1 Introduction

Anisotropic dimensional change on sintering is a critical aspect to be considered designing parts produced by press and sinter. Considering axi-symmetric ring shaped parts, not only dimensional change in the compaction plane may significantly differ from that along the axial direction, but even dimensional change of the internal diameter may be different from dimensional change of the external diameter, as by previous investigation [1]. The phenomenon has been extensively studied by the authors at the University of Trento, dealing with the many different variables, which determine it.

The influence of material is critical on change in volume: shrinking or swelling, and their amount, mainly derive from chemical composition, and anisotropy of dimensional changes is highly affected too. The behaviour of parts made up of Fe–Cu–P [2], Fe–Cu–C [3, 4], Fe–Cr–Mo–C [5] was investigated in previous work, and the influence of copper and phosphorus on pure iron was highlighted and compared in [6, 7]. Process parameters also strongly affect anisotropic dimensional change on sintering, as highlighted by the studies comparing parts sintered at standard sintering temperature and parts high-temperature sintered [8,9,10,11]. As by several authors [12,13,14,15,16,17], material and process parameters determine the different mechanisms occurring during sintering, among them the occurrence of solid state sintering or liquid phase sintering, which also strongly affect anisotropy of dimensional changes and were investigated in depth [18,19,20,21,22,23]. According to Zavaliangos and Bouvard [24, 25], deformation of particles by prior cold compaction has been observed as one of the main causes of anisotropy [26,27,28,29,30]. Olewsky [31] and Bordia et al. [32] proposed a theory for anisotropic shrinkage based on deformation of powder particles, which determines anisotropic orientation of pores. Wakai et al. [33, 34], according to the continuum solid mechanics approach, described anisotropic sintering shrinkage on the basis of the relationships between sintering stress and strain rate, as by the anisotropic particle packing and their rearrangement during sintering.

Aiming at applying the knowledge gained on anisotropy of dimensional changes to real parts, the influence of geometry was investigated, focusing the attention on axi-symmetric parts (disks and rings characterised by different H/D and H/(Dext − Dint), respectively) [6, 32]. On the basis of the experimental results, aiming at predicting dimensional changes, as in Raman et al. [36], an analytical model for the anisotropic dimensional change on sintering of ferrous axi-symmetric parts was developed [1, 37], from which the design procedure [38] used in the present work.

This work applies the design procedure to predict the anisotropic dimensional change of real parts produced in industrial conditions. The parts differ for material, complexity of geometry, green density, and process conditions, thus representing a demanding test-bed for the design procedure and a good opportunity to identify guidelines for future work.

2 Experimental Procedure

Aiming at broadly investigating dimensional change on sintering, the five industrial parts shown in Fig. 1 have been considered in this work, differing for material, geometry, size, and production process. All of them were produced in standard industrial conditions.

Fig. 1
figure 1

The parts studied

Full size image

Table 1 reports the approximate smallest (Dmin, Hmin) and largest dimensions (Dmax, Hmax) recognizable in the different parts, referring to Fig. 2.

Table 1 Main dimensions of the different parts
Full size table
Fig. 2
figure 2

Minimum and maximum dimensions of the parts studied

Full size image

Seven to ten parts for each material/geometry have been measured by a CMM, according to the measurement procedure following described. The same part has been measured in the green and sintered state, aiming at minimizing the influence of noise factors on the evaluation of dimensional changes. Dimensions have been derived from geometrical features, namely: heights from the distance of two parallel planes (the planes were obtained by minimum least square method applied to the points measured by continuous scan [39] on the related flat surfaces), diameters from cylindrical surfaces (the cylinders were obtained by minimum least square method applied to the circles measured by continuous scan at different levels). In some cases the diameters measured at different levels showed a trend, which was related to the influence of the compaction step, as explained in the following. Gear teeth were considered as defined by two cylindrical features, obtained from the points measured at the base and at the top of the teeth at different levels.

The chemical composition and the green density are summarized in Table 2.

Table 2 Chemical composition and green density
Full size table

3 Anisotropy Parameter K

In previous work [6] the anisotropy parameter K has been defined, aiming at quantitatively describing the anisotropy of dimensional changes, main steps are here shortly resumed. Dimensional change on sintering is due to change in volume, as by the relationship in Eq. (1), referring to a ring (see Fig. 3)

Fig. 3
figure 3

Dimensional changes for a ring

Full size image
$$\begin{aligned} & 1 + \frac{{V_{s} - V_{g} }}{{V_{g} }} = \left( {1 + \varepsilon_{h} } \right)\frac{{(1 + \varepsilon_{\phi \,ext} )^{2} - R^{2} (1 + \varepsilon_{\phi \,int} )^{2} }}{{1 - R^{2} }} \\ &\quad R = \frac{{\phi_{{\text{int} \,g}} }}{{\phi_{ext\,g} }} \\ \end{aligned}$$
(1)

Aiming at identifying a reference parameter to quantify anisotropy, the isotropic dimensional change has been considered, which is related to the change in volume by Eq. (2)

$$1 + \frac{{V_{s} - V_{g} }}{{V_{g} }} = (1 + \varepsilon_{iso} )^{3}$$
(2)

The anisotropy parameter K comes from the relationships above, relating the actual anisotropic dimensional changes to the reference isotropic dimensional change, through the change in volume, as by Eq. (3), where the dimensional change in the compaction plane lies on the right and the dimensional change in the axial direction on the left

$$\frac{{\left( {1 + \varepsilon_{iso} } \right)^{3} }}{{(1 + \varepsilon_{h} )}} = \frac{{(1 + \varepsilon_{\phi ext} )^{2} - R^{2} \alpha^{2} (1 + \varepsilon_{\phi ext} )^{2} }}{{1 - R^{2} }} = (1 + \varepsilon_{\phi ext} )^{2} \frac{{1 - R^{2} \alpha^{2} }}{{1 - R^{2} }}$$
(3)

where \(\alpha = \frac{{1 + \varepsilon_{{\phi \text{int} }} }}{{1 + \varepsilon_{\phi ext} }}\).

Plotting the entities in Eq. (3) on the graph shown in Fig. 4, the points relevant the different dimensional changes will lie on the bisector line, and the far they will be from the point relevant to the isotropic dimensional changes, the larger the anisotropy of dimensional changes.

Fig. 4
figure 4

Anisotropic versus isotropic dimensional change

Full size image

Through the distance highlighted in Fig. 4, related to the isotropic dimensional change, the anisotropy parameter K is defined by Eq. (4), where \(\gamma = \sqrt {\frac{{1 - R^{2} \alpha^{2} }}{{1 - R^{2} }}}\)

$$K = \frac{{(1 + \varepsilon_{\phi \,ext} )\gamma - (1 + \varepsilon_{iso} )}}{{\varepsilon_{iso} }}$$
(4)

K has been experimentally derived as a function of the isotropic dimensional change measuring, in the green and sintered state, the dimensions of disks and rings showing different diameters and heights, obtained by different materials in different process conditions. The study has been carried out step by step, so that the functions best fitting the data related to small and large change in volume have been found to be different, as shown in Fig. 5 (see references [1] and [6] for explanation in depth).

Fig. 5
figure 5

Anisotropy parameter K as a function of the isotropic dimensional change

Full size image

4 The Design Procedure

A design procedure aimed at predicting the anisotropic dimensional changes on sintering of ring shaped parts has been proposed, on the basis of the anisotropy parameter K. Main steps are summarized in Fig. 6.

Fig. 6
figure 6

The design procedure aimed at predicting anisotropic dimensional changes

Full size image

The isotropic dimensional change, which is directly related to the change in volume, is the starting point of the design procedure. In this project very different materials and geometries were considered, so that the isotropic dimensional change has been derived from measurement of the green and sintered parts. However, future development aims at identifying classes of materials/reference geometries/production processes to establish a database of isotropic dimensional changes to be directly used in the design procedure. The procedure allows predicting the dimensional changes of a ring providing that K and α parameters are known (see references [1] and [6] for explanation in depth).

The procedure above is related to a single ring, the application on real parts implies the identification of coaxial rings in the real axi-symmetric parts. As shown in Fig. 7, coaxial rings can be identified by a “column based approach” (referring to the powder columns, which can be identified during uniaxial compaction) or by a “maximum section based approach” (referring to the maximum amount of shrinking/swelling material, which drags after the other rings). Congruency among the dimensional changes of diameters common to the different rings is ensured. See Ref. [35] for explanation in depth.

Fig. 7
figure 7

Comparison between “column based approach” and “maximum section based approach”

Full size image

Figure 7 also shows the difference between the dimensional change predicted by the procedure above (as calculated by the two different approaches) and the dimensional change measured on the real parts; each bar corresponds to one of the measured parts. The better effectiveness of “maximum section based approach” is confirmed, so that it will be used to predict the dimensional changes from here on.

5 Results and Discussion

For each part, coaxial rings determining the whole geometry have been distinguished, according to the maximum section based approach, as shown in Fig. 8.

Fig. 8
figure 8

Parts splitting as by maximum section based approach

Full size image

Slightly conical surfaces have been represented by two slightly different coaxial cylindrical surfaces, as shown by parts 1 and 5. Part 3 represents a particular case, being the hole oval, and will be presented singularly, as the peculiar case of application of the design procedure to non-cylindrical surfaces. In the case of gears, the external ring has been considered both by the enveloping cylinder (top of the teeth) and by the cylinder at the base of the teeth, obtaining similar results.

The application of the design procedure allowed determining the predicted dimensional change for all the dimensions defining the coaxial rings in Fig. 8, providing the necessary congruencies among dimensions common to different entities (for example, the dimensional change of the internal diameter is supposed to be equal in all the rings with the same hole). Predicted dimensional changes were then compared with measured dimensional changes.

Results are presented grouping parts showing small change in volume, namely parts 1, 2, and 4 (and consequently small isotropic dimensional change, that is − 0.0018, 0.0005, and 0.0012) and parts showing large change in volume, namely parts 3 and 5 (and consequently large isotropic dimensional change, − 0.024 and − 0.025).

The results will be presented grouping dimensions by size, showing both the difference between the predicted and measured dimensional change, and the same difference referred to the measured dimensional change, to obtain a value in percentage terms.

5.1 Parts Showing Small Change in Volume

The difference between predicted and measured dimensional change of internal diameters, both in absolute values (upper side) and in percentage terms (lower side) is shown in Fig. 9.

Fig. 9
figure 9

Difference between predicted and measured dimensional change of internal diameters, both in absolute values (upper side) and in percentage terms (lower side)

Full size image

The difference increases on increasing the nominal dimension, but in percentage terms it is always lower than 0.15%, thus meaning a generally good precision in predicting dimensional changes.

In some cases a different behaviour is observed when considering dimensions, which are nominally the same, but are related to features closer to the bottom or the upper punch in compaction, features B and T in Fig. 9, respectively. The trend, however, is not necessarily verified, as by Fig. 6, and might be attributed to the compaction strategy and/or to the orientation of parts in the sintering furnace.

Figures 10 and 11 show the difference between predicted and measured dimensional change of external diameters and heights, respectively, both in absolute values (upper side) and in percentage terms (lower side).

Fig. 10
figure 10

Difference between predicted and measured dimensional change of external diameters, both in absolute values (upper side) and in percentage terms (lower side)

Full size image
Fig. 11
figure 11

Difference between predicted and measured dimensional change of heights, both in absolute values (upper side) and in percentage terms (lower side)

Full size image

As shown in Fig. 10, dimensional change of external diameters is predicted even more precisely than that of internal ones, while a slightly lower precision is observed in prediction of dimensional change of heights. However, it must be observed that the largest differences between predicted and measured dimensional changes are related to the dimensions of the part with the lowest isotropic dimensional change (0.0005, practically negligible).

The anisotropy parameter K used in the design procedure is defined by a ratio with the isotropic dimensional change at denominator (Eq. 4), what may imply a strong effect when the isotropic dimensional change is close to zero. As a general remark, the precision of dimensional change prediction of parts with small change in volume is good, apart from two single heights the difference between predicted and measured dimensional change is lower than 0.15%.

5.2 Parts Showing Large Change in Volume

The difference between predicted and measured dimensional change of internal and external diameters, both in absolute values (upper side) and in percentage terms (lower side) is shown in Fig. 12.

Fig. 12
figure 12

Difference between predicted and measured dimensional change of internal and external diameters, both in absolute values (upper side) and in percentage terms (lower side)

Full size image

The difference generally increases on increasing the nominal dimension, but not necessarily. In percentage terms it is always larger than in parts showing small change in volume, around 0.25% rather than 0.15%, thus meaning a decrease of precision in predicting dimensional changes.

Concerning heights, the difference between predicted and measured dimensional change, both in absolute values (upper side) and in percentage terms (lower side), is shown in Fig. 13.

Fig. 13
figure 13

Difference between predicted and measured dimensional change of heights, both in absolute values (upper side) and in percentage terms (lower side)

Full size image

The worsening in predicting dimensional change observed for diameters is even larger for heights, where the difference between predicted and measured dimensional change is up to 0.75%. The design procedure is less reliable in predicting dimensional change of parts with large change in volume. The reason for this worsening has to be ascribed to parameters (both geometrical and process related) determining the anisotropy parameter K, which are experimentally derived. In previous experiments, in fact, the sampling showing large change in volume encountered liquid phase sintering, while in the present work, as well as in previous experiments determining small change in volume, always solid phase sintering occurs. Consequently, the mechanisms occurring during sintering are completely different, and they likely affect the anisotropic dimensional change. Further work will imply a wider sampling, to enlarge and strengthen the reference database.

5.3 Dimensional Change of Non-cylindrical Features

Part shown in Fig. 14 has an oval hole defined by two axes, namely ϕi min and ϕi max.

Fig. 14
figure 14

Dimensions of features in part 3

Full size image

The design procedure has been developed referring to cylindrical features, so that the part above has been firstly approached as it was a cylinder with a cylindrical hole, under the hypothesis of three different diameters, ϕi min, ϕi max, and an average diameter given by (ϕi min + ϕi max)/2, respectively. In all the cases results were disappointing, leading to a strong underestimation of the minimum axis and a strong overestimation of the maximum axis, as shown in Fig. 15.

Fig. 15
figure 15

Difference between predicted and measured dimensional changes as obtained considering part 3 as a ring with a cylindrical hole (different diameters)

Full size image

A function representing the whole profile has thus been considered, derived from the points measured by continuous scan on the green and sintered surfaces of the hole. However, it is not possible to compare the data points by points, since the angular position of the data acquired are not perfectly coincident. This means that the data of the inner scan of the green part cannot be used directly in the model, so that an analytical curve, fitting the points of green and sintered scans was used to compare the values at the same position, as shown in Fig. 16.

Fig. 16
figure 16

Scan points of green and sintered parts and fitting functions

Full size image

The fitting function used is Eq. (5)

$$\left( {\frac{X}{a}} \right)^{n} + \left( {\frac{Y}{b}} \right)^{n} = 1$$
(5)

Same as Eq. (6) in polar coordinates

$$\left( {\frac{\rho *\cos \theta }{a}} \right) + \left( {\frac{\rho *\sin \theta }{b}} \right) = 1$$
(6)

where a and b are the semi axes of the curve previously measured. The parameter determined by the fitting equation is the exponential n, which is very close to 2, so that the oval might be reasonably approximated by an ellipse.

Nevertheless, even using the function representing the profile, instead of the single values of the axes, a nearly negligible improvement was obtained, as by Fig. 17. Again, the minimum axis was strongly underestimated and the maximum axis strongly overestimated.

Fig. 17
figure 17

Difference between predicted and measured internal “diameter” in the different positions

Full size image

However, the dimension and its angular position corresponding to the minimum difference between the predicted and measured dimensions was identified, which corresponds to the equivalent radius Req of the circular hole of the ring having the same volume of the part, as it was verified. Referring to the difference between the inner radius and the equivalent radius, as in the abscissa axis of the graph plotted in Fig. 18, a corrective function well fitting the difference between predicted and measured internal dimension has been identified.

Fig. 18
figure 18

Difference between predicted and measured internal “diameter” as a function of the difference between the “radius” and the equivalent radius—measured data and fitting function

Full size image

The corrective function (Eq. 7) was used to determine a new predicted dimension (as by Eq. 8), and consequently a new difference between the predicted and measured dimension

$$\phi_{int} (Predicted) - \phi_{int} (Measured) = 0.015 \cdot (R - R_{eq} )^{2} + 0.094(R - R_{eq} )$$
(7)
$$\phi_{int} (Pred\_corr) = \phi_{int} (Predicted) - \left[ {0.015 \cdot (R - R_{eq} )^{2} + 0.094 \cdot (R - R_{eq} )} \right]$$
(8)

The corrective function was applied to the whole data, and it allowed to strongly cut the difference between the predicted and the actual dimensions. Examples are shown in Fig. 19.

Fig. 19
figure 19

Examples of the difference between predicted and measured internal “diameter” in the different positions without corrective function (upper) and with corrective function (lower)

Full size image

This difference is always included in the interval (− 0.05/+ 0.05), representing the required dimensional tolerance for the minimum and maximum «diameters» of the oval hole, as summarized in Fig. 20.

Fig. 20
figure 20

Difference between predicted and measured dimensional changes—effect of corrective function

Full size image

5.4 Predicted Dimensional Changes and Tolerance Classes

From the designer point of view, the effectiveness of the procedure can be evaluated from the ISO tolerance class related to the difference between predicted and measured dimension. Figure 21 summarizes the mean values for ISO IT classes derived from the difference between predicted and measured dimensions for all the parts studied, showing both small and large change in volume. The ISO IT classes have been grouped by internal diameter, external diameter, and height, distinguishing the different size intervals.

Fig. 21
figure 21

ISO IT classes derived from the difference between predicted and measured dimensions for the different features

Full size image

It may be observed that there is no direct correlation between the size of the dimension and the allowable dimensional tolerance, neither an influence of the compaction direction can be highlighted.

The ISO IT class related to the difference between predicted and measured dimensions never exceeds ISO IT 9, and in most of the cases it is very low, belonging to the interval IT3/IT7. These results are reasonably good from the designer point of view, considering the tolerance classes generally related to press and sinter (IT10–IT12).

Nevertheless, it must be honestly admitted that in this way part of the allowed tolerance is consumed by the uncertainty in prediction. Work is in progress aiming at furtherly lowering the tolerance classes, so that the procedure actually becomes a design tool ensuring high precision of industrial parts.

6 Conclusions

This work investigated the effectiveness of the design procedure accounting for anisotropic dimensional changes developed in previous work. Five axi-symmetric industrial parts were considered, differing for material, geometry, and production process. Main results are here summarised.

  • The dimensional changes of parts showing small change in volume were predicted with good accuracy, the difference between predicted and measured dimensional change never exceeded 0.15%, being lower in most of the cases. The dimensional changes of parts showing large change in volume were less precisely predicted, particularly concerning dimensions in the axial direction. The difference between predicted and measured dimensional change of diameters did not exceed 0.25%, while it was up to 0.75% for what concerns heights. The reason for this worsening was ascribed to parameters (both geometrical and process related) determining the anisotropy parameter K, which are experimentally derived from sampling encountering liquid phase sintering, while in the present work always solid phase sintering occurs. Consequently, the mechanisms occurring during sintering are completely different, and they likely affect the anisotropic dimensional change.

  • Nevertheless, results are reasonably good from the designer point of view. In terms of tolerances, the difference between predicted and measured dimension never exceeded IT 9, corresponding to the worst cases, while in most of the cases it belonged to the interval IT3–IT6.

  • Influence of compaction on anisotropic dimensional changes has been highlighted, likely due to compaction/ejection strategy and to the consequent density gradient.

  • Concerning non-cylindrical features (oval hole), a corrective function has been identified allowing a very accurate and precise prediction of dimensional change.

This work also highlighted directions for improvement, mainly aimed at enlarging and strengthening the reference database, for example with relationship to large isotropic dimensional change obtained by solid state sintering, and to different geometries, thus improving the effectiveness of the design procedure.