Visco-hyperelastic constitutive modeling of soft tissues based on short and long-term internal variables

Table 1 The material properties of the anterior cruciate ligament found by fitting the constitutive model (Eq. 24 ) to tensile experimental data

Elastic properties	Short-term viscous properties	Long-term viscous properties
a ₂ = 2.213 × 10⁶ Pa a ₃ = 3.879 (dimensionless)	a ₄ = 0.3653 × 10⁶ Pa.s, a ₅ = 0.652 (dimensionless)	τ _i = 3.77, 148.24, 10987.98 s
		w _i = 0.154, 0.161, 0.682 s⁻¹
		(i = 1, 2, 3)

The elastic properties, a ₂ and a ₃ , were found from the near equilibrium loading (1.2%/s); and the short-term viscous properties, a ₄ and a ₅ , were determined using the ramp loading under 25%/s strain-rate (Figure 1). Finally, the long term viscous parameters, τ _i and w _i, were determined from the stress relaxation data (Figure 2). The compressive property of the ligament, a ₁, was neglected in the fit. All parameters were enforced to be positive to satisfy the second law of thermodynamics. Additionally, the constraint of Σw _i = 1 was also applied (Eq. 7) during the optimization.

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