The Ring-Pull Assay for Mechanical Properties of Fibrous Soft Tissues – an Analysis of the Uniaxial Approximation and a Correction for Nonlinear Thick-Walled Tissues


Background: The ring-pull test, where a ring of tissue is hooked via two pins and stretched, is a popular biomechanical test, especially for small arteries. Although convenient and reliable, the ring test produces inhomogeneous strain, making determination of material parameters non-trivial. Objective: To determine correction factors between ring-pull-estimated and true tissue properties. Methods: A finite-element model of ring pulling was constructed for a sample with nonlinear, anisotropic mechanical behavior typical of arteries. The pin force and sample cross-section were used to compute an apparent modulus at small and large strain, which were compared to the specified properties. The resulting corrections were validated with experiments on porcine and ovine arteries. The correction was further applied to experiments on mouse aortic rings to determine material and failure properties. Results: Calculating strain based on centerline stretch rather than inner-wall or outer-wall stretch afforded better estimation of tissue properties. Additional correction factors were developed based on ring wall thickness H, centerline ring radius Rc, and pin radius a. The corrected estimates for tissue properties were in good agreement with uniaxial stretch experiments. Conclusions: The computed corrections improved estimation of tissue material properties for both the small-strain (toe) modulus and the large-strain (lockout) modulus. When measuring tensile strength, one should minimize H/a to ensure that peak stress occurs at the sample midplane rather than near the pin. In this scenario, tensile strength can be estimated accurately by using inner-wall stretch at the midplane and the corrected properties.


The mechanical characterization of biological soft tissue can be done in many ways (for an overview in the cardiovascular space, see [1]). However, as the scale of the biological samples of interest becomes smaller, numerous challenges arise for most testing modalities (e.g. planar uniaxial and biaxial). Most notably, the effects of handling and preparation are amplified as it becomes more difficult to make samples of uniform dimensions. Gripping the sample becomes difficult and removes a large portion of tissue from the test.The necessary precision of testing machines goes up and costs can rise to prohibitive levels.

As a result, there are two mainstays in the mechanical characterization of very small (mm or smaller) biological tissues: indentation (or atomic force microscopy) and ring pull tests. In the case of natural ring shapes, such as arteries, the ring pull test has several distinct advantages: 1. Ring-pull tests load the tissue circumferentially, which is typically in line with the physiologic load state. 2. Ring tests are simple in both gripping and testing, and do not require specialized machinery to do (one can even do such tests on a lab weigh scale). 3. Ring tests are high throughput and repeatable because the sample preparation and test are straightforward.

Ring-pull tests, also called uniaxial ring tests, are appropriate for all types of cylindrical and spherical tissues. A schematic of a ring-pull test is shown in Fig. 1. Ring-pull tests are particularly well suited to small, branched structures because the segmentation of rings allows one to analyze mechanical properties locally without the use of complex inverse finite element (FE) methods. The ring test also lends itself well to assays of passive and active properties of tissues from the cardiovascular, digestive, or respiratory system. A list of systems that ring tests have been used to study, by no means exhaustive, is shown in Table 1 (see Fig. 1 for definition of measures).

Fig. 1

a. Undeformed geometry for the ring pull test where Rc is the centerline radius, Ri is the luminal surface radius, Ro is the outer surface radius, H is the tissue thickness, a is the pin radius, and L) is the centerline length. b. Deformed geometry where d is the pin to pin displacement, a is the pin radius, and ℎ = λtH is the deformed tissue thickness

Table 1 Tissues used in ring pull tests including species, nominal stretch range for tests, and sample geometry. Isometric contraction is entered as Iso under “Nominal Stretch”

Unfortunately, there is no precisely defined strain mapping for the ring-pull test, which means one cannot simply fit a material model in a theoretical framework to an experiment [12]. Van Haaften et al. [13] investigated the validity of the uniaxial approximation and proposed a correction factor in the context of linear, isotropic materials. They concluded that the uniaxial approximation breaks down in the case of thick rings and is correctable for linear, isotropic tissues at <5% strain. Although the linear analysis is informative, soft tissues operate at strains much larger than 5% in almost all cases (see Table 1). Tissues also typically exhibit nonlinear, or at least bilinear, material behavior characterized by a small-strain (toe) region, a distinct transition strain (or stretch), and a large-strain (lockout) region.Footnote 1 The nonlinear behavior of tissues can also affect the results because the ring-pull test is not a pure extension of the tissue due to bending caused by the pin grip. The typical planar strip uniaxial test is a much better approximation of a pure extension. Additionally, if we consider small samples like the mouse carotid artery or very thick tissues like ventricular rings of the heart, we need corrections that are not extrapolated, but are verified by experiment in the extremes.

Furthermore, the ring test is often used to analyze failure properties of samples [14,15,16]. It is therefore important to consider the heterogeneity of the ring-pull deformation, along with geometric factors that may concentrate stress. Particularly, when reporting failure strain, one must be aware that the actual strain in the region of failure initiation is often very different than the calculated nominal strain of the sample. The same is true of the failure stress. Failure stress is reported as the mean stress of the sample at failure, which, for a uniaxial test tends to be a good approximation of the maximum stress. However, for a ring test, the deformation is not homogeneous, and we must account for the stress difference across the sample.

The goal of this work is to create a simple geometry-based strategy for converting force-displacement data from uniaxial ring tests to a corrected uniaxial extension allowing for quantitative assessment of circumferential material and failure properties. To accomplish this goal, we use computational models to assess the uniaxial approximation for rings composed of nonlinear, anisotropic materials under large strains, analyze the effect due to wall thickness and pin (hook) geometry, and propose purely geometric corrections for small-strain modulus, transition strain, and large-strain modulus. We propose a strategy for pin selection and offer an approach for determining failure properties. We then validate our approach using both ring test and uniaxial strip extension experiments on porcine and ovine arterial tissues, and apply our failure approximation to ring tests of murine aortic tissue.


Our approach has three primary components outlined in the subsections below. First, we performed finite element analysis of the ring pull test for a range of ring radii, ring thicknesses, and pin radii to assess the behavior of an anisotropic, nonlinear material under a range of geometric conditions. These computational models were used to construct a geometric correction function for converting nominal material properties assessed from ring-pull tests to the equivalent material properties that would be assessed from strip uniaxial extension. Second, we performed ring-pull experiments on various arterial tissues in several animal models using multiple pin sizes when possible. We used these experiments along with the correction function to predict the uniaxial extension behavior. We then assessed and validated our correction by performing uniaxial extensions of the cut and opened ring strips. Third, the computational and experimental data were analyzed and compared as described in the final subsection.

Computational Model

A finite element model (Fig. 2) of the ring pull experiment was created using PreView 2.1.1, solved using FEBio 2.9.1, and visualized using PostView 2.4.4 [17]. Briefly, the model was created using a ring represented as a quarter of a hollow cylinder sliced in half longitudinally to invoke one-eighth symmetry. A rigid body was created in the ring to mimic the pin. The pin surface was prescribed as a node-on-facet sliding contact with the ring sample inner surface, neglecting friction between the surfaces. An augmented Lagrangian formulation was used with two-pass, automatic penalty for contact. The faces on symmetry planes were held on their respective planes but allowed to slide along the plane. A displacement was prescribed to the ring surface opposite the pin to mimic the experiment. The pin was meshed using 33,600–122,300 hex8 elements depending on pin diameter, and the ring sample was meshed using 42,000–90,000 hex8 elements depending on ring thickness. The material model for the ring was an uncoupled elastic mixture consisting of a nearly incompressible neo-Hookean solid and an exponential fiber family oriented circumferentially around the ring. The strain energy density, W, for this model is:

$$ W={C}_1\left({I}_1-3\right)+\frac{\xi }{2\alpha}\left[\exp \left(\alpha {\left({\lambda}_f^2-1\right)}^2\right)-1\right]+\frac{1}{2}K{\left(\mathit{\ln}(J)\right)}^2 $$

where: C1 is the ground matrix modulus, I1 is the first strain invariant, ξ is the fiber modulus, α is the fiber nonlinearity, λf is the fiber stretch, K is the ground matrix bulk modulus, and J = det(F) where F is the deformation gradient tensor. In the ring simulations, parameters of C1 = 50kPa, ξ = 10kPa, α = 1.0, K = 75 MPa were used. These parameters were taken from assessment of the uniaxial porcine abdominal aortic samples as described in the Experimental Protocol subsection. The value of the bulk modulus, K, was chosen to make the material nearly incompressible. It is noted that since the material behavior is nonlinear, the term “modulus” refers to the modulus in the limit of infinitesimal strain.

Fig. 2

Ring model used for simulations. The ring is shown in purple and the pin in orange. The contact surfaces are indicated by blue arrows. The symmetry planes are shown as red dashed lines. The red arrows show the direction the ring sample is pulled

Sample Preparation

Tissue samples from three animal models were used in this study. Healthy 6-month-old sheep and pig tissue samples (abdominal aorta and carotid) were procured from the Visible Heart Lab (Department of Surgery, University of Minnesota, Minneapolis, MN, USA) immediately following euthanasia for other studies. The samples were trimmed of perivascular connective tissue then sectioned axially into ~6 mm rings. The rings were then equilibrated in 1x PBS for 12 h to ensure investigation of only passive mechanics of the tissue. Immediately prior to testing the ring dimensions were measured optically using a digital camera and analysis software, FIJI [18].

Wild-type (C57BL/6 J) mice were procured from the Provenzano Lab (Department of Biomedical Engineering, University of Minnesota, Minneapolis, MN, USA) immediately following euthanasia as a part of colony maintenance. The heart, lungs, and aorta were excised en bloc. The lungs were removed, and the aorta was trimmed of perivascular connective tissue. The aorta was then placed in 1x PBS and allowed to equilibrate for 12 h as above. Immediately prior to testing, the aorta was sectioned axially into ~1.5 mm rings from the descending arch to the renal artery branches. The dimensions of each ring were then measured using a Leica S-Series dissection microscope (Leica Microsystems Inc., Buffalo Grove, IL, USA) and analysis software, FIJI [18].

Experimental Protocol

Ring-pull tests were performed on all tissues. Sheep and pig tissues were tested using a custom ring-pull grip apparatus where the pin diameter could easily be altered by replacing a sleeve (Fig. 3a). This flexibility allowed investigation of the effect of relative pin to ring size on the apparent tissue behavior. Each sample was placed so that both pins passed through the lumen of the vessel (Fig. 3a). The pins were lubricated using canola oil to reduce friction between the tissue and the stainless-steel pins. Lubrication is particularly important for larger pins where more sliding is necessary as the tissue stretches. Samples were kept hydrated during the tests by pipetting 1x phosphate-buffered saline solution (PBS) across the sample width. The samples were immersed in 1x PBS between tests. A uniaxial testing machine (TestResources Inc., Shakopee, MN, USA) with a 45 N load cell (sheep and pig aorta) or a 5 N load cell (carotid artery and mouse aorta) was used to perform the circumferential uniaxial tests. The samples were preconditioned on the largest radius pin for 5 cycles to 50% nominal engineering strain, then uniaxial ring tests were performed for each subsequent pin size at 0.6% strain per second to a nominal engineering strain of 65%. After completion of ring testing, the rings were cut open and allowed to equilibrate for 60 min in 1x PBS. The opening angle was measured immediately before the sample was placed in a uniaxial setup consisting of two grips that clamped the tissue using low-grit sandpaper to prevent slipping (Fig. 3b). The uniaxial experiments were conducted in the same way as the ring-pull tests with 5 preconditioning cycles to 50% engineering strain and a uniaxial pull to 65% engineering strain at 0.6% strain per second.

Fig. 3

a. Uniaxial ring-pull apparatus showing removable pin sleeves and tissue sample, b. Uniaxial pull apparatus for opened, flattened ring tissue strip

Mouse aortic rings were only tested in the ring test configuration. They were loaded at a strain rate of 1% per second and pulled to failure. Hydration was maintained as above.

For a subset of the sheep aortic rings, ultrasound was used to image the cross-section of the ring samples at several stretches. The samples were submerged in a 1x PBS bath and imaged using a VEVO2100 small animal ultrasound machine (FUJIFILM VisualSonics Inc., Toronto, ON, CA) with a 30 MHz transducer.

Data Analysis

For the ring samples, the centerline lengths were used to analyze the kinematics of the tissue. This approach differs from typical measures which use either the pin-to-pin distance from the flattened state to the stretched state, which equates roughly to using the inner luminal surface stretch, or that proposed by Van Haaften et al. [13], which uses the external surface stretch.The deformed thickness, h, was calculated by assuming that the transverse stretches, λt, (i.e. in the thickness, H, and width, W, directions) were equal to one another, and that the tissue volume was conserved (i. e. the Jacobian, \( J=1={\lambda}_1{\lambda}_2{\lambda}_3={\lambda}_1{\lambda}_t^2 \)) (Eq. 2). Under that assumption, the volume conservation can be simplified to a cubic function of λt (Eq. 3), which is easily solved using any root finding algorithm.

$$ {V}_0=V=\pi \left({\left({R}_i+H\right)}^2-{R}_i^2\right)W=\pi \left({\left(a+{\lambda}_tH\right)}^2-{a}^2\right){\lambda}_tW+2d\left({\lambda}_tH\right)\left({\lambda}_tW\right) $$
$$ 0=\pi {H}^2{\lambda}_t^3+2H\left(\pi a+d\right){\lambda}_t^2-2\pi {R}_cH $$

Once h and L were determined, the stretch of the centerline was calculated as λ1 = L/L0. The stretch was used to assess the engineering strain, ε1 = λ1 − 1 for the corresponding first Piola-Kirchoff (PK1) stress (P11 = F1/A0) where P11 is the circumferential PK1 stress, F1 is the force in the direction of stretch, and A0 is the undeformed sample cross-sectional area. The average PK1 stress was calculated for the midplane region of that sample assuming that each side of the ring held half the total force on the pin.

It is noted that the analysis performed here uses the assumption of equal transverse stretches and incompressibility, which are approximations. Also, a single uniaxial extension test cannot elucidate full-field mechanical properties because of its one-dimensionality. When we refer to the vessel properties in this paper, reference is to the circumferential material properties around the vessel wall.

In the assessment of circumferential mechanical properties for this paper we chose to use the small-strain (toe) modulus, the large-strain (lockout) modulus, and the transition strain as our metrics of material behavior. The distinct benefit of these properties is that they allow for intuitive assessment of material behavior, as well as allowing one to reconstruct rough stress-strain curves for the material. The assessment of these properties was performed by taking the small-strain modulus between 5% to 20% engineering strain for uniaxial cases and 10% to 25% engineering strain for ring tests. The reason for the different limits between the uniaxial and ring tests is that the ring test is characterized by a region of flattening from the ring shape to an ellipsoid where there is very little tissue stretch, and the deformation is mostly bending. This effect typically arises in the range of 0–5% centerline stretch (as shown, for example, in Fig. 9). The large-strain modulus was taken for PK1 stress ranges between 200 kPa and 350 kPa for aortic samples, and between 150 kPa and 300 kPa for the carotid artery. These ranges were selected based on the stress range of roughly linear modulus at large strains as assessed from preliminary data from uniaxial extensions on the different tissue types. The moduli were fit to a line using MATLAB2019a (Mathworks Inc., Natick, MA, USA) where the slope is the modulus. The intersection of the small-strain line and the large-strain line is defined as the transition strain (Fig. 4).

Fig. 4

Fit of the small-strain and large-strain modulus (dashed lines) along with the transition strain (red star) for a ring test


Computational Model

The first question considered was whether the stress-strain curve estimated from the pin force per midplane cross-sectional area and the average tissue strain was a good estimate of the actual circumferential stress-strain behavior of the tissue. We started by analyzing luminal surface strain, centerline strain, and exterior surface strain as metrics for stress-strain behavior for the material (Fig. 5). In Fig. 5, the black line is the true material behavior in uniaxial extension, and the colored lines represent the strain estimate for different tissue thicknesses. Using the luminal surface strain shows a much less stiff material behavior than the true behavior because the luminal strain is higher for any given pin force (Fig. 5a). The centerline strain estimation shows a better approximation, but still consistently underpredicts the tissue stiffness relative to uniaxial (Fig. 5b). The exterior surface strain shows stiffer material behavior than uniaxial because the exterior strains are much lower for any given pin force (Fig. 5c). All strain measures consistently underpredict small-strain modulus due to the importance of bending, particularly as the tissue thickness increases.

Fig. 5

First Piola-Kirchoff stress vs engineering strain curves for uniaxial and ring pull simulations demonstrating the difference between using a. luminal (inner) surface strain, b. centerline strain, and c. exterior (outer) surface strain values for ring tests. All simulations shown here are for Rc = 0.4mm and a = 0.10mm

To analyze the stress-strain estimations further, we examined the strains across the sample, as shown in Fig. 6. There was large variability in the strains from exterior (outer) surface to luminal (inner) surface for a given nominal stress. The inner surface begins stretching as the pin begins to move and the ring, viewed from the top, begins to flatten into an oval shape. The stretching begins generating force even before the mean stretch ratio of the tissue departs from λ1 = 1. The outer surface lags the inner surface because it requires more pin movement to begin stretching. In fact, as shown in Fig. 6a, the exterior surface goes into compression for a short period as the ring begins to flatten (shown by blue arrows). As the ring deforms there is initial bending that transitions to stretching. Thus, we should expect a stress profile that looks like a bending deformation (tension on one surface and compression on the other) followed by uniaxial extension (tension on both surfaces albeit of different magnitudes). Furthermore, a volumetric effect arises due to incompressibility, and the tissue in the midplane region deforms from a rectangle to a C-shape. This happens because the higher stretch on the inner surface causes a larger transverse compression on the inner surface relative to the outer surface. As a result, the ring height decreases more on the inner surface causing the cupping shown in Fig. 6b.

Fig. 6

a. Cauchy stress vs Green strain plots for thin (H = 0.05 mm) and thick (H = 0.15 mm) ring tests. The shaded area represents the limits of strain at the inner and outer surfaces of the tissue with respect to stress. The dashed line represents the calculated centerline stress-strain curve. Shown The red X’s in the stress-strain plots show the position of B on the stress-strain curve. The blue arrows show the outer compression. The strain range across the sample at B is represented by the horizontal purple arrow. b. A visualization of the strain across the midplane of thin and thick rings at a nominal Green strain of 0.35

Another goal of this work was to investigate the maximum stresses the tissue undergoes during deformation in different ring-pull configurations. Stress concentrations are important because one often wishes to estimate failure properties from the ring-pull test [14,15,16]. As shown in Fig. 7, for small pin sizes, the stresses concentrate around the back of the pin. This is intuitively demonstrated using Euler beam theory, which we acknowledge is valid neither for large deformations nor for nonlinear materials, but which serves for illustration of the principle. For a beam of rectangular cross-section, the bend radius is given by Rbeam = EI/M where E is the Young’s modulus, I is the moment of inertia, and M is the internal moment. The maximum stress in a linear beam subjected to such bending is \( {\sigma}^{max}=M\left(\frac{H}{2}\right)/I \). If we then require that σmax < σfailure, we see there exists a minimum bend radius which gives us a theoretical minimum pin radius where our assumptions of stretch are valid: \( a\ge E\left(\frac{H}{2}\right)/{\sigma}^{failure} \) and H/a ≤ 2σfailure/E. Using this approximate expression, failure properties from [19], and our own values for thickness and large-strain modulus in pig aorta, we get H/a ≤ 2(2.75MPa)/(2.42MPa) = 2.3. For larger values of H/a, stress concentrates on the back-side of the pin which would likely cause failure at the pin, and not in the midplane region (Fig. 7a & 7b). If one avoids this condition by using H/a < 1, the stresses do not concentrate around the pin, but instead reach a maximum on the inner surface of the ring at the midplane region (Fig. 7c & 7d). One can also accurately estimate the luminal stretch at that region as λlumen = (2πa + 2d)/2πRi. Once one knows the stretch at failure, one can calculate the stress at the luminal surface using our fitted (corrected) material properties. This approach gives us a much more accurate measure of failure stress than the mean stress over the sample since the initiation of failure happens at the luminal surface.

Fig. 7

A comparison of the stress concentrations for different pin sizes and wall thicknesses. a. Ring with thickness H = 0.10 mm and pin size a → 0 (i.e. the nodes on the symmetry plane on the luminal side are fixed in the direction of the pulling force). b. Ring with thickness H = 0.10 mm and pin size a = 0.05 mm. c. Ring with thickness H = 0.05 mm and pin size a = 0.20 mm. d. Ring with thickness H = 0.20 mm and pin size a = 0.20 mm. All rings have centerline radius Rc= 0.6 mm. The black arrows show the area of stress concentration


The first objective of the experiments conducted was to determine whether our model was consistent with the behavior of the tissue. We confirmed that qualitative deformation features of our model were also evident in tissue using high-resolution ultrasound of a sheep aortic ring sample at different stretches as shown in Fig. 8. Midplane images (Fig. 8, top row) demonstrate how stretching the sample causes the same C-shape cupping as in the simulation (Fig. 6b). The tissue in contact with the pin also compressed in the thickness direction while being stretched circumferentially, forcing the tissue into a trapezoidal shape (Fig. 8, bottom row). The shape change is consistent with what we see near the pins in the model (Fig. 7).

Fig. 8

Ultrasound images of porcine abdominal aortic ring cross sections in the center (belly) region (blue) and at the pin (yellow) for a. undeformed ring, b. flattened ring (unstretched centerline) c. stretched ring (λ”#5″ ≈ 1.33). The red line indicates the position of the pin edge along the tissue sample

To assess the need for correction, we ran experiments on various arterial tissues in several animal models as shown in Fig. 9. Where applicable, the uniaxial pull is shown in black and each pin size is shown in a different color. Our results indicate that thicker, smaller diameter tissues exhibit markedly different stress-strain behavior in the ring-pull relative to the uniaxial experiment, especially in the small-strain region. We also note that, for sheep aorta and carotid, the transition strain is significantly different for the ring compared to the uniaxial (Fig. 9b & 9c). The mouse aortic rings had no uniaxial equivalent as they were stretched to failure (Fig. 9d).

Fig. 9

Experimental first Piola-Kirchoff stress vs. engineering strain plots with mean (solid lines), the upper and lower 95% confidence intervals (dashed lines) for uniaxial and ring pull tests with varied pin diameters for a. Porcine abdominal aorta with Rc = 6.84±0.23 mm and H = 1.51±0.24 mm, b. Ovine abdominal aorta with Rc = 5.69 ± 0.28 mm and H = 1.37 ± 0.16 mm, c. Ovine external carotid artery with Rc = 2.21±0.25 mm and H = 1.11±0.39 mm, and d. Mouse thoracic aorta with Rc = 0.450 ± 0.050 mm and H = 0.072 ± 0.012 mm. The final point is the average stress and strain at failure. All values are mean ± 95% CI. The value n is the number of ring samples per animal, and m is the number of animals used

Material Property Corrections

Correction factors for small-strain modulus, large-strain modulus, and transition strain were obtained from simulations. The rings used in the simulations had dimension in the range of Rc = 0.40–0.70 mm and H = 0.035–0.20 mm. The pins had radii in the range of a = 0.02–0.20 mm. Overall, 50 simulations were performed over this range. For each simulation, the small-strain modulus was calculated. For models using small pins however, the simulation was unable to reach large deformations due to numerical instability. Therefore, we only used 37 data points for the large-strain modulus and transition strain.

To investigate the role of geometry on the mechanical properties, we use two dimensionless quantities. The first is the ratio of ring thickness to centerline radius: H/Rc. As this ratio approaches zero the bending rigidity of the sample goes to zero and the test approaches a uniaxial stretch in the limit of zero friction. The second quantity is the ratio of ring thickness to pin diameter: H/a. This dimensionless parameter relates to the ability of the ring to form around the pin as discussed above.

We introduce a functional limit and two fundamental limits for dimensions of the ring and pin. The functional limit is that two pins of radius a must fit in the ring lumen so that Rc − H/2 ≥ 2a. This limit is shown in Fig. 10 as a dashed line. The fundamental limit is if one used two semi-circular D-shaped pins, we must have Rc − H/2 ≥ a. This limit is shown in Fig. 10 as a greyed area. Since the inner radius of the ring must be greater than zero, an additional limit is that H/Rc < 2.

Fig. 10

Ratios of the material properties assessed from rings to those assed from uniaxial for a. small strain (toe) region modulus, b. large strain (lockout) modulus, and c. transition strain from small strain regime to large strain regime. The dashed line represents the functional limit for fitting two pins in the ring lumen. The grey area represents the fundamental limit of fitting two D-shaped pins (of radius Ri) in the lumen. Data was not obtained in the blue regions due to non-convergence of the contact solution for very small pins

The ratio between ring and uniaxial measurements for the two nondimensionalized parameters, H/Rc and H/a, are shown in Fig. 10. The small-strain modulus (Fig. 10a) is consistently underpredicted especially for values of H/Rc > 0.3, because of the strain inhomogeneity for ring tests. Furthermore, the small-strain modulus demonstrates strong nonlinear dependence on both nondimensional parameters. In contrast, the large-strain modulus (Fig. 10b) is overestimated when H/Rc is greater than 0.3 for moderate H/a values, but is underestimated for moderate H/Rc values and large H/a values. The large-strain modulus shows similarly strong nonlinear dependence on both quantities. The transition strain (Fig. 10c) shows a consistent increase with increasing H/Rc and a decrease with H/a. The transition strain shows more linear dependence on H/Rc. Thus, the effect of a thick-walled sample (relative to radius) depends on the size of the pins being used to perform the ring pull. There is a window (H/a ∈ [0.5,1.5] and H/Rc ∈ [0.15,0.3]) where the estimations are close to the true parameters and the parameter ratios near unity. The ratio H/Rc is out of the experimenter’s control, but the choice of pin size can be optimized to estimate the material parameters accurately. Arteries tend to have H/Rc in the range of [0.15–0.35].

Each contour plot in Fig. 10 was fitted to a bivariate cubic function as given in Eq. 4. The parameters which were nearly zero, or whose range at 95% confidence included zero were removed from the fit. Table 2 shows the fitted coefficients along with their 95% confidence intervals. Root mean squared error (RMSE) is shown for the fit and can be interpreted as the fraction of error not accounted for by the geometric correction.

$$ \kern0.5em f\left(\frac{H}{R_c},\frac{H}{a}\right)=\sum \limits_{i=0}^3\sum \limits_{j=0}^{3-i}{k}_{ij}{\left(\frac{H}{R_c}\right)}^i{\left(\frac{H}{a}\right)}^j $$
Table 2 Parameters fit to the nondimensionalized geometric quantities and the associated 95% confidence intervals of the parameter along with the RMSE for the fit. The other coefficient values were zero and are therefore not included

Thus, we propose the following correction equations for determining material properties from ring-pull tests (Eq. 5A-C).

$$ \frac{E_s^{Ring}}{E_s^{Uni}}\approx 1-0.042{\left(\frac{H}{a}\right)}^2+0.214\left(\frac{H}{R_c}\right){\left(\frac{H}{a}\right)}^2-1.552{\left(\frac{H}{R_c}\right)}^2\left(\frac{H}{a}\right) $$
$$ \frac{E_L^{Ring}}{E_L^{Uni}}\approx 1+0.094{\left(\frac{H}{a}\right)}^2-0.627\left(\frac{H}{R_c}\right){\left(\frac{H}{a}\right)}^2+2.359{\left(\frac{H}{R_c}\right)}^2\left(\frac{H}{a}\right) $$
$$ \frac{\varepsilon_t^{Ring}}{\varepsilon_t^{Uni}}\approx 1+0.358\left(\frac{H}{R_c}\right)-0.054\left(\frac{H}{a}\right) $$


We fit each experimental tissue to our bilinear model, and the results are shown in Fig. 11. The uniaxial extension test was assumed to be the ground truth for circumferential properties. The ring-pull small-strain moduli (Fig. 11a) show large errors when compared to the uniaxial data. Particularly, the aortic samples show that the luminal and exterior stretch estimates are poor with the exterior stretch overpredicting the modulus and the luminal stretch underpredicting the modulus. The centerline estimate, however, is reasonably good. The carotid sample, on the other hand, shows that inner and centerline estimations underpredict the modulus, but the exterior estimate is good. It is important to note these differences because the aortic tissues are thin-walled while the carotid is relatively thick-walled by comparison (i.e. H/Rc is much greater for the carotid than for the aorta). Therefore, no single estimate (luminal, exterior, or centerline) provide correct estimates of small-strain modulus. We were, however, able to correct the small-strain modulus for both the aortic samples and the carotid samples. The correction shows that the error becomes almost constant across all ring sizes indicating that we have removed the geometric effects and are now only left with systemic error. The results for large-strain modulus (Fig. 11b) show that, for aortic samples, the exterior surface stretch tends to overpredict the modulus while the luminal surface stretch tends to underpredict the modulus. In the aortic samples, the centerline tends to be a reasonably good estimate of the large-strain modulus. The carotid data shows that the exterior and centerline stretches tend to overestimate the large-strain modulus, while the luminal stretch serves as a good approximation. The corrected large-strain modulus showed little change from the uncorrected value (<5%) for both the pig aorta and sheep aorta, but showed a marked improvement (>40%) for the sheep carotid (Fig. 11b). For the transition strain, we saw little change comparing the corrected value to the uncorrected value with large error throughout. We did see a large improvement in the sheep carotid data (>15%), but the corrected transition strain was still 51% above the uniaxial value. We presume the transition strain was polluted by residual stress effects due to testing an intact ring versus an opened, flattened uniaxial sample.

Fig. 11

Experimental data for a. Small-strain modulus, b. Large-strain modulus, c. Transition engineering strain estimated from uniaxial (black dashed lines) and ring pull tests using luminal surface stretch (red bars), exterior surface stretch (blue bars), uncorrected centerline stretch (purple bars), and corrected centerline stretch (green bars). Dashed lines and error bars are the 95% confidence interval of the mean. Error bars on the corrected values are the centerline estimate error bars scaled by the correction factor

Across all tissue types there is little change in the correction for aortic samples, but there is a significant improvement in modulus estimation for the sheep carotid artery samples shown by the grey arrows in Fig. 11. Particularly, we see that the small-strain modulus for the sheep carotid artery, when corrected, approaches the exterior surface estimate while the large-strain modulus for the sheep carotid artery approaches the interior surface estimate. This shows that our correction, for a thick-walled tissue such as the carotid, is a vast improvement over any single estimate alone (i.e. using the luminal surface, exterior surface, or centerline without correction).

Failure Property Estimation

The mouse ring-pulls had no uniaxial equivalent because of the challenges arising from uniaxial testing of very small tissue samples as described in the Introduction. All data in this section are reported as mean ± 95% confidence interval for n = 30 samples. The uncorrected values for small-strain modulus and large-strain modulus were 0.304 ± 0.065 MPa and 1.883 ± 0.209 MPa respectively. The uncorrected transition strain was 0.397 ± 0.048. The values for H/Rc and H/a were 0.159 ± 0.039 and 0.516 ± 0.088 respectively. These values are relatively small meaning the tissue has a relatively thin wall (which places the corrections in the lower left corner of Fig. 10), so the ring-pull is close to a uniaxial test. From our correction equations, we would expect the small-strain modulus to be 0.311 MPa, the large strain modulus to be 1.829 MPa and the transition strain to be 0.385.

The ratio of tissue thickness to pin radius (H/a = 0.516) was small enough that we can estimate the failure stress and strain at the midplane (as demonstrated by the computational model and confirmed through experiments). The calculated central strain at rupture was 0.999 ± 0.145 while the luminal strain at rupture was 1.107 ± 0.166. If we look purely at the experiment, we get a mean PK1 stress at failure of 1.254 ± 0.538 MPa. However, if we use the average properties obtained for the mouse aorta as given above along with the luminal strain, we get a PK1 stress at failure of 1.440 ± 0.591 MPa. These results are in agreement with previously reported values of uniaxial failure for mouse descending thoracic aorta [14]. There is a difference of 14% between the centerline and luminal failure stress values, even though the wall thickness to radius ratio is relatively small (H/Rc = 0.159). This would indicate that, even in the case of relatively thin samples, one should use the luminal surface stretch as the metric for failure.


Our model accurately captures the kinematics of the ring pull test, including the unfolding, flattening behavior, and C-shape deformation of the center region shown in ultrasound imaging through the cross section (Fig. 6). Furthermore, we showed that as an approximation for average strain, one can use the centerline stretch estimated using a transverse stretch calculated by assuming incompressibility. This approximation has distinct advantages over using either the inner surface, which vastly underpredicts the uniaxial stress-strain curve while overpredicting the transition strain, or the exterior surface, which underestimates the small-strain properties, overestimates the large strain properties, and underestimates the transition strain (Fig. 5a & 5c). The estimation of centerline stress-strain still underpredicts the small-strain modulus and overestimates the transition strain, but the curves are much closer to the uniaxial stress-strain curve than those for either of the other measures (Fig. 5b). Additionally, any of these stretches can be estimated using the assumption of incompressibility with no additional data beyond the force-displacement data one would ordinarily obtain.

To predict failure of ring samples, one needs to know the maximum stress the sample experiences. We have shown that there exist two possible peak stress locations: the luminal side of the central region and the exterior side directly behind the pin (Fig. 7). Additionally, we posit the existence of a minimum bend radius of the material, which greatly changes the kinematics of the ring pull test as we approach it. The problem of bending around the pin is complicated by the fact that the material has some initial curvature, the material behavior is nonlinear, and the material is stretched over the pin rather than being in pure bending. These three issues mean the neutral axis and minimum bend radius are not easily calculated, but we present a straightforward, albeit oversimplified, analysis that indicates that the minimum bend radius and thus pin size is an important consideration in the testing of ring pull samples.

In this work, we propose several equations for correcting ring-pull test properties to uniaxial properties (Eqs. 5a-5c), which show the unaccounted for error from this geometric correction to be from 3% to 9%. Our data show that the small-strain modulus is the most affected property across the range tested (Fig. 10a). We also note that the corrections are most significant when H/Rc is large (Fig. 10a-c). In the cases of the aorta in pig, sheep, and mouse, the corrected properties are very similar to the uncorrected because we are operating in the range of H/Rc = 0.15 − 0.25 (Fig. 11). We also show that the overall error in small-strain modulus becomes nearly the same across pin sizes for each tissue tested. This indicates that we have mitigated the error due to these geometric effects, but we are still left with some unaccounted-for error. This error could be due to the material properties, especially the nonlinearity, or due to other factors such as residual stresses. The error was larger for the sheep aorta than for the pig aorta before and after correction; the reason for the difference between species is not clear. In the correction for the sheep carotid, where H/Rc = 0.50, significant improvements are made in matching the uniaxial properties for all extracted parameters. Thus, for thick-walled structures like the ventricle or small arteries, one should use such corrections to attain quantitatively correct properties. In addition, we proposed the use of mean properties assessed in this way to determine a corrected failure stress. We have shown that for a pin of large enough size, one can expect failure to initiate from the midplane region of the ring. Thus, one can estimate the strain at failure on the inner surface and use this strain to give a corrected failure stress from the average material properties.

The most relevant and similar work to this is that of van Haaften et al. [13] for ring tests on linear, isotropic materials. Their proposed correction was validated for small strains (<5%), but they did not address transition strain or lockout modulus. They propose using the exterior ring surface stretch as it becomes linear in strain for tests on materials with linear behavior once the ring transitions from the initial bending to the stretch dominated deformation. However, almost all biological soft tissues do not abide by this principle since they are anisotropic and behave nonlinearly. In fact, even the validation tissue used for the experiments of van Haaften et al. behave nonlinearly. Comparing our data to those of van Haaften et al., we see that our small-strain and large-strain modulus corrections bookend their proposed correction (Fig. 12). Additionally, the effect of material nonlinearity is quite pronounced, especially for thick-walled tissues where our correction diverges from van Haaften et al.

Fig. 12

Comparison of the correction for small and large strain proposed in this work compared to that proposed in [13] for H/a = 1.5

Finally, we emphasize that this work is designed for arterial samples, which have broadly similar behavior to one another. If we attempt to apply this relation to samples with different material behavior, we may expect the relation for correction to change with respect to geometric dimensions. However, the trends will remain the same in that as the tissue thickness increases, one will observe altered stress-strain behavior. To probe whether our proposed correction factor relations held for material properties outside the correction factor testing range, ring tests using rings having H/Rc = 0.2 − 0.375 with varied fiber modulus (ξ in Eq. 1) from 0.5 to 5 times the baseline value were analyzed. In these simulations we saw changes in the small-strain modulus correction less than 3%, changes in the large strain modulus correction less than 3.5%, and changes in the transition strain correction less than 11%. The relatively low error introduced for these large changes in material properties indicate that the underlying relation is driven primarily by geometry and is relatively insensitive to the material stiffness. Additionally, this work probes only the passive mechanical behavior of soft biological tissues. We believe that active contractile assays will have similar corrections necessary for ring shaped samples. However, such a study is beyond the scope of the current work.


In this work, we have shown that the uniaxial approximation is only valid in a narrow range of ring geometries. In general, one does not need corrections if the sample falls within the range of H/Rc = 0.15 − 0.3. However, attention should be paid to the selection of pin size in the range of H/a = 0.25 − 1.0 if one intends to investigate failure properties. In these cases, H/a should be near 0.25 because the smaller this value is, the less stress concentrates near the pin. In the limit of H/Rc → 0, we should note that minimal correction is needed provided that the pin size is in the range of H/a = 0.25 − 1.0. In contrast, as H/Rc → 0.5, correction is necessary regardless of the choice of pin radius. However, we again need to be cognizant of the stress concentration from small pins. In this case, one should minimize H/a subject to the constraints of using circular or D-shaped pins. The correction equations are only valid in the range of H/Rc = 0 − 0.5 and H/a = 0.25 − 3. As one attempts to extrapolate this phenomenological correction, one should be aware that the correction may diverge. It is much safer to simply take the value at the extrema in the validated range. It should be emphasized that our proposed correction is for soft collagenous biological tissues that exhibit a bilinear mechanical response to load, and that this correction framework remains valid only under this condition.

Included as a supplement to this work at is a MATLAB script designed to take as inputs the sample geometry (radius, thickness, width), pin radius, and force-displacement data. Using these data, the script calculates the centerline strain vs. PK1 stress, fits the small-strain modulus, large-strain modulus, and transition strain, and calculates the luminal strain vs luminal PK1 stress for the assessment of failure properties.


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    The large-strain portion of the stress-strain curve is often called the linear regime, but we use lockout in this work to avoid confusion with linear material models.


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The authors acknowledge funding from the National Science Foundation Graduate Research Fellowship Program (NSF GRFP) under Grant No. 00039202. The authors also acknowledge funding from the National Institutes of Health under Grant No. U01-HL139471. The authors acknowledge and thank the University Imaging Center (UIC) at the University of Minnesota for the use of the small animal ultrasound system, Dr. Paul A. Iaizzo and the Visible Heart Lab for the porcine and ovine tissue used in this study, and Dr. Paulo P. Provenzano and the Provenzano Research Group for the mice used in this study. The authors also thank Drs. Neeta Adhikari and Jennifer L. Hall for their technical advice on dissection and ring testing of mouse tissues, Shannen B. Kizilski for her assistance in design and fabrication of the ring pull apparatus, and Elizabeth Gacek for her technical assistance in setting up contact in the ring pull simulations.

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Correspondence to V. H. Barocas.

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The authors declare that they have no conflict of interest. All animal tissue used in the current study was obtained from animals previously sacrificed as part of other, institutionally-approved studies; no live animals were used or sacrificed for the sole purpose of the current study. This study involved no human subjects.

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Mahutga, R.R., Schoephoerster, C.T. & Barocas, V.H. The Ring-Pull Assay for Mechanical Properties of Fibrous Soft Tissues – an Analysis of the Uniaxial Approximation and a Correction for Nonlinear Thick-Walled Tissues. Exp Mech 61, 53–66 (2021).

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  • Uniaxial ring-pull
  • Computational biomechanics
  • Large strain
  • Soft biological tissue
  • Arterial biomechanics
  • Cardiovascular biomechanics
  • Nonlinear
  • Anisotropic