]tN] d.s Js

1h: aN- 0 T 0 ..-$oar[B,]= ;

w aNaZ 0 2.. 0;I- 0 3 0.3 0r r r1N, aN, alv, alv, aN* alv,---- --a i?r a ar . a ar

[B,]= %+:[ 2 z ?,I $.. ?$?+~z 1{zP} is the nodal point displacement vector{P} is the nodal point temperature vector.

Temporal discretization. In this paper, we use implicit one-step time integration method forsolving the time dependent problem. It is assumed that the physical values vary linearly in a discretetime integral At, from t,, to tn+ where tn+ t, + A t{d}n+a a(d),+, + (1 - a){d}, {d},+, = {dlniai{dl = (dj.+iL id}, (13)

where (d} is the vector of physical values (including the displacement and temperature vector);tn+a t, + a At; and CYs a parameter, taken to be in the interval [0, 11. Some well-knownintegration methods are identified in Table 1.

Tablea

22131

MethodForward Euler (explicit)Central differenceGalerkin schemeBackward Euler


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Transient thermal contact problem 935By neglecting the dynamic effect and the thermoelastic coupling term, and using the Galerkin

scheme (i.e. a = 2/3), eq. (12) becomes

i.e.Nln+Mn+a = {O,C (14)

An eight-node isoparametric quadrilateral element is used for both displacement andtemperature calculations [3-51. The finite element mesh used for simulation is shown in Fig. 2a.The problem of this paper is axisymmetric about the z-axis, and therefore only one quarter of thehollow circular cylinder needs to be modeled. With the temperature distribution in the untrackedcylinder now known, it is easy to determine the transient thermal stress by the classical finiteelement formulation.2. The contact problem in an isothermal cracked hollow cylinder

When the inner surface of a hollow cylinder with an internal axial crack is subjected to heating,it may induce very high compressive thermal stresses near the inner surface of the cylinder(4

transitionlement

quarter ointelementFig. 2. a) Finite element mesh used for simulation of part 1 uncraclced cylinder 99 elements, 338 nodes).b) Finite element mesh used for simulation of part 2 cracked cylinder 150 elements, 503 nodes).


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936 CHAO-KUANG CHEN and BOR-LIH KU0(4

(c)

Fig. 3. (a) Cracked hollow cylinder suddenly heated on the inner surface with crack contact length c.(b) The concept of contact double force. (c) Determination of compliance matrix C,, for contact pairs.

0.0 0.2 0. 4 0.6 0.8 1. 0Fig. 4. Transient temperature distributions as a function of R* for Biot number /I = a, and Ri/Ro = 0.3[T* = (T(R*, t*) - TO)/(T, - T,), R* = (R - Ri)/(Ro - Ri), f* = tct/pC,Rz, j? = ~L/K].

0. 6G

0. 4

Fig. 5. Thermal stress distributions as a function of RR for fi = m and Ri/Ro = 0.3[o = (1 -v). a$#*, 1*)/5x,. (T, - To),R* = (R - Ri)/(Ro - Ri)].


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Transient thermal contact problem 937

Fig. 6. Ttmsicn t temperature dis~butio~ as a function of R+ for B = m and RijRo = 0.5(T, = (T(R* , t* ) - T,)/(T, - T,), R* = (R - Ri)/(Ro - Ri), t* = xf/pC,Rz].

which force the crack surface together over a contact length, and a region of tensile stress nearthe tip of the crack which tends to open the crack as shown in Fig. 3s. However, the crack contactlength E in Fig. 3a is an unknown. Therefore, we introduced a modified elimination finite elementmethod [13 to solve this contact problem, as follows.The contact forces and area are determined iteratively by eliminating the nodes having tensilenormal contact forces and imposing the compatibility conditions along the contact surface.The concepts of contact-node-pairs relative displacement or penetration, contact double forceand compliance matrix are introduced in this scheme. The compliance matrix is defined as therelative displacement induced by two unit contact forces, i.e. unit double force. In Fig. 3b, alongthe crack contact area, the forces and displ~ments at node pair are denoted by pi and 6,.As the configuration reaches an equilibrium state, the relations between contact forces anddisplacements will be

Fig. 7. Thermal stress distributions as a function of R* for fl= co and Ri/Ro -0.5[u& = (1 v) -u& (R*, I )/I& , T, - T,), R* = (R - Ri)/(Ro - Ri)l.


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0.6T

0. 4

.I3Fig. 8. Transient temperature distributions as a function of R* for /I = co and Ri/Ro = 0.7

[T* = (T(R* , t* ) - T ,)/(T, - T ,), R* = (R - Ri)/(Ro - Ri), t* = u/pC,Ra].

where 112s the total number of contact node pairs, {s;} is the vector of the total penetration atthe pair due to external load, [Cij] is the compliance matrix, and (p;} is the vector of contactforces. The compliance coefficient C, represents the relative displacement of node pair producedby a unit contact double force acting at node pair j as shown in Fig. 3c. Because the contact areais not known in advance, the iterative procedure has to be used in the finite element scheme.A procedure of the computer implementation is presented in Table 2. By using this scheme, wecould obtain the contact length and the contact pressure Pj along the contact surface which isexpressed as follows:

where pj is the contact force in node pair j, and Aj is the sum of the areas of the surface elementssurrounding the pair j.The problem of an isothermal cracked elastic body may be conducted using the stressesobtained in part 1. These stresses, with opposite sign, become the crack surface tractions in theboundary value problem and can be solved by finite element method with a triangular quarterpoint isoparametric singular element [5]. The finite element mesh is adopted as shown in Fig. 2b.The first and second modes of stress intensity factors can be expressed as follows:

K, = [4(* - d) - (C e )I

Table 2. The steps of modified FEM for solving the contact pressure1. Establish a set of contact nodal points.2. Assume that contact will not take place, then calculate the relative displacement or penetration

along a possible contact area.3. Form the compliance matrix [C,,] which is defined as in the text.4. Delete all the separated node pairs (i.e. S, > 0) and the corresponding columns and rows in thecompliance matrix.5. Referring to eq. (17), calculate the contact double forces needed to cancel the penetration.6. Delete all the node pairs where the contact forces are in tension (i.e. p, > 0), and delete thecorresponding columns and rows in the compliance matrix.7. Repeat procedures 5 and 6 until all the contact forces are. in compression (i.e. p, c 0).8. Calculate the normal contact pressure along the contact surface.

(17)


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Transient thermal contact problem 939

0.8

0.6

Fig. 9. Thermal stress distributions as a function of R* for /3 = co and Ri/Ro = 0.7[a& = (1 v) . a (R*, P /Ea, .(To Z,),R* = (R - Ri)/(Ro - Ri)].

where rc = 3 - 4v for the plane strain problem; ub, uC, u,, u, and trb,v,, u,+ v, are nodal displacementsin tangential and normal directions with respect to the crack surface, respectively; G is the shearmodulus, and 1, s the element length of the triangular element. Since the accuracy of the stressintensity factor is strongly influenced by the element length of the triangular singular element,a transition element between regular and singular elements is also adopted to smooth the changeof nodal displacements.

R SULTSFor the thermal shock problems, it is convenient to specify results in terms of the followingdimensionless variables:

R* = (R - Ri)/(Ro - Ri), t* = Kt/pC,Rf, T* = (T(R* , t* ) - T,)/(T, - TO),a&=(1 -v).t~& (R*,t* )/Ea;(T,- To).

Ri/Ro=O.J

-1

Fig. 10. The ratio of the crack surface contact length to the overall crack length as a function of 1* forji = co and Ri/Ro = 0.3 (t* = ~r/pC,Ra).


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Rn/ Ro=OS

Fig. 11. The ratio of the crack surface contact length to the overall crack length as a function of t * forB = co and RilRo = 0.5 (I* = rct/pC,R:).

The numerical results detailing the transient temperature and thermal stress in the untrackedhollow circular cylinder have been obtained for the radius ratios of Ri/Ro = 0.3,0.5, and 0.7, andare shown in Figs 4-9; the results are in good agreement with the analytical solutions for the crackopening case [6,9].If the inner surface of the hollow cylinder is suddenly exposed to a heat source, it mayinduce high compressive thermal stress near the inner surface which in turn may force the crackedsurfaces to close together. The crack surface contact length and normal pressure can be determinedfrom eqs (15) and (16) by using a modified elimination finite element scheme. Figures lo-12show the ratio of the crack surface contact length L to the overall crack length a, as a functionof non-dimensional time t* for various crack length ratios a/(Ro - Ri) and the radius ratiosRi/Ro = 0.3,0.5, and 0.7. It can be seen that the larger the crack length ratio the smaller the contactlength ratio ~/a, and the larger the radius ratio the larger the contact length ratio. Figures 13-15,respectively, show the contact pressure distribution with a/(Ro - Ri) = 0.4, 0.5, and 0.6. In thesefigures, the dimensionless contact pressure, P*, is defined as (1 - v)P/Ecr,(T, - To). The contactpressure distributions for Ri/Ro = 0.5 and 0.7 are similar to Figs 13-15. The normalized stressintensity factors, obtained from eq. (17), are plotted in Figs 16-18. It can be seen that the largerthe crack length ratio, the higher the stress intensity factor.

0.6E/O

0.4

Ri / Ro=0.7

:-:-_ 0.6 : -

Fig. 12. The ratio of the crack surface contact length to the overall crack length as a function of r* for1 = cc and RilRo = 0.7 (t * = ~t/pC, Ra).


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Transient thermal contact problem

0. 6P

0. 4

0. 00.0 0. 2 0. 4 0. 6 0. 8 1

RFig. 13. Normal&d contact pressure distributions along contact surface as a function of R* for the cracklength ratio a/(Ro - Ri) = 0.4 and Ri/Ro = 0.3 [P* = (1 - v)P/Ea,(T, - TO), P = tcf/pC,R fl = 001.

0. 8

0. 6P

0. 4

0. 2

0. 00. 0 0. 2 0. 4 0. 6 0.8 1

R

941

Fig. 14. Normalized contact pressure distributions along contact surface as a function of R* for the cracklength ratio o/(Ro - Ri) = 0.5 and Ri/Ro = 0.3 [P* = (I - v)P/Ea,(T, - T,), t* = ~r/pC,Rt, /I = co].

0.8

0.6P

0. 4

0. 2

0. 00. 0 0.2 0. 4 0. 6 0.6 1

Fig. 15. Normali& contact pressure distributions along contact surface as a function of R* for the cracklength ratio a/(Ro - Ri) = 0.6 and Ri/Ro = 0.3 [P = (1 - v)P/Ea,(T, - T,), I = ~f/pc,R , 1 = co].


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942 CHAD-KUANG CHEN and BOR-LIH KU0Ri/Ro-0.3

0.05

Fig. 16. Transient &rem intensity factor at the crack tip of cylinder for b = a3 and Ri/Ro = 0.3[K* = (1 - v) .K,/&ka, * T. - T,), I * = xr/pC,R:].

Ri/Ro=O.S0.05

0.04

0.03K

0.02

0.01

0.00

- o/ Ro-Ri)

:-

Ri/Ro=0.7

Fig. 17. Transient stress intensity factor at the crack tip of cylinder for /I = Q) and Ri/Ro = 0.5[KS = (1 - v) *K,/,/&a, * T, - To), r* = rct/pC,R:].

Fig. 18. Tmuien t StrsU iDuty factor at the cmck tip of cylinder for j = Q) and Ri/Ro = 0.7[K+ - (1 - v) . K, /,/ii&~, * T,, To), t+ = u/pC,R;].


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Transient thermal contact problem 943CONCLUSIONS

In the present paper, a powerful numerical analysis which is called implicit finite elementmethod is employed to analyze the transient thermal stress problem of an internal axial-crackedhollow circular cylinder subjected to heating. The calculated temperature and thermal stressdistributions are in good agreementwith the values predictedby the analytical method. A simplenumericalprocedure s used to analyze the crackedsurface contact problem by using the conceptsof contact node pairs and contact double forces. The numerical results of the contact length andthe normal contact pressureare obtained. The larger he crack length ratio the smallerthe contactlength ratio, and the larger the radius ratio the larger the contact length ratio. Finally, we alsoobtain the normalized stress intensity factor for the crack tip of the cylinder, and it is shown thatthe larger the crack length ratio, the higher the stress intensity factor.

REFERENCES[l]. A. Boley and J. H. Weiner, TIvory of Themnil Stresses. John Whey, New York (1960).[2]T. Belytschko and T. J, R. Hu&es, C~~~u~~~ Methods for Transient Anufysis, Volutw 1. Non-Holy,Amsterdam (1983).[3] R. S. Barsoum, Gn the use of isoparametric finite elements in linear fracture mechanics. Int. J. nunrer. Meth. Engng10,25-37 (1976).[4] P. P. Lynn and A. R. Inggrafta, Transition elements to be used with quarter point elements. Inf. J. nunrep.Meth. Engrtg17, 1031-1036 (1981).[S] V. Murti and S. Valliappan, A universal optimum quarter point element. &gng Frucfzue Mech. 2 & 237-258 (1986).[q H. F. Nied and F. Erdogan, Transient thermal stress problem for a circumferentially cracked hollow cylinder,J. t&ernt. sfies.res 6 l-14 (1983).[7] H. F. Nied, Thermal shock fracture in an edge-cracked plate. J. therm. Srresses ,217-229 (1983).[8] N. Noda and N. Smni, Stress intensity factor for transient thermal stress of a thin plate with a Gritllth crack.

J. tkrm. Stresses 8, 173-182 (1985).[9] F. Delale and S. P. Kolluri, Fracture of thick-walled cylinders subjected to transient thermal stresses. 1. tkrm. Stresses8.235-248 (1985).[lo] N. Noda, Y. Matsuna8a and H. Nyuko, Stress intensity factor for transient thermal stresses in an infinite elastic bodywith an external crack. J. &erm. Stresses9, 119-131 (1986).[ll] K. Kokini,On the useof the finite element method for the solution of a cracked strip under thermal shock.&gng Frachw e Mech. 24, 843-850 (1986).1121 R. Oliveira and X. R. Wu. Stress intensity factors for axial cracks in hollow cylinders subjected to thermal shock.. Engng Fracture Mech. 27,185-197 (1987).[13] T. C. Chen and C. I. Wen8 Generalized coupled transient thermoelastic plane problems by Laplace transform/finiteelement method. J. ad. h& h. 55.377-382 (1988).u41

115)WIU7jWI

T. C. Chen and C. I. Wcng, Coupled ~~ient.~e~~i~tic response in an edge-cracked plate. &grtg Frucfwe Me&.39,915~925 (1991).H. F. Nied, Thermal shock in an edgecrackad plate subjected to uniform surface heating. &gttg Froefure Mech. 26,239-246 (1987).A. Francavilla and 0. C. Zienkiewicz, A note on numerical computation of elastic contact problems. Int . J. numer.Meth. Engn g 9, 913924 (1975).T. D. Sachdeva and C. V. Ramakrishnan. A finite element solution for the two-dimensional elastic contact problemswith friction. fnt. J. nunter. Meth. Eagtrg 17, 1257-1271 (1981).H. S. Jing and M. L. Liao, An improved finite element scheme for elastic contact problems with friction.Cornput. Srnretures 35, 571-578 (1990).

(Received 25 January 1993)

--transient thermal contact problem for axial crack in a hollow circular cylinder

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