Question 3.1: A chordae tendineae specimen initially 10 mm long is to resi...

We recall that axial stress is computed via the applied force acting over the cross-sectional area. Hence, consider

As seen, such calculations are very easy. A key question to ask, however, is whether a chordae tendineae can sustain a 1.25-MPa stress without tearing, which is to say, How does this value compare to the range of stresses that would be expected to exist in vivo? Toward this end, see Exercise 3.3, which should be attempted only after completion of the next two sections on inflation problems. At this juncture, however, let us recognize another very important issue.
The value of stress of 1.25 MPa in this example was computed using the applied load and the original cross-sectional area A_{\omicron }. Such stresses are called by various names:   the      Piola–Kirchhoff   stress   (named after two nineteenth-century investigators), the nominal stress, or, sometimes, the engineering stress. The important observation though is that the derivation for \sigma _{xx}=f/A in Eq. (3.29) is actually based on A, the current cross-sectional area over which the force actually acts. This definition is often called a Cauchy stress, after the famous mathematician/mechanicist A. Cauchy, or the true stress because one uses the actual area over which the load acts. When the deformation (and thus strain) is small, A\sim A_{\omicron } and the two definitions yield similar values. When the deformation is large, however, as is the case for most soft tissues, the computed values can be very different. To compute the Cauchy stress in Example 3.1, therefore, we must measure A rather than A_{\omicron }. Clearly, the latter is easier experimentally; thus, the wide usage of the nominal stress Σ_{(face)(direction)} by experimentalists. We will in general prefer the Cauchy stress σ_{(face)(direction)} herein, however, which appears naturally in the equilibrium equations. Fortunately, we shall see in Chap. 6 that the various definitions of stress are related. Indeed, Exercise 3.4 shows how the Cauchy and nominal stress are related in the simple case of a 1-D state of stress and incompressibility.

                                        \sigma _{xx}\int{dA}=f\rightarrow \sigma _{xx}=\frac{f}{A},                             (3.29)