Постер_1_Eng

1.

THE COUPLED NONLINEAR PROBLEMS IN THERMOMECHANICS
THE AIMS OF RESEARCH
THE PURPOSE OF CURRENT WORK
The realization of the general approach to solve boundary
value problems of thermomechanics taking into account
finite deformations and anisotropy of material properties.
The list of solutions of the problems about of elastic bodies
loading in non-uniform temperature field with mixed
boundary conditions is supposed to obtain.
I To determine the rate of the mutual influence in strain and temperature fields calculation for particular
class of axisymmetric problems;
II To develop iterative algorithm to take into account contact friction;
III To solve the series of problems about the interaction between hypoelastic bodies and rough plane.
THE FORMULATION OF THE PROBLEM OF FINITE DEFORMATIONS OF
HYPOELASTIC BODIES IN ACTUAL CONFIGURATION
THE SYSTEM OF VARIATIONAL EQUATIONS
OF NONLINEAR THERMOELASTICITY IN
REFERENCE CONFIGURATION
1
R R U 1 R R U 1 R R v dV0
U
V0
P0 v d 0 F v dV0 ;
0
V0
B MT B MT c 0T T dV0
V0
n0 q0 T d 0 0 T T dV0 .
0
V0
The variational statement of equilibrium flow of
deformation process
Constitutive and kinematic equations
S S v S v dV P P n W n vd
V
S N W
Boundary and initial conditions
P P ( x, t ) x P , v v ( x, t ) x v t t0
S t t S0 ( x ), v t t v0 ( x ), x V
0
S S S S
e i i эi
, i 1..3
x
X i
3K T c 0T T dV0
Shape functions of FE
n v 0,
n v v 0.
Conditions for boundary FE
S12
n S22
Step-by-step loading
method for stresses
1
M WR R v v R 1.
2
R N M B(T T0 )
" "
S12(( ik) 1) S22(( ik) 1)
S12(( ik) 1) S12(( ik)) (1 ) S22(( ik) 1) S22(( ik)) S12(( ik) 1)
( k( i )1)
( k 1) ,
S12( i )
t
( k 1)
S
( k 1)
12( i )
( k 1)
S
.
22( i )
(i )
S12(( ik) 1)
" "
v v
(I )
( III )
const , t [ti , ti 1 ] i 1..N
S((ik) 1) S((ik)) S((ik)) dt , где
The algorithm of correction of stresses in contact FE
Constitutive relations and additional
kinematic equations
du
, v,
v
dt
T
U U U U v v ,
2
R K G E 2GWR 3K TE.
3
v j (c j1x1 c j 2 x2 c j 0 )v j , j [ I II III ]
THE MEASURES OF STRESSES AND STRAINS
0
Amonton-Coulomb law
n0 q0 T d 0 T T dV0
0
V0
G
R S R 1;
g
I1(W )
1
W v v
2
THE ACCOUNTING OF CONTACT FRICTION IN THE FORM OF AMONTON-COULOMB LAW
V0
R
R 1 R
k iteration number
i loading step number
Initial approximation is
selected as the solution of the a
problem with the condition
of full sticking
(0)
(0)
v((0)
i ) , S12(i ) , S22(i ) v 0 x
c
0
THE INTERACTION BETWEEN CYLINDRICAL BODY AND ROUGH PLANE
Initial and boundary conditions
v nAB x 0 0,
q qte2 , t [0, t H ]
S t 0 0, v t 0 0,
N n AB x 0 0
2
The temperature changes in elastic
isotropic cylinder in adiabatic shell for various
value of thermal expansion coefficient.
c K 1
0
3 K
T 3 K G 3 G (T T0 )
l l0
e
T0
THE MUTUAL INFLUENCE OF STRESSES, STRAINS AND
TEMPERATURE IN LAME PROBLEM
INTERNAL PRESSURE LOADING
2
max( i ) min
i 1..m
The scheme of a problem
Contact zone change
Calculations without
contact friction
THE RESULTS OF NUMERICAL SOLUTION
EXTERNAL PRESSURE LOADING
The radius of contact zone
Normal and shear stresses in contact zone
Tangential stress component distribution with respect to cylinder radius
Temperature distribution with respect to cylinder axis
The fields of Cauchy true stress tensor components at maximum load value
CONTACTS
Astapov Yurii, +79157861876 ast3x3@gmail.com
Russia, Tula, Tula State University
Author thanks doctor of physical and mathematical sciences professor Markin Aleksey Aleksandrovich
for his consultations during research implementation