# Uniaxial Tension

Mechanical testing is both an art and a science. The science of mechanical testing is grounded in the study of the mechanics of materials which is a mathematically rigorous and highly technical field. Unfortunately, few practitioners of practical mechanical testing are well versed in this science, which results in significant misunderstandings and misuse of testing data. Mechanical testing is also an art. Some researchers who work in the field of mechanics, are unaware of the significant technical challenges in implementing the experiments that their mathematics call for. We seek to use accessible experimental setups to obtain relevant mechanical data that has real physical (and mathematical) meaning and can be used to directly compare different biological and synthetic materials.

## Uniaxial Tension

Assumptions:

1. Incompressible: This assumption is satisfied when the bulk modulus is much higher (>100x) the shear modulus. This has been shown to be true of all soft biological tissues and will not be proven in our experiments.
2. Isotropic: This is false for most biological materials. This means that we must be aware of this limitation, and take note of the loading direction.
 Stretch Ratio $\lambda=\dfrac{l(t)}{l_0}$ Pseudo Traction $\vec{T} =\dfrac{|\vec{F}|}{A_0} \hat{n}$ Deformation Gradient $F=\dfrac{d\vec{x} }{d\vec{X} }$ Green Tensor $C=F^T F$ Finger Tensor $b=FF^T$ 1st Piola Kirchhoff Stress Tensor $P\hat{n} =\vec{T}$ 2nd Piola Kirchhoff Stress Tensor $S=F^{-1} P$ Kirchhoff Stress $s=PF^T$ Green Strain $E=1/2 (C-I)$ Almansi Strain $e=1/2 (I-b^{-1})$ Lodge Strain $\varepsilon =1/2 (I-C^{-1})$ Signorini Strain $\epsilon =1/2 (b-I)$
 Using the figures above, we can produce: $\lambda=\dfrac{l(t)}{l_0 }$, $\delta=\dfrac{d(t)}{d_0 }=\dfrac{w(t)}{w_0 }$ Where, for an incompressible, isotropic material: $det(F) = 1 = \delta\times\delta\times\lambda$, therefore $\delta = \lambda^{-.5}$ $F= \left[ \begin{array}{ccc} \lambda & 0 & 0 \\ 0 & \lambda^{-.5} & 0 \\ 0 & 0 & \lambda^{-.5} \end{array} \right]$ $C = b = \left[ \begin{array}{ccc} \lambda^{2} & 0 & 0 \\ 0 & \lambda^{-1} & 0 \\ 0 & 0 & \lambda^{-1} \end{array} \right]$ $P= \left[ \begin{array}{ccc} T & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right]$ $E = \epsilon = \dfrac{1}{2} \left[ \begin{array}{ccc} \lambda^{2} - 1 & 0 & 0 \\ 0 & \lambda^{-1} - 1 & 0 \\ 0 & 0 & \lambda^{-1} - 1 \end{array} \right]$ $S= \left[ \begin{array}{ccc} T\lambda^{-1} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right]$ $\varepsilon = e = \dfrac{1}{2} \left[ \begin{array}{ccc} 1 - \lambda^{-2} & 0 & 0 \\ 0 & 1 - \lambda & 0 \\ 0 & 0 & 1 - \lambda \end{array} \right]$ $s= \left[ \begin{array}{ccc} T\lambda & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right]$

## Testing Results

 The response can appear to be very different when using different stress and strain measures. For this reason, stress and strain should not be reported as raw data. Instead, we report tractions and stretches which can be used to calculate any strain measure or stress measure. Using the traction and stretch we can calculate a stiffness that has a direct physical meaning. This measure, however, does not necessarily apply in a straightforward manner to loading scenarios other than uniaxial tension.