True stress and Poisson's ratio of tendons during loading

Abstract

Excessive axial tension is very likely involved in the aetiology of tendon lesions, and the most appropriate indicator of tendon stress state is the true stress, the ratio of instantaneous load to instantaneous cross-sectional area (CSA). Difficulties to measure tendon CSA during tension often led to approximate true stress by assuming that CSA is constant during loading (i.e. by the engineering stress) or that tendon is incompressible, implying a Poisson's ratio of 0.5, although these hypotheses have never been tested.The objective of this study was to measure tendon CSA variation during quasi-static tensile loading, in order to assess the true stress to which the tendon is subjected and its Poisson's ratio. Eight equine superficial digital flexor tendons (SDFT, about 30 cm long) were tested in tension until failure while the CSA of each tendon was measured in its metacarpal part by means of a linear laser scanner. Axial elongation and load were synchronously recorded during the test.CSA was found to linearly decrease with strain, with a mean decrease at failure of -10.7 ± 2.8 % (mean ± standard deviation). True stress at failure was 7.1 to 13.6 % higher than engineering stress, while stress estimation under the hypothesis of incompressibility differed from true stress of -6.6 to 2.3 %. Average Poisson's ratio was 0.55 ± 0.12 and did not significantly vary with load.From these results on equine SDFT it was demonstrated that tendon in axial quasistatic tension can be considered, at first approximation, as an incompressible material.

Axial speed of sound is related to tendon's nonlinear elasticity

“…Asymptotic axial speed of sound in 9 superficial digital flexor tendons plotted against the tendon's cross-sectional area at rest. 8 previously shown that such approximation reduces the error introduced by the engineering stress for large deformations (Vergari et al, 2011). SOS A was correlated to tendon's elastic modulus (p = 0.019), confirming that the quasi-static measurement of tendon's stiffness is related to the effective stiffness which determines the SOS (Eq.…”

“…Abrahams (1967) observed that the stress/strain relation of SDFTs became linear beyond 3 % engineering strain, while Riemersma and Schamhardt (1985) fixed a threshold at 2 % engineering strain. In the present article, a threshold was fixed at 2.7 % true strain (the difference with the engineering strain being 0.036 %) since it was previously observed that tendon Poisson's ratio at higher strain does not significantly vary with load (Vergari et al, 2011); this assures that the effects of Poisson's ratio on the SOS variation in the linear region are minimal. The average elastic modulus approximated for small deformations was 1.54 ± 0.17 GPa, which is consistent with previous studies (Crevier-Denoix et al, 2005;Gillis et al, 1995;Riemersma and Schamhardt, 1985).…”

“…For a lossless isotropic medium (Rose, 2004): where ρ is the medium's mass density, ν is its Poisson's ratio (the negative ratio of axial on transversal strain) and E is its effective stiffness; the latter represents the medium's elastic response to the dynamic excitation of the ultrasound waves at the wavelength scale (about 2 mm in current axial SOS applications, which utilize 1 MHz signals). It was previously observed that the average equine SDFT Poisson's ratio is 0.55 (Vergari et al, 2011), and that it does not significantly vary with load. While isotropic materials cannot have Poisson's ratios higher than 0.5, this is quite common in transversely isotropic materials (Jones, 1999;Lempriere, 1968).…”

“…This definition assumes that tendon's Poisson's ratio is 0.55 (Vergari et al, 2011). Each true stress/strain curve was approximated with a 3 rd order polynomial, the derivative of which represents the tendon's tangent modulus.…”

“…2), and it was calculated for each tendon in order to characterize their stiffness (Riemersma and Schamhardt, 1985). The lower strain threshold was fixed at 2.7 % since it was previously observed that tendon's Poisson's ratio at higher strain does not significantly vary with load (Vergari et al, 2011). The elastic modulus approximated for small deformations (E S ), which can be compared with previous literature data, was calculated as the linear slope of the engineering stress (F/CSA 0 ) versus engineering strain ((L-L 0 )/L 0 ) curve, for strains higher than 2.7 %.…”