On residual stresses and homeostasis: an elastic theory of functional adaptation in living matter

Abstract

Living matter can functionally adapt to external physical factors by developing internal tensions, easily revealed by cutting experiments. Nonetheless, residual stresses intrinsically have a complex spatial distribution, and destructive techniques cannot be used to identify a natural stress-free configuration. This work proposes a novel elastic theory of pre-stressed materials. Imposing physical compatibility and symmetry arguments, we define a new class of free energies explicitly depending on the internal stresses. This theory is finally applied to the study of arterial remodelling, proving its potential for the non-destructive determination of the residual tensions within biological materials.

Figure 1

Release of residual stresses after cutting experiments in different living materials: solid tumour ((a) adapted from, freely available online through the PNAS open access option); small intestine ((b) adapted from, distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0)); artery ((c) adapted from, this image is not covered by the CC by license); Eucalyptus log ((d) adapted from37). All images are reprinted with permission of the copyright holders.

Figure 2. Kinematics of the finite deformation of a residually stressed elastic body subjected to a generic external traction load f in the spatial configuration.

Figure 3. Functional adaptation in arteries: change of the unloaded geometry with the internal pressure P (left); resulting distribution of residual (right, top) and optimal Cauchy stresses (right, bottom).

We set r_o = 2.34 mm, r_i = 1.87 mm, P/μ = 2.38 from the experimental data in, giving R_o = 1.92 mm and R_i = 1.3 mm.

Figure 4. Comparison of the results of the proposed model (solid lines) and the opening angle method (dashed lines): distribution of the Cauchy hoop stress (left); and maximum stress difference as a function of the aspect ratio r_i(r₀ − r_i) (right).

As in Fig. 3, we set r_o = 2.34 mm, r_i = 1.87 mm, whilst we show the corresponding curves for P/μ = 0.5, 1, and 2.38.

Figure 5. Left: Dispersion curves showing the pre-stress parameter α/μ for which a cylindrical wave with a given frequency ω can propagate as a function of the aspect ratio R₀/R_i of the tube.

We depict curves for different values of the angular wavenumber m, setting . Right: Resulting wave shape for R_o/R_i = 2 and m = 10. The amplitude is arbitrarily set as 0.15 R_o for the sake of graphical clarity.