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2025-12-15T05:28:34Z

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	==Determination methods==		==Determination methods==

	The flexural modulus ''E''<sub>f</sub> is usually determined in a [[Bend Test#The three-point bending test method\|three-point]] or [[Bend Test#The four-point bending test method\|four-point bending test]] under [[Quasi-static Test Methods\|quasi-static loading]] [1–3] on [[Plastics\|plastics]] or [[Short-Fibre Reinforced Plastics\|short-fibre reinforced]] or filled [[Composite Materials Testing\|plastic composites]]. The test on rigid and semi-rigid plastics, i.e. [[Thermoplastic Material\|thermoplastic]] [[Moulding ~~Compounds~~\|moulding compounds]] or extrusion and casting compounds, is carried out in accordance with ISO 178 [2] in a three-point bending test.		The flexural modulus ''E''<sub>f</sub> is usually determined in a [[Bend Test#The three-point bending test method\|three-point]] or [[Bend Test#The four-point bending test method\|four-point bending test]] under [[Quasi-static Test Methods\|quasi-static loading]] [1–3] on [[Plastics\|plastics]] or [[Short-Fibre Reinforced Plastics\|short-fibre reinforced]] or filled [[Composite Materials Testing\|plastic composites]]. The test on rigid and semi-rigid plastics, i.e. [[Thermoplastic Material\|thermoplastic]] [[Moulding Compound\|moulding compounds]] or extrusion and casting compounds, is carried out in accordance with ISO 178 [2] in a three-point bending test.

	For [[Fibre-reinforced Plastics\|fibre-reinforced plastics]], the three-point or four-point bending test (method A or B) according to ISO 14125 [3] can be used. However, the flexural modulus can also be determined as a complex modulus under dynamic loading (see: [[Dynamic Mechanical Analysis (DMA) – Bend Loading\|dynamic-mechanical analysis (DMA) – bend loading]]).		For [[Fibre-reinforced Plastics\|fibre-reinforced plastics]], the three-point or four-point bending test (method A or B) according to ISO 14125 [3] can be used. However, the flexural modulus can also be determined as a complex modulus under dynamic loading (see: [[Dynamic Mechanical Analysis (DMA) – Bend Loading\|dynamic-mechanical analysis (DMA) – bend loading]]).

Oluschinski: Created page with "{{Language_sel|LANG=ger|ARTIKEL=Biegemodul}} {{PSM_Infobox}} Flexural modulus FORCETOC ==Determination methods== The flexural modulus ''E''_f is usually determined in a three-point or four-point bending test under quasi-static loading [1–3] on plastics or S..."

2025-12-02T08:20:15Z

Created page with "{{Language_sel|LANG=ger|ARTIKEL=Biegemodul}} {{PSM_Infobox}} Flexural modulus __FORCETOC__ ==Determination methods== The flexural modulus ''E''f is usually determined in a three-point or four-point bending test under quasi-static loading [1–3] on plastics or S..."

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{{Language_sel|LANG=ger|ARTIKEL=Biegemodul}}

{{PSM_Infobox}}
Flexural modulus
__FORCETOC__

==Determination methods==

The flexural modulus ''E''f is usually determined in a [[Bend Test#The three-point bending test method|three-point]] or [[Bend Test#The four-point bending test method|four-point bending test]] under [[Quasi-static Test Methods|quasi-static loading]] [1–3] on [[Plastics|plastics]] or [[Short-Fibre Reinforced Plastics|short-fibre reinforced]] or filled [[Composite Materials Testing|plastic composites]]. The test on rigid and semi-rigid plastics, i.e. [[Thermoplastic Material|thermoplastic]] [[Moulding Compounds|moulding compounds]] or extrusion and casting compounds, is carried out in accordance with ISO 178 [2] in a three-point bending test.

For [[Fibre-reinforced Plastics|fibre-reinforced plastics]], the three-point or four-point bending test (method A or B) according to ISO 14125 [3] can be used. However, the flexural modulus can also be determined as a complex modulus under dynamic loading (see: [[Dynamic Mechanical Analysis (DMA) – Bend Loading|dynamic-mechanical analysis (DMA) – bend loading]]).

==Definition of the flexural modulus==

The determination of the [[Elastic Modulus|modulus of elasticity]] under [[Quasi-static Test Methods|quasi-static]] [[Bend Loading|bend loading]] in [[Polymer Testing|polymer testing]] does not differ in principle from other mechanical [[Stress|stress]], such as in [[Tensile Test|tensile testing]] or [[Compression Test|compression testing]]. The modulus of elasticity is determined in the linear elastic and [[Linear-viscoelastic Behaviour|linear-viscoelastic]] deformation range as the secant modulus between 0.05 and 0.25 % [[Peripheral Fibre Strain|peripheral fibre strain]] (see also: [[Elastic Modulus – Examples and Material Values|elastic modulus – examples and material values]]), whereby the [[Test Speed|test speed]] can be identical to that of the actual [[Bend Test|bending test]] ('''Fig. 1'''). In analogy to the tensile test, however, different test speeds can also be used to determine the modulus of elasticity and the bending properties, whereby the switchover point of the test speeds must then be above 0.25 % peripheral fibre strain. The general calculation of the flexural modulus ''E''f is carried out according to '''Eq. (1)''', but in the specific metrological case, the geometric conditions and the measurement location must be taken into account.

{|
|- valign="top"
|width="550px"|<math>E_{f}=\frac{\sigma_{f2}-\sigma_{f1}}{\varepsilon_{f2}-\varepsilon_{f1}}=\frac{\Delta \sigma_{f}}{\Delta \varepsilon_{f}}=\frac{\Delta \sigma_{f}}{0,002}</math>
|width="50px"|(1)
|}

[[file:Biegemodul-1.jpg|500px]]
{|
|- valign="top"
|width="50px"|'''Fig. 1''':
|width="600px"|Determination of the secant modulus in three-point and four-point bend tests
|}

==Metrological Determination of the flexural modulus==

To determine the flexural modulus ''E''f, the quasi-static [[Polymer Testing|polymer testing]] under flexural stress of the [[Bend Test#The three-point bending test method|three-point]] or [[Bend Test#The four-point bending test method|four-point]] flexural test is used with a [[Material Testing Machine|material testing machine]]. When applying the ISO 178 standard, only the three-point flexural test is specified, whereby the measurement of the centre deflection is generally used here. In the case of the bending test, the traverse path measurement ('''Fig. 2a''') can be used to determine the centre deflection ('''Eq. 2'''), whereas a centrally positioned extensometer must be used to determine the flexural modulus ('''Fig. 2b'''). In this equation, ''I''y is the minimum axial moment of inertia (see: [[Bend Test|bend test]]) of the prismatic test specimen.

{|
|- valign="top"
|width="550px"|<math>s=\frac{F\cdot L^{3}}{48\cdot E_{f}\cdot I_{y}}</math>
|width="50px"|(2)
|}

Since the reference point for the deflection measurement is identical for both the traverse path measurement (see also: [[Tensile Test|tensile test]] and [[Compression Test|compression test]]) and when using a centre sensor (support of the bending fins), the calculation '''Eq. (3)''' is the same for both measurement cases.

{|
|- valign="top"
|width="550px"|<math>E_{f}=\frac{F\cdot L^{3}}{s\cdot4b\cdot h^{3}}</math>
|width="50px"|(3)
|}

[[file:Biegemodul-2.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 2''':
|width="600px"|Determination of the flexural modulus in a three-point bending test: (a) by measuring the traverse path and (b) using a extensometer
|}

The advantage of the extensometer lies in the fact that the penetration of the bending fin into the [[Surface|surface]] of the test [[Specimen|specimen]] is not recorded, which means that the bending modulus is slightly higher (up to approx. 10 %) compared to traverse path measurement. Such bending sensors are commercially available from [https://www.zwickroell.com/ ZwickRoell GmbH & Co. KG, Ulm], and formerly Instron Deutschland GmbH, Pfungstadt ('''Fig. 3''').

In conjunction with the Multisens transducer from ZwickRoell, adapters for use in [[Compression Test|compression tests]] and [[Bend Test|bending tests]] are also available in addition to the extensometers for tensile tests (see: '''[[Tensile Test#Tensile test, path measurement technique|tensile test, path measurement technique]]'''). The central sensor can be used to measure the compressive modulus, but also to record the centre deflection for determining the flexural modulus in three- and four-point bending tests (see: '''Fig. 2b'''). If the penetration of the supports in the bending test is not to be recorded in the measurement signal, so-called fork sensors can be used as an alternative, which generate a differential measurement signal between the forks and the central sensor. Due to the fact that this measurement signal is significantly smaller than, for example, the traverse path, high-resolution measurement techniques must be used here ('''Fig. 4''').

[[file:Biegemodul-3.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 3''':
|width="600px"|Extensometer for mid-span deflection (a) Deflectometer W-E401-H from Fa. Instron Deutschland GmbH, Pfungstadt, and (b) bending sensor BTC-EXOMFL.H01 from [https://www.zwick.de/ Fa. ZwickRoell GmbH & Co. KG, Ulm]
|}

[[file:Biegemodul-4.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 4''':
|width="600px"|Application of (a) fork gauges in three-point bending tests and (b) fork gauges for the Multisens transducer from [https://www.zwick.de/ Fa. ZwickRoell GmbH & Co. KG, Ulm]
|}

The deflection depends on the geometric dimensions of the fork sensor and is calculated according to '''Eq. (4)'''. The calculation of the flexural modulus using fork sensors, which is often not included in the test software of the [[Manufacturer Material Testing Machines|test machine manufacturer]], can be programmed and evaluated using external software according to '''Eq. (5)'''.

{|
|- valign="top"
|width="550px"|<math>s=\frac{F\cdot L_{G}^{2}}{96\cdot E_{f}\cdot I_{y}}\cdot(3L-L_{G})</math>
|width="50px"|(4)
|}

{|
|- valign="top"
|width="550px"|<math>E_{f}=\frac{F\cdot L_{G}^{2}(3L-L_{G})}{s\cdot 8b\cdot h^{3}}</math>
|width="50px"|(5)
|}

When determining the flexural modulus in the four-point bending test, there are various measurement variants that place different demands on the measurement technology. The simplest options here are also to use the traverse path ('''Fig. 5a''') and to measure the centre deflection in analogy to the three-point bending test ('''Fig. 5b''').

[[file:Biegemodul-5.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 5''':
|width="600px"|Application of (a) mid-span deflection in four-point bending test and (b) traverse path measurement
|}

In the case of four-point bending, the bending modulus is obtained from the relative movement between the supports (see: [[Support Distance|support distance]]) and an extensometer attached centrally, using the mid-span deflection ''s'' ('''Fig. 5a''') according to '''Eq. (6)'''.

{|
|- valign="top"
|width="550px"|<math>E_{f}=\frac{F\cdot L_{A}}{s\cdot 4b\cdot h^{3}}\left [ 4L_{A}(3L_{F}+2L_{A})+3L_{F}^{2} \right ]</math>
|width="50px"|(6)
|}

With the traverse path ''s'' ('''Fig. 5b'''), in the case of four-point bending, the bending modulus is calculated as the path difference between the fixed bending beam and the supports according to '''Eq. (7)'''.

{|
|- valign="top"
|width="550px"|<math>E_{f}=\frac{F\cdot L_{A}^{2}}{s\cdot b\cdot h^{3}}\left ( 3L_{F}+2L_{A} \right )</math>
|width="50px"|(7)
|}

[[file:Biegemodul-6.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 6''':
|width="600px"|Application of (a) fork gauges in three-point bending tests and (b) Multisens fork gauges from ZwickRoell
|}

The flexural modulus is calculated here from the difference ''s'' between the centre of the reference or flexural beam and the traverse movement according to '''Fig. 6a''' using '''Eq. (8)''', thereby avoiding influences on the deflection due to [[HERTZIAN Pressure|HERTZIAN pressure]] and penetration at the supports.

{|
|- valign="top"
|width="550px"|<math>E_{f}=\frac{3F\cdot L_{F}^{2}\cdot L_{A}}{s\cdot 4b\cdot h^{3}}</math>
|width="50px"|(8)
|}

The relative distance s between the centre of the reference beam system or loosely mounted fork sensors and the centre of the test [[Specimen|specimen]] within the bending beam gives the bending modulus according to '''Fig. 6b''' and '''Eq. (9)'''. With this type of measuring system, influences on the deflection due to pressure on the supports and the test stamp can be avoided, although this requires a high-resolution measuring system.

==Characteristic values of the three-point bending test==

A comprehensive literature analysis of flexural moduli and [[Flexural Strength|flexural strengths]] for numerous [[Plastics|plastics]] is contained in [4], from which the [[Material Value|characteristic values]] for selected materials are shown in Table 1. Due to the importance of the materials, only unreinforced plastics and plastics with a 30 M.-% filler or reinforcement content were included in this list, which were generally determined by measuring the traverse path at room temperature.

{|border="1px" style="border-collapse:collapse; text-align:center"
|+ '''Table 1''': Characteristic values of the [[Bend Test|bend test]] for technical [[Plastics|plastics]] according to [4]
!! style="width: 125px; background:#DCDCDC"|material groupe
!! style="width: 225px; background:#DCDCDC"|modification
!! style="width: 125px; background:#DCDCDC"|Ef (MPa)
!! style="width: 125px; background:#DCDCDC"|σfM (MPa)
|-
|'''ABS'''
|
|1373 – 3792
|47.1 – 95.1
|-
|
|style="text-align:left"|ABS + 30 M.-% GF
|6067 – 9400
|86.9 – 162
|-
|
|style="text-align:left"|ABS + 30 M.-% CF
|16547
|179
|-
|
|style="text-align:left"|ABS / PBT + 30 M.-% GF
|8205 – 10100
|147 – 186
|-
|
|style="text-align:left"|ABS / PC + 30 M.-% GF
|7505 – 8136
|137 – 167
|-
|'''ASA'''
|style="text-align:left"|
|1344 – 3000
|38.6 – 84.8
|-
|'''PA 11'''
|style="text-align:left"|PA 11
|300 – 1241
|51.7 – 75.8
|-
|'''PA 12'''
|style="text-align:left"|
|345 – 2068
|20.0 – 98.6
|-
|
|style="text-align:left"|PA 12 + 30 M.-% GF
|5516 – 7100
|150 – 185
|-
|'''PA 612'''
|style="text-align:left"|
|1724 – 2758
|82.7 – 96.5
|-
|'''PA 6'''
|style="text-align:left"|PA 6 (normfeucht)
|380 – 1400
|75.8 – 127
|-
|
|style="text-align:left"|PA 6 + 30 M.-% GF (normfeucht)
|2600 – 5600
|192 – 200
|-
|
|style="text-align:left"|PA 6 + 30 M.-% GB (normfeucht)
|1400 – 4482
|100 – 138
|-
|
|style="text-align:left"|PA 6 + 30 M.-% MF (normfeucht)
|4000 – 5396
|98.1 – 124
|-
|
|style="text-align:left"|PA 6 + 30 M.-% CF (normfeucht)
|14000 – 17237
|310
|-
|'''PAEK'''
|style="text-align:left"|
|17000
|130
|-
|'''PBI'''
|style="text-align:left"|
|6500
|220
|-
|'''PBT'''
|style="text-align:left"|
|1900 – 2760
|62.1 – 101
|-
|
|style="text-align:left"|PBT + 30 M.-% GF
|6343 – 11500
|150 – 225
|-
|
|style="text-align:left"|PBT + 30 M.-% MF
|3500 – 4500
|90.0 – 110
|-
|'''PC'''
|style="text-align:left"|
|1889 – 2786
|71.7 – 114
|-
|
|style="text-align:left"|PC + 30 M.-% GF
|6180 – 8900
|147 – 220
|-
|
|style="text-align:left"|PC + 30 M.-% CF
|15900 – 16200
|207 – 241
|-
|'''PE-HD'''
|style="text-align:left"|
|689 – 1655
|25.5 – 32.4
|-
|'''PE-LD'''
|style="text-align:left"|
|69.0 – 621
|7.0 – 15.2
|-
|'''PE-LLD'''
|style="text-align:left"|
|207 – 827
| -
|-
|'''PE-MD'''
|style="text-align:left"|
|345 – 900
| -
|-
|'''PE-UHMW'''
|style="text-align:left"|
|276 – 923
|41.4
|-
|'''PEEK'''
|style="text-align:left"|
|2758 – 4300
|103 – 170
|-
|
|style="text-align:left"|PEEK + 30 M.-% GF
|8963 – 12000
|221 – 261
|-
|'''PEI'''
|style="text-align:left"|
|2900 – 3447
|89.6 – 165
|-
|'''PEK'''
|style="text-align:left"|
|4200 - 6205
|207
|-
|'''PET'''
|style="text-align:left"|
|1090 – 2758
|79.3 – 82.7
|-
|
|style="text-align:left"|PET + 30 M.-% GF
|6965 – 11928
|123 – 240
|-
|'''PMMA'''
|style="text-align:left"|
|1200 – 3654
|46.2 – 121
|-
|
|style="text-align:left"|PMMA + 30 M.-% GF
|6481
|106
|-
|'''POM'''
|style="text-align:left"|
|1471 – 3150
|52,0 – 95,1
|-
|
|style="text-align:left"|POM + 30 M.-% GF
|6890 – 9000
|121
|-
|
|style="text-align:left"|POM + 30 M.-% GB
|1900 – 3170
|62.1
|-
|'''PP'''
|style="text-align:left"|
|1500 – 2462
| -
|-
|
|style="text-align:left"|PP + 30 M.-% GF
|4975 – 7001
|136 – 157
|-
|
|style="text-align:left"|PP + 30 M.-% T
|2340 – 3900
|38.6 – 55.2
|-
|
|style="text-align:left"|PP + 30 M.-% MF
|2300 – 3700
|40.2 – 54.9
|-
|
|style="text-align:left"|PP + 30 M.-% GB
|1290 – 1600
| -
|-
|
|style="text-align:left"|PP + 30 M.-% CaCO3
|1896 – 2800
|32.4 – 45.0
|-
|'''PS'''
|style="text-align:left"|
|1724 – 3447
|37.9 – 75.8
|-
|'''PVC'''
|style="text-align:left"|
|1810 – 3378
|33.0 – 93.1
|-
|
|style="text-align:left"|PVC + 30 M.-% GF
|7930 – 9310
|145 – 159
|-
|'''PVDF'''
|style="text-align:left"|
|413 – 3309
|48.3 – 94.1
|-
|
|style="text-align:left"|PVDF + 30 M.-% GF
|6300
|85.0
|-
|'''SAN'''
|style="text-align:left"|
|3447 – 3792
|103 – 135
|-
|'''TPC'''
|style="text-align:left"|
|196 – 3330
|66,2 – 71,7
|-
|'''TPO'''
|style="text-align:left"|
|83,0 – 1569
|15.2 – 43.4
|-
|'''TPU'''
|style="text-align:left"|
|345 – 750
|9.0 – 14.5
|-
|
|style="text-align:left"|TPU + 30 M.-% GF
|2175
|15.2
|-
|'''TPZ'''
|style="text-align:left"|
|689
|6.2
|}

{|
|- valign="top"
|width="450px" style="text-align:center"|GF: glass fibres, GB: glassballs, MF: mineral fibres, MX: unspecified mineral filling, GX: unspecified glass filling
|}

==See also==

* [[Bend Test|Bend test]]
* [[Flexural Strength|Flexural strength]]
* [[Bend Loading|Bend loading]]
* [[Elastic Modulus|Elastic modulus]]
* [[Indentation Modulus|Indentation modulus]]

'''References'''

{|
|-valign="top"
|[1]
|[[Bierögel, Christian|Bierögel, C.]]: Bend Test on Polymers. In: [[Grellmann, Wolfgang|Grellmann, W.]], [[Seidler, Sabine|Seidler, S.]] (Eds.): Polymer Testing. Carl Hanser Munich (2022) 3rd Edition, pp. 133–143 (ISBN 978-1-56990-806-8; see AMK-Library under A 22)
|-valign="top"
|[2]
|ISO 178 (2019-04): Plastics – Determination of Flexural Properties
|-valign="top"
|[3]
|ISO 14125 (1998-03): Fiber-reinforced Plastic Composites – Determination of Flexural Properties (ISO 14125 Technical Corrigendum 1: 2001-07; ISO 14125 AMD 1: 2011-02)
|-valign="top"
|[4]
|Bierögel, C., [https://www.researchgate.net/profile/Wolfgang-Grellmann Grellmann, W.]]: Bend Loading. In: [https://en.wikipedia.org/wiki/Wolfgang_Grellmann Grellmann, W.], [https://de.wikipedia.org/wiki/Sabine_Seidler Seidler, S.]: Mechanical and Thermomechanical Properties of Polymers. Landolt-Börnstein. Volume VIII/6A3, Springer, Berlin (2014) pp. 164–191, (ISBN 978-3-642-55165-9; see AMK-Library under A 16)
|}

[[category:Bend Test]]

Flexural Modulus - Revision history

Oluschinski at 05:28, 15 December 2025

Oluschinski: Created page with "{{Language_sel|LANG=ger|ARTIKEL=Biegemodul}} {{PSM_Infobox}} Flexural modulus __FORCETOC__ ==Determination methods== The flexural modulus ''E''f is usually determined in a three-point or four-point bending test under quasi-static loading [1–3] on plastics or S..."

Oluschinski: Created page with "{{Language_sel|LANG=ger|ARTIKEL=Biegemodul}} {{PSM_Infobox}} Flexural modulus FORCETOC ==Determination methods== The flexural modulus ''E''_f is usually determined in a three-point or four-point bending test under quasi-static loading [1–3] on plastics or S..."