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

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.


Supplemental Material 1 Derivation of a constitutive equation for elastic bodies with residual stresses
Under the assumptions of a macroscopic length-scale at the tissue scale and a characteristic time-scale short enough to neglect dissipative effects on the residual stresses (e.g. ageing phenomena), it is possible to define an elastic free energy Ψ = Ψ(F, τ ). Since we neglect any material anisotropy within the material (e.g. fiber reinforcement), we postulate that Ψ is a scalar-valued function depending on ten invariants: the six principal invariants of C and τ , and the four combined invariants of C and τ . From standard thermo-mechanical arguments, the constitutive equation of the Cauchy stress can be expressed as: where the chain differentiation rule has been used. Recalling the following tensor rules: it is possible to derive the Equation ( 2 Derivation of the initial stress symmetry conditions The initial stress symmetry (ISS) enforces the necessity to recover the residually stressed configuration from the loaded state by reversing the deformation mapping: where p τ is the Lagrange multiplier due to the incompressibility constraint. Hereafter we show that the ISS conditions in Eq. (S4) can be expressed as a set of scalar equations, relating the free energy density Ψ(F , τ ) and the invariants of τ and C. Let us first use Eq. (S4) to write the residual stress as where Ψ σ I k := Ψ ,I k (F −1 , σ), Ψ σ ,Jm := Ψ ,Jm (F −1 , σ), I 1 = tr(C), I 2 = tr(C −1 ) using the Cayley-Hamilton theorem with det C = 1, and comma denotes partial derivative with respect to the following term. The above must hold for every τ and C. Thus, we can substitute the constitutive equation for the Cauchy stress into Eq. (S5), obtaining : where we assume summation for i, k, n over {−2, −1, 0, 1, 2} and j, m over {0, 1, 2}. Dealing with symmetric tensors, we have the symmetries α ijkmn = α njkmi = α nmkji , where the values of α ijkmn are not reported here for the sake of brevity.
Finally, it is possible to further simplify Eq. (S6) to obtain nine scalar equations that hold for any choice of Ψ and of all the invariants. This can be done by varying C and τ while keeping the invariants fixed. Since Eq.(S6) must hold for any C and τ , we obtain nine scalar equations which compactly rewrite where n sums over {0, 1, 2, 3, 4}, Ψ J 0 := 1, and: The other matrices in Eq.(S7) have cumbersome expressions and will be not reported here for the sake of brevity. In summary, Eqs (S7) and (S8) represent respectively 6 and 3 scalar equations, respectively, determining the ISS conditions, the unknowns being the scalar functions Ψ, p and p τ .
For the sake of simplicity, we now look for simplified expression neglecting the functional dependence on J 3 and J 4 , which represent higher order terms in the mixed functional dependence. In this case, the fourth scalar equation of Eq. (S7) rewrites: which is only possible, for every F and τ , if and only if Ψ does not depend on J 2 or on J 1 . In particular, if Ψ does not depend also on J 2 , then Eq. (S7) reduces to: where we have used Eq. (S12) 1 to derive the other three equations. Let us now impose the physical compatibility of the strain energy, by letting F = I, σ = τ and p τ = p. We obtain that Ψ ,J 1 = 1/2, 2Ψ ,I 1 = p and Ψ ,I 2 = 0 for F = I. Accordingly, we derive the class of free energies for residually stressed material expressed by Eq. (5)  [σ 2 rr,r + (rσ rr ) 2 ,rr ]dr = 0 (S13) under the constraint ro r i σ rr,r dr = P given by the boundary conditions. Before solving the corresponding Euler-Lagrange equations, let us introduce the following dimensionless variables: so that and where we have introduced α = r i /(r o − r i ). The functional L in Eq.(S13) therefore rewrites : and Λ is a Lagrange multiplier that appears due to the constraint (S18). The solution of (S19) is given by Finally, we determine Λ from the constraint (S18) to be Finally, we compare our optimal solution for the Cauchy stress against the one obtained using a opened ring as virtual state, also known as the opening angle method [3], as depicted in Figure 1. We highlight that σ θθ for the opening angle method tends to the optimal stress σ θθ only if P/µ tends to zero, while the plots for σ rr do not show significant differences between the two methods.

Wave propagation in a residually stressed tube
Let us deal with the infinitesimal wave propagation of an undeformed tubular tissue with residual stresses. We consider an inhomogeneous infinitesimal wave u of the form: where E R and E Θ are the radial and tangential unit vectors, so that u, v represent the incremental radial and hoop displacement fields, respectively. Indicating with Γ = Grad u the spatial displacement gradient associated with the incremental deformation, the incremental incompressibility condition reads: Following the incremental elastic theory [4], the incremental equations of motion read: with boundary conditions e R s = 0 at the inner and outer radii, R i and R o , respectively.
The components of the incremental nominal stress s for the constitutive theory in Eq. (5) of the article read: where q τ is the incremental pressure, and λ τ is the real root of λ 2 + λI τ 1 + I τ 3 − µ 2 = 0, which is the equivalent of Eq.(6) of the article in plane strain conditions.
Let as now make an educated guess of the solution in the form of a time-harmonic cylin-drical wave, having displacement and stress components defined as: where m is the integer angular wavenumber, ω is the angular frequency, and the amplitudes U, V, S RR , S RΘ , Q are scalar functions of R only. Following [5], the incompressibiity condition Eq.(S24) and the equation of motion Eq.(S25) can be recast in a system of four ordinary differential equation of the first order: also known as the Stroh formulation of the incremental problem. The sub-blocks of the Stroh matrix in Eq.(S28) have the following components Let us now introduce a functional relation between the incremental traction and the displacements vectors as R S(R) = Z(R) U(R), where Z is a surface impedance matrix [6]. Substituting the previous expression into Eq.(S28), we derive the following differential Riccati equation for Z, Let us now clarify how Eq.(S30) can be used to establish a non-destructive method for measuring the residual stress distribution within a pre-stressed tube. An illustrative example is sketched in the following. Imposing the equilibrium equations in the undeformed configuration, a simple expression of residual stress distribution is given by: R o and R i are prescribed. Thus, for a given tube, we adjust the pre-stress parameter α/µ, proportional to the amplitude of the residual stress, until we meet the target condition for a given m. Once α/µ is determined, we integrate the first line of Eq.(S28), i.e.
simultaneously with Eq.(S30) to compute the incremental wave field throughout the thickness of the tube wall.