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.
Highlights
Additive manufacturing technologies, which are often referred to as 3D printing, comprise a promising avenue in modern industrial world [1]
One can find the recent results concerning the problem of residual stresses prediction in [6], the challenge of matching the initial shape compensation, aimed at lowering of their intensity in [7], and optimization problems with account of various parameters of additive processes in [8,9,10]
The above discussion makes it clear that the global stress-free shape of the body, produced with some additive process, can be found on a smooth manifold endowed with non-Euclidean affine connection, which turns it into non-Euclidean space, for instance, into Riemannian or Weitzenböck spaces
Summary
Additive manufacturing technologies, which are often referred to as 3D printing, comprise a promising avenue in modern industrial world [1]. In order to obtain a suitable mathematical formalization for finite incompatible deformations, we use the methods of geometrical mechanics of continuum [25,26,27,28,33,38] To this end, together with body manifold and physical space we define one more smooth manifold, i.e., non-Euclidean reference space R, which can contain global stress-free shapes SR due to additional geometrical degrees of freedom, namely, curvature, torsion and nonmetricity. One can imagine the disassembling process, in which the whole body disintegrates into either finite number of finite parts, namely, layers, or into continual family of surfaces Upon such disassembling, each layer or surface can be deformed into a stress-free shape in physical space. Translation vectors, i.e., elements of V, are denoted by Latin boldface lowercase symbols u, v, w, . . ., and linear mappings from V to V (second order tensors) are denoted by Latin boldface uppercase symbols P, R, S, . . . The symbol I is reserved for the identity tensor
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