Plane Stress and Strain State

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Plane stress and strain state

Generalised HOOKE's Law

Assuming isotropic material properties in all spatial and axial directions, the generalised HOOKE's law (Eq. 1) is as follows:

$\epsilon _{11}={\frac {\sigma _{11}}{E}}-{\frac {\nu }{E}}(\sigma _{22}+\sigma _{33})$ $\epsilon _{22}={\frac {\sigma _{22}}{E}}-{\frac {\nu }{E}}(\sigma _{33}+\sigma _{11})$ $\epsilon _{33}={\frac {\sigma _{33}}{E}}-{\frac {\nu }{E}}(\sigma _{11}+\sigma _{22})$

(1)

It can be seen that the strain in the respective axial direction is primarily caused by the stress in the same axis, but an additional strain component from the other stress directions is added, depending on Poisson's ratio ν and the modulus of elasticity E. In the case of uniaxial stress in a tensile or compression test, in addition to the longitudinal strain ε_x, there is also transverse strain ε_y and ε_z, i.e. in the width and thickness directions of the test specimen, if a plane stress state is present. This is usually the case when a sufficiently slender test specimen is used, which allows the transverse strain to be measured in the width or thickness direction (Fig. 1).

Fig. 1:

Deformation of the test specimen under load in a plane stress state

In addition to the normal stress σ_x (Eq. 2), loading causes longitudinal strain ε_x or ε_L (Eq. 3) and transverse strain in the width and thickness directions (Eq. 4).

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

(2)

$\epsilon _{x}=\epsilon _{L}={\frac {L-L_{0}}{L_{0}}}={\frac {\Delta L}{L_{0}}}$

(3)

$\epsilon _{y}=\epsilon _{q}={\frac {B-B_{0}}{B_{0}}}={\frac {\Delta B}{B_{0}}}$

(4)

In the case of isotropy and a homogeneous material state, the strain in the width direction ε_y and in the thickness direction ε_z (Eq. 5) are identical, with the absolute value being specified for both transverse strains. The stresses σ_y and σ_z are equal to zero in the plane stress state.

$\epsilon _{z}=\epsilon _{q}={\frac {H_{0}-H}{H_{0}}}={\frac {\Delta H}{H_{0}}}$

(5)

Determination of Poisson's ratio

The Poisson's ratio, also known as the transverse contraction number, is then calculated according to Eq. (6):

$\nu =\nu _{B}=\nu _{H}={\frac {\epsilon _{y}}{\epsilon _{L}}}={\frac {\epsilon _{z}}{\epsilon _{L}}}={\frac {\epsilon _{q}}{\epsilon _{L}}}$

(6)

If, instead of the slim test specimens, thicker and/or wider prismatic test specimens are used as shown in Fig. 2, no measurable strain signal will be obtained in either the thickness or width direction, as the test specimen is now in a plane strain state.

Fig. 2:

Deformation of the test specimen under load in a plane state of strain

The resulting longitudinal strain ε_x should be identical, which is why an increased normal stress σ_x is required due to the larger cross-section A₀. To explain this situation, we can assume an identical geometry as in Fig. 1, whereby a possible change in the width of the test specimen is prevented by lateral abutments (Fig. 3). If these supports could also be equipped with a large number of load cells (see: electro-mechanical force transducer and piezoelectric force transducer), the forces in the width and thickness directions of the test specimen could be measured. Under load, this would result in an identical longitudinal strain ε_x according to Eq. (3) and a higher normal stress σ_x. In this case, Poisson's ratio ν can no longer be calculated because the strains ε_y and ε_z are equal to zero. Since forces in the y and z directions would be measured in this simulated state, stresses also arise in these directions, the magnitude of which can be calculated according to Eq. (1) [1].

Fig. 3:

Simulation of the slender test specimen in a plane strain state

See also

• Uniaxial stress state
• Energy elasticity
• Crack model according to IRWIN and Mc CLINTOCK
• Plastic zone
• Plastic hinge model

References

[1]	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 978-1-56990-807-3; see AMK-Library under A 23)