Mechanical Design of AM Fabricated Prismatic Rods under Torsion

We study the stress-strain state of viscoelastic prismatic rods fabricated or repaired by additive manufacturing technologies under torsion. An adequate description of the processes involved is given by methods of a new scientific field, mechanics of growing solids. Three main stages of the deformation process (before the beginning of growth, in the course of growth, and after the termination of growth) are studied. Two versions of statement of two problems are given: (i) given the torque, find the stresses, displacements, and torsion; (ii) given the torsion, find the stresses, displacements, and torque. Solution methods using techniques of complex analysis are presented. The results can be used in mechanical and instrument engineering. 1 Statement of the torsion problem Consider a homogeneous viscoelastic ageing rod manufactured at the initial instant of time and occupying some cylindrical domain 1 . The lateral surface 1 of the body is stress-free until time 0 0 at which forces statically equivalent to a couple with torque ) (t M are applied to the faces of the cylindrical body. At time 0 1 , continuous accretion of the rod by elements manufactured simultaneously with the rod begins. Let t L be the time-varying boundary of the cross-section ) (t , so that 1 1 = L L and 1 1 = . The boundary t L consists of two parts, t L t L t L * = . Here ) ( * t L is the accretion boundary (or the growth boundary) on which the material influx occurs at time t , and * * = ) ( L t L for 1 < ; ) (t L is the stress-free boundary. We also assume that the time ) , ( 2 1 0 x x at which the load is applied to the elements being added coincides with the time 2 1 * , x x at which they are added to the growing rod. The accretion is terminated at time 1 2 , and since then the rod occupies some domain 2 2 = whose cross-section 2 2 = has the boundary 2 2 = L L . The boundary conditions posed on the boundary 2 2 = L L are the same as on the boundary 1 L before the beginning of growth. Somewhat later, at time 2 3 , accretion may begin again; it may well happen that the new growth boundary is not related in any way to the earlier existing one. Then one can assume that accretion terminates at time 4 , etc., thus successively arriving at the problem of piecewise continuous accretion of a solid rod with n growth beginning instants and accordingly n growth termination instants. We proceed to the analysis of main stages of the piecewise continuous accretion process for viscoelastic rods. From now on, we consider sufficiently slow processes, and so inertial terms in the equilibrium equations can be neglected (also see [1–7]). 2 Boundary value problem for the rod of fixed composition Consider the stress-strain state of a viscoelastic ageing rod of fixed composition on the time interval 1 0 , t . We have the boundary value problem (e.g., see [1–3]) consisting of the equilibrium equations 0, = 0, = 0, = 2 23 1 3 1 3 3 2 3 3 1 x x x x (1) the strain–displacement relations 1,2,3, = , 0, = = 0, = 2 1 = 12 j i x u


Statement of the torsion problem
Consider a homogeneous viscoelastic ageing rod manufactured at the initial instant of time and occupying some cylindrical domain 1 3 .The lateral surface 1 3 of the body is stress-free until time 0 0 t W at which forces statically equivalent to a couple with torque ) (t M are applied to the faces of the cylindrical body.
At time , continuous accretion of the rod by elements manufactured simultaneously with the rod begins.Let t L be the time-varying boundary of the cross-section ) (t : , so that is the stress-free boundary.We also assume that the time ) , ( at which the load is applied to the elements being added coincides with the time 2 1 * , x x W at which they are added to the growing rod.
The accretion is terminated at time 1 2

W t W
, and since then the rod occupies some domain

W t W
, accretion may begin again; it may well happen that the new growth boundary is not related in any way to the earlier existing one.Then one can assume that accretion terminates at time 4 W , etc., thus successively arriving at the problem of piecewise continuous accretion of a solid rod with n growth beginning instants and accordingly n growth termination instants.
We proceed to the analysis of main stages of the piecewise continuous accretion process for viscoelastic rods.From now on, we consider sufficiently slow processes, and so inertial terms in the equilibrium equations can be neglected (also see [1][2][3][4][5][6][7]).

Boundary value problem for the rod of fixed composition
Consider the stress-strain state of a viscoelastic ageing rod of fixed composition on the time interval > @ 1 0 , W W t .We have the boundary value problem (e.g., see [1][2][3]) consisting of the equilibrium equations the strain-displacement relations the boundary condition on the lateral surface, and the constitutive equation is the unit outward normal vector on the lateral surface of the body, (5) Relations (1)-( 5) form a problem of the viscoelasticity theory for a homogeneous ageing body.The solution of this problem permits one to describe the stress-strain state of the body on the time interval > @ where The boundary value problem ( 6)- (8), in contrast to problem (1)-( 5), contains time as a parameter and is mathematically equivalent to a boundary value problem of the elasticity theory with parameter t .
By substituting the strain-displacement relations into the equilibrium equations of problem (6), we obtain the equilibrium equations in terms of displacements in the form 0. = 0, = 0, = We use the first three strain-displacement relations in (6) to obtain (10) where the displacement i u is independent of the coordinate i x .In view of ( 9) and ( 10), the equilibrium equations in terms of displacements become It follows from the third equation in (11) . To this end, we integrate the first two equations in (11) twice with respect to 3 x .As a result, we obtain ., , = , , , = By using (12) and the fourth strain-displacement relation in (6), we obtain Equation ( 13) permits us to conclude that By ( 14), for the displacements 1 u and 2 u we obtain The last terms in the expression (15) do not affect the stress state of the twisted cylindrical body.To eliminate the translational displacements of the rod along the 1 x - and 2 x -axes, assume that some point of the rod end is fixed, so that 0 = = 2 1 x x at that point.We place the origin there and require that the additional conditions hold at that point so as to prevent the rigid rotations of the cylindrical body about the 1 x -, 2 x -, and 3 x -axes.Then , and hence the expressions (15) for the stresses 1 x and 2 x acquire the form The function t T is called the torsion angle (or torsion) and is related to the displacements by the formula By analogy with [8], set where M is some function to be determined of the variables 1 x , 2 x , and 1 W .It is called the torsion function and characterizes the deformation of cross-sections.
Then, by ( 16) and ( 18), we have By using relations (19) and the constitutive equations in ( 6), we obtain By substituting (20) into the third equilibrium equation in (3), we find that the function We see that the first integral equilibrium condition in (8) acquires the form and the second and third conditions are satisfied identically.
The constant 1 D is called the torsional rigidity.It only depends on the shape of the cross-section of the body.
The problem in question has two possible statements:  V with the use of (25), and determines t M on the basis of (5).Sometimes it is convenient to study the problem for the main body in terms of rates of the corresponding variables, because information about these rates is needed when solving the problem at the growth stage.To this end, one should differentiate relations (6) with respect to t .(From now on, differentiation is always understood in the sense of distributions.)Then we have 0, = 0, = 0, = By analogy with the preceding, one can show that i v and ij S have the following structure: where the dot stands for differentiation with respect to t (we use this notation in what follows) and t x T is the torsion angular velocity.
By substituting (27) into the third equilibrium equation in (26), we find that the function Thus, we have reduced the original torsion problem for a prismatic body to the Neumann problem for the function  V by formulas (31), and find the torque according to (5).Note that the initial values of the stresses, displacements, and torsion angle in (31) are determined from the solution of the boundary value problem (1)-( 5) or ( 6) for 0 = W t .Thus, the study of the torsion problem for a growing body at the stage preceding accretion is complete.

Initial-boundary value problem for a growing rod
Now consider the process of continuous accretion of a deformable body for . For a growing rod, we have the equilibrium equations the strain rate-displacement rate relations the boundary condition Let us divide the relations containing the stresses 13 V and 23 V in (32)-(37) by G and then apply the operator Then, in view of the notation , we obtain Let us transform the initial-boundary value problem (40) to a boundary value problem for the strain rates, the displacement rates, and the operator stress rates.To this end, we differentiate the equilibrium equations, the constitutive equations, the boundary condition on the immovable boundary t L V , and the equilibrium condition for the end cross-section with respect to t and obtain the boundary value problem x where the second term in the last formula is a contour integral over the boundary t L * .Being supplemented with the initial conditions for the main body at 1 = W t , relations (40), which also contain the initial-boundary condition on the growth boundary, form an initial-boundary value problem with time parameter t .
The formulas for the variables i v and ij S can be obtained by the replacement of the function The torsion function t M can be found from the where t D is the variable torsional rigidity of the growing body and t M is the unknown harmonic function.
At the stage of growth of the twisted body, we also have two possible versions of the statement of the problem.
To  V by formulas (45), and find the torque according to formula (5).
To solve the boundary value problem (43), one can apply methods of the theory of functions of one complex variable, which we will discuss below.
Thus, the study of the torsion problem for a growing body at the stage of continuous accretion is complete.

Deformation of the rod after the termination of growth
By analogy with the preceding, we obtain the boundary value problem To find the solution of the resulting Neumann boundary value problems, one can use methods of complex analysis, which are presented, say, in [9].The true stresses and displacements can be completely reconstructed with the use of the decoding formulas (45).

Conclusions
• A torsion theory of growing viscoelastic prismatic rods is constructed.Statements of the corresponding classical and nonclassical initial-boundary value problems are given.• Solution methods for such problems are suggested based on the reduction on nonclassical problems for growing bodies to elasticity problems with a parameter.The subsequent use of complex analysis and the decoding formulas developed in the paper permits one to reconstruct the true values of stressstrain characteristics of growing rods.• A new effect in mechanics is revealed.In the classical case, the maximum tangential stress intensity is known to be attained on the boundary of the body.In the case of a growing rod, the maximum tangential stress intensity can be attained at an arbitrary point of the grown part depending on the growth mode.• The results obtained can serve as a basis for modeling the processes of crystal growth from melt, manufacturing or repairing machine parts by spraying, weld overlay cladding, or precipitation, and a number of other additive manufacturing processes (see also [10][11][12][13][14][15][16]).
boundary conditions posed on the boundary 2 2 = W L L are the same as on the boundary 1 L before the beginning of growth.Somewhat later, at time 2 3 i.e., the torsion function1 WM is harmonic in the domain 1 : .Let us find the boundary condition for the function 1 W M on the contour 1 L of the cross-section 1 : .By substituting (20) into the boundary condition (3), we obtain have reduced the original torsion problem for a prismatic body to the Neumann problem for the function 1 W M in the domain 1 : , that is, the problem of finding a harmonic function 1 W M in the cross-section domain 1 : (see (1.21)) from the given values of its normal derivative on the contour 1 Relations (26) form an initial-boundary value problem parametrically depending on time t .

1 L 1 :
of the cross-section acquires a form similar to (22), have reduced the original torsion problem for a prismatic body to the Neumann problem for the function t M in the domain t : .Obviously, in view of (42) we obtain

!
Problem (47) is obviously similar to problem (26).In this case, formulas (27), where the function Let us transform the initial-boundary value problem for a continuously growing viscoelastic ageing body into a problem with time parameter coinciding in form with a boundary value problem of the elasticity theory.At the first stage, we transform the accretion problem for a viscoelastic body with constitutive relations (36) to the accretion problem for an elastic body described by Hooke's law.