Low-strain effective Young’s modulus model and validation for multi-layer vocal fold-based silicone specimens with inclusions

Abstract

A model of the effective low-strain elastic Young’s modulus of multi-layer stacked composites is proposed, which is capable to account for an arbitrary stacked inclusion. Geometrical and discretization-based model results are validated against measured effective Young’s moduli (from 10 up to 40 kPa) on 14 molded silicone specimens embedding a stiff (298 kPa) inclusion with variable size, position, and stacking. Specimens without inclusion represent the muscle, superficial, and epithelium layers in a human vocal fold with Young’s moduli between 4 and 65 kPa. The proposed model allows to predict the influence of a stiff inclusion, mimicking a structural abnormality or pathology somewhere within the vocal fold, on the low-strain effective Young’s modulus. Quantifying the influence of an inclusion or local stiffening on the vocal fold bio-mechanics is a necessary step toward the understanding and mitigation of structural vocal fold pathologies and associated voice disorders.

Modelling and validation of the non-linear elastic stress–strain behaviour of multi-layer silicone composites

“…An additional mixture I 6 with greater low-strain Young's modulus so that 4 ≤ E I 6 /E I 1...5 ≤ 150, is considered to represent a local stiffening within the VF as reported for some structural VF abnormalities or disorders [16,17,18]. Low-strain Young's moduli E in Table 1 characterising the linear stress-strain curves σ t (ε t ) at room temperature (21 ± 2 • C, mean and standard deviation) are obtained as the slope of a linear fit (coefficient of determination R 2 ≥ 98%) to the low-strain range with upper low-strain limit ε l of measured stress-strain curves σ t (ε t ) gathered from uni-axial tensile tests on molded specimens [19,20].…”

“…M5, MRI, EPI) are omitted in systematic physical studies on the influence of the VF structure on the FSI due the lack of an a priori mechanical characterisation. Recently, in [19,20], the low-strain Young's modulus of ML silicone composites for which perfectly bounded adjacent layers are stacked either parallel ( ), serial (⊥), a combination of both ( ⊥) or arbitrary (Arb) with respect to the force direction as illustrated in Fig. 1(a), is modeled considering the low-strain effective Young's modulus E ef f of an equivalent homogenised composite.…”

“…The model approach was extensively validated against measured E ef f values using uni-axial tensile testing (at room temperature) on molded ML specimens with 5 ≤ E ef f ≤ 40 kPa. Validation was first done in [19], using I 1...5 , on six two-layer and seven three-layer silicone molded specimens and then in [20], using fourteen specimens obtained as a three-layer composite (I 1 , I 2 and I 5 ) embedding a stiff inclusion (I 6 ) with variable size, position and stacking. That resulted in more complex specimens with at least four layers which are stacked either parallel ( ), serial (⊥), combined ( ⊥) or arbitrary (Arb).…”

“…That resulted in more complex specimens with at least four layers which are stacked either parallel ( ), serial (⊥), combined ( ⊥) or arbitrary (Arb). For each molded specimen in [19,20], the low-strain effective Young's modulus E ef f of the equivalent homogenised composite was estimated (R 2 ≥ 96%) on the measured stress-strain curves σ t (ε t ) as the slope characterising the linear low-strain region ε t ≤ ε l with ε l = 0.30 ± 0.10. It follows that the low-strain effective Young's modulus E ef f , in the strain range up to ε l ≈ 0.3, of an equivalent homogenised ML silicone composite can be accurately modelled (maximum difference of 5.2 kPa [19,20]) from its layers E and stacking geometry.…”

“…Therefore, in this work, the stress-strain relationship of silicone ML composite specimens is investigated beyond the low-strain elastic range. In particular, 63 measured stress-strain curves on 40 molded specimens from uniaxial stretching at room temperature described in [19,20] are further analysed in order to characterise and model the stress-strain curves for ε t > ε l .…”