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{{Language_sel|LANG=ger|ARTIKEL=Randfaserdehnung}}
{{PSM_Infobox}}
Peripheral fibre strain
__FORCETOC__

==Stress and strain in tensile and compression tests==

Provided that the [[Material & Werkstoff|material]] properties are isotropic and homogeneous, the tensile or [[Compression Test|compressive stress]] generates a constant stress and strain in the area of [[Tensile Test Uniform Elongation|uniform elongation]] in the cross-section of the test [[Specimen|specimen]] at any given time ('''Fig. 1'''), the sign of which is positive in the [[Tensile Test|tensile test]] and negative in the [[Compression Test|compression test]]. The [[Stress|stress]] corresponds to the quotient of the measured force ''F'' and the initial cross-section ''A''0, and the strain is the measured elongation relative to the initial measurement length.

[[File:randfaserdehnung1.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 1''':
|width="600px" |Stress and strain in tensile and compression tests
|}

==Stress and strain distribution in bending tests==

In contrast to tensile and compression tests, [[Bend Loading|bending loading]] generates variable stress and strain in the test specimen cross-section [1]. If a [[Specimen|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''').

[[File:Periphal Fibre Strain Fig-2.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 2''':
|width="600px" |Stress and strain in [[Bend Test|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''').

[[File:randfaserdehnung3.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 3''':
|width="600px" |Stress a) and strain distribution b) in the [[SENB-Specimen|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].

{|
|-
|width="20px"|
|width="500px" | <math> \sigma_{f}=\pm \frac{3\ F\ L}{2\ b\ h^2}</math>
|(1)
|}

{|
|-
|width="20px"|
|width="500px" | <math> \varepsilon_{f} =\pm \frac{6\ s\ h}{L^2}</math>
|(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|plastics]], such as polystyrene ([[Plastics – Symbols and Abbreviated Terms|abbreviation]]: PS) with differing [[Yield Stress|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 [[Bend Test|bending test]] are no longer applicable. In this case, different absolute values of the [[Material Value|characteristic values]] apply to the stresses and strains on the tension and compression sides.

[[File:Periphal Fibre Strain Fig-4.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 4''':
|width="600px" |Shift of the neutral fibre with different tensile and compressive behaviour
|}

==See also==

* [[Bend Loading|Bend loading]]
* [[Flexural Strength|Flexural strength]]
* [[Bend Test – Influences|Bend test – Influences]]
* [[Bend Strip Method|Bend strip method]]
* [[Creep Behaviour – Determination|Creep behaviour – Determination]]
* [[Flexural Modulus|Flexural 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; E-Book: ISBN 978-1-56990-805-7; see [[AMK-Library]] under A 22)
|-valign="top"
|[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)
|}

[[Category:Bend Test]]

Peripheral Fibre Strain - Revision history