Influence of Specimen Geometry on the Estimation of the Planar Biaxial Mechanical Properties of Cruciform Specimens

Abstract

It is in general challenging to characterize planar mechanical properties of extremely soft tissues such as cell-seeded collagen gels. One of the difficulties is related to premature failure of specimens. This issue may be resolved by employing fillets on stress-concentrated spots of the specimen. The existence of fillets, however, complicates the estimation of stress at the center of the specimen where stiffness data are collected. In this study, cruciform rubber specimens with two types of fillets (general vs. cut-in fillets) at the intersections of perpendicular arms were prepared and subjected to planar biaxial mechanical testing, aiming at investigating how the fillets affect the estimation of mechanical properties of cruciform specimens. Digital image correlation was used to analyze full-field deformation in the central region of the specimens. Finite element analysis with a Neo-Hookean model was performed to simulate the full-field deformation under the same experimental boundary conditions. The strain distribution for each specimen geometry obtained by finite element analysis was found to be in good agreement with that analyzed by digital image correlation, validating the finite element models. Finite element simulation showed that the greatest value of the maximum principal strain decreased with increasing the fillet radius regardless of the fillet type. Increasing the fillet radius, in general, also reduced the strain field uniformity in the central region. Compared with general fillets, however, the use of cut-in fillets provided greater strain field uniformity given the same fillet radius. Finite element analysis was also used to estimate effective transverse length required to convert tensile force at the boundary to local stress at the center. It was found that the effective transverse length for each specimen geometry remained relatively constant if the specimen was not excessively deformed (i.e., global equibiaxial stretch ≤ 1.2). We suggest using cut-in fillets at the intersections of perpendicular arms when preparing cruciform specimens for testing extremely soft tissues.

Appendix A

A linear interpolation function for a triangular element was used, which is defined as ${N}_i=\frac{a_i+{b}_i{X}_1+{c}_i{X}_2}{2\cdot Area};i=1,2,3,$

where $Area=\frac{1}{2} \det \left|\begin{array}{ccc}\hfill 1\hfill & \hfill {X}_1^1\hfill & \hfill {X}_2^1\hfill \\ {}\hfill 1\hfill & \hfill {X}_1^2\hfill & \hfill {X}_2^2\hfill \\ {}\hfill 1\hfill & \hfill {X}_1^3\hfill & \hfill {X}_2^3\hfill \end{array}\right|=\frac{1}{2}\left({X}_1^2{X}_2^3+{X}_1^3{X}_2^1+{X}_1^1{X}_2^2-{X}_1^2{X}_2^1-{X}_1^3{X}_2^2-{X}_1^1{X}_2^3\right),{a}_1={X}_1^2{X}_2^3-{X}_1^3{X}_2^2,{a}_2={X}_1^3{X}_2^1-{X}_1^1{X}_2^3,{a}_3={X}_1^1{X}_2^2-{X}_1^2{X}_2^1,{b}_1={X}_2^2-{X}_2^3,{b}_2={X}_2^3-{X}_2^1,{b}_3={X}_2^1-{X}_2^2,{c}_1={X}_1^3-{X}_1^2,{c}_2={X}_1^1-{X}_1^3,{c}_3={X}_1^2-{X}_1^1$and, X ₁ and X ₂ are reference coordinates; the subscript indicates coordinate in the 1 or 2 direction and the superscript indicates the three vertices of the triangular element. With the interpolation function, the position vector $\mathbf{x}=\left(\begin{array}{ll}x{}_1\hfill & x{}_2\hfill \end{array}\right)$ of a particle within the triangular element in a deformed configuration was approximated in terms of the position vectors of the three vertices of the triangular element in the deformed configuration ${\mathbf{x}}^j=\left(\begin{array}{cc}\hfill {x}_1^j\hfill & \hfill {x}_2^j\hfill \end{array}\right)$; j = 1, 2, 3, as follows.

$$ \begin{array}{l}{x}_1\left({X}_1,{X}_2\right)\cong {N}_1{x}_1^1+{N}_2{x}_1^2+{N}_3{x}_1^3\hfill \\ {}{x}_2\left({X}_1,{X}_2\right)\cong {N}_1{x}_2^1+{N}_2{x}_2^2+{N}_3{x}_2^3\hfill \end{array} $$

Because the interpolation functioin is a function of reference coordinates, the position vector is also a function of reference coordinates. Then the components of 2-dimentional deformation gradient ${\mathbf{F}}_{2\mathrm{D}}\left(\mathbf{F}=\frac{d\mathbf{x}}{d\mathbf{X}}\right)$ were calculated using

$$ \begin{array}{l}{F}_{11}=\frac{d{x}_1}{d{X}_1}=\frac{b_1}{2\cdot Area}{x}_1^1+\frac{b_2}{2\cdot Area}{x}_1^2+\frac{b_3}{2\cdot Area}{x}_1^3,\hfill \\ {}{F}_{12}=\frac{d{x}_1}{d{X}_2}=\frac{c_1}{2\cdot Area}{x}_1^1+\frac{c_2}{2\cdot Area}{x}_1^2+\frac{c_3}{2\cdot Area}{x}_1^3,\hfill \\ {}{F}_{21}=\frac{d{x}_2}{d{X}_1}=\frac{b_1}{2\cdot Area}{x}_2^1+\frac{b_2}{2\cdot Area}{x}_2^2+\frac{b_3}{2\cdot Area}{x}_2^3,\hfill \\ {}{F}_{22}=\frac{d{x}_2}{d{X}_2}=\frac{c_1}{2\cdot Area}{x}_2^1+\frac{c_2}{2\cdot Area}{x}_2^2+\frac{c_3}{2\cdot Area}{x}_2^3.\hfill \end{array} $$

The deformation gradient is found to be constant within the triangular element; that is, it does not vary with positions within the element. Surprisingly, we found that the deformation gradient calculated based on this algorithm is identical to the one calculated based on the idea of linear transformation with an assumption of homogeneous strain field [28]. In fact, it can be shown that $\left[\begin{array}{ccc}\hfill {x}_1^1-{x}_1^3\hfill & \hfill {x}_1^2-{x}_1^1\hfill & \hfill {x}_1^3-{x}_1^2\hfill \\ {}\hfill {x}_2^1-{x}_2^3\hfill & \hfill {x}_2^2-{x}_2^1\hfill & \hfill {x}_2^3-{x}_2^2\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill {F}_{11}\hfill & \hfill {F}_{12}\hfill \\ {}\hfill {F}_{21}\hfill & \hfill {F}_{22}\hfill \end{array}\right]\left[\begin{array}{ccc}\hfill {c}_2\hfill & \hfill {c}_3\hfill & \hfill {c}_1\hfill \\ {}\hfill -{b}_2\hfill & \hfill -{b}_3\hfill & \hfill -{b}_1\hfill \end{array}\right]$. Nevertheless, although the same results were achieved, the interpolation approach that we used in this study significantly reduced the computing time (1/10 of the linear transformation approach).