One of the most celebrated contributions to the study of the mechanical
behavior of materials is due to J.D. Eshelby, who in the late 50s
revolutionized our understanding of the elastic stress and strain fields due to
an ellipsoidal inclusion/inhomogeneity that undergoes a transformation of shape
and size. While Eshelby's work laid the foundation for significant advancements
in various fields, including fracture mechanics, theory of phase transitions,
and homogenization methods, its extension into the range of large deformations,
and to situations in which the material can actively reorganize in response to
the finite transformation strain, is in a nascent state. Beyond the theoretical
difficulties imposed by highly nonlinear material response, a major hindrance
has been the absence of experimental observations that can elucidate the
intricacies that arise in this regime. To address this limitation, our
experimental observations reveal the key morphogenesis steps of Vibrio cholerae
biofilms embedded in hydrogels, as they grow by four orders of magnitude from
their initial size. Using the biofilm growth as a case study, our theoretical
model considers various growth scenarios and employs two different and
complimentary methods -- a minimal analytical model and finite element
computations -- to obtain approximate equilibrium solutions. A particular
emphasis is put on determining the natural growth path of an inclusion that
optimizes its shape in response to the confinement, and the onset of damage in
the matrix, which together explain the observed behavior of biofilms. Beyond
bacterial biofilms, this work sheds light on the role of mechanics in
determining the morphogenesis pathways of confined growing bodies and thus
applies to a broad range of phenomena that are ubiquitous in both natural and
engineered material systems.
