Shear Modulus

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Shear modulus

General information

The shear modulus G, also known as the sliding or torsion modulus, is, alongside Poisson's ratio, an essential parameter for describing the energy-elastic properties of plastics.

The short-term moduli G determined in quasi-static tests such as the torsion or single- or double-shear test are suitable for quality assurance, material development and optimisation, as well as simple dimensioning tasks (see: plastic component, dimensioning), but cannot be used for demanding design applications, as in this case the creep modulus from long-term experiments, also depending on the test temperature, must be known.

The structural cause of the energy-elastic behaviour of plastics is the change (elongation, compression and distortion) in the average atomic distances and bond angles when subjected to mechanical stress. The mechanical work performed in this process is stored in the form of potential energy (increase in internal energy) and is completely returned when the stress is removed (first law of thermodynamics). As a result of its structural causes, energy-elastic behaviour is therefore limited to the range of very small deformations. The deformation (shear, slip or shear) is completely reversible, whereby the relationship between stress and deformation can be linear or non-linear (see: elasticity). The loading and unloading curve is identical in every case, which means that no hysteresis occurs. If a linear relationship between stress (force or moment) and slip or shear (deformation) is observed, then in the case of a torsional or shear load, the relationship can be described by analogy to a spring or proportionality constant using HOOKE's law for thrust or shear (Eq. 1) for small deflection or slip angles with tan γ ≈ γ (Fig. 1).

Fig. 1:

Elastic deformation of the spring model and the solid body according to HOOKE

$G={\frac {\tau }{\tan \gamma }}={\frac {\tau }{\gamma }}$

(1)

Energy elasticity dominates the behaviour of polymer materials, particularly in the case of small deformations and at low temperatures, as well as at high loading velocities (strain rates), whereby classical energy elasticity theory contributes significantly to our understanding of the deformation behaviour of plastics. In addition, it provides useful approximate solutions for the quantitative description of the stress–strain relationship of a shear stress [1].

In testing practice, three methods are mainly used to determine the shear modulus. These are quasi-static tests for materials and polymers, dynamic-mechanical analysis and ultrasound testing, with short-term mechanical tests (see: tensile test) being the most relevant in practice.

Quasi-static short-term tests

To determine the shear modulus G, torsion and shear tests are performed in quasi-static material and polymer testing using universal testing machines.

Fig. 2:

Torsion test: (a) – Schematic measurement principle, (b) TL1000 torsion testing machine and hybrid universal testing machine from Fa. ZwickRoell GmbH & Co. KG, Ulm

Figure 2a shows a schematic representation of the measurement principle used to determine the shear modulus in a quasi-static torsion test. The round specimens are clamped firmly at the upper end and are rotated through an angle φ by two symmetrical forces via a cylindrical disc, whereby the torsional moment M_t = R · F is applied as a load. For very small reversible angles of rotation, the shear modulus G is then calculated according to Eq. (2) from the geometric conditions and the measured variables F and φ.

$G={\frac {2LR}{\pi r^{4}}}\cdot {\frac {F}{\varphi }}$

(2)

where:	L	–	test specimens length
	R	–	disc radius
	r	–	test specimens radius

In contrast to testing under tensile, compressive or bending stress, there are very few standards for quasi-static torsion testing. There are binding standards for torsion or twisting testing of wires [2, 3] or building materials made of wood [4, 5], and recommendations are given for torsion testing of metallic materials, but these do not usually include the determination of the shear modulus. There are no standards available for plastics in this regard, as the production of round specimens already poses considerable problems due to the moulding seams. If, in the absence of appropriate test standards, the shear modulus under torsional stress is determined using testing machines in accordance with Figs. 2b or 2c on prismatic polymer test specimens, then analogous regulations as for the determination of the modulus of elasticity in tensile or bend tests should be observed. Due to the comparatively low elastic deformations of < 0.1 % strain and the linear-viscoelastic behaviour, which is recorded up to approximately 0.3 %, only very small forces and deformations can be used. The torsion of the test specimen should not exceed 0.5 %, as non-linear viscoelastic deformation components (see: elasticity) may then come into play, and the deformation rate should be limited to approx. 1 %/min. Depending on the shape of the torsional stress--shear diagram, the shear modulus can then be specified as the tangent modulus or secant modulus within the deformation limits of approximately 0.05 and 0.25 % (Fig. 3). The test should be stopped at torsional deformations > 0.3 %. The shear modulus G is then calculated for the tangent modulus according to Eq. (5) or for the secant modulus using Δτ and Δγ according to Eq. (6), whereby the rotations γ₁ and γ₂ are used as default values to determine τ₁ and τ₂.

Fig. 3:

Determination of the tangent modulus (a) and secant modulus (b) in torsion and shear tests on plastics

When calculating shear stresses, it should be noted that in the case of prismatic test specimens, the polar moment of inertia I_p of the circular cross-section according to Eq. (3) must not be used. Since rectangular cross-sections are not free of buckling forces, I_p is then an approximation according to Eq. (4) [6].

$I_{p}={\frac {\pi }{32}}d^{4}=\pi r^{4}$

(3)

$I_{p}=\int _{A}r^{2}\partial A=I_{y}+I_{z}$

(4)

$G={\frac {\tau }{\gamma }}$

(5)

$G={\frac {\tau _{2}-\tau _{1}}{\gamma _{2}-\gamma _{1}}}={\frac {\Delta \tau }{0,002}}$

(6)

If necessary, these tests can also be carried out in a temperature-controlled chamber, allowing the shear modulus G to be represented as a function of temperature.

There are essentially two technically used variants of the shear test, which are referred to as single-cut and double-cut shear [7] (Fig. 4). These tests are only relevant for metallic materials and wood-based building materials [8–10], as illustrated by the number of available standards. For plastics, especially composite materials (see also: composite materials testing) or laminates, a larger number of test standards are available, as shearing is a relevant type of loading for these materials in the automotive and aerospace industries due to their layer structure and the bonding techniques used.

Fig. 4:

Schematic representation of the shear test under tensile and compression loads (a) two-cut and (b) single-cut

The shear tests to determine the shear modulus G can be performed on round or prismatic test specimens. Care must be taken to ensure that the test tool has sharp shear edges and that the supports are positioned precisely in order to minimise the bending component in the test result. The shear stress τ can be determined in a simplified manner by normalising the measured force F to the respective cross-sectional area A = b · h or A = π d²/4. A more accurate solution can be obtained by using the measured crosshead travel. In this case, the area of the prismatic test specimen A = b · Δl must be used, whereas for the round specimen, the circular sector area A_s must be used, which can be found in relevant reference tables.

The shear angle γ can be calculated from the determined crossbeam displacement for single-cut shear according to Eq. (7) and double-cut loading according to Eq. (8).

$\gamma =\arctan {\frac {\Delta l}{s}}$

(7)

$\gamma =\arctan {\frac {2\Delta l}{D-d}}$

(8)

The shear modulus G is then calculated according to Eqs. (5) or (6), depending on whether the tangent or secant modulus is used. Experience has shown that these shear tests are subject to a more or less pronounced run-up behaviour, meaning that an offset must be taken into account when determining the modulus.

Dynamic-mechanical analysis (DMA) under torsional stress

Dynamic-mechanical analysis (DMA) or dynamic-mechanical thermal analysis (DMTA) using torsional loading can basically be carried out using two test methods:

• forced vibrations and
• free damped vibrations

In dynamic-mechanical analysis, a test specimen with a specified geometry is subjected to periodically alternating stress. By varying the frequency, it is possible to characterise the time dependence of the material behaviour at a constant temperature. When these tests are carried out in a temperature-controlled chamber, the test method is referred to as DMTA and the temperature dependence of the dynamic shear modulus for the materials under investigation is characterised [11−13].

To characterise the viscoelastic properties of plastics using forced vibrations, the test specimen is subjected to a sinusoidal mechanical stress with a constant frequency and constant amplitude (Fig. 5). In the case of linear-viscoelastic behaviour, the temporal changes in stress τ and strain γ in the steady state have the same frequency but different phase positions. The value δ is the phase angle, which lies in the range between 0 and π/2. In the case of loading with shear stress, Eqs. (9) and (10) apply.

$\gamma (t)=\gamma _{0}\cdot \sin \omega t$

(9)

$\tau (t)=\tau _{0}\cdot \sin(\omega t+\delta )$

(10)

Fig. 5:

Temporal change in stress and strain during dynamic-mechanical analysis using forced vibrations

As a result of the phase shift δ between loading (stress) and deformation (shear), the modulus must be introduced as a complex quantity G* according to Eq. (11) in order to describe the stress–shear relationship.

$G^{\ast }=G^{\prime }+iG^{\prime \prime }$

(11)

The complex module can be viewed as a vector in the complex number plane (Fig. 6), whose direction is given by the phase angle δ and whose magnitude is given by the ratio of the amplitude values of stress and strain.

Fig. 6:

Representation of the module G* in the complex number plane

The absolute value of the respective module is calculated from the ratio of the initial loading to the initial deformation according to Eq. (12).

$\left|G^{\ast }\right|={\frac {\tau _{0}}{\gamma _{0}}}$

(12)

Using simple trigonometric relationships, it is possible to divide the energy into the real part Gʼ and the imaginary part Gˮ, which is done using Eqs. (13) and (14). The real part Gʼ is referred to as the storage modulus and is a measure of the reversible energy stored during an vibration period. The imaginary part Gˮ corresponds to the energy dissipated during the period and is called the loss modulus.

$G^{\prime }=G^{\ast }\cdot \cos \delta ={\frac {\tau _{0}}{\gamma _{0}}}\cos \delta$

(13)

$G^{\prime \prime }=G^{\ast }\cdot \sin \delta ={\frac {\tau _{0}}{\gamma _{0}}}\sin \delta$

(14)

The forced torsional vibration method is limited to frequencies below the resonance frequency of the test specimen. Commercial devices operate in the range from approx. 10^-2 Hz to 10² Hz, with the power consumption of the drive motor serving as the measured variable. If, for example, the glass transition (see: glass transition temperature) is reached as a result of an increase in the test temperature, the damping increases significantly and the power of the motor must be increased to maintain the test frequency and stress or deformation amplitude.

The measurement can be performed in both strain-controlled and stress-controlled modes, which enables the determination of the complex modulus G* and the complex compliance C* = 1 / G*. This allows the determination of complex shear moduli in a wide stiffness range from approx. 10^-3 MPa to 10⁶ MPa. The biggest disadvantage of the method is its low sensitivity when measuring plastics with very low damping (tan δ < 0.01), i.e. very stiff or high-modulus materials (see: elastic modulus − examples and material values).

Device systems for DMA and DMTA investigations

Due to their wide range of applications, test methods involving forced vibrations are now highly significant in the dynamic-mechanical analysis of polymer materials.

Fig. 7:

(a) (a) Schematic representation of a DMTA system and (b) DMA system from Anton Paar GmbH, Graz, Austria (https://www.anton-paar.com/us-en/)

However, in most applications, tabletop testing systems (stand-alone systems) are used for smaller test forces (Fig. 7). What all methods have in common is that the deformation of the test specimen is very small and should not exceed the linear-viscoelastic range. As a result of these small deformations, high test frequencies of up to 100 Hz can be achieved with DMA or DMTA in the temperature range from approx. –180 °C to 400 °C with mechanical excitation. Depending on the equipment, there is a wide range of commercial testing systems available for tabletop testing systems, which are usually equipped with additional devices.

Determination of modulus–temperature curves

Free damped oscillations are actually only used in measurements with the torsion pendulum, whereby significantly lower measurement frequencies are possible here. If a test specimen is deflected from its resting position by a single impulse-like loading, it returns to its equilibrium state in free damped oscillations. The deflection should not exceed the range of linear-viscoelastic deformation. The eigenfrequency of the vibration and the temporal decrease in the vibration amplitudes (damping) depend on the viscoelastic properties of the material under investigation and the test temperature (Fig. 8).

Free damped vibrations are used at frequencies in the range from 0.1 to 10 Hz, whereby the investigation of materials with low damping of tan δ ≤ 0.1 is preferred. Since investigations dependent on temperature cause a change in the natural frequency of the system due to the change in modulus, modulus–temperature curves are therefore usually measured at a sliding frequency, whereby compensation for the frequency changes is possible in principle by varying the moment of inertia of the vibration mass.

Fig. 8:

Freely decaying damped vibration

The principle of free damped vibrations is used in technical applications in the form of the torsion pendulum method and is standardised in ISO 6721-2 [12].

Schematic configuration of a torsion pendulum

The principle configuration of a torsion pendulum is shown schematically in Fig. 9. A test specimen, preferably prismatic in shape, is clamped firmly at one end, while the other end is connected to a flywheel mass that influences the moment of inertia and thus the natural frequency of the entire system. At the same time, however, this mass also causes expansion as a result of mechanical and thermal stress (Fig. 9a). To avoid these superimposed normal stresses in the longitudinal direction of the test specimen, a weight compensation can be used in the form of a compensation mass (Fig. 9b). An impulse-like initial deflection φ of the flywheel mass excites the test specimen to freely decaying torsional vibrations, as shown schematically in Fig. 9c.

Fig. 9:

Schematic representation of the torsion pendulum setup (a) without weight compensation, (b) with weight compensation, and (c) test arrangement with initial deflection φ and test specimen

The storage module Gʼ can be determined from the eigenfrequency of the vibration according to Eq. (15) [3].

$G^{\prime }=4\pi l_{p}\cdot \left(f_{d}^{2}F_{d}-f_{0}^{2}\right)F_{g}$

(15)

Here, f_d is the eigenfrequency of the pendulum with test specimen and f₀ is the eigenfrequency of the pendulum without test specimen (when working without weight compensation, f₀ = 0). Other influencing factors to be taken into account are the moment of inertia I_p of the flywheel mass with clamping, a damping correction factor F_d and the geometry factor F_g. When using prismatic test specimens with a clamping length L, width b, thickness h and an h/b ratio ≤ 6, the geometric correction factor is F_c = 1 − 0.63 h/b, whereby Gʼ is then calculated according to Eq. (16).

$G^{\prime }=12\pi ^{2}f_{d}^{2}l_{p}\cdot \left(1-\left({\frac {\Lambda }{2\pi }}\right)^{2}-\left({\frac {f_{0}}{f_{d}}}\right)^{2}\right){\frac {L}{bh^{3}F_{c}}}$

(16)

An approximate determination of the shear modulus Gʼ is also possible using the tensile vibration test from the modulus of elasticity Eʼ or with the modulus of elasticity in the tensile test E_{t</sin>, if Poisson's ratio µ is known in the corresponding temperature interval (Eq. 17).}

$G^{\prime }={\frac {E}{2(1+\mu )}}$

(17)

Ultrasound testing (wave propagation)

In the high frequency range, the dynamic shear modulus Gʼ can also be used to determine characteristic values by means of ultrasonic waves (see: ultrasound testing) [14], provided that transverse waves and angle beam sensors are used.

Above the resonance frequency, the wavelength λ of the vibrating mechanical stress becomes small in comparison to the test specimen dimensions. This makes it possible to use the characteristics of wave propagation in the material to determine its viscoelastic properties. The measurements are usually carried out using ultrasound (f > 20 kHz) in the pulse-echo or transmission method [14]. The sound velocities c (Eq. 18) and the sound absorption coefficient α (Eq. 19) are determined from the acoustic path length d and the associated pulse transit time t as well as the amplitudes A₁ and A₂ at different path lengths d₁ and d₂, whereby the angle of incidence β of the ultrasonic sensors must be taken into account.

$c={\frac {d}{t}}$

(18)

$\alpha ={\frac {1}{d_{2}-d_{1}}}\ln {\frac {A_{1}}{A_{2}}}$

(19)

Using transverse waves (c_T, α_T), the shear modulus G can be determined if the density ρ of the material is known. With low damping (α·λ/2π << 1), Eq. (20) applies approximately.

$G^{\prime }=\rho c_{T}^{2}$

and

$G^{\prime \prime }={\frac {2\rho \alpha _{T}c_{T}^{3}}{\omega }}$

(20)

Ultrasonic measurements are usually performed at frequencies between 100 kHz and 100 MHz. The upper end of the frequency range is determined by the strong increase in attenuation. Working in this wide frequency range requires the use of different vibration sensors. Relatively large frequency ranges can be recorded after broadband excitation using Fourier or Wavelet analysis.

A relatively simple method for determining the shear modulus using ultrasound is available when using plate-shaped test specimens of known thickness d and density ρ (Fig. 10).

Fig. 10:

Schematic representation of (a) pulse-echo ultrasonic technique and (b) transmission technique with angle beam sensors

When using angle beam sensors (Figs. 6a and b), transverse waves are used for measurement. It should be noted that with the pulse-echo technique, the runtime of the ultrasound is approximately identical to that of the transmission method. The runtime Δt can then be calculated from the A-scan in direct coupling or immersion bath technique for a known thickness d and incidence angle β according to Eq. (21) to determine the transverse wave velocity c_T.

$c_{T}={\frac {2d}{\Delta t\cdot \cos \beta }}$

(21)

Provided that the test temperature T is known, the shear modulus G can then be calculated using Eq. (22), whereby it should be noted that this characteristic value is frequency-dependent.

$G=c_{T}^{2}\cdot \rho$

(22)

See also

• Deformation
• Poisson's ratio
• Plastic component
• Stiffness
• Elastic modulus
• Ultrasound – Elastic parameters
• Multiaxial stress state

References

[1]	Lüpke, T.: Fundamental Principles of Mechanical Behavior. In: Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 71–86 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-8807-5; see AMK-Library under A 22)
[2]	ASTM A 938 (2018; reapproved 2024): Standard Test Method for Torsion Testing of Wire
[3]	DIN ISO 7800 (2013-09): Metallic Materials – Wire – Simple Torsion Test
[4]	DIN 51212 (1978-09): Testing of Metallic Materials – Torsion Test of Wires (withdrawn)
[5]	DIN EN 408 (2012-10): Timber Structures – Structural Timber and Glued Laminated Timber − Determination of some Physical and Mechanical Properties
[6]	Göldner, H. (Eds.): Arbeitsbuch Höhere Festigkeitslehre. Fachbuch Verlag Leipzig, (1978) pp. 163–167 (see AMK-Library under T 13)
[7]	Wawrziniok, O.: Handbuch des Materialprüfwesens. Springer, Berlin (1923), (ISBN 978-3-642-90524-7)
[8]	DIN 50141 (1982-01): Testing of Metals – Shear Test (withdrawn)
[9]	DIN EN 28749 (1992-10): Determination of Shear Strength of Pins
[10]	DIN EN 789 (2005-01): Timber Structures – Test Methods – Determination of Mechanical Properties of Wood Based Panels
[11]	ISO 6721-1 (2019-04): Plastics – Determination of Dynamic Mechanical Properties – Part 1: General Principles
[12]	DIN EN ISO 6721-2 (2019-09): Plastics – Determination of Dynamic Mechanical Properties − Part 2: Torsion-Pendulum Method
[13]	ISO 6721-8 (2019-04): Plastics – Determination of Dynamic Mechanical Properties – Part 8: Longitudinal and Shear Vibration – Wave-propagation Method
[14]	Matthies, K. u. a.: Dickenmessung mit Ultraschall. DVS-Verlag GmbH, Berlin, 2nd Edition (1998), (ISBN 3-87155-940-7; see AMK-Library under M 44)