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

Table 2 The material properties of articular cartilage (Eq. 28 ) found using data obtained from both confined compression and multistep tension and relaxation tests

Elastic properties	Short-term viscous properties	Long-term viscous properties
b ₁ = 0.425 × 10⁵ Pa	b ₄ = 190 × 10⁶ Pa.s	τ _i = 141.00, 3.55, 14303.43 s
b ₂ = 0.5 × 10⁵ Pa		w _i = 0.346, 0.0709, 0.582 s⁻¹
b ₃ = 14.2 × 10⁶ Pa		(i = 1, 2, 3)

The confined compressive test was used to evaluate the stiffness of the non-fibrillar matrix, b ₁ (Figure 3). The tensile test consisted of 5 steps, and 2% tensile strain was applied at 0.15%/s in each step, followed by stress relaxation. The equilibrium data at each step were used to find the elastic stiffness of the fibers, b ₂ and b ₃ (Figure 4). The short-term, b ₄, and the long-term viscous parameters, τ _i and w _i, were determined by fitting the model to the entire ramp loading and stress relaxation data (Figure 5). All parameters were enforced to be positive. Additionally, the constraint of Σw _i = 1 was also applied (Eq. 7) during the optimization.

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