Biaxial Tension of Fibrous Tissue: Using Finite Element Methods to Address Experimental Challenges Arising From Boundary Conditions and Anisotropy

Short abstract

Planar biaxial tension remains a critical loading modality for fibrous soft tissue and is widely used to characterize tissue mechanical response, evaluate treatments, develop constitutive formulas, and obtain material properties for use in finite element studies. Although the application of tension on all edges of the test specimen represents the in situ environment, there remains a need to address the interpretation of experimental results. Unlike uniaxial tension, in biaxial tension the applied forces at the loading clamps do not transmit fully to the region of interest (ROI), which may lead to improper material characterization if not accounted for. In this study, we reviewed the tensile biaxial literature over the last ten years, noting experimental and analysis challenges. In response to these challenges, we used finite element simulations to quantify load transmission from the clamps to the ROI in biaxial tension and to formulate a correction factor that can be used to determine ROI stresses. Additionally, the impact of sample geometry, material anisotropy, and tissue orientation on the correction factor were determined. Large stress concentrations were evident in both square and cruciform geometries and for all levels of anisotropy. In general, stress concentrations were greater for the square geometry than the cruciform geometry. For both square and cruciform geometries, materials with fibers aligned parallel to the loading axes reduced stress concentrations compared to the isotropic tissue, resulting in more of the applied load being transferred to the ROI. In contrast, fiber-reinforced specimens oriented such that the fibers aligned at an angle to the loading axes produced very large stress concentrations across the clamps and shielding in the ROI. A correction factor technique was introduced that can be used to calculate the stresses in the ROI from the measured experimental loads at the clamps. Application of a correction factor to experimental biaxial results may lead to more accurate representation of the mechanical response of fibrous soft tissue.

1. Introduction

Tensile testing of fiber-reinforced soft tissue is widely used to characterize the tissue mechanical response, evaluate functional differences between treatment groups, develop constitutive formulas, and obtain material properties to be used in finite element studies. Uniaxial tension is often chosen because it is easy to implement and the interpretation of the results is straightforward. The one-dimensional configuration ensures homogeneous loading, as the loads applied at the clamps are transferred directly through the entire tissue. Strain is measured using optical imaging in a region of interest in the center of the tissue, avoiding any clamp effects. This allows the researcher to determine material properties by fitting a constitutive equation to the stress-strain response within the region of interest. However, uniaxial tests do not represent the in situ loading of many fibrous tissues. In uniaxial tension, the unconstrained edges contract as the tissue is stretched and the fiber component in the lateral direction is compressed. Therefore, biaxial testing has been adopted as a technique to characterize the mechanics of soft tissue because it prevents fiber compression and therefore better represents tissue in situ strains.

Biaxial tension testing has been conducted on numerous fiber-reinforced tissues, including skin, cardiac, aortic, lung, annulus fibrosus, tendon, ligament, as well as tissue analogs. In contrast to uniaxial tension, biaxial tension produces many challenges in both the implementation of the experimental loading and the interpretation of the results. Especially problematic to the biaxial loading modality is the dependence on the experimental boundary conditions, such as clamping method, sample geometry, and tissue anisotropy. The focus of this paper is two-fold. In the first section, key works in the literature of biaxial tension of soft tissue will be highlighted, especially with respect to the challenges and difficulties associated with the interpretation of experimental results. The second part will focus on new results using a finite element approach to correct for these effects.

2. Review: Planar Biaxial Tension of Fibrous Soft Tissues

2.1. Foundational Studies.

Sacks and Sun elegantly captured the history and evolution of biaxial testing in their 2003 review entitled “Multiaxial Mechanical Behavior of Biological Materials” [1], with a special focus on the challenges related to the testing of biological soft tissue for the development of constitutive theories. This review serves as an excellent reference regarding the challenges and requirements of biaxial testing for biological tissues, including the small sample sizes, tissue heterogeneity, and material anisotropy of biological tissue. Further challenges discussed include the difficulty in gripping the tissue, in applying constant forces from the loading apparatus, and in creating a homogeneous stress and strain profile within the tissue, which is compounded by the need to identify the tissue material axes. It also serves as a reference for the techniques and kinematics of biaxial testing, including applicable concepts of hyperelasticity, the construction of the deformation tensor for planar biaxial tension, and an analysis of the force and strain tensors associated with the biaxial configuration [1].

In addition to the foundational concepts of test design and kinematics, this study provides a comprehensive review of biological testing from the pioneering works of Lanir and Fung in 1974 [2,3] and of Tong and Fung [4], in which the Fung-type constitutive model for soft tissue was introduced. This is still the most widely used constitutive model in the soft tissue biomechanics literature. While this constitutive formula has successfully modeled and predicted the mechanical response of many types of soft tissue, it is limited in its ability to link the mechanical behavior of the tissue to its structural components. Therefore, the review by Sacks and Sun discusses structurally motivated constitutive relationships, which can elucidate structure-function mechanisms [1]. Special attention is given to the incorporation of strain energy functions for collagen fibers, with fiber deformations calculated through a tensor transformation from global tissue deformation to local fiber deformation. These tensor deformations utilize assumptions of initial fiber orientation and affine kinematics. Improvements on these assumptions are shown with the use of a small angle light scattering (SALS) technique, which uses laser light scattering patterns to quantify the spatial distribution of collagen fibers within the tissue throughout the entire tissue test. The experimentally quantified structural data identified through the SALS technique is demonstrated to provide superior fits to the tissue stress-strain loading curves [1].

2.2. Recent Implementation of Planar Biaxial Tension.

In the ten years since the 2003 review by Sacks and Sun, biaxial tension has continued to be advanced and will be described in the following sections. Biaxial tension has been applied to soft tissue for widespread applications, such as tissue characterization; constitutive modeling, including investigation into structure function relationships; testing of tissue analogs; and finite element modeling. In addition, challenges related to boundary conditions have been noted.

2.2.1. Tissue Characterization Using Elastic Modulus.

Biaxial tension is frequently utilized for tissue characterization and evaluation of tissue treatments by calculating the elastic modulus from the slope of the stress-strain response. These types of studies offer a simple but useful means to compare differences in the mechanical response between tissues in different test groups and have been used to study heterogeneous regions of tissue, to compare between species, and to evaluate the effect of various tissue treatments. Holmes et al. compared the regionally dependent mechanical properties for human transverse carpal ligament, showing that the proximal zones of the ligament had higher moduli than distal zones and likewise radial zones had higher moduli than ulnar zones [5]. In a similar comparison study, the validity of using an ovine animal model for percutaneous aortic valve replacement was tested by comparing the biaxial response of ovine and human coronary sinus and ascending aorta [6]. Significant differences in both the structural composition and mechanical response to the biaxial loads were manifest between the species, suggesting that the ovine model is not ideal. Beyond comparison of different regions or species, the effect of loading environment is readily testable using a simple slope analysis of a biaxial stress-strain curve. In one such example, the average modulus of porcine annulus fibrosus was tested at multiple loading rates and did not change in either the disc circumferential or axial directions [7].

Studies that only analyze the modulus of a biaxial response are useful to identify mechanical differences between tissue regions, species, and treatments. However, they do not provide the ability to analyze the details of the mechanical tissue response or offer the predictive tools and structure-function mechanisms that can be provided by constitutive models.

2.2.2. Constitutive Modeling.

A more sophisticated analysis of the mechanical response of biological tissues to planar biaxial tension involves fitting the stress-strain response to constitutive strain energy models to obtain material properties. Constitutive models offer an enhanced insight into biological tissues, as they can describe mechanical attributes, such as nonlinearity, anisotropy, and viscoelasticity. Importantly, they can also serve to predict the mechanical response of a tissue to various types of previously untested loading. This enables them to be useful, not only in mechanical characterization, but also in finite element applications of in vivo tissues, organs, and medical devices.

Constitutive models can be divided into two classifications: phenomenological and structural. Phenomenological models were the first to be implemented for use in biaxial tests of soft tissue, and their usage is still commonplace. The first phenomenological model to be implemented was the Fung-type model [2–4]. This model remains the most widely utilized and has been used to characterize human sclera [8], annulus fibrosus [9], diseased human coronary and carotid arteries [10], porcine duodenum [11], bioprosthetic heart valve [12], mitral valve [13,14], and others. Beyond material characterization, parameters obtained for the Fung model are often implemented in finite element (FE) models of whole tissue systems to predict in vivo loads, which will be discussed in Sec. 2.2.4.

Although phenomenological models can provide excellent fits to tissue mechanics, they are not developed utilizing information about the tissue architecture. They are therefore limited in their ability to elucidate structure-function mechanisms. Structural constitutive models seek to relate the mechanical response of a tissue to the composition and architecture of its material constituents. For fibrosus tissues, this is typically accomplished using a strain-energy function to describe the mechanical contributions of collagen fibers based upon their modulus, nonlinearity, and orientation. A detailed history of the origins of structural constitutive models in planar biaxial tension, including pioneering works of Lanir [15,16] and Humphrey et al. [17,18] can be found in the 2003 Sacks and Sun review [1] and theory of structural modeling of fiber-reinforced biologic material in the work of Spencer [19].

Structural models have recently been successfully applied to biaxial experiments for several tissues, often explaining the underlying mechanisms behind phenomena previously observed and quantified using only modulus measurements or with phenomenological models. For example, a structural model of aortic valve cusp incorporating fiber crimp, stiffness, and distribution was able to distinguish the chemical and mechanical effects of glutaraldehyde treatment. Although previous experiments had noted that treatment caused a tissue-stiffening effect [21], the structural model indicated that this effect was primarily due to alterations in the collagen network as opposed to direct chemical effects. The importance of fiber structure and organization has also been highlighted in other biological tissues. Using a structural model that included parameters for sample-specific fiber orientation enabled Szczesny et al. to show that the planar tensile mechanics of human supraspinatus tendon are principally a function of fiber modulus, crimp, and angular distribution [22]. O'Connell et al. showed that degeneration of the intervertebral disc altered the annulus fibrosus fiber mechanics, resulting in an elongated toe-region of the stress-strain curve and a lower stiffness [23].

2.2.3. Tissue Analogs.

Tissue analogs offer the ability to study the quantitative relationships between tissue structures and their mechanical behavior using constructions where fiber density, orientation, and degree of anisotropy are prescribed. This is useful both to aid in a better understanding of the function of natural tissue and in the development of tissue-engineered constructs. Tissue analogs are often created using aligned collagen fibers in an isotropic media, and are frequently seeded with cells. Because their purpose is to replicate the basic properties of native tissue, it is important that tissue analogs are loaded similarly to the in vivo loading of their representative tissue. Therefore, they are often tested in planar biaxial tension.

As shown in recent studies of tissue analogs tested in biaxial tension, the modulus of collagen-embedded Teflon molds is proportional to the orientation of collagen fibers [24] and the mechanics of collagen scaffolds change with glutaraldehyde crosslinking [25]. For cell-seeded gels, strain transfer to cells in biaxial tension is dependent on cell-matrix interactions and local anisotropy [26]. Extracellular matrix synthesis is dependent on the ratio of biaxial strain and initial fiber alignment, while the increases in modulus are due to increases in collagen content and deposition [27–29].

2.2.4. Finite Element Modeling.

Finite element (FE) modeling has greatly advanced the quantification of tissue stress and strain in biaxial tension and, by applying biaxial tests and FE models, within the whole tissue or organ system. FE modeling relies on the constitutive work discussed in Sec. 2.2.2; therefore, the groups at the forefront of biaxial constitutive modeling are often responsible for major advances in FE modeling of fibrous biological tissues. Holzapfel first implemented a Fung-type constitutive model [30] concurrently with FE theory developments by Lanir [31]. Sun and Sacks later implemented the Fung-type model in the commercial software ABAQUS and addressed issues of computational instabilities by imposing physically based constraints on the form of the FE functions [32]. Raghupathy and Barocas developed a closed-form analytic structural fiber model for FE implementation [33]. Importantly, FEBio, an open-source finite element package with specific focus on the mechanics of biological tissue, has recently been released [34]. This software package integrates several anisotropic phenomenological models, such as the Ogden, Holzapfel, and Fung-type, as well as structurally based constitutive models.

These recent developments have enabled researchers to use material properties determined from biaxial tension to simulate complete biological systems, such as the aortic valve [35] and small intestine [11]. FE models have also supported the interpretation of experimental results, such as predicting the fiber distributions within tissue analogs tested in biaxial tension [23]. Further, FE models aid in determining the mechanisms through which applied boundary conditions affect the stress and strain distributions in biaxial tension, discussed in Section 2.3.

2.3. Challenges Related to Boundary Conditions.

Despite the increasing adoption and implementation of planar biaxial tension, several critical challenges remain. Of primary importance are those related to the dramatic effects of the boundary conditions on the stress and strain fields within the sample. In uniaxial tension, boundary conditions are generally considered not to influence the loads or deformations in the central region of interest (ROI). This is because applied loads are constrained to transfer uniformly throughout the length of the tissue, and deformations in the central ROI, measured optically, are accepted to be free from the influence of loading clamps. In planar biaxial tension, the boundary conditions have considerable influence on both the load and deformation that the ROI experiences. Applied loads and deformations are therefore not uniformly distributed throughout the tissue. The method of loading, the sample geometry, and the material anisotropy may alter the magnitude, distribution, and homogeneity of loads and deformations throughout the sample.

The most common method of loading is to attach several sutures or wire rakes to each edge of a square tissue sample. Typically, 4–16 sutures or rakes are applied per edge. This technique is useful because it allows for each edge to be compliant transverse to its loading axis, which allows for the whole sample to strain biaxially. However, the use of sutures and rakes introduces several problems. In particular, sutures do not fully constrain the entire edge of the sample and the spaces between individual sutures are not fully loaded. For fibrous tissues, this can result in fibers that are not stretched during loading. Additionally, each suture is a point load. Although this is not always problematic for some compliant tissues, for stiffer fibrous tissues, the point loads may cause sutures to tear away from the sample. Further, any nonuniformity in the placement or spacing of sutures disrupts the homogeneity of the loads and deformations throughout the sample [36]. Importantly, the combination of point loading at the specimen edges and the effects of irregularities in suture placement prevents one from being able to determine the stresses within the ROI.

As an alternate loading mechanism, clamps, similar to those used in uniaxial tension, may be used. Clamps are beneficial in that they apply a uniform load across the entire edge, ensuring that all fibers are stretched during the experiment. However, the use of clamps redistributes part of the applied load around the edges of the sample, away from the ROI, effectively stress-shielding the ROI [37]. This may cause difficulties in interpreting experimental data, as the applied loads may be higher than those reached in the ROI and the stresses in the ROI cannot be precisely determined. This can result in artificially high modulus calculations [36]. Experimentally, clamps were shown to nearly double the reported modulus of bovine pericardium tissue compared to sutures, although no tests were performed to determine if this was a result of clamp-induced stress-shielding or due to fibers that were not stretched in the sutured samples [38].

As an alternate to the commonly used square geometry, a cruciform shape is sometimes utilized. This geometry is thought to benefit from the usage of clamps yet prevent some of the load redistribution and stress-shielding observed in square samples. Although each of the four edges are clamped, allowing for all fibers on each edge to be engaged, the cruciform geometry does not have any tissue directly spanning the area between clamps. Thus, more of the applied load transmits to the ROI. However, FE studies confirm that stress concentrations still occur at the corners where the horizontal and vertical arms intersect, which may prevent a portion of the applied load from reaching the ROI [37]. In addition, many tissue sources may not be large enough to provide a sample with a cruciform geometry.

2.4. Summary of Biaxial Tension Review.

Biaxial tension remains an important experimental testing modality and is widely used to characterize tissue mechanical response, evaluate treatments, develop constitutive formulas, and obtain material properties for use in structural finite element studies. Although the application of tension on all edges of the test specimen represents the in situ environment of most fibrous soft tissues, the boundary conditions, including loading technique and geometry, may confound the interpretation of experimental measurements. As a result, an experimentalist does not know the loads within the ROI. This leads to erroneous calculations of material properties when fitting constitutive equations to stress-strain curves with stress obtained from clamp measurements and strain recorded from optical measurements within the ROI.

Because prior experimental and modeling studies have confirmed that biaxial boundary conditions lead to erroneous material properties [36–39], while none have established a method to correct for these behaviors, the objective of the study presented in Sec. 3 was to use finite element analysis to quantify stress-shielding in biaxial tension and to formulate a correction factor that can be used to determine ROI stresses. This FE model can therefore be used to quantify the impact of sample geometry, material anisotropy, and tissue orientation on the correction factor.

3. Finite Element Simulation

A primary difficulty in interpreting experimental studies in planar biaxial tension remains the inability to quantify the loads in the ROI. Applied loads at the sample edges are recorded using load cells, yet the boundary condition effects described above influence the force distribution throughout the sample and result in a shielding of stress in the ROI. The purpose of this study is to quantify the relationship between the forces at the load cell, which are measured experimentally, and the actual stress experienced in the ROI. An FE-based approach to solving this problem is ideal, as stress distributions can be quantified throughout the tissue; thus, a relationship between the experimentally measured forces and the stresses in the ROI can be developed.

3.1. Finite Element Methods.

FE simulations of biaxial tension were conducted with both square and cruciform geometries. For each geometry, multiple levels of material anisotropy were tested, including isotropic (ISO), transversely isotropic (XISO), and orthotropic (ORTH). All FE simulations were performed using the finite element package FEBio [34]. Square geometries had dimensions of 10 × 10 mm², consisting of 7779 elements, while cruciform geometry had arm lengths of 7 × 21 mm and consisted of 3028 elements. Sample size was selected based on previous experimental studies of human intervertebral disc [23]. Symmetry conditions were imposed about the y axis; therefore, only the right half of each geometry was modeled (Figs. 1(a) and 1(b)). Loading clamps, simulated as rigid bodies, were connected to the tissue through a rigid body interface between the element faces of the clamps and adjacent tissue. Biaxial tension was applied by prescribing displacements to the loading clamps.

Fig. 1.

Biaxial tension response for isotropic material. (a) Square and (b) cruciform color map of von Mises stress concentration with color scale representing the ratio of stress to the average ROI stress. Internal arrows point to regions of peak stress concentration. The ROI is denoted by a dotted square, bold arrows indicate loading boundary conditions at the clamps, and the vertical hash line represents the symmetry implemented in the FE simulations. (c) Stress-strain curve for the loading clamp (square and cruciform) and ROI with all responses plotted against the ROI Lagrange strain. Lower stresses are experienced in the ROI than applied at the clamp for both square and cruciform geometries. (d) Stress transfer to the ROI as a percentage of the total stress at the loading clamp for both square and cruciform geometries.

3.2. Constitutive Models.

Structurally based hyperelastic continuum models based on the work of Spencer [19] were used for the finite element simulations. These models are advantageous in that they link the mechanical response of the tissue to its fiber and matrix components and may therefore be used to investigate structure-function relationships. These material models consist of strain-energy functions using the integrity basis of invariants (I _i, i = 1–5, Eq. (1)) as

$$\begin{array}{c}\multicolumn{1}{c}{I_1=\mathit{tr}\mathit{C},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{\mathit{I}}_2=1/2[{\left(\mathrm{tr}\mathit{C}\right)}^2-\hspace{1em}\mathrm{tr}{\mathit{C}}^2],\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{I}_3=det\mathit{C},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{I}_4=\mathit{a}\mathbf{\hspace{1em}}·\mathbf{\hspace{1em}}\mathit{C}\mathbf{\hspace{1em}}·\mathbf{\hspace{1em}}\mathit{a},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{I}_5=\mathit{b}\mathbf{\hspace{1em}}·\mathbf{\hspace{1em}}\mathit{C}\mathbf{\hspace{1em}}·\mathbf{\hspace{1em}}\mathit{b}}\end{array}$$ (1)

where C is the right Cauchy–Green deformation tensor (C = F^TF), F is the deformation gradient, and the unit vectors a and b represent the fiber directions in the reference configuration.

The strain-energy function for the ISO material (W_ISO) was described using a compressible Holmes–Mow constitutive relationship [40]. This formulation has been used to represent the noncollagenous matrix of biological tissues, such as articular cartilage and annulus fibrosus matrix [41,42] and is a function of invariants I _1, I _2, and I ₃ and material properties c₀ (MPa), c₁ (unitless), and c₂ (unitless) as

$$W_{\mathrm{ISO}}=\frac{c_0}{I_3^{\mathrm{\beta }}}{e}^{[c_1(I_1-3)+c_2(I_2-3)]}$$ (2)

where β is an exponential stiffening coefficient calculated as β = c₁ + 2c_2. Material properties c₀–c₂ can be transformed to the more familiar properties of Young's modulus (E) and Poisson's ratio (ν) using the following relationships:

$$\begin{array}{c}\multicolumn{1}{c}{E=4c_0(c_1+c_2)(1+\mathit{\nu })\mathit{\nu }=\frac{c_2}{c_1+3c_2}}\end{array}$$ (3)

Anisotropic materials were modeled by adding a fiber strain energy formulation (W_f) to the strain energy of the ISO material to create the XISO (one fiber population) and ORTH (two fiber populations) materials. Each fiber strain energy term consists of an exponential stress-stretch relationship that is a function of material properties c₄ (MPa), representing fiber stiffness; c₅ (unitless), representing fiber nonlinearity; and invariant I ₄ or I ₅, which is the squared fiber stretch,

$$W_f=\frac{c_4}{2c_5}(e^{c_5{(I_{\mathrm{\alpha }}-1)}^2}-1)$$ (4)

where α = 4 or 5, depending on the fiber population. The complete description for the XISO and ORTH materials was therefore described as

$$\begin{array}{c}\multicolumn{1}{c}{W_{\mathrm{XISO}}=W_{\mathrm{ISO}}+\frac{c_4}{2c_5}(e^{c_5{(I_4-1)}^2}-1)W_{\mathrm{ORTH}}=W_{\mathrm{ISO}}+\frac{c_4}{2c_5}(e^{c_5{(I_4-1)}^2}+e^{c_5{(I_5-1)}^2}-2)}\end{array}$$ (5)

Because the XISO and ORTH materials have the identical matrix strain energy function as the ISO material, any difference in mechanical response between these materials and the ISO material is a result of the increasing level of anisotropy due to the fibers. Fibers were initially aligned parallel to the x-loading axis for the XISO material and parallel to the x- and y-loading axes for the ORTH material (Fig. 3(a)). The role of fiber orientation compared to the loading direction was also determined by modifying the XISO material such that the fiber population was +25 degrees to the x-loading axis and by modifying the ORTH material to align the fibers ±25 degrees to the x-loading axis (Fig. 4(a)).

Fig. 3.

XISO (top) and ORTH (bottom) response for square geometry. (a) Diagram of fiber orientations with solid lines representing fibers. (b) Von Mises stress concentration; color map depicts ratio of stress to ROI stress. (c) Plot of ROI stress versus clamp stress for the complete FE simulation, with all stress values normalized to peak clamp stress. The solid line is obtained through linear regression, and the correction factor is the corresponding slope of the regression. Correction factors closer to 1 indicate ROI stresses that are more similar to the applied clamp stresses.

Fig. 4.

Effect of fiber angle. XISO with fibers oriented 25 deg to the x loading axis (top) and ORTH with fibers aligned ±25 deg to the x and y axes (bottom). (a) Diagram of fiber orientation with solid lines representing fibers. (b) Color map of von Mises stress concentration; color map depicts ratio of stress to ROI stress. (c) Plot of ROI stress versus clamp stress for the complete FE simulation, with all stress values normalized to peak clamp stress.

3.3. Material Properties.

Initial material properties for all of the material symmetry conditions were based on reanalysis of human annulus fibrosus stress-strain data using the hyperelastic continuum models described above. Matrix tensile properties c₀–c₂, which were c₀ = 0.0758 MPa, c₁ = 0.2205, and c₂ = 0.9438, were calculated from experiments that applied a combination of osmotic swelling and confined compression to determine matrix tensile properties (n = 8) [43]. These properties were converted to modulus E = 0.075 MPa and Poisson's ratio ν = 0.22 through Eq. (3). Fiber properties, which were c₄ = 0.296 MPa and c₅ = 65 (unitless), were calculated from uniaxial tensile stress-strain data (Fig. 2(a)) [44]. The average response of five samples was determined and fit using the fmincon function in Matlab (MathWorks, Inc, Natick, MA). To confirm that these material properties were fit correctly and that the constitutive equations were properly input into the FEBio model, the uniaxial tension experiment was simulated in FEBio using these material properties. The FE-predicted stress-strain response matched the average experimental response with excellent agreement (Fig. 2(b)).

Fig. 2.

(a) Uniaxial tension stress-strain data from O'Connell 2009. Filled circles are individual sample responses, while the open circles and solid line represent the average of the experimental dataset (n = 5). (b) Average experimental stress-strain curve from (a) with Matlab fit and FEBio prediction, demonstrating excellent agreement between experimental data and FE prediction for the uniaxial response.

3.4. Data Analysis.

Experimental stresses in biaxial tension are usually calculated by dividing the forces at the load cell by the tissue cross-sectional area. To create an analogous stress measure for the FE simulations, clamp reaction forces, as output from FEBio, were divided by the reference cross-sectional area to create a first Piola–Kirchhoff stress measurement, which is called the “clamp stress.”

The ROI for this study was defined to be the central 25% of the tissue, corresponding to the region where strain is often experimentally measured using optical methods. Cauchy stress and Lagrange strain of the elements within this region were averaged to represent the ROI properties. Cauchy stress was converted to first Piola–Kirchhoff stress for comparison to clamp stress, and this is called the “ROI stress”. The percentage of stress transferred to the ROI was calculated by dividing the clamp stress by the ROI stress. Stress concentrations throughout the tissue were computed by dividing the stress in each element by the ROI stress.

3.5. Correction Factor.

As described in the Sec. 2 literature review, a significant challenge for interpreting a biaxial test is that experimental devices cannot record the actual ROI loads, which can be much lower than the applied loads recorded by the load cells. In contrast, optical imaging methods are routinely used to determine experimental ROI strain. Therefore, the FE results from the current study were used to calculate a correction factor, which can be used to calculate the experimental ROI stresses from the applied loads. This was accomplished by plotting the ROI stress versus the clamp stress over the entire loading duration and applying a least-squares linear regression to determine an equation of the form y = mx. The slope of the regression, which can be used to relate the ROI stress to the clamp stress, was called a correction factor. For anisotropic materials, this process was repeated for both axes, comparing the X ROI stress to the X clamp stress and the Y ROI stress to the Y clamp stress.

3.6. Sensitivity.

The correction factors determined above were calculated using an initial set of matrix and fiber material properties obtained from the average material response of human annulus fibrosus. To test the dependence of the correction factor on the initial choice of material properties, FE simulations were repeated using a range of material properties that span the experimental dataset. The correction factor analysis was repeated for each set of material properties. Matrix properties were evaluated using the ISO material, with material properties ranging from 0.4 to 1.8 times the initial input choice. For XISO and ORTH materials, fiber modulus c₄ was varied between 0.015 to 2.4 times the average material value in uniaxial tension (Table 1). Similarly, fiber nonlinearity c₅ was varied between 0.2 to 2.5 times the average value. Matrix properties were not varied for XISO and ORTH materials, as they were evaluated for the ISO material.

Table 1.

Material parameter inputs for sensitivity study, including initial value and the range of values tested in absolute and factor scales

Material parameter	Initial value	Sensitivity range	Sensitivity factor range
E - modulus	0.0758 MPa	0.00758–7.58	0.1X–10X
ν - Poisson's ratio	0.22	0.1–0.4	0.4X–1.8X
c4 - fiber modulus	0.296 MPa	0.004–0.7	0.015X–2.4X
c5 - fiber nonlinearity	65 unitless	13–161	0.2X–2.5X

4. Results

4.1. Material Response—Isotropic.

Biaxial tension simulations of the ISO material exhibited regions of large stress concentrations for both the square and cruciform geometries (Fig. 1). For the square geometry, stress concentrations were primarily localized in the regions of tissue that span the area between the loading clamps (Fig. 1(a), arrows). In this region, peak stresses reached up to 10 times the average stress in the ROI. For the cruciform geometry, the predominate location of stress concentration was at the intersection of the cruciform arms, where peak stresses reached three times the average ROI stress (Fig. 1(b), arrows).

High stress concentrations in both the square and cruciform geometry redistributed part of the applied loads away from the ROI, resulting in a ROI stress-shielding effect. Thus, forces applied at the loading clamps, which are measured experimentally by the load cells, did not fully transmit to the central ROI, as is typically assumed in an experiment. This resulted in ROI stress-strain relationships that are lower than the measured clamp stress-strain (Fig. 1(c)). This was particularly true for the square geometry where stress-shielding led to average ROI stress transfer of 37% of the applied experimental clamp stress. The cruciform geometry exhibited less stress-shielding, with 73% of applied clamp stress transferring to the ROI (Fig. 1(d)).

4.2. Material Response—Transversely Isotropic and Orthotropic.

The inclusion of reinforcing fibers lowered the concentrations of peak stresses for the XISO and ORTH materials compared to the ISO material (Fig. 3(b)). For the XISO material with square geometry, peak stress concentrations were 1.9 times higher than the average ROI stress (Fig. 3(b)). Similar to the ISO material, stress concentrations were lower for the cruciform geometry than for the square geometry for all levels of anisotropy (Table 2). Also similar to the ISO material, stress concentrations led to ROI stress shielding in both the square and cruciform geometries.

Table 2.

Peak von Mises stress concentration, defined as peak specimen stress divided by average ROI stress

	Square	Cruciform
Material	stress concentration	stress concentration
ISO	10X	3X
XISO	1.9X	1.4X
ORTH	9.6X	1.7X

4.3. Correction Factor.

A correction factor was calculated for each combination of sample geometry and material anisotropy. ROI stress was plotted versus clamp stress, and a regression analysis was applied to calculate their relationship (Fig. 3(c)). The linear relationship facilitates an easy implementation of the correction factor, where ROI stresses may be determined by multiplying the clamp stresses by the correction factor. Correction factors closer to 1 indicate ROI stresses that are more similar to the applied experimental stresses.

The correction factor was dependent on sample geometry (Table 3). For each material model tested, the cruciform geometry was closer to unity (ROI stresses more similar to clamp stresses) than the square geometry. For example, the ISO correction factor was 0.40X for the square geometry and 0.76X for the cruciform geometry.

Table 3.

Correction factor for square and cruciform geometries at all levels of anisotropy. For XISO and ORTH materials, a –X or –Y indicates the correction factor for the respective loading axis.

	Square		Cruciform
Material	Correction factor	R2	Correction factor	R2
ISO	0.40	1	0.76	1
XISO-X	0.87	1	0.97	1
XISO-Y	0.27	0.97	0.59	1
ORTH-X	0.77	1	0.94	1
ORTH-Y	0.77	1	0.94	1

The correction factor was dependent on anisotropy, with correction factors closer to unity in the axes that were reinforced by fibers (XISO-X, ORTH) than for axes without fibers (ISO, XISO-Y) (Fig. 3(c), Table 3). The square XISO-Y correction factor exhibited slight tapering (nonlinearity) towards the end of the curve. This is likely a result of a mechanical coupling effect between the x axis, which is much more stiff, and the y axis, which only contains isotropic matrix and is much less stiff. This phenomenon has been previously described [19,20,45]. While this effect may limit the utility of the correction factor for XISO-Y, at large stresses, the mechanics in the direction perpendicular to the fibers are less important because these matrix properties can be directly measured in other loading modalities [43] and the contribution of nonfiber-reinforced matrix to the strain energy of a tissue is quite minor compared to that of the fibers [22,44,46–48].

4.4. Fiber Orientation.

The effect of fiber orientation relative to the loading direction was quantified by rotating the fiber orientation in the FE simulations to +25 deg to the x axis for the XISO material (Fig. 4(a), top) and by rotating the fibers in the ORTH material to ±25 deg to the x axis (Fig. 4(a), bottom). This is analogous to experimentally rotating the tissue specimen such that its material axis is no longer parallel with the experimental loading axis. This rotation created a large increase in the stress concentrations for both materials (Fig. 4(b)). Further, it limited the range of strains over which the correction factor could be determined using a linear fit (Fig. 4(c)). This indicates that stresses near the loading clamps are increasing exponentially faster than within the ROI. This phenomenon is likely due to the fibers directly spanning between the clamps, where the greatest tissue strain occurs. This causes fiber stretch between the clamps to be much greater than fiber stretch within the ROI. Because the fiber strain energy increases exponentially with fiber stretch (Eq. (4)), the disparity between clamp stress and ROI stress is magnified in this orientation and becomes highly nonlinear (Fig. 4(c)).

4.5. Sensitivity to Initial Material Properties.

To test the dependence of the correction factor on the initial choice of material properties, FE simulations were repeated using a range of material properties. The correction factor for the ISO material was insensitive to the initial choice of modulus (Fig. 5(a)). Thus, the correction factor determined from biaxial simulations would be the same for any matrix modulus value within the entire range of observed experimental modulus values (0.007–7.58 MPa). The ISO material was somewhat dependent on the initial choice of Poisson ratio (Fig. 5(a)). However, the dependence on ν varied by less than 20% over most of the range of ν tested (0.1–0.4). This dependence may not be problematic, as Poisson's ratio can be determined in other loading modalities prior to loading the tissue in biaxial tension.

Fig. 5.

Sensitivity of correction factor to the initial choice of material properties. X-axis is the factor change in initial material property, and y-axis is the resulting factor change in correction factor (new correction factor ÷ initial correction factor). (a) Correction factor for ISO material is independent of modulus (square) and dependent on Poisson ratio (diamond). (b) Correction factor for XISO-X is independent of fiber modulus c₄ (square) and fiber nonlinearity c₅ (triangle). (c) Correction factor for XISO-Y is linearly dependent on both fiber modulus c₄ (square) and nonlinearity c₅ (triangle). (d) Correction factor for ORTH is independent of fiber modulus c₄ (square) and nonlinearity c₅ (triangle).

For anisotropic materials, the correction factor along the fiber direction (XISO-X, ORTH) did not change with variations in fiber modulus (c₄), except for the lowest value tested, c₄ = 0.004, which is outside the range of common experimental values for annulus fibrosus. Similarly, along the fiber direction, no change was observed for fiber nonlinearity (c₅) (Figs. 5(b) and 5(d)). The correction factor transverse to the fiber direction (XISO-Y, Fig. 5(c)) exhibited large relative dependence on changes in both c₄ and c₅. However, the load magnitude in this direction is much smaller than the fiber-aligned x axis, which amplifies percent changes. Further, the tissue properties of this loading direction are assumed to consist only of ISO matrix, which has a low contribution to the total strain energy of the tissue [22,44,46–48].

5. Discussion

This study reviewed the tensile biaxial literature over the last ten years, noting experimental and analysis challenges relating to stress-shielding and dependence on sample geometry and material anisotropy. In response to these challenges, this study used finite element simulations to quantify stress-shielding in biaxial tension and to formulate a correction factor that can be used to determine ROI stresses. Additionally, the impacts of sample geometry, material anisotropy, and tissue orientation on the correction factor were determined. Large stress concentrations were evident in both square and cruciform geometries and for all levels of anisotropy (Figs. 1(a), 1(b), and 3(b)), which is consistent with previous FE studies [37]. In general, stress concentrations were greater for the square geometry than the cruciform geometry. For both square and cruciform geometries, materials with fibers aligned parallel to the sample loading axes reduced stress concentrations compared to the isotropic response, resulting in more of the applied load being transferred to the ROI (Fig. 3(b)). In contrast, fiber-reinforced specimens rotated such that the fibers aligned at an angle to the loading axes produced very large stress concentrations across the clamps and shielding in the ROI (Fig. 4(b)).

Stress concentrations around the loading clamps redistribute the applied loads, giving rise to a stress-shielding phenomenon, wherein the ROI stresses are up to 50% lower than the applied clamp loads. This is problematic when determining material properties, because measurement techniques do not currently exist to directly measure the ROI stresses. Therefore, in current practice, the applied loads are divided by a sample cross-sectional area to calculate experimental stresses. This practice employs the assumption that loads applied at the clamps are fully transferred to the ROI, which is shown in this and previous studies [37–39] to be an inaccurate assumption (Fig. 1(c)). When applied stresses are plotted against strain measured directly in the ROI, an artificially stiffer material response is recorded. This phenomenon has previously been recognized [36–39]; however, a compensatory method to correct for these effects has not previously been proposed.

In this study, we used a FE approach to develop a correction factor that can be used to correct for the ROI stress-shielding induced by the biaxial boundary conditions (Table 3). The correction factor incorporates the relationship between the clamp stress, which is the stress that would be obtained experimentally, and ROI stress, which is the stress measurement desired during an experiment but which is not readily available. Thus, by applying the correction factor to the experimental clamp stresses, one can obtain a suitable estimation of the stresses in the ROI. The ROI stress would otherwise remain unknown. This allows for material properties to be calculated by fitting a constitutive equation to a stress-strain curve of ROI values.

The correction factor was dependent on sample geometry and was closer to unity for the cruciform geometry than for the square geometry (Table 3). This result, coupled with the reduced stress concentrations observed for the cruciform geometry discussed above, provides an indication that the cruciform geometry may be better suited for planar biaxial tension than the square geometry. However, for many fibrous biological tissues, the size requirements of the cruciform geometry are prohibitive.

The correction factor was also dependent on anisotropy and on fiber orientation relative to the loading direction (Table 3). When fibers are aligned parallel to a loading axis, more of the stress transfers to the ROI compared to the isotropic case. Therefore, the ROI stresses are closer to the applied clamp stresses and the correction factors are closer to unity. One of the worst cases occurred when fibers were oriented such that they spanned between adjacent loading clamps (Fig. 4(a)). Fibers orientated between the loading clamps experience large fiber stretch, much larger than the fiber stretch in the ROI. Because the fiber strain energy is an exponential function of the fiber stretch (Eq. (4)), the strain energy in these regions increases much more rapidly in these areas than in the ROI. This leads to increases in clamp stress that occur much faster than in the ROI and causes a breakdown of the linear relationship between the ROI and clamp stresses (Fig. 4(c)). For these cases, the correction factor technique is not recommended outside of the limit of small strains where the linear relationship exists. In the actual experiments, it is likely that yield or failure will occur at these locations, although this was not modeled in this study. These cases demonstrate the importance of the orientation of the biological tissue within the biaxial loading device and also highlight the utility of an FE technique to guide experimental design.

Sensitivity analysis showed that the correction factor was independent on the initial choice of modulus for the ISO material (Fig. 5(a)). Some dependency exists on Poisson's ratio; however, it was within 20% for most ν, and this material property can be determined experimentally using an alternate loading. For the XISO and ORTH materials, the correction factor was independent of c₅ and exhibited only slight dependence on c₄ (Figs. 5(b)–5(d)). Because the dependence on c₄ is linear, it is likely that an iterative scheme can be used to converge on a correction factor when c₄ is not known a priori. The sensitivity of the correction factor to other constitutive models, such as the Fung-type phenomenological model or structural models with different matrix and fiber formulations, was not determined. It is unlikely that the results presented within this study are directly transferrable to other constitutive models. The scope of this study was not to obtain specific correction factors for all soft tissues and all material models but to introduce a correction factor technique and demonstrate its utility using a structural constitutive model, which is commonly implemented for fibrous soft tissue. The general procedure for obtaining a unique correction factor may be repeated as needed for any constitutive model that best suits a particular tissue being tested in biaxial tension.

In summary, planar biaxial tension remains a critical loading modality for fibrous soft tissue and is widely used to characterize tissue mechanical response, evaluate treatments, develop constitutive formulas and obtain material properties for use in finite element studies. While many of the technical complications related to conducting the physical experiment have been overcome, there remains a continued need to address the interpretation of experimental results. Unlike uniaxial testing, the applied forces at the loading clamps do not transmit fully to the ROI, which may lead to improper material characterization if not accounted for. In this paper, FE simulations were used to quantify stress distributions throughout square and cruciform biaxial specimens with multiple levels of anisotropy. A correction factor technique was introduced that can be used to calculate the stresses in the ROI from the measured experimental loads at the clamps. Application of a correction factor technique to experimental biaxial results may lead to more accurate representation of the mechanical response of fibrous soft tissue.