Creep of cyclically loaded bodies and their fracture due to creep mechanisms are studied. Creep damage equations are built by use of tensor parameter for titanium alloy. Similar equations were derived for stress cyclic varying using the method of asymptotic expansions and averaging in a period. 2d plane stress problems were solved  by FEM and fields of stress, strain, displacement, damage parameter components as well as time to rupture values were obtaine

Breslavsky, D.

Korytko, Yu.

Mietielov, V.

Morachkovsky, O.

Tatarinova, O.

Library of National Technical University "Kharkiv Polytechnic Institute"

379 
Proceedings of the 4th  International Conference on Nonlinear Dynamics 
ND-KhPI2013 
June 19-22, 2013, Sevastopol, Ukraine 
Numerical Calculations of Creep Damage at 
Cyclic Loading by Use of Tensor Damage 
Parameter Model 
D.Breslavsky1*, Yu. Korytko1, V.Mietielov1, O.Morachkovsky1, O.Tatarinova1 
Abstract 
Creep of cyclically loaded bodies and their fracture due to creep mechanisms are studied. Creep 
damage equations are built by use of tensor parameter for titanium alloy. Similar equations were 
derived for stress cyclic varying using the method of asymptotic expansions and averaging in a 
period. 2d plane stress problems were solved by FEM and fields of stress, strain, displacement, 
damage parameter components as well as time to rupture values were obtained  
Keywords 
Creep, damage, cyclic loading, tensor damage parameter, FEM, plane stress, numerical simulation 
1
 National Technical University ‘Kharkiv Polytechnic Institute’, Kharkiv, Ukraine 
* Corresponding author: brdm@kpi.kharkov.ua 
Introduction 
New structural materials like light weight alloys and metallic composites are widely used in 
industrial applications. Their creep-damage behavior is strongly anisotropic which demands using of 
tensor models for description of creep strain rate as well as damage accumulation. Such models had 
been suggested by Murakami [1], Cordebois-Sidoroff [2], O.Morachkovsky [3] and others. In recent 
years these approaches were used for the case of static loading.  
Otherwise, very often structural elements are loaded by joint action of static and cyclic loading 
from their forced vibrations. As it is known [4], in this case so-called ‘dynamic creep’ processes are 
developed in the material. They are characterized by essential acceleration of creep strain rate as well 
as of damage accumulation which leads to decreasing of time to rupture values in comparison with 
pure static loading.  
Constitutive equations for description of dynamic creep-damage processes as well 
mathematical procedures for their adequate numerical simulation were suggested by D.Breslavsky 
and O.Morachkovsky [5, 6]. In this paper these approaches were applied for tensor creep-damage 
model [3]. 
1. Anisotropic creep-damage laws 
Let us regard creep-damage laws for anisotropic materials which correspond to general linear 
tensor dependences for principal axes of symmetry of creep and damage properties of anisotropic 
material. Damage accumulation in this case will be described by tensor of second rank ij_ .  
Let as accept that damage tensor ij_ , as an internal parameter, corresponds to the associated 
with it tensor of thermodynamic stresses in damaged media Rij . For materials with initial anisotropy 
equivalent stress VM is determined by mutual invariants of stress tensor ijM with material creep 
properties tensors ijklij ba , , as well as equivalent stress VR is determined by mutual invariants of 
tensor ijR with material damage properties ijklij dg , :  
D. Breslavsky, Yu. Korytko et al. 
 
380 
 
,21 MMM V  ,1 ijija MM   ;2 klijijklb MMM 
 
,21 RRRV   ,1 ijij RgR   .2 klijijkl RRdR 
 
 
Using above mentioned suggestions let us present the dissipation potentials for creep strain and 
damage parameter’s rates in the following forms: 
 
  ijijV cTDD  M_M  ),;( ,   ijijV RTR __   ),;( . (1) 
 
Thereat the flow rule and damage equations can be presented using the gradiental law as 
follows: 
 
ij
ij
D
c
M

7


 ,   
ij
ij R

7_


(

 , 
 
where scalar multipliers 
(
77
,  are determined from eqs (1) after specification of the form of each 
potential. 
For the establishing of failure criterion in invariant scalar form let us consider in a some 
material point the dissipation’s part which is responsible for creep damage, as well as its limit value in 
the rupture time moment: 
 
F

t
dtt
0
)(  ,       
F
(

(
t
dt
0
 . 
 
Let us suppose that limit value of creep damage dissipation of the material 
(
  is completely 
definite material characteristic. In this case, assuming as the scalar damage measure the value of 
(
 /)()( tt , the failure criterion can be written in the following form: 1)( 
(
t , where t* is the 
value of failure time.  
By use of thermodynamic assumptions for materials with initial creep anisotropy we obtain the 
constitutive equations: 
 
  )(),(
22 R
Rd
g
R
b
a
D
c
klijkl
ij
V
ij
klijkl
ij
V
ij 

_
M
M
M



 , (2) 
 
where ijc , ij_  are symmetric tensors of irreversible creep strains and damage; ijijcD  M  is dissipation 
due to creep: ijijR _    is the dissipation due to creep damage, which is referred for its limit  value 
(
  at failure moment )10( == . 
By using of the principle of strain equivalence the influence of damage tensor on effective 
stress is considered from the following equation: )1/( M  ijijR , and besides ijijR _   , 
)10( == , / 0 1
(
t . Keeping of the demands of basic thermodynamic inequality 0> D  
dissipation potentials (1) can be rewritten in the following form: 
 
  )(),;( VV RDTD  M ,   )(),;( VV RT ((   M , (3) 
 
where )1/( M  VVR , ,21 MMM V  ,1 ijija MM   klijijklb MMM 2 ; )1/( M  (( VVR , 
,21 (((  MMM V  ,1 ijijg MM (  klijijkld MMM (2  are effective stress invariants. 
Within the supposed assumptions the principal directions of the material anisotropy are 
considered constant directly to failure, as well as surfaces of the dissipation potentials in the stress 
space are expanded in time proportionally to one parameter, which is connected with damage measure 
10 == . 
Let us regard further constitutive equations (2) for materials with creep transversally isotropy. 
D. Breslavsky, Yu. Korytko et al. 
 
381 
 
Dissipation potentials (3) are chosen as power functions from stress invariants: 
 
  ,)1/(, SkV
N
V RRD   (  (4) 
 
where N, k, S  are the constants. 
Equations (2) are regarded as constitutive ones for materials with initial anisotropy and creep-
damage properties asymmetry, in particular for tension and compression. If these effects are absent, 
then 0 ijij ga . In this case creep-damage equations for materials with creep transversally isotropy 
can be written in the following form: 
 
  
/ 0
4 5
/ 0
4 5M

M
_M

M
DdBbc
Sk
k
k
N
N
N
1
2
22/
1111
1
22/)1(
1111
1
,
1


(






 , (5) 
  
/ 0
/ 0 / 0 1,00,
1
22/
1111 


(

( td
Sk
k
k


M

 ,   (6) 
 
where / 0
Tcccc 122211 2,,   , / 0
T
122211 ,, MMMM  , / 0
T
122211 2,, ____    are the vectors of creep strain 
rates, stresses and damage rates 4 5 ,22 MMM B
T

 4 5MMM DT
(
2
2  are the stress invariants; 
 
4 5 ,
400
0
01
2221
12



B  ,4,,
2
1
1111
1212
1111
2222
22111112
b
b
b
b
b    
4 5 ,
400
0
01
2221
12
1
11
1
D  
1111
1212
1111
2222
22111112 4,,
2
1
d
d
d
d
d  111 . 
 
Further let us regard the cyclic loading, where stress tensor consists from pure static part σ0 
and fast varying part σ1 : σ=σ0+σ1 . The frequency f is the frequency of forced oscillations: 
/ 0fta MM 2sin1  , where M
a
 is amplitude stress. Such process is referred as dynamic creep-damage 
process [4]. It had been mathematically described in [6] for scalar damage parameter by use of 
method of asymptotic expansions jointly with averaging in a period technique. 
Let us use this approach for eqs (5)-(6). After introducing the small parameter μ=Т/t* we 
present processes in two time scales (slow t and fast = /T, where = t/, ) in the following form of 
expansions:  
)()( 10 ctcc 3 ,  ),()( 10  3 t                                             (7) 
 
where c0(t), 0(t), c1(), 1(t,) are the functions which coincide to basic creep-damage process in 
slow (0) and fast (1) time scales.  
Considering the dependence of creep strain and damage parameters only from ‘slow’ time [5], 
after averaging we have: 
 
 / 0 / 0 / 0tcdtcc 0
1
0
00
3
F
 ,    / 0 / 0 0
1
0
11
3
F
 dcc ,           / 0 / 0 / 0tdt 0
1
0
00
 3
F
,  
/ 0 / 0 0
1
0
11
3
F
 d ,      / 0 / 0 / 0tdt 0
1
0
00
___ 3
F
,         / 0 / 0 0
1
0
11
3
F
__ d .  
 
So, by use of asymptotic expansions methods with subsequent averaging in a period of cyclic 
load, after transition to the case of multi-axial stress state, we obtain constitutive equations for 
dynamic creep-damage processes in alloys with orthotropic creep-damage properties: 
 
D. Breslavsky, Yu. Korytko et al. 
 
382 
 
  
/ 0
/ 0
4 5 / 0
/ 0
4 5M

M
_M

M
DAHdBAKbc
Sk
k
k
k
N
N
n
N
1
2
2
1
2/
1111
1
22/)1(
1111
1
,
1


(






 , (8) 
  / 0
/ 0
/ 0 / 0 1,00,
1
2
2
2/
1111 


(

( tAHd
Sk
k
k
k


M

 ,   (9) 
 
where / 0/ 0
F

1
0
2sin1)(  dAAK nnn , 
2
2
M
M
a
nA  , / 0/ 0
F


1
0
1
1 2sin1)(  dAAH
k
kk , 
/ 0/ 0
F

1
0
2 2sin1)(  dAAH
k
kk , 
2*
2*
M
M
a
rA  . / 0 / 0 4 5
aTaa B MMM 
2
2 , / 0 / 0 4 5
aTaa D MMM 
(
2
2  are the 
invariants of cyclic stresses.  
 
2. FEM solution for plane stress problem  
Let us regard the problem statement for structural element, made from material with anisotropic 
creep-damage properties, which is in plane stress state. The volume V is fixed in a surface part S1 and 
is loaded by traction р {р1, р2} on another surface part S2.  In co-ordinate system ОX1X2 the motion of 
material points under the creep conditions is described by use of Lagrange approach. The vectors of 
displacements u={u1, u2}
T 
 and their rates v={v1, v2}
T 
are introduced. In these assumptions creep 
problem for the case of small displacements and strains is described by following boundary – initial 
value problem: 
 
  ,, jijij vf CM     (i,j=1,2)   ;, 21 Vxx I  ,2sin ftppn
a
iijij M     221, Sxx I .  (10) 
  ;111 udtduv   222 udtduv    (11) 
  ,
1
1
11 x
u


D   ,
2
2
22 x
u


D   ,
2
1
1
2
2
1
12
#
#
%
!
"
"
$
 

x
u
x
u




D  (12) 
 
The level of stresses which don’t exceed yield limit will be considered. Let us accept, that 
elastic eij and creep cij
 
strain components as well as their rates are additive: 
 
  ,ijijij ce  D  (13) 
 
Obtained system (10-13) has to be solved jointly with constitutive equations (8-9). Boundary 
conditions on surface parts  S1 :
*
ii uu    and  S2: ijij pn  M  as well initial ones at t=0 considering the 
elastic strain-stress distribution have to be added. Constitutive equations (8-9) are added to above 
system.  
 The system (8-13) is solved by use of two time scales method jointly with averaging in a 
period of forced vibration 1/f. Full description of the procedure can be found in [5,6]. The problem 
derives to the simplified one which is similar to problem of static loading, but with constitutive 
equations (8-9). 
Anisotropic creep-damage behavior is simulated by FEM software had been developed in NTU 
’KhPI’. Triangular linear element is used. The problem which is formulated in rates is solved: 
 
  4 5 ,cFFK 

1  (14) 
 
where 4 5 4 5 4 54 5dVBDBK
T
ee V

F
  is global stiffness matrix; 
4 5 ,
2
`
`


F
dpNF
T
ee
  4 5

F

e e
T
c
V
dVBF *M  are  
the loading vectors from  traction  and forces which determined by creep strains.  
 
 
D. Breslavsky, Yu. Korytko et al. 
 
383 
 
3.  Anisotropic creep-damage in titanium plates 
Developed method and software were used for creep-damage modeling in titanium plates made from 
BT1-0 alloy which is similar to IMI125 or T40 grades. The properties of creep and long term strength 
of the specimens made from the plane billet in three directions were experimentally obtained for 
temperature T=773K by O.Morachkovsky and V.Konkin [7]. The constants values which had been 
obtained after data processing are the following: ,10303,2 41111

b  
,10151,1 41122

b ,10058,2,10924,1 41212
4
2222

 bb  (МPа)-2N/N+1/h2/N+1; ,10771,1 51122

d  
,10127,3,10324,3 51212
5
2222

 dd  (МPа)-2/(h)2/k; k=N=5, s=1. Comparison between 
experimental data at static loading for specimens had been cut in different directions with numerical 
simulations of rectangular plates in tension simulating the above test data were done at first in order to 
verify numerical method. A satisfactory agreement which doesn’t exceed 25-32% had been obtained 
for time variation of strains and time to rupture values. Table 1 contains some examples of these 
comparisons for time to rupture values for net stress 60 MPa.  
 
Table 1.  Comparison of time to rupture values for static loading  
Orientation of 
anisotropy axes, K

 
Stress, MPa 
Experimental value of 
time to rupture t
(
, h 
Numerical value of time 
to rupture t
(
, h 
0 60 24  25.9 
45 60 19  20.5 
90 60 28.7 19.7 
 
Further the static and dynamic creep of titanium plate (0.8mx0.8m) with a central hole (radius 
is equal to 0.1m) were numerically simulated. One quarter of a plate was considered and stress-strain 
state and damage redistribution on time were analyzed. Fig.1 contains the distribution of damage 
measure  for the case of pure static loading by net stress 10MPa at the moment t=220h  which is 
nearest to the failure moment t*=221h. Essential stress concentration near the hole leads to failure in 
this area. The creep –damage behavior of the plate is strongly non-linear: increasing of the traction 
from 10 to 13MPa leads to decreasing of time to failure from 221h to 68.5h. 
 
 
 
Figure 1. Damage parameter at t=220h 
D. Breslavsky, Yu. Korytko et al. 
 
384 
 
Let us present the results of dynamic creep simulation of the similar plate. The axial loading in 
this case consists from static part p0=16.9MPa and the cyclic one which is varying due to sine law 
with frequency 10Hz. Let us introduce the loading parameter L=p0/pa, where pa is amplitude value of 
the loading. The varying of L from 0 to 0.3 was analyzed.  
The time to failure values in this loading program decreases with increasing of L from 0.166 h 
for L=0 to 0.122h for L=0.3. It happens for material with anisotropic creep-damage properties just the 
same in dynamic creep processes of elements with isotropic ones [6]. The character of stress and 
damage redistribution is similar for the static case, but both processes are run faster.  
However, for smaller values of net stress, and, respectively, for smaller stress values in a plate, 
character of a process differs from above analyzed one. Dynamic loading accelerates the stress 
relaxation in area near the hole as well as in conventional dynamic creep [6], wherein the obtained 
stress level became so small, that rate of damage accumulation decrease significantly. For example, 
for p0=13MPa the value of time to failure for L=0 is equal to t*=68.5 h, but adding of small cyclic part 
(L=0.05) leads to it’s increasing and t*=72.6 h. Subsequent increasing of L leads to similar growth of 
time to failure values due to fast dynamic relaxation at first time of a process: for L=0.1 t*=120.1 h.  
 
Conclusions 
An approach for the effective mathematical modeling of dynamic creep-damage processes in 
structural elements made from the titanium alloy with anisotropic properties is presented. The use of 
two time scale method with asymptotical expansions on the small parameter and subsequent 
averaging in a period allows us to avoid the direct integration through a cycle and simulate only static 
problems with flow rule and damage equation of special form. Effects of essential acceleration of 
strain and damage rate growth as well as stress relaxation due to dynamic creep had been established 
for plate with a hole made from material with anisotropic creep-damage properties.   
 
References 
[1]    Murakami S. Notion of Continuum Damage Mechanics and Its Application to Anisotropic Creep 
Damage Theory J. Engng. Mater. Techn, Vol. 105, pp. 99-105, 1983. 
[2]   Cordebois J.P., Sidoroff F. Damage Induced Elastic Anisotropy Mechanical Behaviour of 
Anisotropic Solids (J. P. Boehler, editor), Colloque Euromech 115, Villard-de-Lans, June 19-22, 
Martinus Nijhoff Publishers, pp. 761- 774, 1979. 
[3]    Morachkovsky O.K., Pasynok M.A. Investigation of the influence of the obtained anisotropy due 
to precurcive creep on the creep of materials Proc. KhSPU , KhSPU, Kharkiv, Vol. 27, pp. 197-203, 
1998 (in Russian). 
[4]   Taira S., Ohtani R. Theory of High Temperature Strength of Materials, Metallurgia, Moscow, 
1986. (in Russian, translated from Japan). 
[5]    Breslavsky D., Morachkovsky O. A new model of nonlinear dynamic creep. IUTAM Symposium 
on Anisotropy, Inhomogenity and Nonlinearity in Solid Mechanics, Kluwer Academic Publishers, 
Dordrecht, pp. 161-166, 1995.  
[6]    Breslavsky D., Morachkovsky O. Dynamic creep continuum damage mechanics: FEM-based 
design analysis. Computational Plasticity: Fundamentals and Applications. Proc. of the Fifth 
International Conference on Computational Plasticity held in Barselona, Spain, 17-20 March 1997, 
IMNE, Barselona IMNE, Part 1, pp.1071-1076, 1997. 
[7]    Konkin V.N., Morachkovsky O.K. Creep and long term strength of light weight alloys with 
anisotropic properties. Problems of Strength, Vol. 5, pp. 38-42, 1987. (in Russian) 
 


Numerical Calculations of Creep Damage at Cyclic Loading by Use of Tensor Damage Parameter Model

Electronic National Technical University &quot;Kharkiv Polytechnic Institute&quot; Institutional Repository (eNTUKhPIIR)

379 Proceedings of the 4th  International Conference on Nonlinear Dynamics ND-KhPI2013 June 19-22, 2013, Sevastopol, Ukraine Numerical Calculations of Creep Damage at Cyclic Loading by Use of Tensor Damage Parameter Model D.Breslavsky1*, Yu. Korytko1, V.Mietielov1, O.Morachkovsky1, O.Tatarinova1 Abstract Creep of cyclically loaded bodies and their fracture due to creep mechanisms are studied. Creep damage equations are built by use of tensor parameter for titanium alloy. Similar equations were derived for stress cyclic varying using the method of asymptotic expansions and averaging in a period. 2d plane stress problems were solved by FEM and fields of stress, strain, displacement, damage parameter components as well as time to rupture values were obtained  Keywords Creep, damage, cyclic loading, tensor damage parameter, FEM, plane stress, numerical simulation 1 National Technical University ‘Kharkiv Polytechnic Institute’, Kharkiv, Ukraine * Corresponding author: brdm@kpi.kharkov.ua Introduction New structural materials like light weight alloys and metallic composites are widely used in industrial applications. Their creep-damage behavior is strongly anisotropic which demands using of tensor models for description of creep strain rate as well as damage accumulation. Such models had been suggested by Murakami [1], Cordebois-Sidoroff [2], O.Morachkovsky [3] and others. In recent years these approaches were used for the case of static loading.  Otherwise, very often structural elements are loaded by joint action of static and cyclic loading from their forced vibrations. As it is known [4], in this case so-called ‘dynamic creep’ processes are developed in the material. They are characterized by essential acceleration of creep strain rate as well as of damage accumulation which leads to decreasing of time to rupture values in comparison with pure static loading.  Constitutive equations for description of dynamic creep-damage processes as well mathematical procedures for their adequate numerical simulation were suggested by D.Breslavsky and O.Morachkovsky [5, 6]. In this paper these approaches were applied for tensor creep-damage model [3]. 1. Anisotropic creep-damage laws Let us regard creep-damage laws for anisotropic materials which correspond to general linear tensor dependences for principal axes of symmetry of creep and damage properties of anisotropic material. Damage accumulation in this case will be described by tensor of second rank ij_ .  Let as accept that damage tensor ij_ , as an internal parameter, corresponds to the associated with it tensor of thermodynamic stresses in damaged media Rij . For materials with initial anisotropy equivalent stress VM is determined by mutual invariants of stress tensor ijM with material creep properties tensors ijklij ba , , as well as equivalent stress VR is determined by mutual invariants of tensor ijR with material damage properties ijklij dg , :  D. Breslavsky, Yu. Korytko et al.  380  ,21 MMM V  ,1 ijija MM   ;2 klijijklb MMM  ,21 RRRV   ,1 ijij RgR   .2 klijijkl RRdR   Using above mentioned suggestions let us present the dissipation potentials for creep strain and damage parameter’s rates in the following forms:    ijijV cTDD  M_M  ),;( ,   ijijV RTR __   ),;( . (1)  Thereat the flow rule and damage equations can be presented using the gradiental law as follows:  ijijDcM7 ,   ijij R7_( ,  where scalar multipliers (77,  are determined from eqs (1) after specification of the form of each potential. For the establishing of failure criterion in invariant scalar form let us consider in a some material point the dissipation’s part which is responsible for creep damage, as well as its limit value in the rupture time moment:  Ftdtt0)(  ,       F((tdt0 .  Let us suppose that limit value of creep damage dissipation of the material (  is completely definite material characteristic. In this case, assuming as the scalar damage measure the value of ( /)()( tt , the failure criterion can be written in the following form: 1)( (t , where t* is the value of failure time.  By use of thermodynamic assumptions for materials with initial creep anisotropy we obtain the constitutive equations:    )(),(22 RRdgRbaDcklijklijVijklijklijVij _MMM , (2)  where ijc , ij_  are symmetric tensors of irreversible creep strains and damage; ijijcD  M  is dissipation due to creep: ijijR _    is the dissipation due to creep damage, which is referred for its limit  value (  at failure moment )10( == . By using of the principle of strain equivalence the influence of damage tensor on effective stress is considered from the following equation: )1/( M  ijijR , and besides ijijR _   , )10( == , / 0 1(t . Keeping of the demands of basic thermodynamic inequality 0> D  dissipation potentials (1) can be rewritten in the following form:    )(),;( VV RDTD  M ,   )(),;( VV RT ((   M , (3)  where )1/( M  VVR , ,21 MMM V  ,1 ijija MM   klijijklb MMM 2 ; )1/( M  (( VVR , ,21 (((  MMM V  ,1 ijijg MM (  klijijkld MMM (2  are effective stress invariants. Within the supposed assumptions the principal directions of the material anisotropy are considered constant directly to failure, as well as surfaces of the dissipation potentials in the stress space are expanded in time proportionally to one parameter, which is connected with damage measure 10 == . Let us regard further constitutive equations (2) for materials with creep transversally isotropy. D. Breslavsky, Yu. Korytko et al.  381  Dissipation potentials (3) are chosen as power functions from stress invariants:    ,)1/(, SkVNV RRD   (  (4)  where N, k, S  are the constants. Equations (2) are regarded as constitutive ones for materials with initial anisotropy and creep-damage properties asymmetry, in particular for tension and compression. If these effects are absent, then 0 ijij ga . In this case creep-damage equations for materials with creep transversally isotropy can be written in the following form:    / 04 5/ 04 5MM_MMDdBbcSkkkNNN1222/1111122/)1(11111,1( , (5)   / 0/ 0 / 0 1,00,122/1111 (( tdSkkkM ,   (6)  where / 0Tcccc 122211 2,,   , / 0T122211 ,, MMMM  , / 0T122211 2,, ____    are the vectors of creep strain rates, stresses and damage rates 4 5 ,22 MMM BT 4 5MMM DT(22  are the stress invariants;  4 5 ,400001222112B  ,4,,21111112121111222222111112bbbbb    4 5 ,4000012221121111D  111112121111222222111112 4,,21ddddd  111 .  Further let us regard the cyclic loading, where stress tensor consists from pure static part σ0 and fast varying part σ1 : σ=σ0+σ1 . The frequency f is the frequency of forced oscillations: / 0fta MM 2sin1  , where Ma is amplitude stress. Such process is referred as dynamic creep-damage process [4]. It had been mathematically described in [6] for scalar damage parameter by use of method of asymptotic expansions jointly with averaging in a period technique. Let us use this approach for eqs (5)-(6). After introducing the small parameter μ=Т/t* we present processes in two time scales (slow t and fast = /T, where = t/, ) in the following form of expansions:  )()( 10 ctcc 3 ,  ),()( 10  3 t                                             (7)  where c0(t), 0(t), c1(), 1(t,) are the functions which coincide to basic creep-damage process in slow (0) and fast (1) time scales.  Considering the dependence of creep strain and damage parameters only from ‘slow’ time [5], after averaging we have:   / 0 / 0 / 0tcdtcc 010003F ,    / 0 / 0 010113F dcc ,           / 0 / 0 / 0tdt 01000 3F,  / 0 / 0 010113F d ,      / 0 / 0 / 0tdt 01000___ 3F,         / 0 / 0 010113F__ d .   So, by use of asymptotic expansions methods with subsequent averaging in a period of cyclic load, after transition to the case of multi-axial stress state, we obtain constitutive equations for dynamic creep-damage processes in alloys with orthotropic creep-damage properties:  D. Breslavsky, Yu. Korytko et al.  382    / 0/ 04 5 / 0/ 04 5MM_MMDAHdBAKbcSkkkkNNnN12212/1111122/)1(11111,1( , (8)   / 0/ 0/ 0 / 0 1,00,1222/1111 (( tAHdSkkkkM ,   (9)  where / 0/ 0F102sin1)(  dAAK nnn , 22MManA  , / 0/ 0F1011 2sin1)(  dAAHkkk , / 0/ 0F102 2sin1)(  dAAHkkk , 2*2*MMarA  . / 0 / 0 4 5aTaa B MMM 22 , / 0 / 0 4 5aTaa D MMM (22  are the invariants of cyclic stresses.   2. FEM solution for plane stress problem  Let us regard the problem statement for structural element, made from material with anisotropic creep-damage properties, which is in plane stress state. The volume V is fixed in a surface part S1 and is loaded by traction р {р1, р2} on another surface part S2.  In co-ordinate system ОX1X2 the motion of material points under the creep conditions is described by use of Lagrange approach. The vectors of displacements u={u1, u2}T  and their rates v={v1, v2}T are introduced. In these assumptions creep problem for the case of small displacements and strains is described by following boundary – initial value problem:    ,, jijij vf CM     (i,j=1,2)   ;, 21 Vxx I  ,2sin ftppnaiijij M     221, Sxx I .  (10)   ;111 udtduv   222 udtduv    (11)   ,1111 xuD   ,2222 xuD   ,21122112##%!""$ xuxuD  (12)  The level of stresses which don’t exceed yield limit will be considered. Let us accept, that elastic eij and creep cij strain components as well as their rates are additive:    ,ijijij ce  D  (13)  Obtained system (10-13) has to be solved jointly with constitutive equations (8-9). Boundary conditions on surface parts  S1 :*ii uu    and  S2: ijij pn  M  as well initial ones at t=0 considering the elastic strain-stress distribution have to be added. Constitutive equations (8-9) are added to above system.   The system (8-13) is solved by use of two time scales method jointly with averaging in a period of forced vibration 1/f. Full description of the procedure can be found in [5,6]. The problem derives to the simplified one which is similar to problem of static loading, but with constitutive equations (8-9). Anisotropic creep-damage behavior is simulated by FEM software had been developed in NTU ’KhPI’. Triangular linear element is used. The problem which is formulated in rates is solved:    4 5 ,cFFK 1  (14)  where 4 5 4 5 4 54 5dVBDBKTee VF  is global stiffness matrix; 4 5 ,2``FdpNFTee  4 5Fe eTcVdVBF *M  are  the loading vectors from  traction  and forces which determined by creep strains.    D. Breslavsky, Yu. Korytko et al.  383  3.  Anisotropic creep-damage in titanium plates Developed method and software were used for creep-damage modeling in titanium plates made from BT1-0 alloy which is similar to IMI125 or T40 grades. The properties of creep and long term strength of the specimens made from the plane billet in three directions were experimentally obtained for temperature T=773K by O.Morachkovsky and V.Konkin [7]. The constants values which had been obtained after data processing are the following: ,10303,2 41111b  ,10151,1 41122b ,10058,2,10924,1 4121242222 bb  (МPа)-2N/N+1/h2/N+1; ,10771,1 51122d  ,10127,3,10324,3 5121252222 dd  (МPа)-2/(h)2/k; k=N=5, s=1. Comparison between experimental data at static loading for specimens had been cut in different directions with numerical simulations of rectangular plates in tension simulating the above test data were done at first in order to verify numerical method. A satisfactory agreement which doesn’t exceed 25-32% had been obtained for time variation of strains and time to rupture values. Table 1 contains some examples of these comparisons for time to rupture values for net stress 60 MPa.   Table 1.  Comparison of time to rupture values for static loading  Orientation of anisotropy axes, K Stress, MPa Experimental value of time to rupture t(, h Numerical value of time to rupture t(, h 0 60 24  25.9 45 60 19  20.5 90 60 28.7 19.7  Further the static and dynamic creep of titanium plate (0.8mx0.8m) with a central hole (radius is equal to 0.1m) were numerically simulated. One quarter of a plate was considered and stress-strain state and damage redistribution on time were analyzed. Fig.1 contains the distribution of damage measure  for the case of pure static loading by net stress 10MPa at the moment t=220h  which is nearest to the failure moment t*=221h. Essential stress concentration near the hole leads to failure in this area. The creep –damage behavior of the plate is strongly non-linear: increasing of the traction from 10 to 13MPa leads to decreasing of time to failure from 221h to 68.5h.    Figure 1. Damage parameter at t=220h D. Breslavsky, Yu. Korytko et al.  384  Let us present the results of dynamic creep simulation of the similar plate. The axial loading in this case consists from static part p0=16.9MPa and the cyclic one which is varying due to sine law with frequency 10Hz. Let us introduce the loading parameter L=p0/pa, where pa is amplitude value of the loading. The varying of L from 0 to 0.3 was analyzed.  The time to failure values in this loading program decreases with increasing of L from 0.166 h for L=0 to 0.122h for L=0.3. It happens for material with anisotropic creep-damage properties just the same in dynamic creep processes of elements with isotropic ones [6]. The character of stress and damage redistribution is similar for the static case, but both processes are run faster.  However, for smaller values of net stress, and, respectively, for smaller stress values in a plate, character of a process differs from above analyzed one. Dynamic loading accelerates the stress relaxation in area near the hole as well as in conventional dynamic creep [6], wherein the obtained stress level became so small, that rate of damage accumulation decrease significantly. For example, for p0=13MPa the value of time to failure for L=0 is equal to t*=68.5 h, but adding of small cyclic part (L=0.05) leads to it’s increasing and t*=72.6 h. Subsequent increasing of L leads to similar growth of time to failure values due to fast dynamic relaxation at first time of a process: for L=0.1 t*=120.1 h.   Conclusions An approach for the effective mathematical modeling of dynamic creep-damage processes in structural elements made from the titanium alloy with anisotropic properties is presented. The use of two time scale method with asymptotical expansions on the small parameter and subsequent averaging in a period allows us to avoid the direct integration through a cycle and simulate only static problems with flow rule and damage equation of special form. Effects of essential acceleration of strain and damage rate growth as well as stress relaxation due to dynamic creep had been established for plate with a hole made from material with anisotropic creep-damage properties.    References [1]    Murakami S. Notion of Continuum Damage Mechanics and Its Application to Anisotropic Creep Damage Theory J. Engng. Mater. Techn, Vol. 105, pp. 99-105, 1983. [2]   Cordebois J.P., Sidoroff F. Damage Induced Elastic Anisotropy Mechanical Behaviour of Anisotropic Solids (J. P. Boehler, editor), Colloque Euromech 115, Villard-de-Lans, June 19-22, Martinus Nijhoff Publishers, pp. 761- 774, 1979. [3]    Morachkovsky O.K., Pasynok M.A. Investigation of the influence of the obtained anisotropy due to precurcive creep on the creep of materials Proc. KhSPU , KhSPU, Kharkiv, Vol. 27, pp. 197-203, 1998 (in Russian). [4]   Taira S., Ohtani R. Theory of High Temperature Strength of Materials, Metallurgia, Moscow, 1986. (in Russian, translated from Japan). [5]    Breslavsky D., Morachkovsky O. A new model of nonlinear dynamic creep. IUTAM Symposium on Anisotropy, Inhomogenity and Nonlinearity in Solid Mechanics, Kluwer Academic Publishers, Dordrecht, pp. 161-166, 1995.  [6]    Breslavsky D., Morachkovsky O. Dynamic creep continuum damage mechanics: FEM-based design analysis. Computational Plasticity: Fundamentals and Applications. Proc. of the Fifth International Conference on Computational Plasticity held in Barselona, Spain, 17-20 March 1997, IMNE, Barselona IMNE, Part 1, pp.1071-1076, 1997. [7]    Konkin V.N., Morachkovsky O.K. Creep and long term strength of light weight alloys with anisotropic properties. Problems of Strength, Vol. 5, pp. 38-42, 1987. (in Russian)  

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Numerical Calculations of Creep Damage at Cyclic Loading by Use of Tensor Damage Parameter Model

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Numerical Calculations of Creep Damage at Cyclic Loading by Use of Tensor Damage Parameter Model

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