1 Introduction

Extrusion-based additive manufacturing or fused filament fabrication (FFF) is a processing method for thermoplastics in particular. The process combines many parameters from dynamics, thermomechanics, and rheology. These should be known and must be precisely coordinated with each other. This is where the use of sensor technology helps, which should enable an increase in quality and speed in FFF. A good overview of previous attempts to integrate sensors and to better understand the process can be found in [1] and [2]. Above all, the processes inside the extruder are not yet precisely understood. Models [3,4,5,6,7,8,9,10] have already been developed. Practical experience shows that an optimal process window must be found for each setup consisting of material, feeding unit, internal geometry of the extruder, heating element, and nozzle. Determining the extrusion forces [11,12,13,14,15,16] or the pressure near the nozzle tip [17, 18], are methods to indirectly observe the flow of the material. Most of the methods already presented for determining the extrusion force use a bending beam model [11, 12, 14,15,16]. Here, measurement uncertainties can arise due to induced moments.

This paper presents a new compact methodology to characterise the forces acting inside an extruder in a concentric set-up. With this setup, a more accurate measurement of the extrusion forces is expected. From the force graphs, specific extrusion characteristics can be identified via parameter variation and their effect can be inter-preted in comparison to improve the quality of the printing process. A comparison of different nozzles is one example of the many possibilities of the test set-up.

2 Materials and method

2.1 Method

In FFF, a filament in wire form is fed to an extruder. (Fig. 1) This is driven by a feed wheel attached to a motor with the torque MM. The feeding force FF is generated by converting the rotational movement at the motor into a translational movement at the filament and can be calculated with the radius of the drive wheel rM as follows:

Fig. 1
figure 1

Schematic figure of forces on filament and melt in a control volume (red) for modelling the extrusion force

$${F}_{\mathrm{F}}=2\cdot \frac{{M}_{\mathrm{M}}}{{r}_{\mathrm{M}}}$$

On the opposite side of the filament, a pressure wheel provides the necessary contact force FC on the filament. To enable the filament to be conveyed, the following condition must be met with μ0 as the coefficient of friction.

$$F_{\text{C}} > \frac{{F_{\text{F}} }}{2 \cdot \mu_0 }$$

Current models calculate with perfect friction (μ0 = 1), without taking into account deformation of the filament or the feeding wheels. Investigations [3, 14, 19,20,21] and practical experience show that the power transmission depends on the penetration capacity of the feed wheel into the filament (hardness of the filament), filament diameter variations as well as deposits and wear on the feed wheel. More deposits on the feeding wheels or lower penetration capacity into the filament mean that FC must become larger, as presumably μ0 becomes smaller. In order to compensate for deformations on the filament or to be able to adjust FC the pressure wheel can be moved towards the filament via springs. If FC is caused by a spring, as shown in Fig. 1, it can be calculated with the spring constant k and the spring displacement Δs using Hooke's law:

$${F}_{C}=k\cdot \Delta s$$

The smaller the radius of the filament or the greater the wear on the feeding wheels or smaller their radius rM and rW, the smaller Δs and thus FC becomes.

In the area of the heating element, the material is transferred from the solid to the almost liquid state. At this boundary layer, the solid material acts as a piston and begins to feed the liquid material in negative z-direction. If the filament has sufficient stiffness in relation to the length from the feeding wheel to this boundary layer and is guided tightly in a barrel, bending or even buckling is avoided [5, 20]. The inner diameter of the barrel cannot be equal to the filament diameter due to increasing friction. This creates a gap where friction can be largely neglected.

The feeding force FF can be described as follows via the force equilibrium in a control volume, see red area in Fig. 1, following [10]:

$${F}_{\mathrm{F}}={F}_{p,1}={F}_{p,C}+{F}_{\tau ,C}+{F}_{\tau ,B}+{F}_{\tau ,N}$$

Here Fp,1, Fp,0 und Fp,C are the forces of the melt on the surface of the control volume in the z-direction. In the case of free extrusion and a large distance of the nozzle from a substrate, Fp,0 = 0 can be assumed. Fτ,B, Fτ,C and Fτ,N are the forces due to tangential stresses on the control volume in the z-direction. If no bending occurs on the filament or no moments are induced, Fp,1 is equal to FF.

The forces or the pressure loss in the control room can be described using Hagen-Poiseuille law [3]. Here, a laminar flow is assumed due to the low feeding velocity vF. According to [10], the following equations result for the individual forces:

$${F}_{\mathrm{p},\mathrm{C}}=2\pi {\int }_{{r}_{N}}^{{r}_{B}}p\cdot r \mathrm{d}r$$
$${F}_{\uptau ,\mathrm{C}}=2\pi \mathrm{cot}\left(\frac{\alpha }{2}\right){\int }_{{r}_{N}}^{{r}_{B}}{\tau }_{ns}\cdot r \mathrm{d}r$$
$${F}_{\uptau ,\mathrm{B}}=2\pi {r}_{\mathrm{B}}{\int }_{{l}_{\mathrm{M}}}{\tau }_{\mathrm{rz}}\mathrm{d}z$$
$${F}_{\tau ,\mathrm{N}}=2\pi {r}_{\mathrm{N}}{\int }_{{l}_{\mathrm{N}}}{\tau }_{\mathrm{rz}}\mathrm{d}z$$

Here p is the pressure, τns the shear stress on the surface of the cone and τrz the shear stress component on the barrel surface. According to [10], the forces Fτ,C, Fτ,N have a small influence on FF. At high vF the shear stresses become larger and thus the influence of the shear forces Fτ,B; Fτ,C and Fτ,N on FF.

The gap between the filament and the cylinder can cause a backflow [10, 20] of the melt. This has a particular influence on heat transfer at this point, so that it cannot be neglected, especially at high feeding velocities vF. Serdeczny et al. [10] refer to this as the transition from a stable to an unstable extrusion. Above a certain vF, the reflux causes an oscillation in the feeding force [11, 15].

2.2 Experimental set-up

The test rig (Fig. 2) has a rotary encoder (a) that records the length of the filament fed to the extruder and its actual speed. The feeding unit (c) has an adjustment option for the contact force (b). The feeding wheel on the motor shaft and the pressure wheel are dynamically linked to each other via gear wheels, so that the filament is actively fed into the extruder from two sides. The polymer is plasticised at the heating element (f). A heat break with a compressed air cooler (e) provides the necessary thermal bridge. The molten material is discharged at the nozzle (g). The force FS is detected with a ring load cell "Burster Miniature Ring Load Cell Type 8438" (d) with integrated strain gauges.

Fig. 2
figure 2

Test rig set-up cross-section (left) and front view (right): a rotary encoder; b contact force adjustement; c filament feeder; d annular force sensor; e heatbreak with air cooler; f heater; g nozzle


The weight forces of the extruder GL and the melt plus extrudate GM are recorded with the force sensor and are deducted from the measurement result by taring. After taring the weight forces, the measured extrusion force is FE.


During a measurement, the temperature at the heating element TH and the feeding velocity vf are setting parameters. The average extrusion force \(\overline{{F }_{\mathrm{E}}}\) is calculated over the extrusion time.

2.3 Force sensor calibration

Before recording forces in the test, the ring force sensor was calibrated using standardised weights.

A bolt was placed on the ring force sensor. A wire attached to the bolt was passed through the centre of the extruder. The measured force at the sensor FS was zeroed with the masses of the bolt and the wire. Outside the die, a standardised weight was attached to the wire. The mass of the weight was previously checked with a precision balance. FS was recorded and the mean value of the recorded forces \(\overline{{F }_{\mathrm{S}}}\) was calculated.

Using the gravitational acceleration g = 9.81 mm/s2, the weight force can be calculated as the nominal force from the mass of the weight. The corresponding values can be seen in Table 1. The maximum difference between the nominal and the measured force is 264 mN. The mean deviation is maximum 8 mN and the standard deviation maximum 13 mN.

Table 1 Force sensor calibration data

2.4 Heatbreak factors

In the test rig, the heatbreak is not only necessary for conveying by separating the solid and liquid phases of the plastic, but also serves to protect the ring force sensor from overheating. To check the influence of the cooling by the compressed air cooler on the temperature at the ring force sensor, the temperature at the ring force sensor was measured while simultaneously recording the force. The temperature of the heating element was varied from 25 to 350 °C, the pressure of the cooling air pA from 50 to 150 kPa.

As Fig. 3a shows, the temperature at the force sensor increases as the temperature of the heating element increases. A higher cooling pressure pA causes less heating of the ring load cell and improves the accuracy of the force sensor (lower slope of the force graph in Fig. 3b). According to these results, the measurement error with pA = 150 kPa would be a maximum of 80 mN in a series of tests over the range of 250–350 °C. The following tests are carried out with a cooling pressure of 150 kPa.

Fig. 3
figure 3

Sensor temperature TS (a) and average initial force \(\overline{{{\varvec{F}} }_{\mathbf{S}}}\) (b) as functions of heater temperature TH at different cooling pressure pA

2.5 Nozzles

For comparison, the nozzles "E3D V6", "Gühring PKD Printer Nozzle" and "3DVerkstan Olsson Ruby", each with a diameter of 0.4 mm, were used.

2.6 Materials

The “filamentworld PLA (blue)” was used as the reference material and the “extrudr GreenTEC Pro Carbon”, a PLA with 10% carbon short fibres. All the filaments tested had a diameter of 1.75 mm.

3 Results and discussion

3.1 Extrusion height dependency

As the weight force of the melt and the extrudate GM increases over the duration of a measurement Eq. (9), this may have an influence on the measured force. To investigate this, the nozzle was positioned at a height of 5–45 mm from the substrate. Mathematically, at most a negligible weight force of 0.02 mN should be added at a height of 45 mm. In the test, 500 mm PLA filament, the "filamentworld PLA (blue)", was extruded at TH = 210 °C and vF = 3 mm/s. The smaller the distance, the greater the force. The smaller the chosen distance, the more often material had to be removed from the die area during extrusion. At a height of 5 mm, the material could not always be removed in time. This led to a jamming of material at the nozzle entrance and thus to an increase in pressure in the extruder or to an increase of \(\overline{{F }_{\mathrm{E}}}\). For heights of 15 mm and more, values of \(\overline{{F }_{\mathrm{E}}}\) were determined with a mean value deviation of approx. 80 mN. This deviation of the mean values of \(\overline{{F }_{\mathrm{E}}}\) is about one-tenth smaller than the mean deviation within a measurement of approx. 850 mN. GM thus has no significant influence on the measurement of FE in the selected measuring range. In the following, a nozzle height of ≥ 15 mm was selected.

3.2 Influence of filament contact force

An important assumption in the calculation models is that the contact force on the filament FC must always be greater than FE/2 at perfect friction Eq. (2). This assumption is validated in the following.

The "filamentworld PLA (blue)" was extruded at a heating element temperature TH = 210 °C and feeding velocities vF of 2 – 6 mm/s and FE was recorded. In the experimental setup, FC is generated via two identical springs mounted in parallel. Two pairs of springs with spring rates of k = 2.842 and k = 7.164 and a maximum force tolerance of 1 N and 2 N were used. The spring displacement Δs and thus FC was adjusted by turning the knurled head screw at the feeding unit (Fig. 2b). According to Eq. (3), the parallel mounting of the springs means an addition of the spring forces. The resulting contact forces FC can be seen in Table 2.

Table 2 Spring data and used contact forces

Since the extreme case of slipping on the filament is considered first, the maximum extrusion forces FE,max are considered. Accordingly, in Fig. 4, half of FE,max (FE,max/2) is compared with the contact force according to Eq. (2). The more FE,max/2 is converging FC, in this case the as the force limit, the less constant the filament is fed. A strong exceeding of the force limit of FE,max/2 was not measured. It is assumed that in this case μ0 from Eq. (2) falls below 1 and thus a slipping occurs on the filament. Thereby, the equation would continue to be valid, only feeding no longer takes place. The further the distance of FE,max/2 from the force limit, the more constant the feeding becomes.

Fig. 4
figure 4

Half of maximum extrusion forces FE,max/2 at different feeding velocities vF and contact force FC itself as force limit (orange, dashed line) as functions of contact force

This relationship can also be illustrated by observing the mean deviation of the extrusion force FE,md in Fig. 5. Up to FC ≤ 20.9 N, FE,md and thus the spread of FE initially decreases. From FC > 20.9 N the spread increases, although the corresponding FE,max moves away from the force limit. One assumption here is a decreasing contact pressure due to the penetration of the feed wheels into the filament and the resulting reduction of Δs. Macro images (Fig. 6) of the filament fed with vF = 3 mm/s confirm the assumption. At FC = 9.5 N, no notch is visible on the contact side of the feed wheel (1). This confirms the assumption of μ0 < 1. A peek FE,max at FC = 9.5 N can, therefore, be explained by non-uniform friction of the feed wheel. In addition, characteristic elevation of material (2) can be seen on the surface of the filament on the pressure side, which could represent shearing of the material due to a stop of the filament in the z-direction. In this case, no extrusion force would be measurable. A notch on the engagement side of the feed wheel (3), among others, can be seen at FC = 20.9 N. Deep notches (4), without significant elevation, are produced FC = 40.0 N. In order to quantify the depth of the notches dN using the images, the software “LAS X” from “Leica Microsystems” was used. The mean value of dN as a function of the contact force is shown in Fig. 5. The measurement shows a greater notch depth at low and high contact forces. The notch depth is smallest at FC = 20.9 N. A comparison of FE,md and dN in relation to the contact force suggests a correlation. A reduction of Δs by dN results in a reduction of FC. If this is added to FE,md, the result is a corrected form of the spread FE,md,cor (Fig. 5). This graph is consistent with the expectation that the spread of corresponding extrusion forces decreases while distancing from the force limit in Fig. 4. Considering the corrected spread a mean deviation limit of approx. 1 N has been identified. Above the limit extrusion seems to be unstable, resulting in higher frequent fluctuations of FE over time. The unstable region is shown in Fig. 4, red area. In summary, at TH = 210 °C the PLA shows stable flow at feeding velocities of 2–3 mm/s and contact forces of about 20–40 N.

Fig. 5
figure 5

Notch depth dN (green), mean deviation of extrusion force FE,md (blue) and corrected mean deviation FE,md,cor (grey) as functions of contact force FC

Fig. 6
figure 6

Microscope image of “filamentwold PLA” filament fed at contact force FC of 9.5 N (a), 20.9 N (b) and 40.0 N (c)

Current extruders have fixed feeding wheels or gradual adjustments of the contact force at the feeding unit. Compared to the use of springs, on one hand, this allows a high contact force and constant values of μ0 near 1; on the other hand, the system does not react to variations in the filament diameter. In this case, either very deep or relatively small notches can be formed in the filament. In relation to the observations presented here, this can lead to larger differences in the material output.

3.3 Full extrusion characterization

To determine the boundary conditions of a PLA filament, the average extrusion forces \(\overline{{F }_{\mathrm{E}}}\) were compared to the feeding velocities vF of 2– 6 mm/s and the contact force FC as well as the heating element temperature TH of 190–230 °C were then varied.

The influence of the contact force at constant TH = 210 °C is shown in Fig. 7. With increasing vF, \(\overline{{F }_{\mathrm{E}}}\) increases. With FC of 20.9 N, 29.9 N and 40.0 N, there is a transition (Fig. 7, red area) to a steep increase of \(\overline{{F }_{\mathrm{E}}}\) at vF von 3.5–4 mm/s. Using a higher FC, \(\overline{{F }_{\mathrm{E}}}\) increases significantly at vF > 3.5 mm/s. Here, a higher \(\overline{{F }_{\mathrm{E}}}\) can be achieved with a higher FC, since at the same time the force limit is higher and higher FE,max can be achieved. The mean deviation of the extrusion force FE,md also increases more strongly from vF > 3.5 mm/s than at lower feeding velocities. At FC = 40.0 N it increases non-corrected from FE,md ≈ 1.9 N to FE,md ≈ 8.9 N, which means significant extrusion fluctuations.

Fig. 7
figure 7

Mean extrusion force \(\overline{{{\varvec{F}} }_{{\varvec{E}}}}\) as a function of feeding velocity vF at different contact forces FC and tightening torques τC (dashed lines)

As a comparison, the springs on the feeding unit were removed and a screw was used to press the pressure wheel onto the filament with a tightening torque MC. Two torques were selected with MC = 0.3 Nm (approximately the maximum holding torque of the extruder motor of 0.32 Nm) and MC = 0.6 Nm. The contact force FC could thus be calculated with the thread pitch of the screw P via the formula of frictionless work.

$$F_{\text{C}} = \frac{{2 \cdot \pi \cdot M_{\text{C}} }}{P}$$

This results in FC = 2.693 N at P = 0.7 mm and MC = 0.3 Nm. This value seems very high compared to the contact force generated with the springs. It suggests that friction cannot be neglected and therefore the actual contact forces are lower. The torques were set with a torque screwdriver before each extrusion, while the pressure wheel was in contact with the filament. Figure 7 shows \(\overline{{F }_{\mathrm{E}}}\) using the two torques as a function of vF. Up to the transition speed vF = 3.5 mm/s, they differ little from those of the feeding unit with spring. The strong increase of \(\overline{{F }_{\mathrm{E}}}\) occurs at approx. vF = 5 mm/s. The mean deviation of the extrusion force FE,md also remains relatively low up to this speed. At vF = 6 mm/s the two forces evolve in opposite directions. The forces at MC = 0.6 Nm showing a high-frequent oscillation. In addition, an occasional clocking of the extruder motor can be observed. At vF = 6 mm/s there is no feeding and continuous clocking. As long as the filament is moving, feeding can take place. Otherwise, exceeding the maximum holding torque of the extruder motor leads to a stop in feeding.

The results shown here show a strong correlation to the results of [11, 15], which also describe the transition from stable to unstable extrusion above a certain feeding velocity. In addition, an independence of this transition zone from the contact force on the filament is shown here. In stable extrusion, the contact force has little significance. In this study, the moment-controlled contact force allows a higher feeding velocity under stable conditions.

Figure 8 shows the influence of the heating element temperature TH. The higher TH, the lower \(\overline{{F }_{\mathrm{E}}}\). At faster vF the processing material can absorb less heat due to a shorter heating time in the heating zone and FE remains greater. Here the transition zone (Fig. 8, red area) is shown in relation to TH. Using TH and vF below the transition a stable \(\overline{{F }_{\mathrm{E}}}\) and thus a stable extrusion is ensured.

Fig. 8
figure 8

Mean extrusion force \(\overline{{{\varvec{F}} }_{{\varvec{E}}}}\) as a function of feeding velocity vF at different heater temperatures TH

This dependence of FE on vF has already been proven several times [11, 15]. The investigations with the experimental set-up described here confirm this relationship.

4 Nozzle characterization

Extrusion characterisation can be used as a tool for evaluating the effect of changes to a printhead. This is demonstrated below using three different nozzles.

After the test series, the “E3D V6” was cut in the centre showing the cross-section in radial direction. The interior geometry was measured using a macroscopic image (Fig. 9). The overview in Table 3 shows the data of the three nozzles, including the data of the technical drawing of "Guehring PKD-Druckduese" and "3D Verkstan Olsson Ruby" measured by Nienhaus et al. [15]. Large differences in the geometry of the nozzles were measured in the cone angles α and the capillary length lN. According to Eqs. (6) and (8), these have a theoretical influence on Fτ,C, and Fτ,N.

Fig. 9
figure 9

Macroscopic image of cross section of “E3D V6” nozzle

Table 3 Nozzle geometry data

First, an analysis of the force limit was carried out. The "extrudr GreenTEC Pro Carbon" was extruded at a temperature of 210 °C and under variation of vF and FC. All values of FE,max/2 are far below the force limit, although an unstable extrusion is present. The cause is assumed to be a less deep penetration of the pressure wheel into the filament. The carbon fibres contained in the "extrudr GreenTEC Pro Carbon" make the filament harder and the penetration depth shallower. This could lead to lower friction μ0. On closer examination, a maximum for FE,max/2 at corresponding FC was detected for all three nozzles. If these maxima are averaged with each other, a potential friction of μ0 = 0.53 can be calculated via the ratio of FE,max/2 und FC. FC reduced by this factor as a new force limit was drawn into the charts in Fig. 10. Here it can be seen that almost all FC = 9.3 N lead to slippage of the pressure wheel. Comparing the "E3D V6" with the "Guehring PKD-Druckduese" up to the "3D Verkstan Olsson Ruby", all FE,max/2 approach the force limit. The "3D Verkstan Olsson Ruby" therefore leads to more frequent slipping of the pressure wheel. By focussing on the mean deviation of the extrusion force FE,md, an unstable region of the extrusion can be defined for each nozzle. These regions cannot be defined as clearly as with PLA. The fibres in the melt are another possibility of producing back pressure and thus also influence FE,md. Thus, the mean deviation limit of the "E3D V6" is approx. 1–1.5 N and for the other two nozzles approx. 1.5–2.5 N. Plotting the region of unstable extrusion results in a decreasing number of parameter sets from the "E3D V6" to the "Guehring PKD-Druckduese" to the "3D Verkstan Ols-son Ruby" resulting in stable extrusion.

Fig. 10
figure 10

Half of maximum extrusion forces FE,max/2 at different feeding velocities vF and force limit (orange, dashed line) as functions of contact force using “GreenTEC Pro Carbon” material: “E3D V6” (a), “Guehring PKD-Druckduese” (b) and “3D Verkstan Olsson Ruby” (c)

Subsequently, an extrusion characterisation was carried out over the heating temperature range TH. For this purpose, the extrusion of the "extrudr GreenTEC Pro Carbon" was varied with the respective nozzle over TH from 210 to 260 °C and vF from 2 to 6 mm/s. The contact pressure on the filament was set using the screw method and a tightening torque of MC = 0.3 Nm. Figure 11 shows the average extrusion forces \(\overline{{F }_{E}}\) as a function of vF. Overall, \(\overline{{F }_{E}}\) increases when using the „Guehring PKD-Druckduese” compared to the „E3D V6 “and increases again with the "3D Verkstan Olsson Ruby". With the “E3D V6”, the transition region from \(\overline{{F }_{E}}\) to the unstable region occurs at TH = 210 °C and vF = 5 mm/s, similarly with the “Guehring PKD-Druckduese”—but at a higher level. The "3D Verkstan Olsson Ruby" causes a transition region at TH = 210 °C already at vF = 4 mm/s. The "E3D V6" thus offers the widest processing range with low extrusion force and spread over a large heating temperature range. However, the use of carbon fibres will cause the brass nozzle "E3D V6" to wear out more quickly, resulting in an increase in the nozzle diameter and thus distorting the interpretation of the extrusion forces.

Fig. 11
figure 11

Mean extrusion force \(\overline{{{\varvec{F}} }_{\mathbf{E}}}\) as a function of feeding velocity vF at different heater temperatures TH for “E3D V6” (a), “Guehring PKD-Druckduese” (b) and “3D Verkstan Olsson Ruby” (c)

According to the calculations in Eqs. (6) and (8), a larger cone angle α and a longer capillary length lN should result in a greater extrusion force. In comparison, the nozzles each show a reduction in α with a simultaneous increase in lN. An increase of lN by approx. 267% with the „3D Verkstan Olsson Ruby” compared to the "E3D V6″ shows a greater effect than the reduction of α by approx. 67%. Furthermore, the statement from the literature [10] can be confirmed that Fτ,C, and Fτ,N only show significance at higher feeding velocities and lower heating temperatures, since the graphs of the "Guehring PKD-Druckduese" and the "3D Verkstan Olsson Ruby" nozzle show a larger slope and an increasing distance from each other.

5 Conclusions

In this study, a concentric setup for measuring the extrusion force is presented. In addition, a new method is shown with which a feeding force limit can be determined on the basis of the maximum force and the range of unstable extrusion can be determined with the mean deviation of extrusion force. This limit value is an adequate supplement to the method of the averaged extrusion force over a temperature range. The averaged extrusion force gives a good overview of the feeding boundary conditions as a function of feeding velocity and heating element temperature. Transition areas to unstable extrusion can also be determined here. In addition, nozzles can be compared with each other and their performance can be classified. In the same way, now other materials can be examined.