An Introduction to Biomechanics by Jay D. Humphrey & Sherry L. O’Rourke

Author:Jay D. Humphrey & Sherry L. O’Rourke
Language: eng
Format: epub, pdf
Publisher: Springer New York, New York, NY

6.5.2 Theoretical Framework

Like many other soft tissues, normal arteries often exhibit a nonlinear, pseudoelastic, heterogeneous, and anisotropic behavior over large physiologic strains. Moreover, they tend to behave incompressibly in many cases. It can be shown that in the case of incompressibility, the general constitutive equation embodied in Eq. (6.23) must be modified. For example, for incompressible behavior, we have

(6.76)

where p is a scalar, pressure-like quantity (actually a Lagrange multiplier) that enforces the incompressibility constraint. That the –p[I] contribution is needed is seen easily by noting that the second term on the right-hand side of Eq. (6.76) represents the stress due to deformation, which is zero in the absence of a deformation.

Imagine then a cube of incompressible material subjected to a hydrostatic pressure P (Fig. 6.18). Clearly, σ 11 = −P, σ 22 = −P, and σ 33 = −P, with all shear stresses zero with respect to (x, y, z) ≡ (1, 2, 3), even though there is no deformation because of incompressibility. Hence, in this case, and this case alone, the Lagrange multiplier p equals the hydrostatic pressure P and Eq. (6.76) correctly describes the state of stress in the absence of a deformation. For a formal derivation of Eq. (6.76), see Humphrey (2002).

Figure 6.18A small cube of incompressible material subjected to a hydrostatic pressure experiences stress but not strain. Indeed, the components of stress (equal and opposite the pressure) are the same regardless of the coordinate system, (cf. Exercise 2.​9).

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