Scaling Laws for the Response of Nonlinear Elastic Media with Implications for Cell Mechanics

We show how strain stiffening affects the elastic response to internal forces, caused either by material defects and inhomogeneities or by active forces that molecular motors generate in living cells. For a spherical force dipole in a material with a strongly nonlinear strain energy density, strains change sign with distance, indicating that, even around a contractile inclusion or molecular motor, there is radial compression; it is only at a long distance that one recovers the linear response in which the medium is radially stretched. Scaling laws with irrational exponents relate the far-field renormalized strain to the near-field strain applied by the inclusion or active force.

Figure 1

(a) Spherical force dipole generates radial displacement $u_0$ at radius $R_0$. (b),(c) Constitutive relations for simple shear: Eq. (1) with $n=-1$, $b=100$, and $\mu =1$. (b) Shear stress vs strain diverges at ${\gamma }_{\max }$. (c) Differential modulus vs shear stress asymptotes to $G\propto {\sigma }^{\beta }$.

Figure 2

Normalized effective far-field displacement vs dimensionless parameter $A\equiv b{u}_0^2/R_0^2$ quantifying the nonlinearity in the system. Colored lines indicate numerical solutions of Eq. (2) for different Poisson ratios, and $n=-1$. Dotted lines are theoretical solutions: Eq. (5) at small $A$ and Eq. (9) at large $A$.

Figure 3

Normalized displacement vs normalized radius for $n=-1$, Poisson ratio $\nu =0.4$, and various values of the dimensionless parameter $A\equiv b{u}_0^2/R_0^2$ characterizing the nonlinearity in the system. Colored solid lines are results of numerically solving the equation of mechanical equilibrium (2). Stars denote the locations of the maxima. The dotted blue line is Eq. (6), the dashed magenta line is Eq. (7), and the solid dark line is a higher-order correction [20], for $A=30$.