Elastic Modulus – Examples and Material Values

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Elastic modulus – Examples and material values plastics

Introduction

The quasi-static modulus of elasticity (elastic modulus) is, alongside the Poisson's ratio, an essential parameter for describing the energy-elastic properties of plastics. The short-term moduli E_t, E_f and E_c determined in tensile, bending or compression tests are suitable for quality assurance, material development and optimisation, as well as simple dimensioning tasks (see: plastic component, dimensioning). However, the modulus of elasticity can also be determined by means of dynamic-mechanical analysis using tensile or bending stress as the dynamic modulus of elasticity E_d, separated into the storage modulus E´ and the loss modulus E´´ relevant for engineering. With instrumented hardness testing, an elasticity modulus called the indentation modulus E_IT can also be determined if the Poisson's ratio of the material is known. The non-destructive testing method ultrasound can also be used to determine material parameters such as the modulus of elasticity or density, e.g. in bone density measurements, whereby the Poisson's ratio in the temperature range under investigation must also be known.

In the following examples for the materials polymethyl methacrylate (abbreviation: PMMA), polypropylene ([[Plastics – Symbols and Abbreviated Terms|abbreviation: PP) and polyvinyl chloride (abbreviation: PVC), the moduli of elasticity determined using different test methods are compared as a function of the test temperature.

The test methods selected were the tensile and bending tests, the instrumented macrohardness measurement and the torsional vibration test, as well as the ultrasound test, for which the test conditions and measurement technology are briefly described below.

Test methods for determining the modulus of elasticity

Quasi-static short-term tests

To determine the elasticity modulus E_t and E_f, tensile and bend tests were performed in the quasi-static polymer testing using an INSTRON 5507 universal testing machine (Eqs. 1 and 2). In accordance with the standard for testing plastics, the secant modulus was determined in both tests within the limits of 0.05 and 0.25 % strain on five test specimens each, in accordance with [1–3] (Fig. 1). The test speed was 1 mm/min in the tensile test and 2 mm/min in the bending test. To simultaneously determine the Poisson's ratio µ (Eq. 2) in the tensile test, a transverse strain sensor was used within the limits of 0.05 % and εy (see: yield point). All quasi-static tests were carried out in a connected temperature control chamber at temperatures ranging from –20 to 60 °C [4].

Fig. 1:

Determination of the secant modulus in the tensile test (a) and bending test (b)

$E_{t}={\frac {\sigma _{2}-\sigma _{1}}{\epsilon _{2}-\epsilon _{1}}}={\frac {\Delta \sigma }{0,002}}$

(1)

$E_{f}={\frac {\sigma _{f2}-\sigma _{f1}}{\epsilon _{f2}-\epsilon _{f1}}}={\frac {\Delta \sigma _{f}}{0,002}}$

(2)

$\mu =\left|{\frac {\epsilon _{q}}{\epsilon _{L}}}\right|=\left|{\frac {\Delta b}{\Delta L}}\right|$

(3)

Determination of the indentation modulus

To determine the indentation modulus E_IT (Eq. 4), the instrumented macro hardness measurement system ZHU2.5 from Fa. ZwickRoell GmbH & Co. KG was used, which is equipped with a closed temperature control chamber, an extended contact foot and a second displacement sensor. After positioning the test plates in the chamber, it was heated to the test temperature over an appropriate period of time. Force-controlled macrohardness measurements were then carried out in the load range from 0.5 to 20 N at a feed rate of 0.5 N/s in accordance with ISO 14577-1 [5–7].

Fig. 2:

Zwick ZHU2.5 macro hardness testing machine with temperature control chamber for performing instrumented hardness measurements from –100 to 100 °C

$E_{IT}={\frac {1-\mu ^{2}}{0,5{\sqrt {{\frac {24,5}{\pi }}\cdot \left({\frac {dh}{dF}}\right)_{F_{max}}}}\cdot \left(4h_{t}-3F_{max}\left({\frac {dH}{dF}}\right)_{F_{max}}\right)-8,73\cdot 10^{-13}}}$

(4)

The resulting load (F)-indentation depth (h) curve is continuously recorded during the loading and unloading process (Fig. 3) and the characteristic values of the instrumented hardness measurement are determined in accordance with [5]. The indentation modulus corresponds to the increase S of the tangent at F_max of the respective unloading curve. For each test temperature, five indentation modules were determined by a defined feed of the positioning device, knowing the temperature-dependent Poisson's ratio µ, which were then averaged.

Fig. 3:

Load–penetration depth curve (a) load curve and (b) unloading curve of the instrumented hardness measurement [6]

Dynamic-mechanical analysis (DMA)

In dynamic-mechanical analysis or spectroscopy, the test specimen is subjected to periodically alternating sinusoidal stress. The advantage of this method when using a temperature control chamber (see: Dynamic-mechanical Thermal Analysis – DMTA|dynamic-mechanical thermal analysis – DMTA]]) is that the dynamic mechanical characteristics are available as a function of temperature. For these measurements with the Mark III test system (Fig. 4), the forced torsional vibration test was selected, where the energy absorption of the motor serves as the measured variable. The frequency was varied in 11 steps in the range between 0.03 and 50 Hz, with a heating rate of 2 K/min in the temperature interval from –40 to 100 °C. For selected measurement frequencies, the dynamic storage modulus G´ was then determined at identical temperatures, as in the tensile test, for example [8].

The approximate determination of the dynamic modulus of elasticity E_d was carried out using the shear modulus G´, since the Poisson's ratio µ is known in the corresponding temperature range (Eq. 5).

$E_{d}=2G^{\prime }\cdot (1+\mu )$

(5)

Fig. 4:

DMTA-System Mark III from Rheometrics Scientific

Ultrasound testing

In the high frequency range, the dynamic modulus of elasticity can also be determined by the propagation of ultrasonic waves in the test specimen, provided that the density ρ and the Poisson's ratio µ are known [9].

The measurements were carried out using the USPC 3040 ultrasonic testing system from Dr. Hillger, Braunschweig, in the transmission technique (f = 2 MHz) in direct coupling of the ultrasonic standard sensors using heat-resistant coupling oil in a temperature-controlled chamber of the FRANK universal testing machine (Fig. 5).

Fig. 5:

Ultrasonic transmission arrangement (a) and application in the temperature control chamber (b)

The longitudinal sound velocity c_L is determined from the running time Δt of the ultrasound through the test piece when the thickness of the plastic plate is known (Eq. 6).

$c_{L}={\frac {d}{\Delta t}}$

(6)

Provided that the Poisson's ratio µ and the density for the test temperature T are known, the modulus of elasticity E_U can then be calculated using Eq. (7), bearing in mind that these characteristic values are frequency-dependent.

$E=c_{L}^{2}\cdot \rho \cdot {\frac {(1+\mu )\cdot (1-2\mu )}{(1-\mu )}}$

(7)

The density of the different materials was determined using the buoyancy method (ethanol) with a Sartorius CP analytical balance with a density measurement attachment in accordance with ISO 1183-1, method A (immersion method [10]), with all measurements being carried out at room temperature.

An overview of the density values determined and the temperature-dependent Poisson's ratios is shown in Table 1. As expected, the Poisson's ratios also increase with increasing temperature.

Table 1:

Overview of Poisson's ratios in tensile testing and density for all materials examined

material	value	μ (-)						ρ (kg/m3)
	T (°C)	–40	–20	0	20	40	60	23
PMMA		0.34	0.35	0.36	0.37	0.38	0.40	1.178
PP		0.29	0.35	0.36	0.41	0.44	0.47	0.912
PVC		0.33	0.34	0.35	0.36	0.37	0.38	1.393

Fig. 6:

Modules of elasticity of polymethyl methacrylate according to various test methods (Ef – bend test, Et – tensile test, Ed – DMTA, EIT – indentation modulus, EU – ultrasound test)

It can be seen that the two amorphous materials, polymethyl methacrylate (abbreviation: PMMA) and polyvinyl chloride (abbreviation: PVC), have a comparable sequence of moduli of elasticity. As expected, all moduli of elasticity decrease with increasing temperature, as the mobility of the chain segments (see: polymers & structure) increases significantly. Due to the dynamic stress during DMTA (here 50 Hz test frequency) and the multiaxial stress state during the instrumented hardness test, the dynamic modulus of elasticity and the indentation modulus show the highest values for both amorphous materials (Figs. 6 and 7). At low temperatures (–40 and –20 °C), the differences between the elastic modulus from the tensile and bending tests are negligible. For PMMA, the differences increase significantly from 20 °C and for PVC from –20 °C, which is due, on the one hand, to the different normal stress state in the bending test and the different material behaviour of the two materials. The modulus of elasticity E_U from the ultrasound test could only be determined in the temperature range from 20 to 60 °C, as severe icing of the test specimens occurred even at 0 °C, which impaired accurate measurement. The curve at negative temperatures was therefore extrapolated (dashed line in Figs. 6 and 7). However, it can be seen that the non-destructively determined modulus of elasticity for both materials shows the same trend and a comparable increase to the other moduli of elasticity. This modulus of elasticity is in the range of the indentation modulus and the dynamic modulus of elasticity from the dynamic-mechanical thermal analysis.

Fig. 7:

Modules of elasticity of polymethyl methacrylate according to various test methods (Ef – bend test, Et – tensile test, Ed – DMTA, EIT – indentation modulus, EU – ultrasound testing)

In contrast to the amorphous materials PMMA and PVC, the semi-crystalline material polypropylene (abbreviation: PP) has the highest characteristic value for dynamic modulus of elasticity (Fig. 8). At low temperatures (–40 and –20 °C), the differences between the indentation modulus and the modulus of elasticity from the tensile test are small, but increase as the test temperature rises. The lowest level is shown by the modulus of elasticity from the bending test. The non-destructively determined modulus of elasticity E_U lies within the scatter range of the various moduli of elasticity.

Fig. 8:

Modules of elasticity of polyvinyl chloride according to various test methods (Ef – bend test, Et – tensile test, Ed – DMTA, EIT – indentation modulus, EU – ultrasound test)

See also

• Elastic modulus
• Elastic modulus – Ultrasonic measurement
• Indentation modulus
• Energy elasticity
• HOOKE´s law

References

[1]	ISO 527-1 (2019-07): Plastics – Determination of Tensile Properties – Part 1: General Principles
[2]	ISO 527-2 (2025-06): Plastics – Determination of Tensile Properties – Part 2: Test Conditions for Moulding and Extrusion Plastics
[3]	ISO/DIS 178 (2026-03): Plastics – Determination of Flexural Properties (Draft)
[4]	Bierögel, C.: Quasi-static Test Methods. In: Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 101–143 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-5; see AMK-Library under A 22)
[5]	ISO 14577-1 (2026-06): Metallic Materials – Instrumented Indentation Test for Hardness and Materials Parameters – Part 1: Test Method
[6]	Grellmann, W.: Härteprüfverfahren. In: Grellmann, W., Seidler, S. (Eds.): Kunststoffprüfung. Carl Hanser, Munich (2025) 4th Edition, pp. 186–207 (ISBN 978-3-446-44718-9; E-Book: ISBN 978-3-446-48105-3; see AMK-Library under A 23)
[7]	Schöne, J., Lach, R., Bierögel, C., Grellmann, W.: A New Generation of Testing Machine: Recording Macroindentation Techniques for Fast Assessment of Temperature-dependent Material Properties. Polymer Testing, 32 (2013) 1479–1486 ; https://doi.org/10.1016/J.POLYMERTESTING.2013.09.016
[8]	ISO 6721-4 (2019-05): Plastics – Determination of Dynamic Mechanical Properties – Part 4: Tensile Vibration – Non-resonance Method
[9]	Matthies, K. u. a.: Dickenmessung mit Ultraschall. DVS-Verlag GmbH, Berlin, 2. Auflage (1998) (ISBN 3-87155-940-7; see AMK-Library under M 44)
[10]	ISO 1183-1 (2025-0): Plastics – Methods for Determining the Density of Non-celluar Plastics – Part 1: Immersion Method, Liquid Pycnometer Method and Titration Method