Uniaxial Stress State

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Uniaxial stress state

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 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 notches, and if there is no tendency to demould, the applied load is distributed as a surface load over the test specimen cross-section A₀ 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 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].

Fig. 1:

Uniaxial stress state in the tensile test (a) and in the compression test (b)

$\sigma _{x}=\sigma _{N}={\frac {F_{N}}{A_{0}}}$

(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 shear stress τ acting in the test specimen cross-section are derived from the equilibrium conditions [3].

	$\sigma _{x}={\frac {1}{2}}\sigma _{\alpha }\cdot (1+\cos 2\alpha )$	(2)
	$r={\frac {1}{2}}\sigma _{\alpha }\cdot \sin 2\alpha$	(3)

Fig. 2:

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].

$\left(\sigma _{N}-{\frac {1}{2}}\sigma _{\alpha }\right)^{2}+\tau ^{2}=\left({\frac {1}{2}}\sigma _{\alpha }\right)^{2}$

(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 or 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, also known as Lüders lines. In plastics, so-called 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 ductile plastics that constrict, the flanks of the constriction fronts often have an angle of approximately less than 45°.

Fig. 3:

Shear bands in acrylonitrile butadiene styrene (abbreviation: ABS) in tensile testing

Stress distribution in three-point bending

A special case of uniaxial stress occurs in the case of pure 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) of the test specimen is calculated according to Eq. (5) for three-point bending, and the stress distribution in the cross-section is symmetrical with the neutral or stress- and strain-free axis (Fig. 4a).

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

(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 bending test on plastics if the condition 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]:

$\tau _{max}={\frac {3\cdot F}{4\cdot b\cdot h}}$

(6)

Fig. 4:

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 and the potential risk of delamination, these 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
• Tensile test
• Bend test
• Plane stress and strain state
• 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 978-1-56990-807-5; see AMK-Library under A 22)
[2]	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)
[3]	Szabo, I.: Einführung in die Technische Mechanik. Springer, Berlin Heidelberg (1984) 8th Edition (ISBN 3-540-13293-7)
[4]	Erhard, G.: Konstruieren mit Kunststoffen. Carl Hanser, Munich (2008) 7th Edition, pp. 189–198 (ISBN 978-3-446-41646-8)