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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3

Integrating for a rectangular cross-section of width b : − ∫ d + =∫ d + ,− ∫ + =∫ + ,(1+ ) − =1+ .

(7)

Solving for w :

= ((1 + ) − −1).

(8)

The updated width and area become: poi ( ) = + = (1 + ) − , poi ( ) = poi 2 , poi = (1+ ) −2 ,

(9)

and the tensile stiffness is then: poi ( )= (1+ ) −2 . Integrated along the elongation:

(10)

−2 − 1−2 .

poi ( ) = ∫ poi ( )d = 0 ( + )(1+ )

(11)

2.4. Combination of both geometric changes Combining true strain and transverse contraction gives enhanced stiffness: cmb ( )= + (1+ ) −2 . This leads to an expression:

(12)

cmb ( ) = ∫ cmb ( )d = 0 1−(1+ ) −2 2 .

(13)

The relationship (13), and surprisingly all previous relationships too, can be then generalized as: gen, ( ) = 1−(1+ ) − , gen, ( ) = 1−(1+ ) − , (14) where p and q are exponents. Note that the equation N eng (1) corresponds with =−1 , N ln (3) to the limit , → 0 , N poi (11) with = 2 − 1 and N cmb (13) with = 2 . Following F igure 2 presents a dimensionless comparison of the derived functions for Poisson’s ratio = 0.2 over a wide range of loading displacement. The engineering normal force exhibits a linear response, while the other models display increasing nonlinearity. The most pronounced deviation from linearity is observed in equation (13), which accounts for both geometric effects — true strain and transverse contraction. Notably, both logarithmic expressions N ln (4) and N cmb (13) have a vertical asymptote at = − as expected due to approaching the singularity when the bar is reaching zero length in compression.