Functionally graded prismatic triangular rod under torsion

The paper considers the problem of torsion of a growing viscoelastic prismatic rod with integral boundary conditions at the ends. The process of continuous growth under the influence of torque is studied. The distributions of the intensity of tangential stresses at various stages of the building process are investigated. The calculations of the torsion problem of a prismatic rod with a section in the form of a regular triangle are presented.


Introduction
The problem of computation and simulation of residual stresses occurring in objects manufactured by additive technologies is actual one for Mechanics of Solids. Such objects are usually demonstrate the properties inherent for the functionally graded materials [1,2,3].
Additive manufacturing technologies are a particular case of growth processes. Mathematical modeling of additive manufacturing technologies is aimed at improving the performance of device, machine, and mechanism parts. The fundamentally new mathematical models considered in the paper describe the evolution of the end product stress-strain state in additive manufacturing and are of general interest for modern technologies in engineering, medicine, electronics industry, aerospace industry, and other fields (see, e.g., [4,5,6]).
A solution of applied problem of growing solids mechanics is a sophisticated and timeconsuming procedure [7,8,9,10,11,13,12]. An substantial feature of the formulation of boundary value problems of the mechanics of growing solids is the formulation of boundary conditions on the interface between the source material and the added part [7,16,14,15].

Growth Problem
Consider a homogeneous viscoelastic aging material manufactured at initial time, occupying a certain prismatic region Π 1 with a cross section Ω 1 , having a boundary L 1 . The load τ 0 is applied to the ends of the rod. These forces are statically equivalent to the pair with the moment M (t). The lateral surface of the body Π 1 is assumed free of stress.
At the moment τ 1 ≥ τ 0 , a continuous growing process of the solid begins by attaching to it elements manufactured simultaneously with it. At the same time, the new incremental elements are not strained. Denote by L(t) the boundary of the cross section Ω(t) changing over time, while L(τ 1 ) = L 1 and Ω(τ 1 ) = Ω 1 . The boundary L(t) of the section Ω(t) consists of two subsections where is the growing boundary to which the material is attached at the actual moment, and L * (t) = L * at τ ≤ τ 1 , L σ (t) is the stress free boundary.
We assume that the moment of the load application to incremental elements τ 0 = τ 0 (x 1 , x 2 ) coincides with the moment of their attachment to the growing solid τ * = τ * (x 1 , x 2 ).
At the moment τ 2 ≥ τ 1 the growing solid stops, and from that moment it occupies the region Π 2 = (τ 2 ) with the transverse section Ω 2 = Ω(τ 2 ), having the boundary L 2 = L(τ 2 ). Note that everywhere below fairly slow processes are considered, such that inertial terms can be neglected in the equations of motion.

Boundary Value Problem
The boundary value problem for the main (fixed) viscoelastic aging solid on the time interval [τ 0 , τ 1 ] is a traditional torsion problem like considered in [7].
The initial-boundary value problem for a continuously growing solid on the time interval t ∈ [τ 1 , τ 2 ] is determined of equilibrium equations in Cartesian coordinate Cauchy relations between strain rates and displacement rates are furnished by Constitutive equations can be assumed as follows Boundary condition on the fixed part of the boundary is stated as Boundary condition on growing surface L * (t) Equilibrium conditions of end sections Ω(t) In equations (1)-(6) the following notation is used: n = {n 1 , n 2 } is the unit vector of the external normal of the side surface; p = {p 1 , p 2 } is the traction vector; Ω * (t) = Ω(t)\Ω 1 -the part of the solid formed during the growth (additional body); E = E(t) and G = G(t) are the tensile and shear elastic moduli; ω(t, τ ) -shear creep measure; K 1 (t, τ ) is the creep kernel; Poisson's ratios of elastic strain and creep strain coincide and are equal to v, I is the identity operator. The values of all functions at time τ 0 ≤ t ≤ τ 1 are known from the solution of the problem for the main solid.
Distinctive features of the initial boundary value problem for a growing solid (1)-(6), which take it beyond the framework of the classical problems of mechanics of solids, are: violation of the compatibility condition for deformations in the region occupied by the additional body, and only it analogue and analogue of the Cauchy relations in the velocities of the corresponding quantities (this circumstance allows us to take into account this fact that the incremented elements can be subjected to deforming influences until they attaching in the main solid depending on the processes occurring in the solid); dependence of the constitutive equations on the function τ 0 = τ 0 (x 1 , x 2 ), which may have discontinuities of the first kind. Taking account of the notation σ 0 ij = (I − L τ 0 )σ ij G −1 , we can transform the problem of a growing viscoelastic solid with the constitutive equations (1)-(6) to the problem of growing elastic solid described by Hooke's law.

Torsion of a growing triangular prismatic rod
As an example, consider the torsion problem of a growing prismatic rod with a cross section in the form of a regular triangle under the action of a torque M (t). The boundary of the cross section L(t) is the growth boundary, i.e. L(t) = L * (t). The material of the medium is viscoelastic and aging, i.e. its properties are time-dependent. Consider the core extension according to the similarity law, in which the side of an equilateral triangle doubles during the extension a 2 = 2a 1 .
We assume that the new incremental elements are free of stress. In virtue of the mathematical equivalence of the obtained problems at each stage under consideration, it is sufficient to directly consider the growing stage. Then the initial-boundary-value problem in terms of velocities takes the form Here S ij are the stress rates, D ij are the strain rates. For the given torsion θ (t), θ t (t), we can find values υ i , S 13 and S 23 : Actual stresses and displacements are restored using the formulas Finally, one can compute M (t) based on the first formula from (6). For the given momentum M (t) one can find derivative dM 0 (t)/dt Thus the torsion rate θ t (t) is calculated by Velocities υ i can be found by (8), and stress rates S 13 and S 23 are reads by Actual stresses, displacements and torsion are restored by the formulas (9).
To build solutions at the stages before and after growing, it is enough to take t = τ 1 and t = τ 2 respectively.

Conclusion
The paper developed the theory of surface growth to study the torsion problem of growing solids in the case when the rate of deformation of the solid surface due to the loads and interference of incremented elements can be neglected compared to the velocity of the boundary due to the influx of new material to this surface.
The formulation of the arising classical and nonclassical initial boundary value problems is given. Methods are proposed for solving such problems, based on the reduction of nonclassical problems of growing viscoelastic aging solids to problems of the theory of elasticity with a certain parameter, using the theory of analytical functions to solve the latter, and restoring the true characteristics of the stress strain state using the obtained decryption formulas.
It is established that during torsion in the finished body without taking into account the process of growth, the maximum intensity of tangential stresses is reached at the boundary of the body. When building, the maximum intensity of tangential stresses can be achieved at the interface between the main and additional bodies, at the border of the finished body and at an arbitrary point of the additional body.
The results obtained can serve as the basis for solving important applied problems for parts and structural elements manufactured using modern additive technologies.