PSI - Issue 73

Petr Frantík et al. / Procedia Structural Integrity 73 (2025) 14–18 Author name / Structural Integrity Procedia 00 (2025) 000 – 000

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2. Analytical solutions Consider an idealized prismatic bar of length L , area of cross-section A, composed of a linear isotropic material characterised by Youngs ’s modulus E and Poisson’s ratio ν . We seek the relationship between the bar ’s elongation u and the axial force N , which is the integral of the assumed constant normal stress on the bar ’s faces, see Fig. 1.

Fig. 1. Prismatic bar under tension/compression (negative N )

2.1. Geometrically linear solution

The well-known relation for small deformations is based on engineering measures: eng ( ) = ,eng ( ) = eng ( ) = = eng , eng = ,

(1)

where σ x, eng is approximation of the stress, ε eng is the engineering strain, and k eng is the bar initial stiffness. 2.2. True strain The linear relation above does not consider the continuous change in the reference length of the bar. By refining the model, we obtain the true strain: ln =∫ d 0 , d = , d = d + . (2) After substitution and integration: ln ( ) = ∫ +1 d 0 = (1 + ), ln ( )= + . (3) The expression is commonly associated with original expression derived by Hencky, 1928: ln ( ) = true ( ), true ( ) = (1 + eng ( )). (4) 2.3. Transverse contraction As seen above, the transverse contraction of the bar is not yet accounted for. During loading, not only does the bar lengthen, but its cross-section also changes. This effect can be isolated. The relationship is well- known from the definition of Poisson’s ratio υ , well explained in common sources (e.g. Wikipedia). With tension along the x -axis: d = d ,d = d , (5) and Poisson ’s ratio is defined as: − = . (6)