Investigation of fibre orientation and notch support of short glass fibre reinforced thermoplastics

International Journal of Fatigue 131 (2020) 105284

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International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue

Investigation of fibre orientation and notch support of short glass fibre reinforced thermoplastics

T

⁎

Gabriel Stadlera, , Andreas Primetzhoferc, Gerald Pinterb,c, Florian Grüna a

Montanuniversität Leoben, Chair of Mechanical Engineering, Franz-Josef-Strasse 18, 8700 Leoben, Austria Montanuniversität Leoben, Chair of Polymer Testing, Otto-Glöckel-Strasse 2, 8700 Leoben, Austria c Polymer Competence Center Leoben, Roseggerstrasse 12, 8700 Leoben, Austria b

A R T I C LE I N FO

A B S T R A C T

Keywords: Short fibre reinforced polymer Fibre orientation Notch support Fatigue Stress analysis

In this paper, the effect of anisotropy on notches for a PPA-GF50 is discussed. For this, fatigue tests on specimens with boreholes at the centre, extracted from injection moulded plates, have been performed. Different, orientated layers, lead to anisotropic material behaviour along the notch. Simulations highlight the local stresses, including stress gradients. Nevertheless, the stress gradient along notches evoked by anisotropy, isn’t considered in common gradient models for notch support. Therefore, a novel method to take the effect of the local fibre orientation into account is proposed. However, experiments show a notch radius influence in fatigue strength and fibre orientation effect on notch support. The consideration of the new findings allows a more precise lifetime assessment for notched, anistoropic materials.

1. Introduction The continuous need for lightweight constructions, especially in automotive applications, requires novel usages and combinations of materials for specific purposes. More often short fibre reinforced polymers (SFRP) serve as a substitutive material for metals. Consequently, fatigue behaviour for component optimisation has to be investigated. The description of influence factors for the consideration in a component design process is one aim of this research, while the scheme of a design process, including lifetime calculations, is proposed in [1,2]. Based on the local S/N-curve concept, different influence factors on local material fatigue behaviour are considered in the lifetime assessment. The local S/N-concept perform lifetime assesments for areas with critical loading conditions. Various environmental factors, like moisture, temperature [3–5], loading conditions, such as multiaxial loadings [6,7] and fibre orientations [8,9] influence the S/N-curves. Thus, the methods to consider for these influences for lifetime estimation are proposed in [2]. This is especially needed for geometrically critical areas, like notches, as these factors have to be considered for lifetime assessment [10,11]. Furthermore, the interaction between notches and other influence parameters is significant for a component load-bearing capacity. Therefore, the influence of local fibre orientations at different notch radii have been widely investigated [12–16]. Bernasconi et al. highlighted in [13] the influence of notch sharpness to lifetime fatigue strength. This refers to moulded, lateral V-shaped ⁎

notches on injection moulded plates. Schaaf et al. propagated in [17] a decreasing failure strength with an increasing notch factor for injection moulded notches. However with injection moulded notches, a transversal fibre orientation in the notch root, which is necessary to determine the interaction between the orientation and the notch, can’t be realized. Therefore, notches needed to be manufactured subsequently in specimens. To investigate two different fibre orientations at comparable specimens, they are milled from injection moulded plates. These dry specimens are tested under cyclic, tensile load with a stress ratio of R = 0.1. The aim of this examination is to gain the influence of the main fibreorientation around the notch, the effect of anisotropy along the notch root and the effect on the notch support. 2. Theory To describe the behaviour of the material around notches under a cyclic load, certain effects have to be considered. Notches cause stress superelevations in the notch root. What we know about stresses in notches for the consideration in lifetime assessment is primarily based on isotropic materials. According to [18,19], a stress maximum can also be (beside FEM-calculations) estimated analytically for different notch shapes. With cyclic loadings, another effect occurs, namely an increase of the local fatigue strength related to the stress maximum, known as the notch support effect [20]. This effect is provoked i. a. by different deformation resistances in the material, which lead to local supporting

Corresponding author. E-mail address: [email protected] (G. Stadler).

https://doi.org/10.1016/j.ijfatigue.2019.105284 Received 29 May 2019; Received in revised form 14 September 2019; Accepted 16 September 2019 Available online 27 September 2019 0142-1123/ © 2019 Published by Elsevier Ltd.

International Journal of Fatigue 131 (2020) 105284

G. Stadler, et al.

areas and can be described with a notch support number. This number is the relationship between the bearable stress amplitude σa, max of a notched specimen and the bearable stress amplitude of an unnotched specimen σa, unnotched [21].

n=

σa, max σa, unnotched

(1)

Eichlseder proposed in [22] a method to consider the notch support effect based on the local stress gradient and the relationship between bending σa, bf and tension-pressure strength σa, tf ,whereby b is the reference specimen width. The exponent KD describes the relationship between the stress gradient and the notch support number. This parameter acts very sensitively on the notch support model. Therefore, it is the main factor for the description of the behaviour depending on the local fibre orientation.

χ′ ⎞ ⎛ σa, bf ⎞ n=1+⎜ − 1⎟ ∗ ⎛ σa, tf 2/ b ⎠ ⎝ ⎝ ⎠ ⎜

KD

⎟

(2) Fig. 1. Tangential plane of the stress field.

This equation can also be written in a more general form:

⎞ ⎛ χ′ ⎞ ⎛ σa, ref n=1+⎜ − 1⎟ ∗ ⎜ ⎟ ′ χref ⎟ ⎜ σa, tf ⎠ ⎝ ⎠ ⎝

KD

By substituting the relationship

the stress gradient along the notch root to the notch support. Therefore, a determination of the stress gradient, evoked by anisotropy, has to be done. As a result, a tangential plane can be created between the highest loaded point, and the points (mesh nodes) next to it (Points 2 and 3) (Fig. 1). These points represent the node stresses from the points along the path (d1 and d2 in Fig. 7). This plane is described with Eq. (9), → → whereby the a and b are the direction vectors of the stress gradients, meaning the vectors between Points 1–2 and the Points 1–3, respec→ represents the stress in the highest loaded point σmax tively. The vector ⎯⎯⎯⎯⎯⎯ and has only entry in the z-direction.

(3) σa, ref σa, tf

with a reference support

′ = 1, we got the number nref at an obtained reference gradient of χref method described by Hück et al. [23]:

n = 1 + (nref (χ ′ = 1) − 1) ∗ χ ′ KD

(4)

For the used model, only the stress decrease component along the path distance x, (direction d1 in Fig. 7) of the stress gradient χ is considered to be described with the term dσ . Outward, the stress gradx dient obtained the stress gradient χ ′ is determined in the following form:

χ′ =

1 dσ ∗ σmax dx

→ → σ ⎛⎜x , y ⎞⎟ = → σmax + x ∗ → a +y∗ b ⎝ ⎠

(5)

→

Kt , a

3+4 1+2

a r a r

⎯→ ⎯ n s ′ = ⎯→ χcom ⎯n ∗ σ g max

(6)

(10)

(11)

3. Methods (experimental/simulation) The material used for the tests is a polyphtalamide (PPA) with a fibre content of 50% by weight of short glass fibre. Bone shaped specimen (Fig. 4) for fatigue tests have been milled from an injection moulded plate with a size of 100 × 100 × 2 mm (Fig. 2). Different flow and temperature conditions along the plate’s crosssectional area led to a layer structure (Fig. 3) with a (averaged) main fibre orientation. This phenomenon is well known and is also shown in several publications [26,13,27–29]. The boundary zone is the result of a rapidly solidified melt due to a high cooling rate of the cavity wall. The local shear rate is a reason for the higher fibre orientation in the shear zone. This rate decrease to the core zone, which leads to a less oriented area. To get different fibre orientations for the tests, specimens along (longitudinal specimens) and perpendicular (transversal specimens) to the main fibre orientation were extracted. The positions of these specimens on the plate are shown in Fig. 2.

(7)

Heywood suggested in [25] a Kt , a calculation for centre holed specimen (Eq. (8)), where d is the hole diameter and W the specimen width.

d ⎞3 Kt , a = 2 + ⎛1 − W ⎝ ⎠

→

In order to get the distance for the stress slope, the normal vector → n ⎯n . The normal is projected to the ground plane (Ω) resulting in vector ⎯→ g ⎯ ) represents the corresponding vector (→ n ) projected to the z-axis (⎯→ n s ⎯ ) stress value. Thus, the stress slope is the magnitude of the vector (⎯→ n s ⎯→ ⎯ concerning the magnitude of the vector ng . The calculation of the re′ is finally done with Eq. (11). sulting gradient χcom

Filippini proposed in [24] a stress gradient calculation method (Eq. (7)) for consideration of the notch curvature a/ r , whereby r represents the curvature radius and a the half-axis length of an elliptical notch. In case of a circle, the distances r and a are equal.

χ ′a r =

→

n =a ×b

whereby σmax represents the largest occurring stress in the notch root. To get a comparable one dimensional stress solution for the gradient calculations, the Von Mises stress has been considered. These methods are already successfully applied on short fibre reinforced polymers [1,11] and implemented in fatigue life prediction software, like FEMFAT®. The calculations of the stress gradient and the stress superelevation are performed analytically. There, the stress superelevation is defined as the local stress maximum σmax with regard to the nominal stress in the specimen cross section σnom . Due to [18], Kt , a is for round boreholes in a disc, which is exactly 3.

σ = max = 3.00 σnom

(9)

The normal vector of the plane → n is calculated by using the cross → → product of the direction vectors ( a and b ) in Eq. (10).

(8)

The effect of anisotropy around the notch to the stress superelevation and stress gradient leads to the assumption of an influence on 2

International Journal of Fatigue 131 (2020) 105284

G. Stadler, et al.

Fig. 2. Positions of the extracted specimens on the plate.

Fig. 5. Positions of the extracted μ CT-specimens on the injection moulded plate.

centre, at Position 2 to the fibre orientation at the centre of the longitudinal specimens, respectively. Test results of transversal specimens, quasistatic as well as cyclic, are equal for both transversal extraction positions (Fig. 1). Therefore, it is acceptable to assume that the differences in the fibre orientations are also negligible for this material and the examined positions. Fig. 6 presents the main entries of the resulting fibre orientation tensor at position 1 and 2 determined with μ CT-scans. The experiments were performed on an electromechanical testing machine, BOSE AT3550 with a load cell limited by 15 kN. The stress ratio for these tests has been defined with R = 0.1 at the frequency of 10 Hz. The tests were carried out at a temperature of 23 °C and a room humidity of 50%. To monitor the temperature development, a thermocouple has been applied on the specimen surface. During the tests a maximum temperature increase due to self-heating of 3 °C has been detected. This increase has, at the investigated temperature, hardly had any effect on the mechanical properties of the polymer. The tests stop if rupture occurs or a cycle number of N = 106 has been reached.

Fig. 3. Layer structure in the plate.

4. Simulations The multi-stage simulation procedure starts with an injection moulding simulation with MOLDFLOW® to get the local fibre orientation along the plate. Further, the fibre orientation is mapped on a hexagonal meshed specimen for structure simulations. Moreover, the local material behaviour is defined in this step [2]. A linear elastic material model has been used both for matrix and the glass fibres,while the mean field homogenisation has been realized with a second-order Mori-Tanaka scheme. The glass fibres are spheroidal shapes with an aspect ratio of 30 and the fibre orientation tensor for the simulation equals the CT-measured tensors. In the structure simulation, performed in ABAQUS®, the surface nodes of one clamping area encased and the nodes of the second are applied with a concentrated force. The resulting stress distribution serves on the one hand as basic for the lifetime assessment in FEMFAT®, on the other hand, to get the highest loaded point. Starting at this point, the stress gradient for the notch support calculations, using Eq. (4), is determined along a path (d1), normal to the notch surface (Fig. 7). Therefore, the stress value at every node along the path (Fig. 7) is required. The simulation results show an increase of the stress gradient toward the notch root. Thus, the stress

Fig. 4. Bone shaped specimen with centred boreholes.

Boreholes with diameters of 0.5, 1, 1.5 and 2 mm have been drilled at the specimen centre (Fig. 4). The aim of drilling the holes instead of moulding them is to get different fibre orientations in the notch root. The holes were manufactured with several drilling steps. This manufacturing approach is similar to the milling process of the specimen and guaranties a high surface quality. However, several fibres in the notches are cut, which influences the local properties. This is especially true if the fibres are separated into two parts or a reduced transverse section of the fibres decreases the local fatigue strength. The local fibre orientations have been determined by μ CT-scans at specimens with a size of 2 × 2 × 2 mm from representative positions [30] (Fig. 5). Therefore, the investigated hole diameters are included. Thus the fibre orientation is known at every notch root. The fibre orientation at Position 1 is equal to those of the transversal specimen’s 3

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Fig. 6. Fibre orientations through plate thickness [30]. dσ

gradient ( χ = dx ) at the highest loaded point is taken into account for the notch support calculations. A comparable mesh is necessary for the different notch radii. Therefore, a meshing concept following the IIW (International Institute of Welding [31]) guideline and according to [32,33] has been utilized (Fig. 8). Due to this meshing concept there should be 12 elements used over a quarter circle (see Fig. 9). Furthermore, the relationship between the element length along the normal vector (ln in direction d1) to the circuit and the circumference (lc ) should be between 0.5 and 1.

a=

ln = 0.5…1 lc

(12)

Additionally, the stress distribution along the notch root was investigated. Similar to path d1, the stresses along path (d2) (Fig. 7) are considered. The results show that there is a correlation between the local fibre orientation (Fig. 6) and the fibre orientation provoked stress superelevation, as shown in Figs. 10 and 6. Significant are the differences of the stress gradient at various notch radii, as well as compared to the specimen without a notch (infinity notch radius; inf) (Fig. 11). This behaviour results from the combination of a small, high loaded area around the highest loaded point and an increased local stress along longitudinal orientation, highlighted in Table 1. To consider the effect of the stress gradient in both directions (d1 and d2), the resulting stress field has been analysed. An exemplary presentation of the stress field, starting at the highest loaded point, for a notch radius of 0.25 mm is shown in Fig. 12.

Fig. 8. Meshing guideline.

The course along the notch (Fig. 12) is equal to those in Fig. 10, with the origin in the centre for the transversal and the surface in the longitudinal direction. Since only the stress gradient, starting at the highest loaded point, is considered for these calculations, it is not necessary to evaluate the whole stress field along the notch. There is a major drop in the stress outward on the notch root. The influence on the stress due to anisotropy is significantly lower in longitudinal as well as transversal specimens according to the notch inducted stress drop. This stress distribution is provoked by a continuous change of the fibre orientation in the layers, outward the highest loaded point in direction d2 .

Fig. 7. Paths for stress gradient determination. 4

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Fig. 9. Schematic course of node stresses along the path outward to the notch root according to [22].

Analytical calculations have been performed and compared with simulation results to check the significance of analytical methods for anisotropic composites. The performed simulations demonstrated very different stress superelevations (Kt , s ) than expected by analytical calculations for a hole in a sheet (Eq. (6)). The results of these calculations, compared with the simulation results (Kt , s , χs′, including the gradient evoked by material anisotorpy χm′ ) for the different notch radii and fibre orientations are summarized in Table 1. It can be shown, that the simulation results for the stress superelevation are higher (except transversal at r = 1.0 mm, which can be explained with the interaction between mesh, notch and material behaviour) compared to the analytical solutions. The stress gradients are underestimated for longitudinal specimens and overestimated for transversal specimens. Thus, the analytical methods are inappropriate for the calculation of anisotropic materials. The results show a 0.5% increase of the stress gradient compared to those without considering the second direction ( χm′ in Table 1) for a notch radius of 1 mm for longitudinal specimens. This value shrinks with a decreasing notch radius. For transversal specimens the difference is even lower than 0.5% with a notch radius of 1 mm. Consequently, calculations using the combined gradient did reveal a negligible influence on the notch support number at the investigated notch radii and fibre orientations.

Fig. 11. Obtained stress gradients at investigated notch radii.

also shown in general for different notch radii (Fig. 13). Further, the performed tests on notched specimen show an influence due to notches in the S/N-curves (Fig. 4). However the differences in the S/N-curves for notch radii of 0.5, 0.75 and 1 mm are not significant. Especially for transversal oriented specimen there is hardly any effect neither on the slope nor on the position of the S/N-curves. Nevertheless, the slope of the notched specimen curves is higher compared to the unnotched specimen for longitudinal specimens. In the transversal case, there is a low difference in the slope. It can be derived that the influence of the notches increase with a decreasing load. This behaviour can be explained with the crack initiation resistance, as the stress superelevation in the notch root may exceed the local resistance, even at a low stress level. This effect is more pronounced for longitudinal compared to transversal oriented specimen due to a higher strength. The most remarkable result shows the S/N-curve of the tests with a notch radius of 0.25 mm, especially the position for transversal orientation. There are several possible explanations for these results including the notch radius, which is similar to the fibre length. Therefore, an increasing influence, caused by imperfections in the material compared to the external notch, decreases the notch effect as a source of failure. Further, the material in the high loaded area will be plasticised, which leads to a reduction of stress peaks. Thus the load with a constant tensile mean stress reinforces stress redistribution in the notch root. The differences of the stress gradients, depending on the fibre orientation, also manifests in the notch support number. These differences are

5. Results and discussion Basically, there is a significant influence of the primary fibre orientation on the fatigue strength of the composite. This influence can be seen at the test results of specimens without a notch (Fig. 4), which are normalized to the tensile strength. Therefore, these results serve as a reference for the test results for notched specimen and this influence is

Fig. 10. Stress distribution along notch root. 5

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Table 1 Notch stress concentration. Fibre orientation

Notch radius

longitudinal

r = 0.25 mm

longitudinal

r = 0.5 mm

longitudinal

r = 0.75 mm

longitudinal

r = 1.0 mm

transversal

r = 0.25 mm

transversal

r = 0.5 mm

transversal

r = 0.75 mm

transversal

r = 1.0 mm

K t,s

Kt , a

χs′

χa′

′ χm

4.30

2.86

11.66 mm−1

9.3 mm−1

0.09 mm−1

4.04

2.73

4.79 mm−1

4.67 mm−1

0.27 mm−1

3.98

2.61

3.93 mm−1

3.1 mm−1

0.17 mm−1

3.36

2.51

2.75 mm−1

2.3 mm−1

0.13 mm−1

3.01

2.86

4.86 mm−1

9.3 mm−1

0.21 mm−1

2.95

2.73

2.45 mm−1

4.67 mm−1

0.12 mm−1

2.89

2.61

1.99 mm−1

3.1 mm−1

0.10 mm−1

2.36

2.51

0.94 mm−1

2.3 mm−1

0.08 mm−1

significant for certain notch radii at given fibre orientations (Fig. 13). Unfortunately, it was not possible to produce smaller holes and to get specimens with a higher fibre orientation. However, it can be shown that there is a higher notch support number for a longitudinal fibre orientation compared to a transversal orientation for every specific notch radius. Moreover, the differences decrease with increasing notch radii. A reason for this behaviour may be a micro notch support effect which is caused by deformation differences between the fibres (small, elastic deformations) and the polymer matrix (greater, elastoplastic deformation). The higher deformation of the matrix leads to plasticisation including stress rearrangement and a residual elongation in the microstructure. As a result, the matrix is supported by the surrounding fibres with an unloading and reloading cycle. This effect is more pronounced with a longitudinal fibre orientation, since transversal oriented fibres offer only a small difference between fibres and matrix in tension, as well as short distances (fibre diameters) in load direction to support the matrix. These behaviours effect the exponent KD in the notch support model. The determination of the exponents KD is done by Eq. (11) with KD = 0.38 for transversal specimen and KD = 0.43 for longitudinal specimen as a result.

Fig. 13. S/N-curves of tests performed at specimen without notches.

area) and local (considering the stress superelevation) system are normalized to the tensile strength. Simulation results show a shift of the S/ N-curves to a more conservative direction for longitudinal as well as transversal specimens. The consideration of a notch exponent, determined by anisotropic specimen, has a higher effect with a smaller notch radius. Moreover, it can be shown, that the effect provoked by the notch exponent (KD ) is more pronounced for local stresses. Therefore, the fibre orientation should be considered for the stress calculation and for the determination of the notch support number or rather the exponent KD . As a result, the lifetime prediction for additional manufactured notches can be improved by considering the fibre orientation in the support model. The main advantage of this method (drilling holes) is to get the sensitivity to notches at certain fibre orientations.

6. Validation Lifetime calculations have been performed with the determined notch exponents KD = 0.38 for transversal specimen and KD = 0.43 for longitudinal specimen. The exponent KD is estimated for that material based on notched specimen [2] with a value of KD = 0.66, which serves as a reference. Figs. 14 and 15 show test results compared to simulation results for the smallest (0.25 mm) and the largest investigated notch radius (1 mm). The values for the nominal (the applied load concerning the cross area, meaning a constant stress distribution across the loaded

Fig. 12. Stress field in notch root for a notch radius of r = 0.25 mm. 6

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G. Stadler, et al.

Fig. 14. S/N-curves of tests with different notch radii.

7. Conclusion In this paper, we investigated the influence of different notch radii, the fibre orientation and material anisotropy along the notch root on fatigue lifetime. It has been shown that the notch support effect also depends on the local fibre orientations and anisotropy along the notch root. The resulting differences can be described with micro notch support effects. Therefore, an investigation of the local fibre orientation should be completed during the design process of short fibre reinforced polymer components. If the notch radius gets very small, e.g. near the fibre length, crack initiating and stopping mechanism around the fibres are dominate. Furthermore, the results of the analytical models for these special cases have been discussed. There is an influence evoked by anisotropy along a notch root to the local stress superelevation. This is the reason why analytical models underestimate the stress superelevation. Also, the stress gradient is highly dependent on the anisotropy and therefore, the results of an analytical calculation should be examined. The stress gradient in a second direction, due to anisotropy, is negligible low compared to the notch stress gradient. Due to this low effect on the gradient, it doesn’t have to be considered for calculation in those cases. However, the local anisotropy along the notch root is needed to be considered in the notch support model with the notch exponents KD . Thus, further investigations that additional influence factors are needed. Especially, temperature or mean stress may have strong effects on the fatigue strength of notched structures with different fibre orientations. Moreover, the influence of weld lines next to notches would be another factor that should be investigated.

Fig. 15. Gradient dependent notch support number [34].

For more information about the interaction between fibre orientation and notch sensitivity, specimen with fibre orientation angles between 0 and 90 degrees (e.g. 45 degrees) should be tested. The main disadvantage of this test approach is the specimen preparation as well as the test and simulation effort. Since most of the notches are injection moulded directly, a calculation with parameters, determined on moulded notched specimen, is often sufficient for a lifetime assessment (see Figs. 16 and 17).

Fig. 16. Validation of S/N-curves with different notch exponents KD for a notch radius of r = 1 mm. 7

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G. Stadler, et al.

Fig. 17. Validation of S/N-curves with different notch exponents KD for a notch radius of r = 0.25 mm.

Acknowledgements The research work of this paper was performed at the Chair of Mechanical Engineering at the Montanuniversität of Leoben in collaboration with the Polymer Competence Center Leoben GmbH (PCCL, Austria) within the framework of the COMET-program of the Austrian Ministry of Traffic, Innovation and Technology with contribution by Borealis Polyolefine GmbH, Engineering Center Steyr GmbH & CoKG (MAGNA Powertrain ECS), EVONIK Industries AG, Schaeffler Technologies AG & Co KG, Volkswagen AG as well as EMS-Chemie AG (EMS-GRIVORY) as associated partner. The PCCL is founded by the Austrian Government and the State Governments of Styria, Lower and Upper Austria.

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