Mathematical and Numerical Modeling of the Coupled Dynamic Thermoplastic Problem

Using the strain space and deformation thermoplasticity theories the coupled dynamic thermoplastic boundary value problems are formulated. Strain space thermoplasticity theory, in contrast to the deformation one, allows formulating the coupled thermoplastic boundary value problems for the displacement and temperature increments. As an example, in one-dimensional case, the explicit and implicit types of finite difference equations are constructed. The numerical solution of the explicit finite difference equations reduced to the application of the recurrent formulas, whereas the implicit scheme reduced to the application of the elimination method. Comparison shows that the numerical results obtained using explicit and implicit scheme coincides.


Introduction
Thermo-mechanical coupling is the most common class of coupled problems, in which the mechanical response of the structure depends on its thermal behavior and vice versa. Investigations the in the field of coupled thermoelasticity and thermoplasticity are enormously developed due to their many applications in the advanced structural design problems.
In case of coupled thermoelasticity and thermoplasticity boundary value problems, the motion equation, constitutive relations are considered in conjunction with the heat conduction equation. It should be noted, that the temperature and its derivative are included in the motion equation, and the heat equation contains a strain tensor. The heat conduction equation, under certain conditions, can be considered and solved separately from the motion equations.
Coupled thermoelasticity problems investigated by Biot [1], Lord & Shulman [2], Yousuf [3], where has been introduced a generalized coupled theory with a wave-type heat equation. The coupled thermoplasticity problems for anisotropic materials at finite deformations are considered in [10].The thermoplasticity coupling problems are considered in Slodrbach [4], Canadjia and Brnik [5]. The coupled visco-plasticity is considered by Stainier and Ortiz [9]. In [6] the coupled thermoplasticity problem is formulated using the strain space plasticity theory, which has advantages comparison to stress space [19] one. Thermomechanical coupled and uncoupled boundary value problems are solved in following works [7,8,11,12,13,14]. This paper deals with the numerical solution of the coupled thermoplastic boundary value problems formulated using the deformation thermoplasticity theory [16] and strain space thermoplasticity [7]. Coupled thermoplasticity boundary problem based on deformation theory may be solved as a thermoelasticity problem i.e. is not required the partition of the problem. Usually in numerical solution of strain space thermoplasticity problems the original problem is partitioned into several smaller sub-problems, which are solved sequentially. Note that, strain space coupled thermoplasticity problem in difference from the stress space one is not depend on stress tensor.
The coupled thermoplasticity boundary value problem consists of hyperbolic and parabolic type of nonlinear partial differential equations with a corresponding initial and boundary conditions. The discrete equations are constructed using a finite difference method [18]. In the discretization process, all the derivatives are approximated by their corresponding difference quotients, and it turns out that two kinds of schemes appear, i.e. explicit and implicit schemes. The explicit scheme is solved with the help of recurrent formulas. For the solution of implicit scheme the elimination method is used [18]. Comparison of two results shows a good coincidence.

Formulation of the Coupled Thermoplasticity Boundary Value Problems
Coupled thermoplasticity problems depending on which theory of thermoplasticity deformation or flow theory is used, can be formulated two types of coupled thermomechanical boundary value problems. Usually the coupled thermoplasticity boundary value problems consist of motion equation, constitutive relations and heat equations with a corresponding initial and boundary conditions. Usually in numerical solution of coupled thermoplasticity problems the original problem is partitioned into several smaller sub-problems, which are solved sequentially. In case of coupled thermoplasticity boundary problems, based on deformation theory, is not required the partition of the problem. Coupled thermoplasticity boundary value problems may be formulated using the stress space and strain space flow theories. Note that the strain space thermoplasticity theory [7] in comparison with stress space one [19] more suitable for the formulating and numerical solution of the coupled boundary value problems.

The Coupled Boundary Value Problem Using Deformation Theory of Thermoplasticity
The coupled thermoplastic boundary value problem using deformation theory consists of the motion equations , , ij j i i X u σ ρ Constitutive relation of the deformation theory [16] ( 3 ) , heat conduction equation [15] 0 , 0 and Cauchy relations ( ) , , with the corresponding initial Substituting the Eq. (8) and Eq. (9) into Eq. (7) gives the motion equation in terms of displacements where 3 2 The heat equation in one dimensional case has the form with the appropriate initial and boundary conditions ( ) where −  the rod length. eq. (10-13) present the coupled boundary value problem for deformation theory of thermoplasticity in one dimensional case.

The Coupled Thermoplasticity Boundary Problem Using the Flow Theory
The coupled boundary problem based on the strain space thermoplasticity theory consists of motion equation written for increments of unknown quantities and constitutive relation of the strain space thermoplasticity theory, which in case of piecewise linear deformation diagram has the following form [6] ( ) differentiated heat conduction equation for isotropic materials [15] and differentiated Cauchy relations with initial and boundary conditions 1 1 where ij e − deviator of the strain tensor, T − temperature, C ε − heat at a constant temperature, α − thermal expansion coefficient, 0 λ − the heat flow coefficient and F − loading function in the strain space [7] ( ) Eq. (14)(15)(16)(17)(18)(19) in one dimensional case for piecewise line of deformation diagrams take following form 11 11 11 11 ( 2 ) and then inserting into Eq. (21) gives motion equation for differentials of displacements and temperature The following heat equation together with a Eq. (25) with an appropriate initial and boundary conditions ( ) presents the 1D coupled thermoplasticity boundary value problem for strain space theory.

Explicit and Implicit Finite Difference Equations
For the formulated in the preceding paragraphs coupled thermoplasticity boundary problems using the finite difference method an explicit and implicit schemes are constructed. Note that, in case of strain space thermoplasticity theory the discrete equations are written for displacement and temperature increments. The explicit schemes consist of two recurrent formulas displacement u and temperature T increments, whereas the implicit schemes take the suitable form for the application the elimination method [18]. Note that in the case of strain space thermoplasticity theory the external thermomechanical forces are gradually applied with a small increments and the results are found as a sum of the increments of corresponding values.

Finite Difference Equations for 1D Coupled Boundary Problem on Deformation Theory of Thermoplasticity
Let us consider, in the area 0, 0 t Replacing the derivatives in eq. (11)(12) by the difference quotients, we obtain In the first term of the finite difference Eq. (29), replacing j into 1 j + , the equation can be reduced to the form Similarly, we can write the finite difference Eq. (30) in the form The values of functions ( , ) u x t and ( , ) T x t are calculated on two primary layers at 0 j = from the initial conditions and at 1 j = from Eq. (33) and. Eq. (34), respectively. The values of these functions on the other layers can be find from the Eq. (35) and Eq. (36) using the elimination method [18]. Numerical results the system of thermoplasticity motion and heat equations allows to describe the deformation process of bodies more precisely.

Finite Difference Equations for 1D Coupled Strain Space Thermoplasticity Problem
Substituting the finite difference quotients into Eq. (25-26) give 1 1 1 1 2 Solving the discrete Eq. (37-38) for the 1 j i du + and 1 j i dT + we can find the following explicit recurrent formulas 1 1 2 Eq. (39-40) allows to find the value of Replacing index j by 1 j + for the first term of the Eq. (37) the finite difference equation can be rewritten in the following form By analogy, the Eq. (38) takes the form In case of the strain space thermoplasticity theory the considered rod problem may be described by the Eq. (25-28). Note, in case of strain space thermoplasticity theory the original problem is partitioned into several smaller sub-problems, which are solved sequentially. Therefore, Eq. (25-28) are obtained depending on the differentials (increments) of the temperature and displacement.
Discrete equations for coupled boundary value problems formulated on the basis of two thermoplastic theories were constructed using the finite difference method. In the previous section, the explicit and implicit finite difference schemes have been drawn up for each task. The explicit schemes are solved using the recurrent formulas for displacement u and temperature T. For solving the implicit schemes are applied the elimination method [18]. For the numerical solution of the finite difference equations was developed software in language C # integrated with Mathcad [20].
In the Tables 1-2 are given the values of ( , ) u x t and ( , ) T x t for the coupled problem for deformation theory of thermoplasticity using the implicit finite scheme. At the Fig. 1 is plotted the graph of the displacement ( , ) u x t using the values given in the Table 3 and shows the distribution of the displacement depending on x and t.  x  t  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14 Table 3.
The numerical results of the 1D coupled boundary value problems formulated using the deformation (Tables 1-2) and strain space (Tables 3-4) thermoplasticity theories show a good coincidence and thus establish the validity of the used modeling equations for describing the coupled thermoplasticity processes.

Conclusions
Using the strain space and deformation thermoplasticity theories two types of coupled boundary value problems are formulated. The coupled strain space thermoplastic boundary value problem more adequate describes the deformation process of the solids. In case of the strain space thermoplasticity problems the original problem is partitioned into several smaller sub-problems and the total deformation is composed from the strain increments corresponding to the increments of the external loads. Coupled thermoplasticity boundary problem based on deformation theory may be solved as a thermoelasticity problem i.e. is not required the partition of the problem. The explicit and implicit finite difference schemes are constructed and solved using the elimination method and recurrent formulas for displacement u and temperature T.
For the numerical solution of the finite difference equations was developed software in language C # integrated with Mathcad [20].
Comparison of the numerical results received from the solving of the two types of 1D coupled thermoplasticity boundary value problems shows a good coincidence.