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A direct automated procedure for frictionless thermoelastic contact problems
A DIRECT AUTOMATED PROCEDURE FOR FRICTIONLESS THERMOELASTIC CONTACT PROBLEMS F. F. MAHMOUD* and A. G. EL SHAFEIt Department of Mechanical Engineering, Zagazig University, Zagaxig, Egypt Abstract-_The objective of this paper is to present an automated direct procedure to solve the problems of two bodies in elastic frictionless thermal contact. The proposed scheme of solution determines incremently the equilibrium configuration of the system by solving uncoupled thermo-elastic stationary problem for each increment of thermal loads. The contact area is unknown a priori and the increment of load is calculated in such a way to yield just one additional contact pair. Conventional thermal boundary conditions are adopted throughout the contact interface. Results of two examples of different nature are presented for illustration.
INTRODUCTION THE ANALYSIS of thermo-elastic contact problems is one of the important engineering problems where thermal effects plays an important role in the performance of wide variety of machine elements. Most of the available models treat the problem mathematically and limited only for specific geometry and boundary conditions[l-31. Those mathematical models adopt, mostly, the conventional thermal boundary conditions which assume that the contact interface has no resistance to heat flow, while no heat transfer takes place between the two solids in the separation zone. As might be suspected, the difficulties encountered with the conventional boundary conditions arise from the discontinuity in the boundary conditions at the interface, where there is an abrupt change from no resistance to heat flow in the contact zone to complete thermal insulation in the separation zone. Barber[4] proposed a modification on the conventional thermal boundary conditions. He introduced a zone of imperfect contact located in-between the contact and separation zones. The new zone is not fully insulated, but offers finite resistance to heat flow, however the contact pressure is vanished on it. The analytical solutions for the mathematical models require considerable mathematical effort, especially when the geometry of the two bodies are complicated. Also, should the boundary conditions be changed then the problem requires extra effort to solve. For the sake of simplicity and generality we propose a numerical model to motivate easily application on the problems having general geometry and boundary conditions. The proposed model exploits the finite element method in the frame work of direct automated procedure[5,6] which treats the nonlinear problem as a sequence of piecewise linear ones. For simplicity the proposed model adopts the conventional thermal boundary conditions, however, Barber boundary conditions could be considered. STATEMENT
OF THE PROBLEM
Consider two smooth elastic bodies are not only pressed together by external loads but also exchange heat by conduction. By applying loads, or thermal effects, contact regions between the two bodies will advance or recede according to the type of contact[7]. The problem is non-linear in the sense that the kinematic boundary conditions, and consequently the total number of constraints, are depending on the external load and thermal effect capacity. For the sake of simplicity, the present study is concerned only with steady state problems where time independency of functions are assumed. The attention is centered on homogeneous and isotropic materials whose elastic and thermal properties do not depend on the temperature. According to the aforementioned assumptions, the problems is considered uncoupled one such that the * Professor, on leave to Kuwait University. t Graduate student. 1.57
15x
F. F. MAHMOUD
and A. G. EL SHAFEI
temperature field and displacement field could be solved independently. Conventional thermal boundary conditions[3] are adopted where there is no resistance to heat flow in the contact regions, while the bodies in the separation regions do not exchange heat. Shortly, perfect contact and perfect insulation thermal boundary conditions are adopted. The assumption of perfect contact is based on the smoothness of the contact surfaces, where, the thermal contact resistance depends basically on the roughness of surfaces[8]. Consequently the temperature distribution and heat flux should be continuous across contact zone. Furthermore, mechanical boundary conditions should satisfy the continuity of displacement field and contact tractions along the contact zone, while, separation zone surface are free tractions. Thermoelastic contact problem is less comfortable to handle mathematically as a boundary value problem for the difficulties in presenting the contact boundary conditions. In general, the contact area is unknown a priori and by increasing the thermal loads monotonically, the contact region may advance or recede. In the present paper, a new finite element model is submitted to simulate the thermo-static behavior of elastic bodies in contact. The model exploits the direct procedure developed by Mahmoud, Salamon et al. [5,6] for the solution of frictionless thermo-elastic contact problems. The model is based upon an equilibrium formulation which is piecewise linear over small increment of thermal load. Over each increment, the contact configuration is considered constant. For each increment, the uncoupled thermo-elastic boundary value problem is tackled. Firstly, the heat conduction boundary value problem is solved to obtain temperature distribution and consequently, the equivalent thermal stresses exerted on the elastic bodies are calculated. Conventional thermal boundary conditions are adopted. Once, the external thermal loads are determined, the displacement field problem is solved considering the permanent and contact boundary conditions. According to the automated direct procedure[5], the contacts are detected through calculation of the interpenetration between the two bodies. The interpenetration function play a unique role in defining the next pair. Once the new contact pair is detected, an additional boundary condition is activated representing one more constraint. The new contact pair is encountered at the site produce maximum value of the interpenetration function. A scaling factor is calculated in such a way to prevent the interpenetration at the detected new contact pair. Once the scale factor is determined, the residual of the applied thermal loads is calculated and modified uncoupled thermo-elastic model need to be set for the next increment by considering the new contact pair. The procedure is continuing until no additional contact pair to yield or the full capacity of thermal loads is completely used. The procedure is adaptive in the sense that the increment of load is just sufficient to avoid the maximum interpenetration at the site of the new detected contact pair. MATHEMATICAL
FORMULATION
Consider two linear elastic bodies shown in Fig. 1 in the undeformed state. Following Nowinski[9], the thermo-elastic contact problem is simulated by two uncoupled boundary value models. The first one representing the heat conduction between the two bodies taking into consideration the conventional boundary conditions where no resistance to heat flow in the contact regions while the separation regions do not exchange heat.
Fig. 1.
Frictionless thermoelastic
The governing following form:
differential
equation
contact problems
and boundary
conditions
V*tI+Qlk=O.
IS9
of the first model take the
(1)
The boundary r, is maintained at a temperature
(2)
ftk y),
8=
where the portion of the boundary P2 is subjected to heat flux
at3 4
=
k -
an
=
Y).
fib,
By applying the conventional thermal boundary conditions on the contact and separation zones, P, and P, respectively, the following boundary conditions are given kfz),=
k2($$))*:
per,.
Where the subscripts 1 and 2 stands for the first and the second body and p is an arbitrary position along contact zone r,, =M(p):
Per,,
i=l,2
(5)
where, 8 is the temperature increment field, k is the thermal conductivity, h is the convective heat transfer coefficient, Q is the heat generated internally per unit area of the material, and n denoted the outward normal to the surface. The governing eq. (1) and the associated boundary conditions (2)~(5) represent a complete formulation of a two dimensions steady state conduction and simple convection heat transfer problem of two bodies in contact. The displacement field mode1 of the elastic bodies subject to thermal load is represented by the following differential equations and boundary conditions,
(6) where y is Poisson’s ratio, E is Young’s modulus. Here A and p are the familiar Lame elastic constants which are represented by E
EY
cL=- 2(1+y)’
*=(l+y)(l-27).
The interface boundary conditions of the elastic field are given by U:(P) = d(p):
PE
rc,
(8)
where the superscripts 1, 2 stand for the first and second body but the subscripts 2 denotes the normal component of displacement, s;*(p)
=
SWAP)
=
0:
s:*(p) = s;*(p) 4 0: u:(p) - u:(p) - G(p) S 0:
Pa
PEP, p+
(9)
IhO
F. F. MAHMOUD
By neglecting the surface tractions, boundary conditions
F
81,
the rest of boundary
=At?.&+ @Ui,,
Where (Y is the coefficient normal and e = at+.
and A. G. EL SHAFEI
+
i=l,2,
~aUj.iij~
of thermal expansion,
THE NU~~ICAL
rather than I,,
has the following
j= 1.
li is the direction
cosines of the outer
MODEL
To solve the problem, the two bodies are descretized into finite elements. Reftning the mesh in critical area and throughout the potential contact region. The model is an incremental one which exploits the procedure developed by Mahmoud, Salamon et al. [5,6]. The procedure is a piecewise linear treatment for the uncoupled thermo-elastic contact problem. For each increment “nr” of the external thermal effects the permanent and boundary conditions are set for the both uncoupled models, heat conduction and thermoelastic models. The response of the heat conduction model 0 follows from the solution of the linear problem by using the conventional procedure of the finite element method
Where [keJm represents the equivalent stiffness matrix of tbe heat conduction problem for the increment m. It should be mentioned that [kJ’ satisfies all permanent and current contact boundary conditions. Moreover,
Hence, {f#}” is the residual thermal load vector yet available to apply on the system. Once the temperature increment vector {d}‘” is calculated, the thermo-elastic model is ready to handle. The temperature (O}m is translated to equivalent thermal stresses and consequently to force vector by exploiting the fundamentals of thermo-elasticity. The response of the elasto-static contact model follows from the solution of the inequality linear problem
42
-
Uj2 -
Gj S 0,
i, j
E
rp,
(141
Where fk,]” is the equivalent stiffness matrix of the elastostatic problem taking into consideration the permanent kinematic boundary conditions, {u}” is the displacement vector, {fJ}” is the static force vector, Ui2, uj2 are the normal components of displacements of the potential contact pair ij as shown in Fig. 1, and Gij is the gap between the node of pair ij which belongs to the potential contact regions I,. The elasto-static contact model eqs (13) and ( 14) is solved by using the direct automated procedure[5] which adopts the interpenetation measure as a criterion to detect a new contact pair. The induced interpenetration, rr, of the portion of surfaces along the potential contact region IP is given by
The inte~enetration function plays a unique role. We can notice from (15) that if rij is negative, zero or positive, node pairs i, j are either separated, in contact or interpenetrating, respectively. From the set of node pairs in rP associated with positive value of Tij that with the maximum value is selected as the next contact pair to close. Designating this node pair as ZJ, the scale factor S” required to establish contact is
Frictionless thermoelastic
Ihl
contact problems
S” = Gg/(u;“z - u;),
(16)
TrJ > 0,
so that the static load required to close gap GrJ is {AK}‘” = Sm{F,}“, and consequently
(17)
the thermal effect required to generate this static load is
and the residual thermal effects yet to be applied in the next increment
- Sm{fe}m. W m+’= {fe}rn In the process, the accumulated
displacement
temperature
(19)
field is computed
{u}+-(u)+ Sm{u)m. and the accumulated
is
W)
field is
Sm{elm.
{e}+-(e)+
(21)
The remaining gaps updated to G$‘+’ = Gij'--S"(u%u,$,
i,j~r~.
(22)
Prior to the following new increment, an additional constraint should be considered for the new activated contact pair and both [ke]“’ and [%I” should be augmented by using elementary matrix transformations. The procedure continues with application of the new residual load until either no further interpenetration is detected or all candidate node pairs are closed. To satisfy the heat balance across the contact interface, the temperature of the consolidated nodes Z and J should be the same. For the sake of simplicity and as an accepted degree of approximation, the temperature is taken as the average of both nodes Z and .Z.
EXAMPLE
PROBLEMS
The results of two examples will be presented. The examples are solved using the proposed incremental procedure which is coded in a computer model appended to an in-house finite element code. One of the problems is chosen as a skeletal structure while the other one represents a Hertzian problem. Example 1 Two symmetrical pin-jointed frames in contact. Two trusses are loaded by thermal loads. The geometry and boundary conditions of the system are shown in Fig. 2. The thermal loads on the members are prescribed as follows,
e,, = e,4 = e,6 = e,8 = 8, = e3 = e, = es= ioo~c, e12 = e,, = & = 8, = 50°C. The remaining members are free of thermal effect. The history of solution of the system is illustrated in Table 1. Example 2 Two symmetrical elastic hollow cylinders under symmetrical thermal loads. This example represents the thermo-elastic frictionless contact between two symmetrical elastic hollow
862
/------ typicaL -+-+--
5ots--+-5oc+ Dim.in mmr Fig. 2.
cylinders under the action of symmetrical thermal loads. The initial geometry and permanent boundary conditions are shown in Fig. 3a. The problem can be viewed as a cylinder on a rigid isolated plane due to its symmetry. Furthermore, the entire cyhnder need not be modelied[I0]. Only a 1.925” portion is modelled and the finite element mesh is ilhtstrated in Fig. 3b. The input load is a constant heat flux distributed over the inner surface boundary,
The thermal and mechanical
properties
of the materiat of each cyfinder are:
k = 0.043 W/mm ‘C, E = 200 x IO3 N/mm2,
h = IO x I Vh W/mm’ “C
y = 0.25 and LY= 15 x IO-” l/“C.
The incremental growth of the contact region vs thermal toad is shown in Fig. 4a. The contact pressure distribution is illustrated in Fig. 4b.
First node displacement (mm)
Second node displacement (mm)
Contact decision
2.0 1.0 2.0
0.57188 0.42906 LO
-0.57188 -0.42906 -1.0
No No Yes
0.856 0.142 0.0
0.73776 0.50 1.0
-0.73776 -0.50 -1.0
No Yes
3
0.525 0,o 0.0
1.O 0.5 1.0
-1.0 -0.5 -1.0
Yes Yes Yes
4
0.0 0.0 0.0
I.0 0.5 1.0
-1.0 -0.5 -1.0
Yes Yes Yes
Increment number 1
2
Gaps* (mm)
*Initial gaps {mm). C& = 2; C& = 1; G11.,2= 2,
% of load
Contact force W)
38.1
0.0 0.0 0.0
50.8 Yes 74.1
100
0.0 0.0 7113 0.0 7326 18651 26266 6725 31499
Frictionless themoelastic
cantact problems
(a) t
Y
(b)
X
Dim. in mms
Fig. 3.
ih4
.f .1~ .
and A. G. EL SHAFEI
F. F. MAHMOUD
‘.O
(a)
l
0.9 3
0.8 -
_* O.?! B
E
.o “u e L
.\ 1~
;‘;I 0.4
/
0.30.2-
q,,- 1.0 x IO+W/mm’ q,,- I.3x10-3W/mm2
.
-
/ i
0.2 0.4 0.6 0.6 1.0 1.2 1.4 Contact half- length
0.1 0.3 o.5 0.7 0.9 I.1 I.3 I.5
(mm)
Contact
half-length
(mm)
Fig. 4.
CONCLUSIONS The problems of thermoelastic frictionless contact are solved by using a modified automated direct method. The procedure consider the problem as uncoupled thermo-elasticity problem and tackled it in an incremental manner. Throughout each increment the heat conduction problem is solved at once to give temperature distribution and consequently equivalent thermal stresses and forces are obtained, Once the static forces are yielded the direct automated method developed by Mahmoud and Salamon is used to detect a new contact pair. The common frictionless constraints and conventional thermal boundary conditions are adopted through the contact interface . The proposed model is applied on two examples of different nature to detect the growth of contact region vs thermal load. Furthermore the contact pressure distribution is also determined. The results have an evidence that, the system is in complete heat balance and complete mechanical equilibrium. REFERENCES [I]
M. Comninou, J. Dunders and J. Barber, Heat conduction through a Rat punch. ASMEJ. Appl. Me&. 48, X7 l-875 (IV&l). [2] M. Comninou, J. Dunders and J. Barber, Planer Hertz contact with heat conduction. ASME J. Appl. Me&. 48. 549-558 (1981). [3] C. Panek and J. Dunders, Thermo-elastic contact between bodies with wavy surfaces. ASME .I. Appt. Me&. 46, 8.54-860 (1979). [4] J. Barber, Indentation of semi-infinite solid by a hot sphere. Int. J. Mech. Sci. 15, X13-819 (1973). [S] F. F. Mahmoud. N. J. Salamon and R. Marks, A Direct automated procedure for frictionless contact problems. Inl. _I. numer. Method. Engng 18, (1982). [6] F. F. Mahmoud, N, J. Salamon and T. P. Pawlak, Simulation of structural elements in receding/advancing contact. Camp. Structures 22 ( 1086). [7] J. Dunders, Properties of elastic bodies in contact. in The Mechanics of Contac? Between Deformable Bodies, (Edited hy J. T. tilker and A. D. dePaten). Delft University Press ( 1975). [X] J. P. H&man, Hear Trunsfer, Fifth edition. McGraw-Hill, New York (1% I). 191J. L. Nowinski, Theory of T~r~el~s~fcf;y with A~~jCQ~nS. Sijthoff and N~~~rdho~ (1978). [IO] W. R. Marks, Solution of frictionless contact problems by a conjugate gradient technique. M. SC. thesissubmitted to the University of Wisconsin-Milwuakee (1970). (Received
2 July 1987)
INTRODUCTION THE ANALYSIS of thermo-elastic contact problems is one of the important engineering problems where thermal effects plays an important role in the performance of wide variety of machine elements. Most of the available models treat the problem mathematically and limited only for specific geometry and boundary conditions[l-31. Those mathematical models adopt, mostly, the conventional thermal boundary conditions which assume that the contact interface has no resistance to heat flow, while no heat transfer takes place between the two solids in the separation zone. As might be suspected, the difficulties encountered with the conventional boundary conditions arise from the discontinuity in the boundary conditions at the interface, where there is an abrupt change from no resistance to heat flow in the contact zone to complete thermal insulation in the separation zone. Barber[4] proposed a modification on the conventional thermal boundary conditions. He introduced a zone of imperfect contact located in-between the contact and separation zones. The new zone is not fully insulated, but offers finite resistance to heat flow, however the contact pressure is vanished on it. The analytical solutions for the mathematical models require considerable mathematical effort, especially when the geometry of the two bodies are complicated. Also, should the boundary conditions be changed then the problem requires extra effort to solve. For the sake of simplicity and generality we propose a numerical model to motivate easily application on the problems having general geometry and boundary conditions. The proposed model exploits the finite element method in the frame work of direct automated procedure[5,6] which treats the nonlinear problem as a sequence of piecewise linear ones. For simplicity the proposed model adopts the conventional thermal boundary conditions, however, Barber boundary conditions could be considered. STATEMENT
OF THE PROBLEM
Consider two smooth elastic bodies are not only pressed together by external loads but also exchange heat by conduction. By applying loads, or thermal effects, contact regions between the two bodies will advance or recede according to the type of contact[7]. The problem is non-linear in the sense that the kinematic boundary conditions, and consequently the total number of constraints, are depending on the external load and thermal effect capacity. For the sake of simplicity, the present study is concerned only with steady state problems where time independency of functions are assumed. The attention is centered on homogeneous and isotropic materials whose elastic and thermal properties do not depend on the temperature. According to the aforementioned assumptions, the problems is considered uncoupled one such that the * Professor, on leave to Kuwait University. t Graduate student. 1.57
15x
F. F. MAHMOUD
and A. G. EL SHAFEI
temperature field and displacement field could be solved independently. Conventional thermal boundary conditions[3] are adopted where there is no resistance to heat flow in the contact regions, while the bodies in the separation regions do not exchange heat. Shortly, perfect contact and perfect insulation thermal boundary conditions are adopted. The assumption of perfect contact is based on the smoothness of the contact surfaces, where, the thermal contact resistance depends basically on the roughness of surfaces[8]. Consequently the temperature distribution and heat flux should be continuous across contact zone. Furthermore, mechanical boundary conditions should satisfy the continuity of displacement field and contact tractions along the contact zone, while, separation zone surface are free tractions. Thermoelastic contact problem is less comfortable to handle mathematically as a boundary value problem for the difficulties in presenting the contact boundary conditions. In general, the contact area is unknown a priori and by increasing the thermal loads monotonically, the contact region may advance or recede. In the present paper, a new finite element model is submitted to simulate the thermo-static behavior of elastic bodies in contact. The model exploits the direct procedure developed by Mahmoud, Salamon et al. [5,6] for the solution of frictionless thermo-elastic contact problems. The model is based upon an equilibrium formulation which is piecewise linear over small increment of thermal load. Over each increment, the contact configuration is considered constant. For each increment, the uncoupled thermo-elastic boundary value problem is tackled. Firstly, the heat conduction boundary value problem is solved to obtain temperature distribution and consequently, the equivalent thermal stresses exerted on the elastic bodies are calculated. Conventional thermal boundary conditions are adopted. Once, the external thermal loads are determined, the displacement field problem is solved considering the permanent and contact boundary conditions. According to the automated direct procedure[5], the contacts are detected through calculation of the interpenetration between the two bodies. The interpenetration function play a unique role in defining the next pair. Once the new contact pair is detected, an additional boundary condition is activated representing one more constraint. The new contact pair is encountered at the site produce maximum value of the interpenetration function. A scaling factor is calculated in such a way to prevent the interpenetration at the detected new contact pair. Once the scale factor is determined, the residual of the applied thermal loads is calculated and modified uncoupled thermo-elastic model need to be set for the next increment by considering the new contact pair. The procedure is continuing until no additional contact pair to yield or the full capacity of thermal loads is completely used. The procedure is adaptive in the sense that the increment of load is just sufficient to avoid the maximum interpenetration at the site of the new detected contact pair. MATHEMATICAL
FORMULATION
Consider two linear elastic bodies shown in Fig. 1 in the undeformed state. Following Nowinski[9], the thermo-elastic contact problem is simulated by two uncoupled boundary value models. The first one representing the heat conduction between the two bodies taking into consideration the conventional boundary conditions where no resistance to heat flow in the contact regions while the separation regions do not exchange heat.
Fig. 1.
Frictionless thermoelastic
The governing following form:
differential
equation
contact problems
and boundary
conditions
V*tI+Qlk=O.
IS9
of the first model take the
(1)
The boundary r, is maintained at a temperature
(2)
ftk y),
8=
where the portion of the boundary P2 is subjected to heat flux
at3 4
=
k -
an
=
Y).
fib,
By applying the conventional thermal boundary conditions on the contact and separation zones, P, and P, respectively, the following boundary conditions are given kfz),=
k2($$))*:
per,.
Where the subscripts 1 and 2 stands for the first and the second body and p is an arbitrary position along contact zone r,, =M(p):
Per,,
i=l,2
(5)
where, 8 is the temperature increment field, k is the thermal conductivity, h is the convective heat transfer coefficient, Q is the heat generated internally per unit area of the material, and n denoted the outward normal to the surface. The governing eq. (1) and the associated boundary conditions (2)~(5) represent a complete formulation of a two dimensions steady state conduction and simple convection heat transfer problem of two bodies in contact. The displacement field mode1 of the elastic bodies subject to thermal load is represented by the following differential equations and boundary conditions,
(6) where y is Poisson’s ratio, E is Young’s modulus. Here A and p are the familiar Lame elastic constants which are represented by E
EY
cL=- 2(1+y)’
*=(l+y)(l-27).
The interface boundary conditions of the elastic field are given by U:(P) = d(p):
PE
rc,
(8)
where the superscripts 1, 2 stand for the first and second body but the subscripts 2 denotes the normal component of displacement, s;*(p)
=
SWAP)
=
0:
s:*(p) = s;*(p) 4 0: u:(p) - u:(p) - G(p) S 0:
Pa
PEP, p+
(9)
IhO
F. F. MAHMOUD
By neglecting the surface tractions, boundary conditions
F
81,
the rest of boundary
=At?.&+ @Ui,,
Where (Y is the coefficient normal and e = at+.
and A. G. EL SHAFEI
+
i=l,2,
~aUj.iij~
of thermal expansion,
THE NU~~ICAL
rather than I,,
has the following
j= 1.
li is the direction
cosines of the outer
MODEL
To solve the problem, the two bodies are descretized into finite elements. Reftning the mesh in critical area and throughout the potential contact region. The model is an incremental one which exploits the procedure developed by Mahmoud, Salamon et al. [5,6]. The procedure is a piecewise linear treatment for the uncoupled thermo-elastic contact problem. For each increment “nr” of the external thermal effects the permanent and boundary conditions are set for the both uncoupled models, heat conduction and thermoelastic models. The response of the heat conduction model 0 follows from the solution of the linear problem by using the conventional procedure of the finite element method
Where [keJm represents the equivalent stiffness matrix of tbe heat conduction problem for the increment m. It should be mentioned that [kJ’ satisfies all permanent and current contact boundary conditions. Moreover,
Hence, {f#}” is the residual thermal load vector yet available to apply on the system. Once the temperature increment vector {d}‘” is calculated, the thermo-elastic model is ready to handle. The temperature (O}m is translated to equivalent thermal stresses and consequently to force vector by exploiting the fundamentals of thermo-elasticity. The response of the elasto-static contact model follows from the solution of the inequality linear problem
42
-
Uj2 -
Gj S 0,
i, j
E
rp,
(141
Where fk,]” is the equivalent stiffness matrix of the elastostatic problem taking into consideration the permanent kinematic boundary conditions, {u}” is the displacement vector, {fJ}” is the static force vector, Ui2, uj2 are the normal components of displacements of the potential contact pair ij as shown in Fig. 1, and Gij is the gap between the node of pair ij which belongs to the potential contact regions I,. The elasto-static contact model eqs (13) and ( 14) is solved by using the direct automated procedure[5] which adopts the interpenetation measure as a criterion to detect a new contact pair. The induced interpenetration, rr, of the portion of surfaces along the potential contact region IP is given by
The inte~enetration function plays a unique role. We can notice from (15) that if rij is negative, zero or positive, node pairs i, j are either separated, in contact or interpenetrating, respectively. From the set of node pairs in rP associated with positive value of Tij that with the maximum value is selected as the next contact pair to close. Designating this node pair as ZJ, the scale factor S” required to establish contact is
Frictionless thermoelastic
Ihl
contact problems
S” = Gg/(u;“z - u;),
(16)
TrJ > 0,
so that the static load required to close gap GrJ is {AK}‘” = Sm{F,}“, and consequently
(17)
the thermal effect required to generate this static load is
and the residual thermal effects yet to be applied in the next increment
- Sm{fe}m. W m+’= {fe}rn In the process, the accumulated
displacement
temperature
(19)
field is computed
{u}+-(u)+ Sm{u)m. and the accumulated
is
W)
field is
Sm{elm.
{e}+-(e)+
(21)
The remaining gaps updated to G$‘+’ = Gij'--S"(u%u,$,
i,j~r~.
(22)
Prior to the following new increment, an additional constraint should be considered for the new activated contact pair and both [ke]“’ and [%I” should be augmented by using elementary matrix transformations. The procedure continues with application of the new residual load until either no further interpenetration is detected or all candidate node pairs are closed. To satisfy the heat balance across the contact interface, the temperature of the consolidated nodes Z and J should be the same. For the sake of simplicity and as an accepted degree of approximation, the temperature is taken as the average of both nodes Z and .Z.
EXAMPLE
PROBLEMS
The results of two examples will be presented. The examples are solved using the proposed incremental procedure which is coded in a computer model appended to an in-house finite element code. One of the problems is chosen as a skeletal structure while the other one represents a Hertzian problem. Example 1 Two symmetrical pin-jointed frames in contact. Two trusses are loaded by thermal loads. The geometry and boundary conditions of the system are shown in Fig. 2. The thermal loads on the members are prescribed as follows,
e,, = e,4 = e,6 = e,8 = 8, = e3 = e, = es= ioo~c, e12 = e,, = & = 8, = 50°C. The remaining members are free of thermal effect. The history of solution of the system is illustrated in Table 1. Example 2 Two symmetrical elastic hollow cylinders under symmetrical thermal loads. This example represents the thermo-elastic frictionless contact between two symmetrical elastic hollow
862
/------ typicaL -+-+--
5ots--+-5oc+ Dim.in mmr Fig. 2.
cylinders under the action of symmetrical thermal loads. The initial geometry and permanent boundary conditions are shown in Fig. 3a. The problem can be viewed as a cylinder on a rigid isolated plane due to its symmetry. Furthermore, the entire cyhnder need not be modelied[I0]. Only a 1.925” portion is modelled and the finite element mesh is ilhtstrated in Fig. 3b. The input load is a constant heat flux distributed over the inner surface boundary,
The thermal and mechanical
properties
of the materiat of each cyfinder are:
k = 0.043 W/mm ‘C, E = 200 x IO3 N/mm2,
h = IO x I Vh W/mm’ “C
y = 0.25 and LY= 15 x IO-” l/“C.
The incremental growth of the contact region vs thermal toad is shown in Fig. 4a. The contact pressure distribution is illustrated in Fig. 4b.
First node displacement (mm)
Second node displacement (mm)
Contact decision
2.0 1.0 2.0
0.57188 0.42906 LO
-0.57188 -0.42906 -1.0
No No Yes
0.856 0.142 0.0
0.73776 0.50 1.0
-0.73776 -0.50 -1.0
No Yes
3
0.525 0,o 0.0
1.O 0.5 1.0
-1.0 -0.5 -1.0
Yes Yes Yes
4
0.0 0.0 0.0
I.0 0.5 1.0
-1.0 -0.5 -1.0
Yes Yes Yes
Increment number 1
2
Gaps* (mm)
*Initial gaps {mm). C& = 2; C& = 1; G11.,2= 2,
% of load
Contact force W)
38.1
0.0 0.0 0.0
50.8 Yes 74.1
100
0.0 0.0 7113 0.0 7326 18651 26266 6725 31499
Frictionless themoelastic
cantact problems
(a) t
Y
(b)
X
Dim. in mms
Fig. 3.
ih4
.f .1~ .
and A. G. EL SHAFEI
F. F. MAHMOUD
‘.O
(a)
l
0.9 3
0.8 -
_* O.?! B
E
.o “u e L
.\ 1~
;‘;I 0.4
/
0.30.2-
q,,- 1.0 x IO+W/mm’ q,,- I.3x10-3W/mm2
.
-
/ i
0.2 0.4 0.6 0.6 1.0 1.2 1.4 Contact half- length
0.1 0.3 o.5 0.7 0.9 I.1 I.3 I.5
(mm)
Contact
half-length
(mm)
Fig. 4.
CONCLUSIONS The problems of thermoelastic frictionless contact are solved by using a modified automated direct method. The procedure consider the problem as uncoupled thermo-elasticity problem and tackled it in an incremental manner. Throughout each increment the heat conduction problem is solved at once to give temperature distribution and consequently equivalent thermal stresses and forces are obtained, Once the static forces are yielded the direct automated method developed by Mahmoud and Salamon is used to detect a new contact pair. The common frictionless constraints and conventional thermal boundary conditions are adopted through the contact interface . The proposed model is applied on two examples of different nature to detect the growth of contact region vs thermal load. Furthermore the contact pressure distribution is also determined. The results have an evidence that, the system is in complete heat balance and complete mechanical equilibrium. REFERENCES [I]
M. Comninou, J. Dunders and J. Barber, Heat conduction through a Rat punch. ASMEJ. Appl. Me&. 48, X7 l-875 (IV&l). [2] M. Comninou, J. Dunders and J. Barber, Planer Hertz contact with heat conduction. ASME J. Appl. Me&. 48. 549-558 (1981). [3] C. Panek and J. Dunders, Thermo-elastic contact between bodies with wavy surfaces. ASME .I. Appt. Me&. 46, 8.54-860 (1979). [4] J. Barber, Indentation of semi-infinite solid by a hot sphere. Int. J. Mech. Sci. 15, X13-819 (1973). [S] F. F. Mahmoud. N. J. Salamon and R. Marks, A Direct automated procedure for frictionless contact problems. Inl. _I. numer. Method. Engng 18, (1982). [6] F. F. Mahmoud, N, J. Salamon and T. P. Pawlak, Simulation of structural elements in receding/advancing contact. Camp. Structures 22 ( 1086). [7] J. Dunders, Properties of elastic bodies in contact. in The Mechanics of Contac? Between Deformable Bodies, (Edited hy J. T. tilker and A. D. dePaten). Delft University Press ( 1975). [X] J. P. H&man, Hear Trunsfer, Fifth edition. McGraw-Hill, New York (1% I). 191J. L. Nowinski, Theory of T~r~el~s~fcf;y with A~~jCQ~nS. Sijthoff and N~~~rdho~ (1978). [IO] W. R. Marks, Solution of frictionless contact problems by a conjugate gradient technique. M. SC. thesissubmitted to the University of Wisconsin-Milwuakee (1970). (Received
2 July 1987)