Control of thermal stresses and displacements in thermoelastic bodies

We study the problem of optimal control for rapidity of the heating of a heat-sensitive layer under constraints on the control (the temperature of the heating medium or the heat flux) and maximal values of the stress intensity in the plastic region of deformation of the material. We propose an algorithm for solving the problem that presumes it has been reduced to the inverse problem of thermoplasticity. For the case of one-sided heating we give a numerical analysis of the direct and inverse problems of thermoplasticity.

Section: Ot T~(ot) --Hi(t)(t(o~-mentioning

confidence: 99%

“…On the basis of the method of the inverse problem of thermomechanics [2] we assume that the control that is optimal for rapidity is either equal to the limiting possible value…”

Section: Ot T~(ot) --Hi(t)(t(o~-mentioning

confidence: 99%

Optimization of the heating of a heat-sensitive layer under constraints on the stresses in the plastic region of deformation of the material

Yasinskii¹

1996

“…with the boundary conditions (3), (4) which according to the equilibrium equation (1) and relations (10), (19) must be replaced by the equivalent integrals (11), (21) and the boundary conditions (4), (5) as a --* 0% taking account of formulas (7) and (18) for a one-dimensional temperature field T(x), one can obtain a known formula [1,2] for the normal stresses in an unbounded layer when the other stress components vanish:…”

Section: (29)mentioning

confidence: 99%

Solution of two-dimensional problems of elasticity and thermoelasticity for a rectangular region

Vigak¹

1997

Self Cite

We propose a separation-of-variables method for the biharmonic equation and construct a complete system of orthogonal functions for constructing exact solutions in the form of non-periodic trigonometric series for two-dimensional problems of the theory of elasticity and thermoelasticity for a rectangular region.Methods of solving problems of elasticity for bounded bodies with corners have been discussed in [4][5][6]. The absence of exact closed-form solutions for elasticity problems for such bodies is connected with the separation of variables in the key biharmonic differential equations. In the present paper we propose a method of constructing exact solutions of two-dimensional problems of elasticity and thermoelasticity in the stresses for a rectangular region under prescribed forces on the boundary. The method is based on the integration of the differential equations of equilibrium, making it possible to express two of the components of the stress tensor in terms of one of the normal components and equivalently replacing the eight boundary conditions for the different components by six conditions for a single component of the normal stresses. This makes it possible to separate the variables in the key bihaxmonic equation and to give the solution of the problems of elasticity and thermoelasticity just posed in the form of expansions in nonperiodic trigonometric series using a complete system of orthogonal functions of the corresponding nonclassical spectral problem.We consider the two-dimensional quasi-static problem of thermoelasticity for a rectangular region G = {(x, y) E [-1, 1] • [-a, a]} with prescribed forces on the boundary. From the determinate system of equations that describes the thermoelastic state in a homogeneous isotropic body we take the equations of equilibrium with boundary conditions (TXlx=l---'Pl(Y), O'xix=_l--~P2(Y), xylx:l=p3(y),auly:a = ql(x), Crylu=_ a = q2(x), a yly:a = q3(x), aXy]y:_ a = q4(x).

“…The symbols a* and a. denote admissible values of the distending and compressive stresses respectively; conditions (7) and (8) do not contradict each other, and in the case when the admissible values are equal, they assume the form max la(T)l <_ a* = la.I. In determining the optimal heating regime for the problem (1)- (8) we apply a known optimization method [3,4], in accordance with which the optimal control for rapidity at each stage of the heating is assumed equal to the limiting possible value and provides the maximum permissible values of the bounding parameters. To control the process it is necessary to assume conditions (7) and (8) at the initial time, and hence when r _> 0, we assume…”

Section: A Nonlinear Optimal Control Problem For the Nonsteady Tempermentioning

confidence: 99%

“…Finding the solution of this optimization problem by use of the method of [3,4] reduces to solving the inverse problem of thermoelasticity for the unknown control and temperature field of the layer on the basis of prescribed thermal stresses at the most highly stressed points of the body.…”

mentioning

confidence: 99%

A nonlinear optimal control problem for the nonsteady temperature of a layer with constraints on the thermal stress

Vigak¹,

Zasadna²

1996

Self Cite

We propose a numerical method of constructing the optimal heating regime for a thermally stressed unbounded layer with constraints on the control and thermal stresses. Solving the nonlinear optimization problem for~'apidity~s reduce&to~olvingAheAnver~qe~problem of therrnoelasticity. The~resulta of~numerical studies are presented.Let us consider the nonlinear problem of rapidity-optimal control of the nonsteady one-dimensional temperature regime of an unbounded layer under constraints on the control (the thermal flux at the boundary) and a maximum for the absolute values of the thermal stresses. Finding the solution of this optimization problem by use of the method of [3,4] reduces to solving the inverse problem of thermoelasticity for the unknown control and temperature field of the layer on the basis of prescribed thermal stresses at the most highly stressed points of the body.The temperature field T(x, 7") of an unbounded layer satisfies the one-dimensional nonlinear heat equation [8] 0 [A(T)~x ]r ~