Study: temperature and residual stress in an injection moulded gear

Journal of Materials Processing Technology 108 (2001) 328±337

Study: temperature and residual stress in an injection moulded gear Gayatri Kansala,*, P.N. Raob, S.K. Atreyac a Indira Gandhi National Open University, Maidan Garhi, New Delhi 110068, India School of Mechanical Engineering, MARA Institute of Technology, Shah Alam 40450, Malaysia c Indian Institute of Technology, Hauz Khas, New Delhi 110016, India

b

Accepted 6 July 2000

Abstract In recent years plastics have found extensive use in the fabrication of important engineering components like gears, piping, etc. The wide application of plastics has led to a rapid development of computer aided design (CAD) and computer aided manufacturing (CAM) techniques for product development. Their application to the design and production of injection moulds has the potential to reduce the mould costs substantially, with improved moulding quality. It is also possible to obtain a `®rst time right' mould without any tool tryouts. Thermal analysis is one of the ®rst step towards developing a CAD system for injection moulds. Once the temperatures at different points inside the cavity are known, cooling channel design can be optimised. Also stresses and shrinkages inside the cavity can be studied knowing the temperature pro®les. In this paper the problem of determining the temperature distribution and thermal residual stresses which are developed due to the non-uniform cooling of the molten plastic inside the mould cavity in the injection-moulded polystyrene gear has been studied. So to reduce the scrap at the mould design phase such a modular system can be very helpful for the Indian tool manufacturers. # 2001 Published by Elsevier Science B.V. Keywords: Temperature and residual stresses; Moulded gear; Fabrication

1. Introduction With the emerging technologies and applications of polymers, polymer processing has become an important ®eld [1]. Litman [2] described a step-by-step procedure for selecting plastic parts for CAE analysis. CAE analysis can provide critical information for optimising design and processing parameters to ensure the successful moulding of plastic parts. It is a cost effective alternative to the conventional ``Cut and Try'' methods. The importance of computer aided techniques and software for analysis of plastic parts and computer aided mould design to enhance productivity and part quality are being developed and advocated [3,4]. An investigation of the literature showed that a number of models have been written. The model which was developed by Kenig and Kamal [5] considered the effect of pressure drop on the energy balance and solved the unsteady state heat conduction equation for high density polyethylene. Injection stage was ignored and under this assumption changes in polymer temperature and pressure were determined, by solving the unsteady state heat conduction equa*

Corresponding author. E-mail addresses: [email protected], [email protected] (G. Kansal). 0924-0136/01/$ ± see front matter # 2001 Published by Elsevier Science B.V. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 6 5 9 - 2

tion for a mould ®lled with polymer. Numerical method was employed to obtain the solution since no analytical method is available for the variable properties. Chang and Chiou [6] studied ¯ow and thermally induced residual stresses during injection moulding of a thin part with complex geometries. Bushkoand and Stokes [7] presented detailed parametric results on shrinkage and residual stresses in plaque like geometries in terms of normalised variables covering wide range of material and processing parameters. Santhanam [8] used a thermoviscoelastic model for the assessment of residual stresses and post moulding deformations in injection moulded parts. Chang and Tsaur [9] developed an integrated theory and computer program for the simulation of shrinkage, warpage and sink marks of crystalline polymer injection moulded parts. Chij and Sunderland [10] have determined the ejection temperature and cooling time in injection mould. They have performed one-dimensional thermal analysis as most of the injection moulded components are thin walled. To solve the governing equations, Crank±Nicolson scheme is used. Numerical method employed is ®nite difference method (FDM). Boldizar and Josef [11] studied cycle time in injection moulding of ®lled thermoplastics. The cooling process was treated theoretically starting from the heat conduction equations. While the measurement of cooling

G. Kansal et al. / Journal of Materials Processing Technology 108 (2001) 328±337

time was based on the assessment of recovery of the ejector marks for samples ejected after varying times of cooling. Agassant et al. [12] have studied injection moulding of thermoplastic materials using a two-dimensional model which solves at each time step the ®nite difference forms of the continuity, momentum and energy equations. A ®ne grid is used near the mould walls to take into account the great temperature gradients. Residual stress level and distribution in quenched amorphous polymers [13] as well as in melt processed [14] and cold rolled polymers have already been reported. In general, residual compressive stresses were measured at the surface layers of quenched and melt processed materials while tensile residual stresses were found at the inner layers. Residual stresses were found to be affected by the thermal history. The effect of thermal history on thermal stresses has been discussed by Siegmann et al. [15]. A number of mechanisms have been identi®ed as cause of part warpage, including difference in cavity pressures, different orientation, thermal stresses and the stresses frozen during packing. The one cause of warpage studied in the present case is the thermal stress caused by thermal gradient while the plastic is solidifying. Calculation of thermal stresses is basically a two pass procedure, that is given as the following: 1. the thermal history of the part must be predicted; 2. based on this thermal history, thermal stresses must be calculated. Siegmann et al. [15] made an extensive study of the effect of injection moulding conditions on the residual stresses in moulded square slabs from PPO (noryl). Sandilands and White [16] investigated the effect of injection pressure and crazing on the residual stresses in moulded bars of polystyrene using layer removal technique. Varying the pressure in the range 37±143 MPa was shown to have essentially no effect on the observed residual stresses, which were parabolic in shape. The introduction of surface crazes by bending around a cylindrical form was found to increase the levels of residual stresses in the moulded bars. Injection moulding with the molten polymer solidifying in the cavity is a major problem for which straight forward analytical solutions can only be obtained for very simple cases [17]. The study described here deals with the unsteady state transfer of heat under pressure in a melt, that exhibits variable properties. Since heat transfer is accompanied by phase transformation, the resulting non-linear partial differential equations that describe the system are not amenable to explicit analytical solutions. Hence numerical methods are employed for the solution. Of the number of methods available, the most commonly used methods are FDM [18] and ®nite element method (FEM). The principal mathematical problem encountered during the solidi®cation of the polymer in the cavity is due to the presence of solid liquid interface within the polymer and the need to calculate its position throughout the cooling cycle. An extensive

329

amount of work has been done in solving the heat transfer problems associated with phase change. A number of different methods has been developed for tackling the phase change phenomenon (see references). The concept of enthalpy, de®ned as the sum of the sensible heat and the latent heat for a phase change is frequently used in numerical modelling of phase change problems. An important advantage of introducing enthalpy is that the phase condition can be implicitly taken into account. As a result, easier numerical solutions of the problem can be achieved since it is no longer necessary to track the moving phase boundary. One of the limitations of the enthalpy method is that it is advisable to use only for those materials which melt over a temperature range. It cannot be used for pure substances which have in®nite speci®c heat at the melting point. However, the enthalpy method can be conveniently used for thermoplastics, as most of them melt over a range. From the literature survey, it is evident that to solve the thermal analysis problem, use of analytical method is not advisable and numerical methods are preferable. Also the simulation of the model can be carried out ignoring the injection stage. During the injection moulding process the polymer undergoes simultaneous mechanical and thermal in¯uences in ¯uid, rubbery and glassy states. Such effects introduce residual stresses and strains into the ®nal product, resulting in highly anisotropic mechanical behaviour and warpage and shrinkage [19]. Thus, understanding the factors governing the residual stress development during moulding is of great importance. Residual stresses in an injection moulded part can be attributed to two main sources. They arise due to the non-isothermal ¯ow of the polymer in the mould cavity during ®lling and packing. The other possibility is due to difference in cooling rate of the polymer near the boundary and inside the cavity. The knowledge of residual thermal stresses can be of great use in the case of injection moulded components in view of the fact that it is directly responsible for the amount of distortion. As moulded parts cool, the speci®c volume decreases along with the temperature. Generally the outer layers of the moulding cool and solidify prior to the interior. This phenomenon gives rise to residual thermal stresses. Rigdahl [20] performed calculations for residual stresses by using ®nite element techniques for thin elements. In the proposed model, relaxation effects are not taken into account, and as a consequence, the results obtained give the maximum stresses. Accordingly a formula from elasticity theory for the residual stress, s ˆ EaDt, has been used. The governing differential equation for thermal stress is from the consideration of equilibrium and compatibility together with the material stress±strain relationship. In general, each strain component consists of an elastic component, a thermal and a plastic component. Numerical method used is the FDM, which allows the calculation of thermal stress pro®le across the different grid points, at various times, until the temperature of the solidifying melt and that of the mould become equal. The results give the maximum

330

G. Kansal et al. / Journal of Materials Processing Technology 108 (2001) 328±337

stresses that are induced since the relaxation effects are not taken into account. The results of the case study developed using FDM are veri®ed by modelling them on a commercially available ®nite element package called P3/PATRAN. Due to the nonavailability of thermal analysis module in this particular installation of P3/PATRAN, it is not possible to model the thermal analysis. However, the stress pattern can be veri®ed qualitatively by using the temperature distribution results of the developed package on the PATRAN advanced-FEA module.

variables have been de®ned as Z F K Tˆ @F 0 K0 Z F Cp @F Hˆ

To develop the system, the following assumptions are made: 1. At zero time, temperature at all the points inside the cavity was assumed to be equal to the melt temperature [5]. 2. Half a teeth of the gear is analysed for symmetry reasons. 3. Half of the component is analysed, where the product is symmetric with respect to X-axes. 4. The heat transfer increases when liquid plastic is injected under pressure, due to convection. To take care of this effect, an effective thermal conductivity of the melt is de®ned. In the present case it is taken to be ®ve times, to match the published and experimental results. Such an approach has been used previously by Mizikar [21], Sharir et al. [22], and Westby [23]. The value of effective thermal conductivity used by Westby is 0.25 in CGS units, while, Mizikar [21] used a value of seven times the thermal conductivity of the liquid. Sharir et al. [22] used a value of ®ve times the thermal conductivity of liquid alloy and Brody [24] used 25 times the thermal conductivity of the liquid. 5. It is assumed that there is no adhesion between the mould cavity and the polymer. 6. Change in the volume is also not considered, i.e., relaxation effects are assumed to be negligible. The general three-dimensional equation for heat conduction for cylindrical co-ordinate system in a solid body is given by       1@ @f 1 @ @f @ @f rK K K ‡ 2 ‡ r @r @r r @y @y @Z @Z @T (2.1) ‡ Q ˆ rCp @t where r is the radius, Q the angle made by radius with positive X-axis, T the temperature (8C), K the thermal conductivity (cal/cm 8C) at temperature 08C, Cp the speci®c heat (cal/g 8C), r the density (g/cm3), t the time (s). In order to avoid solving the above non-linear equation, and to eliminate the tracking of the moving boundary, two

(2.3)

0

where T is the modi®ed temperature (8C), H the heat content or enthalpy (cal/g), K0 the thermal conductivity of solid at 08C. The substitutions applied are given below in the form of equations: K ˆ K0

2. Mathematical formulation

(2.2)

dT ; df

Cp ˆ

dH df

If Eq. (2.1) is used without any modi®cations, then K has to be differentiated at every step making the development of the system impossible. Hence the concept of modi®ed temperature and enthalpy is used [25]. After substituting Eqs. (2.2) and (2.3) in Eq. (2.1) the ®nal equation has the form        K0 1 @ @T 1 @ @T @ @T Q @H r ‡ 2 ‡ ‡ ˆ @r r @y @y @Z @Z r @t r r @r (2.4) For polystyrene the following values were used (Krevelen, 1972). mould temperature ˆ 20 C;

material temperature ˆ 220 C

Speci®c heat Cp …T† …25 C† ˆ 0:105 ‡ 0:029T Cp Cp …T† …25 C† ˆ 0:64 ‡ 0:021T Cp

when T < Tg when T > Tg

Thermal conductivity K…T† 0:034T ˆ 0:66 ÿ when T < Tg ; K…Tg † Tg K…T† 0:20T ˆ 1:2 ÿ when T > Tg K…Tg † Tg where Tg is the glass transition temperature (8C). 2.1. Explicit formulation The central difference technique and the mesh shown in Fig. 1 are used to modify Eq. (2.4) to the form given below: H t‡1 …I; J; L† ˆ H t …I; J; L†  kDt T t …i ‡ 1; j; L† ÿ 2T t …i; j; L† ‡ T t …i ÿ 1; j; L† ‡ r r 2 dy2 t t T …i; j ‡ 1; L† ÿ T …i; j ÿ 1; L† ‡ 2r dr T t …i; j ‡ 1; L† ÿ 2T t …i; j; L† ‡ T t …i; j ÿ 1; L† ‡ dr 2  t t T …i; j; L ‡ 1† ÿ 2T …i; j; L† ‡ T t …i; j; L ÿ 1† ‡ (2.5) dz2

G. Kansal et al. / Journal of Materials Processing Technology 108 (2001) 328±337

331

2.2. Thermal stresses The equation used for solving the residual thermal stresses is similar to the one given by Mendelson [28]: 1 4 …D F† ˆ ÿ…D2 …aT† ‡ G ‡ DG† E

The left-hand side of the equation for cylindrical co-ordinates can be further expanded as given by Williams [29]:  2  2  d 1d 1 d2 d f 1 @f 1 @ 2 f 4 ‡ ‡ ‡ ‡ D fˆ dr 2 r dr r 2 dy2 dr 2 r @r r 2 @y2

Fig. 1. Cylindrical coordinates.

where dr is the increment in radial direction, dy the increment in angular direction, dz the increment along the height, r the radius from the centre, K the thermal conductivity, dt the time increment, r the density. The mould has been divided into four parts to study the temperature inside the mould as follows: 1. 2. 3. 4.

cavity; mould and cavity interface; steel mould; mould and the air (ambient atmosphere).

(2.6)

where h is the surface heat transfer coef®cient, dT/dn the normal gradient of temperature, TB the boundary temperature, TA the ambient temperature. Eq. (2.6) is substituted in Eq. (2.5) to study temperature at the boundaries. H t‡1 is calculated for all the grid points. At time t, all other terms are known except the left-hand side term. These enthalpy values are converted back to modi®ed and actual temperatures. For calculations in the next step, these values are used as input. The numerical calculations are stable only when the following condition is satis®ed [25]: Dt 

1 …2K0 =r†…@T=@H†…1=DX 2 ‡ 1=DY 2 †

(2.9) Considering Uˆ

d2 f 1 df 1 d2 f ‡ ‡ dr 2 r dr r 2 dy2

Ur‡1;y ÿ 2Ur;y ‡ Urÿ1;y Ur‡1;y ÿ Urÿ1;y ‡ dr 2 2r0 dr Ur;y‡1 ÿ 2Ur;y ‡ Ur;yÿ1 ‡ r0 2 …dy†2

(2.12)

By solving Eq. (2.11), value of U is obtained. Substituting the value of U in Eq. (2.10), stress function f can be calculated. Eq. (2.9) is solved in two steps to simplify the problem. Once the value of s at different points is obtained, normal stresses in the direction of radius and angle, i.e., Prr, PQQ and shear stress PrQ, as shown in Fig. 2 can be obtained by using the following equations: @2f 1 @f 1 @ 2 f ÿ ; ; Pry ˆ 2 2 @r r @y r @r@y 1 df 1 d2 f ‡ Prr ˆ r dr r 2 dy2 Pyy ˆ

(2.7)

DY ˆ 0:1 cm

results are not stable if the time step is increased further.

(2.11)

Central difference technique and the mesh shown in Fig. 1 is used to give ®nite difference form of the left-hand side of the equation as given below:

where Dt is the time step, DX the distance between two nodes in X-direction, DY the distance between two nodes in Y-direction. By executing the program with different time steps, it was found that maximum time step of 0.007 can be used when: DX ˆ 0:1 cm;

(2.10)

Eq. (2.8) becomes d2 U 1 dU 1 d2 U ‡ ‡ ˆ ÿE…D2 …aT† ‡ G ‡ DG† dr 2 r dr r 2 dy2

Inside the mould cavity, only conduction has been considered. It has been observed that the temperature of the external surfaces of the tool remain close to ambient temperature and it can be assumed that there is zero heat transfer to the surrounding air. At the cavity wall where the polymer is in contact with the mould wall, following condition has been implemented as given by Croft and Lilley [26] and Convery and Miller [27]: @T h ˆ ÿ …TB ÿ TA † @n K

(2.8)

Fig. 2. Stress components.

(2.13)

332

PrY ˆ

G. Kansal et al. / Journal of Materials Processing Technology 108 (2001) 328±337

FY‡1;r ÿ FYÿ1;r 2r0 2 Dr FY‡1;r‡1 ÿFYÿ1;r‡1 ÿFY‡1;rÿ1 ‡FYÿ1;rÿ1 ÿ (2.14) 4dYDYr0

where dy is the increment in angle, dr the increment in radius.

Trochoidal pro®le makes ®llet at the bottom of the gear tooth. If no undercut is present, then the involute is tangential to the trochoidal pro®le at the base circle. When undercut is present, involute and trochoidal pro®les meet each other above the base circle. 1. Equations used for developing the involute pro®le: Rb ˆ R1 cos f1

3. Case study

where Rb is the radius of the base circle of the involute

Involute spur gear is chosen as one of the cases for study. Data of the gear is given as under module equal to 1:

cos f ˆ

number of teeth ˆ 12;

pressure angle ˆ 14:5

Half a teeth of the gear is considered for symmetry reasons to analyse temperature distribution, stress distribution as shown in Fig. 3. The gear is analysed over a length of 7 cm in the radial direction and at an angle of 158 in the angular direction. Thickness of the gear is 0.5 cm. In the radial direction gear geometry is divided into two sections: 1. From the centre of the gear to the dedendum circle radius. Distance between the nodes is 0.8 cm. 2. From the dedendum circle radius to the addendum circle radius. Distance between the nodes is 0.25 cm. The total number of points along the radial direction are 15. In the angular direction geometry is analysed in steps of 18. Hence the number of points to be analysed are 16. For analysis in the Z-direction, cavity is divided into three layers, i.e., one at 0 cm, second at 0.25 cm and the third at 0.5 cm from the bottom of the cavity. The geometry of the gear is divided and stored in the form of different points as explained above. Thermal analysis, thermal stress analysis and effect of cooling channels can be studied on these points as explained in the following sections.

Rb r

where R1 is the pitch circle radius, M1 the pressure angle at radius R1, r any radius of the pro®le, M the pressure angle at radius r   T T1 ˆ ‡ inv f1 ÿ inv f 2R1 2r where T is the tooth thickness at any radius r. To obtain the Cartesian co-ordinates of the involute pro®le with reference to either the centre line of the tooth or the centre line of the space, ®rst determine 100 from the speci®ed line using the equations as given: 100 ˆ vectorial angle of profile when calculated from centre line of tooth y00 ˆ

T 2r

when calculated from centre line of space p  T  ÿ y00 ˆ N 2r where N is the number of teeth in gear. Value of Cartesian co-ordinates are calculated from the following equation for the involute: X ˆ r sin y00 ;

Y ˆ r cos y00

Equations for the trochoid:

3.1. Thermal study Using Eq. (2.3) thermal analysis is carried out. Geometry of the gear is stored ®rst, using the following equations. Involute spur gear geometry can be divided into two parts [30]: 1. involute (between base circle and the addendum circle); 2. trochoidal pro®le (above the root).

Fig. 3. Gear geometry.

A A ÿ ; R1 ÿ b R1 …R1 ÿ b†tan f dˆfÿ R1 yt ˆ tanÿ1

Aˆ

q rt2 ÿ …R1 ÿ b†2 ;

where rt is any radius of trochoid (mm), b the distance from pitch line to the corner of the rack tooth, yt the vectorial angle of trochoid when d is the angle between origins of trochoid and involute, f the pressure angle of involute at R. The Cartesian co-ordinates of the trochoid in relation to the gear tooth pro®le, can be calculated from the following equations: Xt ˆ rt sin y00t ;

Yt ˆ rt cos y00t ;

y0 ˆ

pM 2N

G. Kansal et al. / Journal of Materials Processing Technology 108 (2001) 328±337 Table 1 Data table storing the points of the involute spur gear tooth above dedendum X (cm)

Y (cm)

Radius (cm)

Angle (rad)

0.3918 0.5327 0.6467 0.7317 0.7831 0.7360 0.6565 0.6146 0.6262 0.8440

6.989 6.728 6.467 6.207 5.948 5.796 5.554 5.307 5.0543 4.768

7 6.75 6.5 6.25 6 5.84 5.592 5.342 5.092 4.842

0.0560 0.0790 0.0996 0.1173 0.1308 0.1263 0.1176 0.1152 0.1232 0.1751

where M is the module of the gear y00t ˆ y0 ÿ d  yt where y0 is the original vectorial angle at the centre of tooth or space, yt the calculated vectorial angle of the trochoid, y00 the vectorial angle of the trochoid with Y-axis at the centre of tooth or space. Pro®le geometry is calculated for half of the gear tooth and rest of the pro®le can be generated by rotating the same geometry around the centre of the gear. Different points along the gear geometry are stored in terms of radius and angle as given in Table 1. Boundary conditions given by Eq. (2.6) are applied. Temperature at different points is recorded after every 5 s. The results of thermal analysis after 20 s given in Table 2 indicates that the corner of the teeth cools ®rst while the centre is still molten. Temperature difference between the hottest and the coolest point as a function of time is shown in Fig. 4. From the ®gure it is observed that the temperature difference ®rst rises with time and then starts reducing and after some time almost becomes constant. This temperature difference causes thermal stresses. The cooling pattern at the top and bottom layer is same. Top and bottom layers are cooling faster as compared to the middle layer. Table 2 Temperature at different points of the gear inside the cavity on the middle layer after 20 s Temperature (K)

Points in the angular direction

399.757 399.533 398.781 397.271 394.606 390.177 382.880 370.148 356.138 347.295 341.882 338.599 336.657

1 2 3 4 5 6 7 8 9 10 11 12 13

Points in the radial direction 6 6 6 6 6 6 6 6 6 6 6 6 6

333

Table 2 (Continued ) Temperature (K)

Points in the angular direction

Points in the radial direction

335.555 334.997 334.826 398.238 397.736 396.092 392.890 387.414 378.464 364.027 348.480 397.704 397.182 395.492 392.299 387.136 379.369 367.898 397.465 396.953 395.269 392.015 386.636 378.413 366.453 397.076 396.511 394.590 390.623 383.245 369.043 353.461 336.971 396.068 395.354 392.929 387.911 378.562 361.841 346.002 331.029 393.318 392.177 388.317 380.206 364.284 348.138 332.037 385.035 383.108 375.523 361.352 346.081 330.243 353.765 351.572 345.355 335.899 324.111 314.821 314.229 312.633

14 15 16 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 1 2 3 4 5 6 1 2 3 4 5 1 2 3

6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 13 13 13 13 13 13 14 14 14 14 14 15 15 15

334

G. Kansal et al. / Journal of Materials Processing Technology 108 (2001) 328±337

Fig. 4. Temperature difference between the hottest and coolest point.

3.2. Thermal stresses The method for calculating the thermal stresses is already explained in Section 2.2. In the case of gear, matrix [A] is stored point by point starting from radius 0.8 to 7 cm, i.e., from point 1 to 15 in the radial direction. In the angular direction number of points in the cavity above the dedendum circle radius are not ®xed because of the changing angle of trochoid and involute. However, up to dedendum circle radius, number of points are 16 at a step of 18 in the angular direction. Thermal stress results along the boundary of the teeth with respect to time are given in Table 3. From Table 3 it is observed that when the cooling starts, stresses are maximum at the corner of the teeth, but as the cooling progresses they turn out to be maximum at the root of the teeth. It is also observed that the radial and hoop stresses are

tensile in nature at the root of the teeth initially but then they change to compressive stresses in nature as shown in Fig. 5 once the point reaches the glass transition temperature zone, i.e., near 365 K. Thermal stresses with respect to time along the centre line of the gear are reported in Table 4. Analysis of this data shows that the thermal stresses, i.e., radial and hoop stresses are tensile in nature at the centre of the gear. It is also seen that these stresses have very low value as compared to the stresses observed above the dedendum of the gear. Shear stresses are negligible as compared to hoop and radial stresses along the boundary of the tooth and they are zero along the centre line of the tooth. In case of gears it is seen that the radial stress magnitude is not always less than the hoop stresses. At some points it is more than the hoop stress.

Table 3 Stresses along the boundary of the gear tooth Time (s)

Shear stress (N/m2)

Radial stress (N/m2)

Hoop stress (N/m2)

Points in angular direction

Points along radius

50

ÿ4.42077 2.22832 1.87626 1.51184 1.13959 0.762242 0.381868 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 2.14542 3.55307 0.000000

ÿ8.92105 ÿ14.0166 ÿ13.2006 ÿ12.4740 ÿ11.8575 ÿ11.4210 ÿ11.134 ÿ11.0500 ÿ154.230 ÿ74.5341 ÿ94.2298 15.5881 ÿ40.9812 22.3826 ÿ25.0575 ÿ7.93156 ÿ9.00447 ÿ9.59611 ÿ10.9542 ÿ28.2986

ÿ22.5753 ÿ24.8432 ÿ25.5493 ÿ26.1065 ÿ26.5268 ÿ26.8200 ÿ26.9930 ÿ27.0503 ÿ88.2282 ÿ28.3564 5.26299 ÿ8.84311 8.84311 ÿ19.3349 15.9266 ÿ12.8176 ÿ27.1999 ÿ24.6960 ÿ18.0098 ÿ10.8467

9.00000 10.0000 11.0000 12.0000 13.0000 14.0000 15.0000 16.0000 8.00000 7.00000 7.00000 8.00000 8.00000 7.00000 6.00000 5.00000 1.00000 2.00000 3.00000 4.00000

6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 6.00000 7.00000 8.00000 9.00000 10.0000 11.0000 12.0000 13.0000 14.0000 15.0000 15.0000 15.0000 15.0000

G. Kansal et al. / Journal of Materials Processing Technology 108 (2001) 328±337

335

Fig. 5. Stresses at the root of the involute spur gear.

3.3. Comparison of results with FEM package To verify the validity of the results obtained from the thermal stress analysis, gear is modelled on FEM P3/ PATRAN package using the geometry (line command) option and the points to model the gear are obtained using equations given in Section 3.1 (Table 1). Advanced FEA module is used for the analysis of stresses. Finite element mesh and the boundary conditions are shown in Fig. 6. The above problem is solved as two-dimensional plane stress problem. For thermal loading in the present case, temperatures at different nodes are given as obtained from the thermal analysis after 20 s. Stress pattern obtained from the FEM package is compared with the results obtained from the proposed analysis. It is seen that the hoop stresses are tensile at the centre of the gear. However, hoop stresses are compressive at the top of the teeth and along the boundary of the teeth. They are compressive in nature at the centre of the teeth as well, as shown in Fig. 7. The presence of these compressive stresses

Fig. 6. Finite element mesh and boundary conditions.

at the centre of the teeth can be explained due to the changing geometry of the gear. Gear has a lesser width at the top and bottom of the teeth as compared to the width at the centre of the teeth. This pattern of the stresses for the

Table 4 Stresses along the centre line of the gear teeth Time

Shear stress

Radial stress

Hoop stress

Points in angular direction

Points along radius

50

0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

0.475664 0.453362 0.516776 0.549123 0.804163 1.68734 2.68642 2.08603 1.58877 0.779378 ÿ12.4049 ÿ2.63883 ÿ7.76445 ÿ4.93317 ÿ9.00447

0.767251 0.493394 0.455126 0.339983 ÿ0.057488 ÿ1.31231 ÿ2.96964 ÿ3.85340 3.33584 ÿ3.86119 ÿ11.4699 ÿ8.84760 ÿ1.60970 ÿ11.1804 ÿ27.1999

1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

1.00000 2.00000 3.00000 4.00000 5.00000 6.00000 7.00000 8.00000 9.00000 10.0000 11.0000 12.0000 13.0000 14.0000 15.0000

336

G. Kansal et al. / Journal of Materials Processing Technology 108 (2001) 328±337 Table 5 Stresses at the corner of the gear teeth

Fig. 7. Hoop stresses in gear.

above thermal loading is matching with results of developed package based on FDM. From the FEM results shown in Fig. 8, it is also seen that the shear stresses are negligible as compared to the radial (see Fig. 9) and hoop stresses. Same results are obtained

Time (s)

Shear stress (N/m2)

Radial stress (N/m2)

Hoop stress (N/m2)

5 10 15 20 25 30 35 40 45 50

0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

ÿ2.11237 ÿ17.4197 ÿ32.4387 ÿ30.9171 ÿ28.4493 ÿ28.4788 ÿ28.5083 ÿ28.4437 ÿ28.3710 ÿ28.2986

ÿ3.54924 ÿ7.17355 ÿ9.24186 ÿ9.25569 ÿ10.4893 ÿ10.5996 ÿ10.7099 ÿ10.7675 ÿ10.8116 ÿ10.8467

Table 6 Stresses at the root of the gear teeth Time (s)

Shear stress (N/m2)

Radial stress (N/m2)

Hoop stress (N/m2)

5 10 15 20 25 30 35 40 45 50

0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

ÿ0.235972Eÿ2 ÿ0.184305Eÿ2 5.29988 ÿ52.2684 ÿ136.048 ÿ140.870 ÿ145.692 ÿ149.063 ÿ151.871 ÿ154.230

ÿ0.134365Eÿ2 ÿ0.213131Eÿ2 4.02100 ÿ24.4026 ÿ76.9495 ÿ79.8490 ÿ82.7486 ÿ84.8868 ÿ86.6949 ÿ88.2282

from the developed package as given in Tables 3 and 4. Stresses at the corner of the teeth are given in Table 5. It is also seen that the stresses are maximum at the root of the teeth (Table 6). 4. Conclusions Fig. 8. Shear stresses in gear.

Fig. 9. Radial stresses in gear.

From the studies on temperature distribution and thermal stress analysis it can be concluded that initially, i.e., when the cooling starts the stresses in case of gears are maximum at the corner of the teeth but as the cooling progresses they turn out to be maximum at the root of the gear. It is also observed that the radial and hoop stresses are tensile in the beginning but changes their sign as the cooling progresses and the glass transition zone is approached. It is observed that the hoop stresses are tensile at the centre of the gear but they are compressive at the top of the teeth, boundary and at the centre of the teeth. It is also observed that the shear stresses are negligible along the boundary of the gear but they are zero along the centre line. This shows that the axis along which the component is symmetrical shear stresses are zero. In case of gear is also seen that at some places hoop stresses are larger than the radial stresses but this is not true for all the points because at few points even radial stresses are more than the hoop stresses reason which can be attributed for this is changing geometry of the gear.

G. Kansal et al. / Journal of Materials Processing Technology 108 (2001) 328±337

References [1] N.P. Cheremininoff, Emerging technologies and applications for polymers, Polym. Plast. Technol. Eng. 30 (1) (1991) 1±26. [2] A.M. Litman, C.R. Mohan, F.E. Norman, Selecting parts for computer aided engineering analysis, Polym. Plast. Technol. Eng. 26 (1987) 333±345. [3] G.G. Trantina, M.D. Minmichelli, T.W. Paro, Integrated analysis of plastic parts and mold stress, ¯ow and heat transfer, in: Proceedings of the ASME International Computers in Engineering Conference and Exhibition, Boston, MA, 1985. [4] K. Schauer, Computer aided molded cooling design raises productivity, Plast. Technol. 63 (1983). [5] S. Kenig, M.R. Kamal, Cooling moulded parts Ð a rigorous analysis, SPE J. 26 (1970) 50±57. [6] R.Y. Chang, S.Y. Chiou, A uni®ed K-BKZ model for residual stress analysis of injection moulded three-dimensional thin shapes, Polym. Eng. Sci. 35 (22) (1995) 1733±1745. [7] W.C. Bushkoand, V.K. Stokes, Solidi®cation of thermoviscoelastic melts. Part II. Effects of processing conditions on shrinkage and residual stresses, Polym. Eng. Sci. 35 (4) (1995) 365±383. [8] N. Santhanam, Analysis of residual stresses and post-moulding deformation in injection moulded components, Ph.D. Thesis, Cornell University, Ithaca, NY, 1992. [9] R.Y. Chang, B.D. Tsaur, Experimental and theoretical studies of shrinkage warpage and sink marks of crystalline polymer injection moulded parts, Polym. Eng. Sci. 35 (1995) 1222±1230. [10] Yu. Chij, J.E. Sunderland, Determination of ejection temperature and cooling time in injection moulding, Polym. Eng. Sci. 32 (3) (1992) 191±197. [11] A. Boldizar, K. Josef, Measurements of cycle time in injection molding of ®lled thermoplastics, Polym. Eng. Sci. 26 (12) (1986) 877±885. [12] J.F. Agassant, H. Alles, S. Philipon, M. Vincent, Experimental and theoretical study of the injection molding of thermoplastic materials, Polym. Eng. Sci. 28 (7) (1988) 460±468. [13] J.R. Saffell, A.H. Windle, The in¯uence of thermal history on internal stress distributions in sheets of PMMA and polycarbonate, J. Appl. Polym. Sci. 25 (1980). [14] L.D. Coxon, J.R. White, Residual stresses and aging in injection molded polypropylone, Polym. Eng. Sci. 28 (7) (1988) 460±468.

337

[15] A. Siegmann, A. Buchman, S. Kenig, Residual stresses in polymers. 1. The effect of thermal history, Polym. Eng. Sci. 22 (L) (1982) 40± 47. [16] G.J. Sandilands, J.R. White, Effect of injection pressure and crazing on internal stresses in injection molded polystyrene, Polymer 21 (1980) 338±343. [17] J.R. Ockendon, W.R. Hodgkins, Moving Boundary Problems in Heat Flow and Diffusion, Oxford University Press, Oxford, 1975. [18] O.C. Zienkiewicz, The Finite Element Method in Engineering Science, McGraw-Hill, New York, 1971. [19] M.S.T. Jacques, An analysis of thermal warpage in injection moulded ¯at parts due to unbalanced cooling, Polym. Eng. II. Sci. 22 (4) (1982) 241±246. [20] M. Rigdahl, Calculation of residual thermal stresses in injection moulded amorphous polymers by the ®nite element method, Int. J. Polym. Mater. 5 (1976) 43±57. [21] E.A. Mizikar, Mathematical heat transfer model for solidi®cation of continuously cast steel slabs, Trans. Metall. Soc. AIME 239 (1967) 1747±1753. [22] Y. Sharir, A. Grill, J. Pelleg, Computation of Temperatures in Thin Tantatum Sheet Welding, Vol. 118, American Society for Metals, Cleveland, OH, Metallurgical Society of AIME, 1980, pp. 257±265. [23] O. Westby, Temperature distribution in the workplace by welding, A Report of Marine Technology Centre, University of Trondheim, Norway. [24] H.D. Brody, Needs and possibilities for process modelling research in process modelling, Proceedings of the American Society for Metals, 1980, pp. 27±55. [25] R.J. Sarjant, M.R. Slack, Internal temperature distribution in the cooling and reheating of steel ingots, J. Iron Steel Inst. 177 (1954) 428±444. [26] D.R. Croft, D.G. Lilley, Heat Transfer Calculations Using Finite Difference Equations, Applied Science Publishers, London, 1977. [27] R.O. Convery, P.P. Miller, Improved mould design by simulation of heat transfer, Plast. Rubber Int. 7 (3) (1982) 138±141. [28] A. Mendelson, Plasticity: Theory and Application, Macmillan, New York (Chapters 9 and 10). [29] J.G Williams, Stress Analysis of Polymers, Ellis Horwood, Chichester, UK, 1973. [30] E. Buckingham, Analytical Mechanics of Gears, Dover, New York, 1963.