Folding is a fundamental process shaping epithelial sheets into 3D architectures of organs. Initial positioning of folds is the foundation for the emergence of correct tissue morphology. Mechanisms forming individual folds have been studied, yet the precise positioning of the folds in complex, multifolded epithelia is an open question. We present a model of morphogenesis, encompassing local differential growth, and tissue mechanics to investigate tissue fold positioning. We use Drosophila melanogaster wing imaginal disc as our model system, and show that there is spatial and temporal heterogeneity in its planar growth rates. This planar differential growth is the main driver for positioning the folds. Increased stiffness of the apical layer and confinement by the basement membrane drive fold formation. These influence fold positions to a lesser degree. The model successfully predicts the emergent morphology of wingless spade mutant in vivo, via perturbations solely on planar differential growth rates in silico.

still recurring subjects of debate (Daniel, 2007). Cohesive elements have become very popular as discontinuities may efficiently be simulated, e.g. splitting due to delamination (Camanho and Mathews, 1999;Camanho et al., 2003;Tvergaard, 2004;Iannucci and Willows, 2006). An up-todate review can be found in Wisnom (2010). The proposed technique admits linkage to cohesive elements, although this is not attempted at this time.The proposed technique is framed on the thermodynamics of irreversible process following Chaboche (1992). Other interesting thermodynamical approaches for damage mechanics of composites are also found in literature, e.g. Simo and Ju (1987), Turon et al. (2006), amongst others. From a purely damage mechanics point of view, the main features of the proposed technique are the computation of paths of damages which provide an effective localisation of the different damage modes and, second, the computation of damage on the strain space through the generation of a mapping between the strain and stress spaces where the so-called normalised energy release rates are readily computed. Thus, the undamaged domain is defined on the strain space bounded for a set of quadratic damage surfaces. The approach is based on computing the damage at quadrature points, i.e. gauss points, within each finite element, forming the mesh of the composite which is performed in the strain space. Moreover, fracture energy -modes I and II -can be added to the model and mapped onto to the strain space following an analogous strategy. This last point permits to treat all the variables associated to failure on the composite in a progressive manner which provides higher stability and convergence in the explicit finite element method (XPFEM) procedure, as the removal of a finite element often causes oscillations when using XPFEM, creating instability, and, eventually, divergence (Camanho et al., 2001).This article is outlined as follows. First, the thermodynamical background, in which the technique is framed, is briefly presented. Second, definition of the measurements of stress utilised are provided. Third, damage components of the technique are described. Fourth, the integration within XPFEM is presented in some detail. Finally, numerical results and discussion are provided. Additionally, an appendix containing relevant tensors is included.

During development, organs must form with precise shapes and sizes. Organ morphology is not always obtained through growth; a classic counterexample is condensation of the nervous system during Drosophila embryogenesis. The mechanics underlying such condensation remain poorly understood. Here, we combine in toto live-imaging, biophysical and genetic perturbations, and atomic force microscopy to characterize the condensation of the Drosophila ventral nerve cord (VNC) during embryonic development at both subcellular and tissue scales. This analysis reveals that condensation is not a unidirectional continuous process, but instead occurs through oscillatory contractions alternating from anterior and posterior ends. The VNC mechanical properties spatially and temporally vary during its condensation, and forces along its longitudinal axis are spatially heterogeneous, with larger ones exerted between neuromeres. We demonstrate that the process of VNC condensation is dependent on the coordinated mechanical activities of neurons and glia. Finally, we show that these outcomes are consistent with a viscoelastic model of condensation, which incorporates time delays due to the different time scales on which the mechanical processes act, and effective frictional interactions. In summary, we have defined the complex and progressive mechanics driving VNC condensation, providing insights into how a highly viscous tissue can autonomously change shape and size.