• Creep-recovery behaviors are explored at different stress levels and biaxial stress ratios.

Articular cartilage has the functions to absorb shocks, transmit loads and sustain complex mechanical loading histories, thus maintaining dynamic mechanical equilibrium of knee joint in daily activities. As a kind of fluid saturated porous material, articular cartilage exhibits viscoelastic responses under mechanical loads. And creep is observed in cartilage when the mechanical load is held constant during standing, single-leg lunge (Hosseini et al., 2010; Choi et al., 2016; Qiu et al., 2016), the stance phase of gait (Liu et al., 2010; Li et al., 2014) and so on. Creep deformation under physiological loads may serve as a functional indicator for cartilage healthy. The majority of creep deformation in cartilage can recover timely. However, long term overloading may cause irreversible deformation, further induce damage in cartilage and lead to early osteoarthritis (Roos et al., 1998; Felson, 2013; Martel-Pelletier et al., 2016; Georgiev and Angelov, 2019; Peters et al., 2022 ). Studying the creep and creep-recovery behaviors of articular cartilage are helpful for understanding the normal joint function and cartilage pathology.

In vivo creep tests were usually conducted by applying 50%–150% body weight on the joint for 300–900 s (Herberhold et al., 1998; Herberhold et al., 1999; Hosseini et al., 2010; Choi et al., 2016; Uzuner et al., 2020), and the creep deformations of cartilage were measured in vivo by using imaging techniques like computed tomography (CT), magnetic resonance (MR), dual fluoroscopic imaging (DF). Herberhold et al. found that the creep deformation had high inter-individual variability (Herberhold et al., 1998; 1999). Hosseini et al. found the cartilage deformation of human tibiofemoral increases sharply in the first 20 s and almost held constant beyond 50 s when a constant full body weight was quickly applied and maintained for 300 s (Hosseini et al., 2010). Choi et al. studied the time-dependent creep behavior of tibial cartilage by applying a load of 75% body weight on limb and they reported that the creep deformation increased rapidly over the first minute and kept stable after 5 min (Choi et al., 2016). Uzuner et al. reported a substantial increase in cartilage deformation when joint loading increased from nil to 75% body weight and a continued small increase over time when the joint loading was held constant (Uzuner et al., 2020). In vivo experimental data help to understand the deformation characters of cartilage under physiological loads. However, technical limitations impede the accurate measurement of creep recovery in the physiological range and it non-etheless has significant difficulty to quantitatively analyze the creep-recovery deformation law of cartilage.

To cover the shortage of in vivo tests, in vitro creep tests are widely used to investigate the creep and creep-recovery behaviors of cartilage. Athanasiou et al. and Stolberg et al. employed the indentation test system to characterize the creep-recovery behavior of cartilage and they found that the initial creep rate is larger than the initial recovery rate and more than 95% of the creep deformation could recovered within 90 min (Athanasiou et al., 1991; Stolberg-Stolberg et al., 2018). Boettcher et al. found that the cartilage deformation increased with the increasing stress. (Boettcher et al., 2016). Cutclife and DeFrate validated the use of the recovery response for mechanical characterization of cartilage in a controlled, ex vivo environment ( Cutcliffe and DeFrate, 2020). Reuter and Hurschler used a biphasic 3D-FE-based method to determine the biomechanical properties of equine articular cartilage (Reuter and Hurschler., 2021). In our previous study, the depth-dependent creep strain of cartilage under uniaxial compressive load was investigated by an optimized digital image correlation (DIC) technique (Si et al., 2022). And the creep deformation could accelerate the fatigue damage of cartilage in knee joint (Gao et al., 2019). It should be mentioned that these studies on the creep-recovery behaviors of articular cartilage were probed under uniaxial loading.

The cartilage with freely squeezing and stretching in vivo is subjected to biaxial or even multiaxial loads, and however literature provides limited data on these. Thus, it is of great significance to study the biaxial or multiaxial creep-recovery behaviors of cartilage. The goal of this study is to investigate the creep-recovery behaviors of articular cartilage under biaxial tensile load. The hypothesize is that changing the stress state from uniaxial loading to biaxial loading has a significant effect on creep deformation of cartilage. To verify the hypothesize, both the uniaxial and biaxial creep-recovery experiments were conducted by using a biaxial cyclic testing system. Effects of stress level and biaxial stress ratio on cartilage creep-recovery were characterized. Creep strain components were analyzed and compared at different loading conditions. Additionally, the creep-recovery behaviors of cartilage were predicted by mathematical models. This study is original because it provides the first insight into the effect of biaxial constraint on tensile creep behavior of cartilage. The results could help to explore intrinsic mechanical properties of articular cartilage and to prevent cartilage damage.

The viscoelastic constitutive model Eq. 2 introduced by Lou and Schapery is used to describe the unconfined mechanical behavior of cartilage (Lou and Schapery, 1971). The constitutive model under uniaxial loading can be expressed as ε = g 0 A 0 σ + g 1 ∫ 0 t Δ A ψ − ψ ′ d g 2 σ d τ d τ (2) Where A 0 is initial components of linear viscoelastic creep compliance, and Δ A ψ is transient components of linear viscoelastic creep compliance. t is the loading time. g 0 , g 1 and g 2 are three stress-dependent material parameters and their changes reflect the third and higher order stress-dependence of Gibbs free energy. The converted time ψ is expressed as ψ = ψ t = ∫ 0 t d t ′ a σ ,   ψ ′ = ψ τ = ∫ 0 τ d t ′ a σ (3) Here a σ is the time-scale factor, which reflects the relevance of free energy, stress and temperature. In this study we assume that a σ depends only on the applied stress.

When the step-stress input σ H t − H t − t 1 (Figure 2B) is applied, Eq. 2 yields the creep strain ε c = g 0 A 0 σ + g 1 g 2 Δ A t a σ σ ,   0 < t < t 1 (4) And the recovery strain can be expressed as ε r = g 2 Δ A ψ − Δ A ψ − ψ 1 σ ,   t > t 1 (5) Where in Eq. (5) ψ 1 = t 1 / a σ ,   ψ = t 1 / a σ + t − t 1 (6)

The creep strain at t ₁ (just before unloading) can be described as follows in Eq. (7). At the moment after unloading the initial strain in recovery phase can be described as follows in Eq. (8). ε c 1 = ε t 1 = g 0 A 0 σ + g 1 g 2 Δ A ψ 1 σ (7) ε r t 1 = ε 1 ′ = g 2 Δ A ψ 1 σ (8)

Therefore, the strain jump value at t = t 1 can be expressed as δ ε 1 = ε c 1 − ε 1 ′ = g 0 A 0 σ + g 1 − 1 g 2 Δ A ψ 1 σ (9)

The transient component of linear viscoelastic creep compliance is expressed as Δ A ψ = C ψ n (10) Where C is creep coefficient, n is creep index.

By substituting Eq. (10) into Eq. (4), we obtain the non-linear viscoelastic creep model. ε c = g 0 A 0 σ + C g 1 g 2 a σ n t n σ (11)

By substituting Eq. (10) into Eq. (5), we obtain the expression of recovery strain: ε r = Δ ε c g 1 1 + a σ λ n − a σ λ n (12) Where in Eq. (12) λ = t − t 1 t 1 (13) and Δ ε c = ε t 1 − ε 0 = g 1 g 2 c ψ 1 n σ (14)

It is noted that the parameters in creep model Eq. 11 and recovery model (Eqs. 12–14 were obtained by fitting the test data with the non-linear least square method in the software origin 2018. Thus, the uniaxial creep-recovery strains of cartilage can be fitted by the built model.

In daily activities, articular cartilage is subjected to long time biaxial loads or multiaxial loads. The creep deformation is induced in cartilage under mechanical loads due to the alternation of unequal fluid exudation and imbibition within its matrix. Once unloaded, most of the creep deformation will gradually recover with time. Uniaxial creep-recovery of cartilage have been well described in previous studies (Gao et al., 2019; Cutcliffe and DeFrate, 2020). However, seldom has any study focused on biaxial or multiaxial creep-recovery of cartilage. Thus, we probed the creep-recovery behaviors of articular cartilage by comparing uniaxial and biaxial tensile loading results.

The creep strain evolution of cartilage consists of two parts, namely initial creep stage and the steady creep stage. In initial creep stage, the uniaxial and biaxial creep strain showed a similar evolution that the creep strain increased rapidly at the beginning, and gradually tended to be stable with time (Figure 4; Figure 6; Figure 9). The rapid increase of strain in initial creep stage for cartilage is due to the rapid outflow of interstitial fluid. When the unbound fluid in cartilage is squeezed out, the burden of supporting tensile load is gradually shifted to the collagen network. The cartilage becomes more resistant to deformation and the next creep stage starts. In steady creep stage, the increase rate of creep strain stays constant. The creep strain evolution observed in this study is similar with that reported by Hosseini et al. and Choi et al. as they found that the creep deformation in vivo kept stable after a rapid growth (Hosseini et al., 2010; Choi et al., 2016). However, it should be mentioned that there is no accurate cut-off point between the first creep stage and the steady creep stage since the creep deformation of cartilage varies with different physiological loads (Schoenbauer et al., 2015; Wang et al., 2015; Chan et al., 2016; Choi et al., 2016). Compared with uniaxial creep, biaxial creep exhibits smaller strain at the same stress level, especially at high stress level of 1.5 MPa. It is found that biaxial loading reduces the creep strain through retarding all creep strain components (Figure 5; Figure 7), especially through retarding delayed elastic strain and viscous flow strain (Figure 8). The smaller strain under biaxial creep is ascribed to the higher constraint imposed by the equi-biaxial stress state, as higher constraints can increase the stiffness of the cartilage and restrict the rapid outflow of interstitial fluid.

Generally, the deformation of articular cartilage can be fully recovered within a sufficient period of unloading time due to the viscoelastic properties of articular cartilage (Torzilli, 1984; Athanasiou et al., 1994). Eckstein et al. suggested at least 90 min of rest might be required to allow articular cartilage to fully recover according to the imaging based studies (Eckstein et al., 1999). Erisken et al. reported that 1,000 s was sufficient for the compressive stress to relax in bovine cartilage (Erisken et al., 2010). In this study, it is found that both uniaxial and biaxial strain have not been fully recovered with the residual strains left when the recovery time is 3,600 s. Compared with uniaxial loading, biaxial loading shows a smaller residual strain (Figure 4; Figure 6; Figure 9), due to the significant decrease in the viscous flow strain at equi-biaxial stress state (Figure 5C; Figure 7C). The residual strain, as viscous flow strain, is slow to recover. Thus, more time may be needed for the residual creep deformation to recover.

It is very likely for cartilage to experience non-uniform stress distribution in daily activities due to its curvature structure, thickness variation, diffusion kinetics and other factors (Kamalanathan and Broom, 1993). Kamalanathan and Broom reported that the tissue was significantly stiffer, i.e. more strain limiting, along the split-line direction than across it. They found that the tissue was significantly stiffer, i.e. more strain limiting, along the split-line direction than across it and the anisotropic strain-limiting response of the fibrillar array will result in fibrils acting more collectively along the less extensible direction than across it. Obviously, the stress state of cartilage under physiological load is not equi-biaxial. To better understand how biaxial constraint influences biaxial creep behavior, we studied the effect of biaxial stress ratio on creep-recovery behavior of cartilage (Figure 8). Both creep strain and recovered strain decrease with the increase of biaxial stress ratio due to the reason that the larger the biaxial stress ratio, the stronger the biaxial constraint. The equi-biaxial stress state (B = 1) imposes the largest biaxial constraint. Thus, when the biaxial stress ratio B is 1.0, the lowest creep and recovery strain is induced in cartilage. Increasing the biaxial stress ratio from .3 to 1.0 causes a decrease in all creep strain components, especially in delayed elastic strain (Figure 9; Figure 10). Furthermore, less proportion of delayed elastic strain results in less time to reach steady recovery state (Figure 8) since delayed elastic strain is slow to recover. In addition, the higher constraint was imposed by the larger biaxial stress ratio, which in turn increases the stiffness of cartilage (Lin et al., 2017; Zhang et al., 2019) resulting in the smaller creep strain and residual strain.

Viscoelastic (Leipzig and Athanasiou, 2005; Si et al., 2022), biphasic (Huang et al., 2003) and poroviscoelastic models (Wang et al., 2001) are employed to describe mechanical behaviors of cartilage in literature. Leipzig and Athanasiou used the viscoelastic model to describe the mechanical behaviors of cartilage cells (Leipzig and Athanasiou, 2005). Gao et al. employed the non-linear viscoelastic constitutive model to predict the depth-dependent creep behaviors of cartilage under compression (Gao et al., 2014). The viscoelastic constitutive model employed in this study is proposed by Lou and sharper (Lou and Schapery, 1971). It has the outstanding advantage of retaining the single time integral form even in the non-linear range. And the non-linear effects are expressed by means of stress-dependent material functions determined experimentally. It has been applied to predict the mechanical behaviors of cracked cartilage and exhibits good results (Si et al., 2022). In this study, the proposed viscoelastic model with the consideration of irrecoverable deformation was developed to describe the uniaxial creep-recovery behaviors of cartilage and a good consistence between test data and fitted data is obtained.

The study probed the uniaxial and biaxial creep-recovery behaviors of articular cartilage. Changing the stress state from uniaxial loading to biaxial loading, creep strain decreases significantly under the same stress level. Increasing biaxial stress ratio (from B = .3–1) causes notable decrease in creep strain. During the recovery process of creep, it takes less time for the biaxial strain than the uniaxial strain to reach the steady recovery state. The cartilage with higher stress level or smaller biaxial stress ratio shows larger residual strain, indicating that the cartilage damage may be accelerated at higher stress level or at smaller biaxial stress ratio. These results provide a new sight for cartilage creep and are significant to improve the durability of cartilage in daily activities.