Young’s modulus
Evaluation of Young’s modulus by three-point flexure test
Nicole Schai Assisted by Claudia Müller
Fig. 1 Substech, [1]
Report
ETH Zürich
29th of October 2011
TABLE OF CONTENT
Page 2 / 15
TABLE OF CONTENTS
1 ABSTRACT ........................................................................................................................................................... 3 2 INTRODUCTION ................................................................................................................................................... 4
2.1 YOUNG’S MODULUS .................................................................................................................................................... 4 2.2 HYPOTHESIS .............................................................................................................................................................. 5
3 MATERIALS AND METHODS ................................................................................................................................. 6 3.1 PREPARATION OF COMPOSITE MATERIALS ........................................................................................................................ 6 3.2 PREPARATION OF THE SAMPLES ..................................................................................................................................... 6 3.3 THREE-‐POINT FLEXURE TEST .......................................................................................................................................... 6
4 RESULTS .............................................................................................................................................................. 8 4.1 PREPARATION ............................................................................................................................................................ 8 4.2 BENDING LINES .......................................................................................................................................................... 9 4.3 YOUNG’S MODULUS AND BENDING STIFFNESS ................................................................................................................. 12
5 DISCUSSION ....................................................................................................................................................... 13 5.1 BENDING LINES ........................................................................................................................................................ 13 5.2 ELASTIC MODULI AND BENDING STIFFNESS ..................................................................................................................... 13
6 REFERENCES ....................................................................................................................................................... 15 6.1 LITERATURE ............................................................................................................................................................. 15 6.2 WEBSITES ............................................................................................................................................................... 15
1. ABSTRACT
Page 3 / 15
1 ABSTRACT
The Young’s modulus is a material constant that describes the elastic properties. In the following experiment, a three-point flexure test is used to measure the elastic modulus and the bending stiffness of steel, copper, aluminium and PVC. The bending stiffness of the homogeneous materials is compared to the bending stiffness determined for three different composite materials. The results show the importance of the exact measuring of the unloaded bending lines in order to get correct values for bending lines, Young’s modulus and bending stiffness. Additionally, the results show some inaccuracies due to shifts of the samples during measuring and inhomogeneous application of glue in compound materials.
2. INTRODUCTION
Page 4 / 15
2 INTRODUCTION
2.1 Young’s modulus
The Young’s modulus is an often-used term to describe materials. According to Hooke’s law it is defined as the ratio between tension and elongation and is therefore the slope of the stress-strain curve during the elastic deformation of a material. As this curve is a straight line during elastic deformation, the Young’s modulus is constant for different tensions applied to the material. On the other hand, Physicians derive the Young’s modulus form the bonding-potential. During this experiment we will see that both explanations are useful in different applications. The size of the Young’s modulus depends on various factors:
• Type of bonding: Stronger bonds lead to a higher Young’s modulus. Primary bonds are much stronger than secondary bonds such as Van der Waals or hydrogen bonds. Within the subgroup of primary bonds, covalent bonds are stronger than ionic bonds. Metallic bonds are the weakest as they are the least direct.
• Structure: Single crystals are anisotropic. Depending on what direction the tension is applied to, the Young’s modulus can vary. However, as many bulk materials are polycrystalline, the factor of direction is statistically cancelled out. Composite materials are not homogenous polycrystalline. It is not possible to evaluate a Young’s modulus for the material.
• Temperature: As temperatures go close to the melting point of a certain material, the Young’s modulus drops due to higher inner energy. The temperature at which a Young’s modulus is measured has therefore a direct influence on the values. However, is the surrounding temperature far lower than the melting point, the influence is negligible.
Metals are usually not used as a pure material but as an alloy. Depending on the size of the atoms of the alloy partners, they build substitutional crystal structures (the metal with the lower percentual content takes places of the main metal, e.g. Cu-Ni) or interstitial crystal structures (small alloy partners with low percentual content sit between the main metal atoms, e.g. Fe-C). The Young’s modulus is directly proportional to the percentual alloy content. Beyond the limit of solubility the alloy consists of different phases and the elastic modulus can vary strongly. Ceramics have – in comparison to polymers and metals – high Young’s modulus. They might not be useful for application where stretching is required, but they are often used. Polymers chains contain covalent bonds. We therefore expect the Young’s modulus to be higher than for metals and ceramics. However, the main influence on the Young’s moduli are the Van der Waals bondings. The resistance of the polymer against elongation depends on the amount of crosslinks. The Young’s modulus can be evaluated with different such as the commonly used tension test. In the following, another way to evaluate Young’s modulus is described - the three-point flexure test. A rectangular piece of testing material is laid on two supporting pins. Force is applied to the middle of the rectangular bar and the bending of the material measured at different points between the supporting pins and the centre of the material where the load is applied. The three-point flexure test will be described more in detail in .Materials and Methods’.
2. INTRODUCTION
Page 5/ 15
Table 1 shows the in literature stated Young’s modulus [2].
Tab. 1 Shows the Young’s modulus in Gigapascale taken from literature [2].
Material Young’s modulus [GPa]
Steel St37 206.0 [3] Aluminium (Al) 70.6 Copper (Cu) 129.8 Polyvinylchloride (PVC) 4.1
2.2 Hypothesis
We expect the materials to behave correspondingly to literature. The weak Van der Waals bondings of Polymers should lead to a low Young’s modulus of Polyvinylchloride (PVC). Concerning the compound materials, we expect the bending stiffness of aluminium comb core with aluminium cover to be higher than the one with Styrofoam core. This assumption is based on the strength of bee combs in nature.
3. MATERIALS AND METHODS
Page 6 / 15
3 MATERIALS AND METHODS
There were two different types of material tested, homogeneous materials like steal, copper, aluminium and PVC as well as composite and therefore heterogeneous materials. For the composite materials the Young’s modulus of Styrofoam with aluminium cover and aluminium comb with aluminium cover was measured.
3.1 Preparation of composite materials
The styropofoam-aluminium composites as well as one of the aluminium-aluminium composite were self-made. Six aluminium plates were cut to a size of 50cm x 5cm. One side was roughened with sandpaper to remove the oxidation layer and afterwards cleaned with acetone. They were put on a plastic foil. A styrofoam plate was cut to the same size as the aluminium plates. The size of the aluminium comb was about one centimetre in each direction bigger than the aluminium plates. It was cut to the right size after the gluing process. Araldite 2011 AD 80456700 was mixed by hand in a bowl and then applied to the scratched side of the aluminium plates with a spatula. The styrofoam pieces and the piece of aluminium comb were put onto the glued aluminium plates and were topped again with an aluminium plate. The composites were wrapped into plastic foil and pressed for 23 hours at room temperature using four bricks.
3.2 Preparation of the samples
The dimensions (length, width and thickness) of all samples were measured before being tested. The height a as well as the width b were measured with a calliper at three different points. For further calculations, the average was taken. The length was measured once with a metal ruler. With a waterproof marker, every two centimetres a mark was drawn. The first mark was two centimetres form the border, the last mark two centimetres from the opposite border.
3.3 Three-point flexure test
For the Three-point flexure test, the samples were placed onto two supporting pins. First the bending curve was measured with a dial gauge without any to the middle of the sample. In a second and third measuring, load was applied to the middle of the piece. Consequently, the material was bending. The bending curve without any load was subtracted from the bending curves under load in order to cancel out any bending due to the own weight of the material or unsmooth surface. The setup is shown in Figure 1 where the bending curve is v(x) and where x is the distance between the first supporting pin to the point of measuring. L is the measured length l minus 4cm, G the force applied to the material, a = the height of the rectangular piece, b = the width.
3. MATERIALS AND METHODS
Page 7 / 15
Fig. 1 Set up of an three-point flexure test. L implies the distance between the two supporting pins. It corresponds to length l of the testing piece minus 40mm. G = force applied, a = height, b= width of the samples. [2]
The bending curve ( v(x) ) can be used to calculate the Young’s modulus (E) (1) as well as the bending stiffness (2).
E = G
12 ∗ v x ∗ I!∗ x! −
G ∗ L!
16 ∗ v x ∗ I!∗ x
E ∗ I! = G
12 ∗ v x ∗ x! − G ∗ L!
16 ∗ v x ∗ x
with the area moment of inertia 𝐼! =!!∗!!"
(3) The set-up of the experiment was for all tested materials the same and was run at room temperature. The weights used to apply the load were measured on a scale (Mettler PC 16) and marked in order to eliminate any confusion.
(1)
(2)
4. RESULTS
Page 8 / 15
4 RESULTS
4.1 Preparation
The dimensions of the samples were measured. Table 2 shows the measured dimensions for each piece and the calculated planar moment of inertia Iz.
Tab. 2 Measured dimensions of tested pieces of material. a = height, b = width, l = length. Styrofoam 1/ 2 are composites made of styrofoam and Aluminium cover. Al-Comb 1/2 are composites too, made of an aluminium comb core and aluminium cover. Al-Comb 1 is self-made, Al-Comb 2 industrially made.
a [mm] ± 0.05 b [mm] ± 0.05 l [mm] ±0.05 Iz [mm4]
Steel 6.0 60.0 900.0 1080 Aluminium 6.0 60.0 900.0 1080 Copper 6.1 62.1 901.0 1193 PVC 6.8 60.1 901.0 1553 Styrofoam 1 11.6 53.3 500.0 6993
Styrofoam 2 11.8 51.4 500.0 7038 Al-Comb 1 11.8 50.9 500.0 6965 Al-Comb 2 10.0 50.0 500.0 4167
The load applied to the different materials is shown in Table 3.
Tab. 3 Applied loads for different materials in grams.
1. real [g] 2. real [g]
Steel 1987.7 2490.2 Aluminium 996.5 1494.9 Copper 995.0 1989.8 PVC 494.8 Styrofoam 1 2484.6 2981.8
Styrofoam 2 2490.2 2987.0 Al-Comb 1 2484.6 2981.8 Al-Comb 2 2490.2 2987.0
4. RESULTS
Page 9 / 15
4.2 Bending lines
Theoretical bending curves were calculated using the elastic moduli known from literature. The theoretical and the experimental curves for Steel, Aluminium, Copper and PVC are shown in Figures 3 to 8.
Fig. 2 Bending lines of Steal St37. Applied loadings to the material: 1987.7g and 2490.2g. The bending lines are plotted against the distance to the first supporting pin [mm].
Fig. 3 Bending lines of Aluminium. Applied loadings to the material: 996.5g and 1494.9g. The bending lines are plotted against the distance to the first supporting pin [mm].
-‐0.50
0.00
0.50
1.00
1.50
2.00
2.50
3.00
0 50 100 150 200 250 300 350 400 450
De#lection v(x) [m
m]
distance from supporting pin x [mm]
Bending lines of Aluminium
experiment 1kg
theory 1kg
experiment 1.5kg
theory 1.5kg
-‐0.50
0.00
0.50
1.00
1.50
2.00
2.50
0 50 100 150 200 250 300 350 400 450
De#lection v(x) [m
m]
distance from supporting pin x [mm]
Bending lines of SteelSt37
experiment 2kg
theory 2kg
experiment 2.5kg
theory 2.5kg
4. RESULTS
Page 10 / 15
-‐2.00
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
0 50 100 150 200 250 300
De#lection v(x) [m
m]
distance from supporting pin x [mm]
Bending lines of PVC
experiment 0.5kg
theory 0.5kg
Fig. 4 Bending lines of Copper. Applied loadings to the material: 995.0g and 1989.8g. The bending lines are plotted against the distance to the first supporting pin [mm].
Fig. 5 Bending lines of PVC (Polyvinylchloride). Applied loadings to the material: 494.8g. The bending lines are plotted against the distance to the first supporting pin [mm].
-‐1.50
-‐1.00
-‐0.50
0.00
0.50
1.00
1.50
2.00
2.50
0 50 100 150 200 250 300 350 400 450
De#lection v(x) [m
m]
distance from supporting pin x [mm]
Bending lines of Copper
experiment 1kg
theory 1kg
experiment 2kg
theory 2kg
4. RESULTS
Page 11 / 15
Fig. 6 Bending lines of Styrofoam core with aluminium cover. Applied loadings to the material: Sample 1: 2484.6g and 2981.8g; Sample 2: 2490.2g and 2987g. The bending lines are plotted against the distance to the first supporting pin [mm].
Fig. 7 Bending lines of Aluminium comb core with aluminium cover. Applied loadings to the material: Sample 1: 2484.6g and 2981.8g; Sample 2: 2490.2g and 2987g. The bending lines are plotted against the distance to the first supporting pin [mm].
-‐0.50
0.00
0.50
1.00
1.50
2.00
2.50
3.00
0 50 100 150 200 250
De#lection v(x) [mm]
distance from supporting pin x [mm]
Bending lines of Styropor-Aluminium Composite
Styropor 1 -‐ 2.5kg
Styropor 1 -‐ 3kg
Styropor 2 -‐ 2.5kg
Styropor 2 -‐ 3kg
-‐0.10
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0 50 100 150 200 250
De#lection v(x) [mm]
distance from supporting pin x [mm]
Bending lines of Aluminium Comb-Aluminium Composite
Al-‐Comb 1 -‐ 2.5kg
Al-‐Comb 1 -‐ 3kg
Al-‐Comb 2 -‐ 2.5kg
Al-‐Comb 2 -‐ 3kg
4. RESULTS
Page 12 / 15
4.3 Young’s modulus and bending stiffness
The bending stiffness was calculated for all materials, homogeneous as well as compound materials. Additionally, Young’s modulus values were calculated for all homogeneous materials like steel, aluminium, copper and PVC. These values were compared to literature. Table 4 shows all values of bending stiffness with the corresponding standard deviation.
Tab. 4 bending stiffness EIz in Pam4. The indicated weights for the loading are values from theory. For exact values see table 3. Standard deviation (σ) calculated for both, average and single loadings.
weight [kg] EIz [Pam4] σ [Pam4] ∅ EIz [Pam4] ∅ σ [Pam4]
Steel 2.0 257.33 81.18 210.39 75.96 2.5 163.44 25.05
Aluminium 1.0 82.44 3.33 82.85 2.55 1.5 83.25 1.40
Copper 1.0 140.05 11.21
140.00 9.76 2.0 139.96 8.41
PVC 0.5 3.74 0.22 3.74 0.22
Styrofoam 1 2.5 34.40 2.00
27.82 15.33 3.0 21.23 20.11
Styrofoam 2 2.5 37.19 8.26 36.56 7.40 3.0 35.92 7.03
Al-Comb 1 2.5 81.16 3.99 82.89 4.36 3.0 84.63 4.26
Al-Comb 2 2.5 97.50 3.87 97.02 3.48 3.0 96.55 3.28
Table 5 shows the calculated Young’s modulus with standard deviation for all homogeneous materials. Additionally, the Young’s moduli are compared to values from literature, shown in Table 1.
Tab. 5 Calculated Young’s moduli (E in Gigapascal) values for all homogeneous materials. Indicated weight (kg) corresponds to theoretical loading. For exact values see table 3. Average Young’s moduli are given too. The average standard deviation indicates the spread of values over all measurements and loadings. The row E Literature shows the Young’s modulus given by literature [2] & [3] and the standard deviation to the averaged calculated values.
weight [kg] E [GPa] σ[GPa] ∅ E [GPa] ∅ σ[GPa]
E [GPa] Literature σ[GPa]
Steel 2.0 238.27 75.17
194.80 70.34 206.00 7.92 2.5 151.34 23.20
Aluminium 1.0 76.33 3.09
76.71 2.36 70.6 4.32 1.5 77.09 1.29
Copper 1.0 117.36 9.40
117.32 8.18 129.8 8.82 2.0 117.28 7.06
PVC 0.5 2.41 0.14 2.41 0.14 4.1 1.19
5. DISCUSSION
Page 13 / 15
5 DISCUSSION
Most of the results correspond to the roughly to the values given by literature [2]&[3]. However, there are some values that are far away form any expected values. We would like to discuss the results in two parts. First the bending lines and later the calculated Young’s modulus and bending stiffness.
5.1 Bending lines
For copper and aluminium, the measured bending lines correspond well with the calculated values. The values measured for copper are generally a bit higher than the calculated values, whereas the measured bending lines for aluminium lie a bit above the calculations. As for both of them the trends for both loadings go in the same direction, one can assume that the difference has occurred due to inexact measuring of the unloaded bending line. The measured values for PVC are much lower than expected. This could be for several reasons. As there is no big jump in the line, the reason for the difference could be the influence of the gauge, which was visually increasing the bending of the PVC sample once it touched it. The bending lines for steal seem to correspond well to the calculated values. However, there are to points, which are far off the calculations. As these points come up in the bending lines for both loadings, it has to be assumed that the unloaded bending line wasn’t measured properly. For the composite materials we do not have any values form literature and therefore no calculated values. However one can see trends. Both of the aluminium core composites have shown similar behaviour when being loaded. Bending lines for higher loadings are higher. Differences between the two samples could come from different amounts of glue. The bending lines for styrofoam core with aluminium cover are not as smooth. For sample 2, they behave as expected. However, there can be a cut-off observed in the line of sample 1, loading 3kg. In general, there are many reasons for possible differences and unlinearities. Most important seems the exact measuring of the unloaded bending line. Furthermore, movements of the measuring piece can lead to slightly different values as well as the incorrect handling of the measuring watch. It is really important to fix the watch tightly and eliminate any movements of the pieces to be measured. The above listed deviations can be seen again in the calculated elastic moduli as for these calculations, the bending lines are necessary.
5.2 Elastic moduli and bending stiffness
As already mentioned in 5.1, the calculated elastic moduli cannot be more exact than the bending lines. Steel for example has two points in the bending line, which do not match well with the rest of the line. In general, however, the line matches well with the calculated values. Correspondingly, the average Young’s modulus of 194.80 GPa has a high standard deviation of 70.34 GPa but only 7.92 standard deviation to the value listed in literature of 206.00 [3]. The bending stiffness of steel behaves correspondingly. The calculated values for the elastic moduli of aluminium correspond with a standard deviation of 4.32 GPa very well with literature [2]. The measured values are lower than the expected.
5. DISCUSSION
Page 14 / 15
The Young’s modulus of PVC is with 2.41 GPa and standard deviation of 0.14GPa lower than expected. However, as the standard deviation is low, the measurement was probably fairly exact. The values of copper reflect several inaccuracies. First of all, the calculated Young’s moduli are lower than expected. This is probably due to inaccuracies of the measurement of the unloaded bending line. This leads to a slightly heightened standard deviation form the values of literature of 8.82GPa. Furthermore, the bending lines are not nice and straight. This can have several reasons such as movement of the measuring watch or the material during the measurement or uneven surface. These inaccuracies lead to a high standard deviation within the measurement of 8.18GPa The bending stiffness was calculated for all materials, homogeneous and heterogeneous (table 4). Calculations showed, that the bending stiffness of styrofoam core with aluminium cover is far lower than the one of homogenuous metals such as aluminium (27.82-36.56 Pam4 vs. 180 Pam4). A high standard deviation of 15.33 (sample 1) and 7.40 (sample 2) was registered. The inaccuracies are probably due to uneven level of glue used. Additionally for Sample one, a cut off in the bending line for 3kg was registered. Possible reasons for this are listed in 5.1. Inaccuracies can also come form the outer dimensions. The styrofoam and aluminium layers were not matched well on top of each other. It was therefore hard to determine the exact width of the rectangular piece of sample 1. Different to the styrofoam core, an aluminium comb core makes the bending stiffness goes up and is even slightly higher than homogeneous aluminium (82.89-97.02 Pam4 vs. 82.85 Pam4). Standard deviations are low (Sample 1: 4.36 Pam4; Sample 2: 3.48 Pam4). This supports the assumptions made concerning Sample 2 in the previous section. Inaccuracies because of glue are not present. Due to holes in the comb, there was enough room for spare glue. As the comb was cut to the right size after the gluing process, there was no shift of layers present. The measuring of the outer dimensions was therefore exact (not like the composite with styrofoam core). All in all there are several points where inaccuracies occur. In general, we can say: high standard deviations within the measurement are due to inaccurate measuring like movement of the material. High standard deviations in comparison to literature are often due to inaccurate measuring of the unloaded bending lines. Finally, the production of the compound materials had a direct influence on their bending stiffness. Especially for styrofoam core compounds, it is important to use as little glue as possible in order to eliminate uneven surfaces. Additionally, the layers must be fixed during drying process. Otherwise the layers will shift and an exact measuring of outer dimensions becomes impossible. However, as the aluminium sample shows, even as there are many points to be inaccurate, it is possible to measure exact values of bending lines and calculate values for Young’s modulus and bending stiffness which correspond with literature.
6. REFERENCES
6 REFERENCES
6.1 Literature
[2] STUDIENGANG MATERIALWISSENSCHAFT ETH ZÜRICH SKRIPT VERSUCHE 3 + 4; E-MODUL
6.2 Websites
[1] SUBSTECH (2010) DIMITRI KOPELIOVICH WWW.SUBSTECH.COM/DOKUWIKI/DOKU.PHP?ID=FLEXURAL_STRENGTH_TESTS_OF_CERAMICS (17.08.2001)
[3]
WIKIPEDIA - BAUSTAHL WWW.WIKIPEDIA.ORG (04.05.2011)
Recommended
