The experimental evidence that a feedback exists between growth and stress in tumors poses challenging questions. First, the rheological properties (the "constitutive equations") of aggregates of malignant cells are still a matter of debate. Secondly, the feedback law (the "growth law") that relates stress and mitotic-apoptotic rate is far to be identified. We address these questions on the basis of a theoretical analysis of and experiments that involve the growth of tumor spheroids. We show that solid tumors exhibit several mechanical features of a poroelastic material, where the cellular component behaves like an elastic solid. When the solid component of the spheroid is loaded at the boundary, the cellular aggregate grows up to an asymptotic volume that depends on the exerted compression. Residual stress shows up when solid tumors are radially cut, highlighting a peculiar tensional pattern. By a novel numerical approach we correlate the measured opening angle and the underlying residual stress in a sphere. The features of the mechanobiological system can be explained in terms of a feedback of mechanics on the cell proliferation rate as modulated by the availability of nutrient, that is radially damped by the balance between diffusion and consumption. The volumetric growth profiles and the pattern of residual stress can be theoretically reproduced assuming a dependence of the target stress on the concentration of nutrient which is specific of the malignant tissue.

Understanding the mechanics of brain embryogenesis can provide insights on pathologies related to brain development, such as lissencephaly, a genetic disease which cause a reduction of the number of cerebral sulci. Recent experiments on brain organoids have confirmed that gyrification, i.e. the formation of the folded structures of the brain, is triggered by the inhomogeneous growth of the peripheral region. However, the rheology of these cellular aggregates and the mechanics of lissencephaly are still matter of debate. In this work, we develop a mathematical model of brain organoids based on the theory of morpho-elasticity. We describe them as non-linear elastic bodies, composed of a disk surrounded by a growing layer called cortex. The external boundary is subjected to a tissue surface tension due the intercellular adhesion forces. We show that the resulting surface energy is relevant at the small length scales of brain organoids and significantly affects the mechanics of cellular aggregates. We perform a linear stability analysis of the radially symmetric configuration and we study the post-buckling behaviour through finite element simulations. We find that the process of gyrification is triggered by the cortex growth and modulated by the competition between two length scales: the radius of the organoid and the capillary length due to surface tension. We show that a solid model can reproduce the results of the in-vitro experiments. Furthermore, we prove that the lack of brain sulci in lissencephaly is caused by a reduction of the cell stiffness: the softening of the organoid strengthens the role of surface tension, delaying or even inhibiting the onset of a mechanical instability at the free boundary. * davide.riccobelli@sissa.it † giulia.bevilacqua@polimi.it arXiv:1905.03659v1 [cond-mat.soft]

Active materials are media for which deformations can occur in absence of loads, given an external stimulus. Two approaches to the modeling of such materials are mainly used in literature, both based on the introduction of a new tensor: an additive stress Pact in the active stress case and a multiplicative strain Fa in the active strain one. Aim of this paper is the comparison between the two approaches on simple shears.Considering an incompressible and transversely isotropic material, we design constitutive relations for Pact and Fa so that they produce the same results for a uniaxial deformation along the symmetry axis. We then study the two approaches in the case of a simple shear deformation. In a hyperelastic setting, we show that the two approaches produce different stress components along a simple shear, unless some necessary conditions on the strain energy density are fulfilled. However, such conditions are very restrictive and rule out the usual elastic strain energy functionals. Active stress and active strain therefore produce different results in shear, even if they both fit uniaxial data.Our results show that experimental data on the stress-stretch response on uniaxial deformations are not enough to establish which activation approach can capture better the mechanics of active materials. We conclude that other types of deformations, beyond the uniaxial one, should be taken into consideration in the modeling of such materials. Date

In this work, we study the mechanics of metamaterial sheets inspired by the pellicle of Euglenids. They are composed of interlocking elastic rods which can freely slide along their edges. We characterize the kinematics and the mechanics of these structures using the special Cosserat theory of rods and by assuming axisymmetric deformations of the tubular assembly. Through an asymptotic expansion, we investigate both structures that comprise a discrete number of rods and the limit case of a sheet composed by infinite rods. We apply our theoretical framework to investigate the stability of these structures in the presence of an axial load. Through a linear analysis, we compute the critical buckling force for both the discrete and the continuous case. For the latter, we also perform a numerical post-buckling analysis, studying the non-linear evolution of the bifurcation through finite elements simulations.(1) SISSA -International School for Advanced Studies, Trieste.