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Created page with "{{Language_sel|LANG=ger|ARTIKEL=Einachsiger Spannungszustand}} {{PSM_Infobox}} Uniaxial stress state __FORCETOC__ ==Stress state in tensile and compression test== If a test specimen, which is supposed to be in a plane stress state, is loaded by a tensile or compressive force ('''Fig. 1'''), then, according to the cut reactions with the cut angle ''α'' = 0, a normal..."

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{{Language_sel|LANG=ger|ARTIKEL=Einachsiger Spannungszustand}}
{{PSM_Infobox}}
Uniaxial stress state 
__FORCETOC__

==Stress state in tensile and compression test==

If a [[Specimen|test specimen]], which is supposed to be in a [[Plane Stress and Strain State|plane stress state]], is loaded by a tensile or compressive force ('''Fig. 1'''), then, according to the cut reactions with the cut angle ''α'' = 0, a normal force ''F''N corresponding to the external load ''F'' is generated in the test specimen. If there are no internal or external inhomogeneities such as cavities, inclusions or [[Notch|notches]], and if there is no tendency to demould, the applied load is distributed as a surface load over the test specimen cross-section ''A''0 and is specified as a normalised force or normal stress in accordance with '''Eq. (1)'''. This normal stress ''σ''x or ''σ''N has a positive sign in the case of [[Tensile Test|tensile stress]] and a negative sign in the case of compressive stress, and is constant across the test specimen cross-section under the conditions specified [1, 2].

[[File:Uniaxial_Stress_State-Fig1.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 1''':
|width="600px" |Uniaxial stress state in the [[Tensile Test|tensile test]] (a) and in the [[Compression Test|compression test]] (b)
|}

{|
|-
|width="20px"|
|width="500px"|<math>\sigma _{x}=\sigma _{N}=\frac{F_{N}}{A_{0}}</math>
|(1)
|}

==MOHR's stress circle==

If the cut reactions are determined at an angle ''α'' > 0, a parallelogram of reaction forces is obtained ('''Fig. 2a''') and, according to '''Eqs. (2)''' and '''(3)''', the normal stress ''σ''N and the [[Bend Test – Shear Stress|shear stress]] ''τ'' acting in the test specimen cross-section are derived from the equilibrium conditions [3].

{|
|-
|width="20px"|
|width="500px"|<math>\sigma _{x}=\frac{1}{2}\sigma _{\alpha} \cdot (1+\cos 2\alpha )</math>
|(2)
|-
|width="20px"|
|width="500px"|<math>r=\frac{1}{2}\sigma _{\alpha}\cdot \sin 2\alpha</math>
|(3)
|}

[[File:Einachsiger_Spannungszustand_2.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 2''':
|width="600px"|Cutting reactions at angle ''α'' (a) and MOHR's stress circle (b)
|}

'''Equations (2)''' and '''(3)''' yield '''Eq. (4)''' of MOHR's stress circle (named after Christian Otto Mohr), in which the normal and shear stresses associated with the angle of cut are represented [3].

{|
|-
|width="20px"|
|width="500px"|<math>\left ( \sigma _{N}-\frac{1}{2}\sigma _{\alpha } \right )^{2}+\tau^{2}=\left ( \frac{1}{2}\sigma _{\alpha } \right )^{2}</math>
|(4)
|}

The illustration in '''Fig. 2b''' shows that the maximum shear stress occurs at an angle of 45°, and is therefore ''τ''max = ''σ''α/2. Macroscopically, the shear stress component manifests itself in [[Tensile Test|tensile]] or [[Compression Test|compression tests]], for example, through slip or shear fracture and deformation cones in ductile metals, as well as through flow lines visible on the [[Surface|surface]], also known as Lüders lines. In [[Plastics|plastics]], so-called [[Shear Band Formation|shear bands]] can be observed on the surface of the test specimen in tensile tests under certain test conditions, which represent one of the dominant deformation processes ('''Fig. 3'''). In [[Ductility Plastics|ductile plastics]] that constrict, the flanks of the constriction fronts often have an angle of approximately less than 45°.

[[File:Einachsiger_Spannungszustand_3.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 3''':
|width="600px"|Shear bands in acrylonitrile butadiene styrene ([[Plastics – Symbols and Abbreviated Terms|abbreviation]]: ABS) in tensile testing
|}

==Stress distribution in three-point bending==

A special case of uniaxial stress occurs in the case of pure [[Bend Test|bending]] about one axis, whereby, however, an inhomogeneous stress state occurs here due to the simultaneous occurrence of tensile, compressive and shear stresses [2, 4]. In the case of identical tensile and compressive properties of the material under investigation, the maximum stress ''σ''f in the peripheral fibre ( see: [[Peripheral Fibre Strain|peripheral fibre strain]]) of the test specimen is calculated according to '''Eq. (5)''' for [[Bend Test#The three-point bending test method|three-point bending]], and the stress distribution in the cross-section is symmetrical with the neutral or stress- and strain-free axis ('''Fig. 4a''').

{|
|-
|width="20px"|
|width="500px"|<math>\sigma _{f}=\frac{3\cdot F\cdot L}{2\cdot b\cdot h^{2}}</math>
|(5)
|}

Due to the shear force bending, additional shear stresses occur in the cross-section, which are distributed parabolically and reach their maximum in the neutral fibre or axis ('''Fig. 4b'''). These shear stresses are negligible in the [[Bend Test|bending test]] on plastics if the condition [[Support Distance|span]] ''L''/specimen thickness ''h'' ≥ 16 is fulfilled.

In simplified terms, the maximum shear stress can be calculated for a rectangular cross-section according to '''Eq. (6)''' [3]:

{|
|-
|width="20px"|
|width="500px"|<math>\tau_{max}=\frac{3\cdot F}{4\cdot b\cdot h}</math>
|(6)
|}

[[File:Einachsiger_Spannungszustand_4.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 4''':
|width="600px"|Normal stress distribution (a) and shear stress distribution (b) in the cross-section of a test specimen under three-point bending
|}

Due to the shear sensitivity of laminates or layered [[Composite Materials Testing|composite materials]] and the potential risk of delamination, these [[Material & Werkstoff|materials]] must satisfy the condition ''L''/''h'' ≥ (20−25) in bending tests. If the material exhibits different tensile and compressive behaviour, a displacement of the neutral fibre occurs, resulting in a non-linear and asymmetrical stress distribution in the cross-section.

==See also==

* [[Compression Test|Compression test]]
* [[Tensile Test|Tensile test]]
* [[Bend Test|Bend test]]
* [[Plane Stress and Strain State|Plane stress and strain state]]
* [[Multiaxial Stress State|Multiaxial stress state]]

==References==

{|
|-valign="top"
|[1]
|Lüpke, T.: Fundamental Principles of Mechanical Behavior. In: [[Grellmann,_Wolfgang|Grellmann, W.]], [[Seidler,_Sabine|Seidler, S.]] (Eds.): Polymer Testing. Carl Hanser, Munich (2022), 3rd Edition pp. 71–86 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-5; see [[AMK-Library]] under A 22)
|-valign="top"
|[2]
|[[Bierögel,_Christian|Bierögel, C.]]: Quasi-static Test Methods. In: [https://www.researchgate.net/profile/Wolfgang-Grellmann 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)
|-valign="top"
|[3]
|Szabo, I.: Einführung in die Technische Mechanik. Springer, Berlin Heidelberg (1984) 8th Edition (ISBN 3-540-13293-7)
|-valign="top"
|[4]
|Erhard, G.: Konstruieren mit Kunststoffen. Carl Hanser, Munich (2008) 7th Edition, pp. 189–198 (ISBN 978-3-446-41646-8)
|}

[[Category:Deformation]]
[[Category:Compression Test]]
[[Category:Tensile Test]]

Uniaxial Stress State - Revision history