Incompatible Deformations in Additively Fabricated Solids: Discrete and Continuous Approaches

Abstract

The present paper is intended to show the close interrelationship between non-linear models of solids, produced with additive manufacturing, and models of solids with distributed defects. The common feature of these models is the incompatibility of local deformations. Meanwhile, in contrast with the conventional statement of the problems for solids with defects, the distribution for incompatible local deformations in additively created deformable body is not known a priori, and can be found from the solution of the specific evolutionary problem. The statement of the problem is related to the mechanical and physical peculiarities of the additive process. The specific character of incompatible deformations, evolved in additive manufactured solids, could be completely characterized within a differential-geometric approach by specific affine connection. This approach results in a global definition of the unstressed reference shape in non-Euclidean space. The paper is focused on such a formalism. One more common factor is the dataset which yields a full description of the response of a hyperelastic solid with distributed defects and a similar dataset for the additively manufactured one. In both cases, one can define a triple: elastic potential, gauged at stress-free state, and reference shape, and some specific field of incompatible relaxing distortion, related to the given stressed shape. Optionally, the last element of the triple may be replaced by some geometrical characteristics of the non-Euclidean reference shape, such as torsion, curvature, or, equivalently, as the density of defects. All the mentioned conformities are illustrated in the paper with a non-linear problem for a hyperelastic hollow ball.

Deformations of Single-Crystal Silicon Circular Plate: Theory and Experiment

“…Constants of integration, defined by the expression in parentheses (20), are chosen in such a way that the boundary conditions (18) are satisfied. They are given by the following expressions…”

“…To implement the above task in all the described methods, it is important to know the laws of evolution of mechanical stresses in the membrane structure depending on the conditions of its formation-technological regimes (operation temperature, growth rate, elemental composition, etc. ), the history of the fabrication and mechanical properties of the underlying layers, as well as the physical parameters of the crystalline substrate [17][18][19]. In particular, one of these parameters that influences the residual mechanical stresses and the resulting deformation of the membrane under the action of an applied external (mechanical or electromagnetic) force is the crystallographic orientation of the substrate or its crystalline anisotropy.…”

Finite extension of accreting nonlinear elastic solid circular cylinders

“…Metlov [1985] proposed the first finite deformation theory of accretion and introduced a time of attachment map. There are many more works in the literature of the mechanics of accretion [Drozdov, 1998a,b, Arutyunyan et al, 1990, Manzhirov, 1995, Ong and O'Reilly, 2004, Kadish et al, 2005, Epstein, 2010, Lychev and Manzhirov, 2013, Manzhirov, 2014, Lychev, 2017, Lychev et al, 2021, Tomassetti et al, 2016, Sozio and Yavari, 2017, Sozio et al, 2020, Abi-Akl et al, 2019, Abi-Akl and Cohen, 2020, Truskinovsky and Zurlo, 2019, Bergel and Papadopoulos, 2021. For a detailed literature review of the mechanics of accretion see [Naumov, 1994, Sozio andYavari, 2017].…”

Incompatible Deformations in Hyperelastic Plates

“…These factors turn out to be especially important in micronscale elastic bodies, particularly in the elements of MEMS structures. The key concepts of incompatible deformation theory are described in [45,46]. The basic idea is that the uniform state can be achieved only locally by local deformation K, which in general does not meet any compatibility conditions, i.e., curl K ̸ = 0.…”

“…However, a discussion of these issues is beyond the scope of this article. We only mention that evolutionary problems are described in more detail in [36,45,53].…”