A discrete model for layered growth

We present a discrete model that captures the fundamental properties of layered deposition in a minimal setting. The model is based on simplified kinematics and allows for the onset of incompatible deformations between discrete layers of a growing stack. Thanks to the discrete nature of the model, we obtain an averaged formulation of mechanical equilibrium for the growing stack, leading to closed -form solutions that are both analytically simple and physically transparent. In particular, we are able to explain the origin of residual stress by the accumulation of incompatible deformations between adjacent layers. The model can also be regarded as an elementary approximation of additive manufacturing. Within this context, we are able to formulate the technologically relevant inverse problem that provides the deposition protocol required to produce a desired state of internal stress in an additively-manufactured stack. Another important aspect analyzed in the work is the role played by an ideal "glue"between the layers, whose presence is fundamental to prevent their sliding and whose mechanical behavior can quantitatively influence the final stress distribution in the stack. The simplicity of the model makes it possible to highlight how the controls exerted during deposition will have qualitative or quantitative effects on the final stress state of the stack.