Peripheral Fibre Strain

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Peripheral fibre strain

Stress and strain in tensile and compression tests

Provided that the material properties are isotropic and homogeneous, the tensile or compressive stress generates a constant stress and strain in the area of uniform elongation in the cross-section of the test specimen at any given time (Fig. 1), the sign of which is positive in the tensile test and negative in the compression test. The stress corresponds to the quotient of the measured force F and the initial cross-section A₀, and the strain is the measured elongation relative to the initial measurement length.

Fig. 1:

Stress and strain in tensile and compression tests

Stress and strain distribution in bending tests

In contrast to tensile and compression tests, bending loading generates variable stress and strain in the test specimen cross-section [1]. If a test specimen is subjected to a bending moment M_b on both sides, this results in symmetrical deflection, which is greatest in the middle of the test specimen (Fig. 2a).

Fig. 2:

Stress and strain in bending test

This creates compressive stress on the upper side and tensile stress on the lower side, to which the test specimen reacts with compression or elongation of the peripheral fibre. If targets are placed at identical distances on the top and bottom sides in their initial state, then after deformation, compression will occur on the compression side and elongation on the tension side, the absolute value of which is identical under certain assumptions (Fig. 2b).

Fig. 3:

Stress a) and strain distribution b) in the bending test specimen

These prerequisites are the validity of the linear-elastic bending theory of the first order, homogeneous and isotropic material behaviour, and identical tensile and compressive properties of the material under investigation. In this case, the stresses and strains are distributed linearly as shown in Fig. 3. This symmetrical triangular distribution means that the stress and strain in the plane of symmetry are zero. For this reason, the centre line under bending stress is called the “neutral fibre” and the maximum strain is called the “peripheral fibre strain,” which are mathematically described by Eq. (1) for stress and Eq. (2) for strain [1, 2].

$\sigma _{f}=\pm {\frac {3\ F\ L}{2\ b\ h^{2}}}$

(1)

$\varepsilon _{f}=\pm {\frac {6\ s\ h}{L^{2}}}$

(2)

Application of the evaluation equation for peripheral fibre strain

The application of evaluation Eqn. (1) and (2) therefore requires a symmetrical stress and strain distribution across the cross-section, so that the zero line of the stress or strain is identical to the neutral fibre of the bending beam. Due to the sometimes very different tensile and compressive behaviour of plastics, such as polystyrene (abbreviation: PS) with differing yield stresses σ_ty and σ_cy, a shift k of the neutral fibre (Fig. 4) may occur, which means that the evaluation equations of the bending test are no longer applicable. In this case, different absolute values of the characteristic values apply to the stresses and strains on the tension and compression sides.

Fig. 4:

Shift of the neutral fibre with different tensile and compressive behaviour

See also

• Bend loading
• Flexural strength
• Bend test – Influences
• Bend strip method
• Creep behaviour – Determination
• Flexural modulus

References

[1]	Bierögel, C.: Bend Test on Polymers. In: Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 133–143 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-805-7; see AMK-Library under A 22)
[2]	Szabó, I.: Einführung in die Technische Mechanik. Springer, Berlin, Heidelberg (1984) 8th Edition, (ISBN 3-540-13293-7; see AMK-Library under T 15)