JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

42, 4, pp. 719-738, Warsaw 2004

MODELLING OF CREEP-DAMAGE IN THICK PLATE

SUBJECTED TO THERMO-MECHANICAL LOADING

CYCLES

Artur Ganczarski

Paweł Foryś

Institute of Applied Mechanics, Cracow University of Technology

e-mail: artur@cut1.mech.pk.edu.pl, pforys@cut1.mech.pk.edu.pl

This paper is an extension of the previous authors’ papers dealing with the
formulation of coupled thermo-creep-damage in 3D rotationally-symmetric
structures in the case of combined reverse cyclic mechanical and thermal lo-
ads. The thermo-damage coupling is described by the modified Fourier heat
flux equation, where the second-rank tensor of thermal conductivity with the
damage tensor as an argument is defined. The crack closure/opening effect is
incorporated by new effective stress definitions for tension or compression in
constitutive equations. It allows for description of incomplete damage deacti-
vation on reverse loading cycles by a new diagonal crack-closure second-rank
tensor.Damage evolution is governedby themixed isotropic/anisotropicMu-
rakamimodel,modified in order to eliminate non-uniqueness in description of
damage growth in the case of rotational symmetry. The creep process coupled
with damage is controlled by the Murakami-type equation adapted to rever-
se cyclic loads. The damage analysis in a 3D plate under thermo-mechanical
loadings, which consists in a simultaneous non-homogeneous temperature di-
stribution in the plate over the upper plate originated from the point heat
source positioned over the upper plate surface, and the uniform reverse cyclic
pressure, is presented as an example.

Key words: thermo-creep-damage coupling, 3D structure, thermo-mechanical
loading cycles

List of symbols

A1,A2,n1,n2,α – constants of Murakami’s anisotropic creep law
B,k,l – constants of Murakami’s anisotropic damage evolu-

tion law



720 A.Ganczarski, P.Foryś

Bijkl – fourth-rank transformation tensors between eigen
and current directions

D,D̃ – second-rank and fourth-rank damage tensor, respec-
tively

Dcrit – critical damage
E,G,ν – Young’s and Kirchhoff’s moduli, Poisson’s ratio
E – Hooke’s constitutive tensor
h – thickness of plate
h – crack closure second-rank tensor
H – Heaviside function
I – fourth-rank unit tensor
n – unit vector normal to wall

n
(i),νk – unit vector associated with ith or kth principal di-

rection, respectively
p – pressure
P – fourth-rank projection tensor
q̇v – intensity of external heat souce
R – radius of plate
t – time
t1 – cycle length
T,Tref,Tf,Ts – temperature, reference temperature, temperature of

transition layer and gas temperature of heat source,
respectively

u= [u,w] – vector of displacement in cylindrical coordinates
x – distance
x= [r,z] – vector of cylindrical coordinates
1 – second-rank unit tensor
α – coefficient of thermal expansion
β0 – coefficient of free convection
ε – strain tensor
λ0,λg – coefficient of thermal conductivity, thermal conducti-

city of gas
λ – second-rank tensor of thermal conductivity
η – damage isotropy to anisotropy ratio
χ(σ) – Hayhurst-type isochronous rupture function
σ,s – stress tensor and stress deviator
σ0,σeq – yield and effective stress
θ – temperature change
ζ – material parameter of unilateral damage response



Modelling of creep-damage in thick plate... 721

and (̃·) denotes quantity affected by damage; superscripts e,c,m,th refer to
the elastic, creep, mechanical or thermal part of the quantity, respectively;
(·)∗ – unilateral quantity in which the negative eigenvalues become partly or
completely inactive; (·)± – positive or negative eigenvalue of the quantity,
respectively; (·)I – Ith eigenvalue of the quantity; (·)eq stand for the effective
quantity; (·)ε,σ refer to a quantitywhose principal directions coincidewith the
principal directions of strain or stress, respectively.

1. Introduction

Thecrucial questionarisingwhenamaterial is subjected to reverse tension-
compression cycles is the proper description of a phenomenon of unilateral da-
mage, also called the damage deactivation or the crack closure/opening effect.
To this end, the decomposition of stress or strain tensors into positive and ne-
gative projections is usually used (cf. Ladeveze and Lemaitre, 1984; Litewka,
1991; Mazars, 1986; Krajcinovic, 1996). In the simplest one-dimensional case,
if a loading is reversed from tension to compression, cracks will completely
close such that the material behaves as uncracked, or in other words, its ini-
tial stiffness is recovered. In a three-dimensional case, modified stress or strain
tensors are used based on the concept of the Heaviside functions, where the
negative principal components are ruled out

ε
∗

I = 〈εI〉=H(εI) or σ
∗

I = 〈σI〉=H(σI) I =1,2,3 (1.1)

where 〈a〉 denote McAuley brackets

〈a〉=

{
a for a­ 0
0 for a< 0

The above means that the negative principal strain or stress components
become completely inactive in the further damage process as long as the lo-
ading condition can render them active again (cf. Litewka, 1991). The positive
parts of the strain or stress tensors can also be expressed by the use of fourth-
order positive projection operators written in terms of the principal directions

n
(i)
ε or n

(i)
σ (Krajcinovic, 1996; Hansen and Schreyer, 1995) as follows

P
+
ε =

3∑

i=1

〈〈ε(i)〉〉n(i)ε ⊗n
(i)
ε ⊗n

(i)
ε ⊗n

(i)
ε (1.2)



722 A.Ganczarski, P.Foryś

or

P
+
σ =

3∑

i=1

〈〈σ(i)〉〉n(i)σ ⊗n
(i)
σ ⊗n

(i)
σ ⊗n

(i)
σ (1.3)

The angular brackets are defined as (Lubarda et al., 1994)

〈〈a〉〉=

{
1 for a­ 0
0 for a< 0

(1.4)

whereas ε(i) and σ(i) are principal strain or stress components. Hence

ε
+ =P+ε : ε ε

− =P−ε : ε P
−

ε = I−P
+
ε (1.5)

or
σ
+ =P+σ :σ σ

− =P−σ :σ P
−

σ = I−P
+
σ (1.6)

where the corresponding negative or positive eigenvalues of the strain or stress
tensors are removed. A more realistic description of the damage deactivation
should allow for some influence of the negative principal components of the
strain or stress tensors for damage evolution, as observed in brittle materials
(cf. Murakami and Kamiya, 1997; Hayakawa and Murakami, 1997). For this
purpose, modified strain or stress tensors are defined in principal coordinate
systems as follows

ε
∗

I = 〈εI〉− ζ〈−εI〉= kε(εI)εI kε(εI)=H(εI)+ ζH(−εI) (1.7)

or

σ
∗

I = 〈σI〉− ζ〈−σI〉= kσ(σI)σI kσ(σI)=H(σI)+ ζH(−σI) (1.8)

The additionalmaterial parameter ζ in (1.7) and (1.8) describes the unilateral
damage response in such a way that for ζ = 1 the unilateral effect is not
accounted for, whereas for ζ = 0 the negative principal components do not
affect the damage growth.When the general coordinate systems are used, the
modified strain or stress tensors are defined in terms of the actual ones by the
following mappings

ε∗ij =
3∑

i=1

k(εI)n
(ε)
iI
n
(ε)
jI
n
(ε)
Ik
n
(ε)
Il
εkl =B

(ε)
ijkl
εkl (1.9)

or

σ∗ij =
3∑

i=1

k(σI)n
(σ)
iI
n
(σ)
jI
n
(σ)
Ik
n
(σ)
Il
σkl =B

(σ)
ijkl
σkl (1.10)



Modelling of creep-damage in thick plate... 723

where the fourth-rank tensors B
(ε)
ijkl
or B

(σ)
ijkl
are built of direction cosines be-

tween theprincipal andcurrent spatial systems (cf. SkrzypekandKuna-Ciskał,
2003). The above concept is more general than the pure positive and negati-
ve projection operators as defined by (1.2) through (1.6). In equations (1.7)
and (1.8), themodified strain and stress tensors account for both positive and
negative eigenvalues with appropriate weights 1 or ζ. Hence, the equivalent
mappingmay be furnished by the use of generalized projection operators

P
∗

ε =P
+
ε + ζεP

−

ε or P
∗

σ =P
+
σ + ζσP

−

σ (1.11)

and

ε
∗ =P∗ε : ε or σ

∗ =P∗σ :σ (1.12)

In the case when ζε = 1 or ζσ = 1, then P
∗

ε = I or P
∗

σ = I, such that the
unique mappings ε∗ = ε or σ∗ = σ hold. If, on the other hand, ζε = 0 or
ζσ =0, then P

∗

ε =P
+
ε or P

∗

σ =P
+
σ , and ε

∗ = ε+ or σ∗ =σ+ hold.

The limitations of the consistent unilateral damage condition applied to
continuum damage theories have been discussed by Chaboche (1992, 1993),
Chaboche et al. (1995). It was shown that in the existing theories developed
by Ramtani (1990), Ju (1989) or Krajcinovic and Fonseka (1981), either non-
symmetries of the elastic stiffness or non-realistic discontinuities of the stress-
strain responsemay occur in general multiaxial non-proportional loading con-
ditions. It is easy to show that if the unilateral condition does affect both the
diagonal and off-diagonal terms of the stiffness or compliance tensor, a stress
discontinuity takes place when one of the principal strains changes the sign
and the other remain unchanged (cf. Skrzypek andKuna-Ciskał, 2003). In the
model proposedbyChaboche (1993), onlydiagonal components corresponding
to negative normal strains are replaced by the initial (undamaged) values. The
consistent description of the unilateral effect was recently developed byHalm
and Dragon (1996, 1998). The authors introduced a new fourth-rank damage
parameter D̃, built upon the eigenvalues of eigenvectors of D, that controls
the crack closure effect with the continuity requirement of the stress-strain
response fulfilled

D̃=
3∑

k=1

Dknu
k⊗nuk⊗nuk⊗nuk (1.13)

where the crack-system of the normals nuk is considered open if the corre-
sponding normal strain is positive (strain-controlled effect).



724 A.Ganczarski, P.Foryś

2. Concept of effective stress accounting for unilateral damage

When the concept of projection operators is applied to constitutive mode-
ling of materials under thermo-creep-damage conditions, the unilateral dama-
ge effect may be controlled either by stress or by strain. In what follows, the
stress-controlled unilateral damage mechanism is applied.

The distinction between tension and compression in in general 3D case
requires the following decomposition of the stress tensor represented by its
eigenvalues

σ= diag{σ1,σ2,σ3}=σ
++σ− = 〈σ〉−〈−σ〉 (2.1)

When anisotropic damage is described by the second rank damage tensor
represented by its eigenvalues

D= diag{D1,D2,D3} (2.2)

and the eigendirections of damage do coincide with the eigendirections of
stress, then, in accordance with the concept of the effective stress, the uni-
lateral conditions are written by means of (1−D) for the positive term and
(1−D·h) for the negative termof (2.1), respectively, then the followingmodi-
fied effective stress for unilateral conditions written in the principal directions
(cf. Foryś and Ganczarski, 2002) holds

σ̃
∗ =±〈±σ̃±〉±

ν

1−2ν
(tr〈±σ̃±〉−〈±trσ̃±〉)1

σ̃
+ =
1

2
[σ · (1−D)−1+(1−D)−1 ·σ] (2.3)

σ̃
− =
1

2
[σ · (1−D ·h)−1+(1−D ·h)−1 ·σ]

in which h stands for the crack closure second-rank tensor represented by its
eigenvalues h= diag{h1,h2,h3}.

Note, that in the formulation presented above the effect of damage on the
Hooke constitutive tensor Ẽ(D) has been ignored. In the general case, ap-
proaching failure (Dimax ¬Dcrit), this effect may be essential and cannot be
neglected. It leads, however, to the necessity of application of damage induced
elastic anisotropy, instead of a simplified case of elastic isotropy in themecha-
nical state equations. The derivatives of the elastic tensor with respect to the
space coordinates must be taken into account.



Modelling of creep-damage in thick plate... 725

3. Equations of a thick plate subjected to thermo-mechanical

loading

In order to apply the above described unilateral damage concept, let us
examine the coupled thermo-creep-damage response of a 3D rotationally-
symmetric structure subjected to reverse thermo-mechanical loading cycles.

3.1. Assumptions

The displacement and temperature fields fulfil the general rotational sym-
metry

∂ϕu=0 ∂ϕT =0 (3.1)

It is postulated that the linearized total strain tensor ε=(∇u+u∇)/2 may
be decomposed into elastic, creep and thermal parts

ε= εe+εc+εth εth =αθ1 (3.2)

where θ=T −Tref and, additionally, the creep part is incompressible

trεc =0 (3.3)

3.2. Equations of the mechanical state

A coupled thermo-creep-damage formulation is used to describe evolution
of a 3D rotationally-symmetric structure subjected to combined thermal and
cyclic mechanical loads. The system of mechanical state equations in the di-
splacement formulation is used, where both the creep and temperature terms
are included

G∇2u+
G

1−2ν
grad(divu)=

Eα

1−2ν
gradθ+2Gdivεc

(3.4)

∇2 = grad(div)− rot(rot)

where the stress is defined as follows

σ=2G
[
ε−εc−αθ1+

ν

1−2ν
(trε−3αθ)1

]
(3.5)

It is convenient to decompose the displacement field into purely mechanical
and purely thermal parts u = um +uth. This leads to separation of the



726 A.Ganczarski, P.Foryś

problemdescribedbyEqs (3.4) into two sub-problems in the incremental sense

G∇2um+
G

1−2ν
grad(divum)= 2Gdivεc

(3.6)

G∇2uth+
G

1−2ν
grad(divuth)=

Eα

1−2ν
gradθ

3.3. Modified Fourier equation

The quasi-stationary Fourier equation coupled with damage is applied to
describe the damage effect on theheat flux.The effective second-rank tensor of
thermal conductivity contains the second-rank damage tensor as an argument

div(λ̃ · gradT)= 0 λ̃=λ0diag{1−D} (3.7)

where λ0 is the coefficient of thermal conductivity of the initially isotropicma-
terial. Only diagonal terms of λ̃ are taken into account (cf. Kaviany, 1995). In
this formulation, the effect of unilateral damage is not considered indefinitions
of the effective tensor of thermal conductivity.

4. Constitutive equations of creep and unilateral damage

A creep process coupled with unilateral damage is controlled by the
McVetty-type equation developed by Murakami. It includes the damage ef-
fective stress and is extended here to unilateral damage conditions

ε̇
c =
3

2
[A1σ

n1−1
eq αexp(−αt)s+A2σ̃

n2−1
eq s̃

∗] (4.1)

where the modified effective stress deviator

s̃
∗ = σ̃∗−

1

3
(trσ̃∗)1 (4.2)

includes the crack closure/opening effect Eqs (2.3).
Damage evolution is governed by the anisotropic Murakami model (Mu-

rakami et al., 1988), modified by Ganczarski and Skrzypek (2001) in order to
eliminate non-uniqueness of the damage growth description in case of rotatio-
nal symmetry

Ḋ=B[χ(σ)]k{tr[(1−D)−1(n(1)⊗n(1))]}l[(1−η)1+ηn(1)⊗n(1)]
(4.3)

η= η0
[
1−
(
1−
〈σ3〉

2〈σ2〉

)〈σ2〉+ 〈σ3〉
〈σ1〉

]



Modelling of creep-damage in thick plate... 727

The damage rate is expressed in (4.3) as a linear combination between the
isotropic criterion (η = 0) and the direction determined by the maximum
principal stress (η=1). In other words, the damage rate is controlled by the
reduction of the net area in a plane perpendicular to the direction n(1) of
the principal (tensile!) stress σ1 (exclusively!) combinedwith the reduction of
the isotropic area governedby theHayhurst-type isochronous rupture function
χ(σ) = aσ1 + btrσ+(1−a− b)σeq. For particular cases η = 0 and η = 1,
Eqs (4.3) reduces to purely isotropic damage evolution and purely orthotropic
microcrack growth in a plane perpendicular to themaximum tensile stress, re-
spectively, whereas for 0<η< 1 amixed isotropic/maximumprincipal stress
mechanism of damage growth occurs. Consistently, a simplified representation
of the Hayhurst function χ(σ)=σ1 is used.

5. Thermo-mechanical boundary problem

5.1. Mechanical boundary problem

Asimple support is assumed for the plate under cyclicmechanical load ±p
(Fig.1).This condition, however, essentially constraints the thermal transverse
expansion at the supported edge, so, for the thermal problem, separate boun-
dary conditions are proposed (Fig.2). Finally, two sets of boundary conditions
are used to solve themechanical problem separately for puerlymechanical and
purely thermal loadings (Table 1).
Table 1. Boundary conditions for purely mechanical and purely thermal

loads

Boundary Mechanical load Thermal load

top surface σz =±p σrz =0 σz =0 σrz =0

bottom surface σz =0 σrz =0 σz =0 σrz =0

symmetry axis umr =0 ∂ru
m
z =0 u

th
r =0 ∂ru

th
z =0

supported edge σr =0 u
m
z =0 σr =0 σrz =0

lower corner uthz =0

5.2. Thermal boundary problem

Assume the point heat source at a distance H above the upper surface.
Hence, the temperature distribution of a gas over the upper plate surface
is spherically symmetric. Assuming spherical symmetry of the temperature
front propagation (Fig.3a), one may find that the temperature distribution



728 A.Ganczarski, P.Foryś

Fig. 1. Boundary conditions for cyclic mechanical ±p load

Fig. 2. Boundary conditions for thermal T(r) load

of the gas neighbouring the top surface reveals rotational symmetry and is
given by the parabola Tf(x)=Ts− [q̇v/(6λg)]x

2, where Ts stands for the gas
temperature at the heat source, λg denotes thermal conductivity of the gas,
whereas x is a distance between the heat source and the particle of the gas
neighbouring the current point at the top surface.

Fig. 3. Temperature front propagation (a) and schematic heat flow through plate (b)

Aschematic heat flux through theplate is sketched inFig.3b. It is assumed
that the whole heat flux, which enters the upper plate surface, goes towards



Modelling of creep-damage in thick plate... 729

the supported edge, whereas at the lower surface adiabatic conditions hold.
The above assumptions allow one to formulate the mixed and the Neumann
integral type boundary conditions for Eqs (3.7) in a following form

upper surface β0(1−D ·n)(Tf −T)=−λ0(1−D ·n)∂nT

lower surface ∂nT =0

symmetry axis ∂nT =0

supported edge

∫

A

[λ0(1−D ·n)∂nT ] dA=C

(5.1)

where β0(1−D ·n) denotes the damage coupled coefficient of the Newtonian
convection between the gas and the upper surface of the plate and n is the
unit vector, normal to the wall. The heat flux that enters the top surface of
the plate and is sent through the supported edge is kept constant throughout
the process (C = const).

6. Numerical algorithm for thermo-creep-damage problem

To solve the initial-boundary problem by FDM, we discretize time by in-
serting N time intervals ∆tk, where t0 = 0, ∆tk = tk − tk−1 and tN = tI
(macrocrack initiation). Hence, the initial-boundary problems are reduced to
a sequence of quasistatic boundary-value problems, the solution to which
determines unknown functions at a given time tk, e.g., T(x, tk) = T

k(x),
u(x, tk) =u

k(x) with an appropriate thermo-elastic solution taken as initial
conditions. To account for primary and tertiary creep regimes, a dynamically
controlled time step ∆tk is required, the length of which is defined by the
boundedmaximum damage increment

∆Dlower ¬max
(x)
{‖Ḋ

k
(x)− Ḋ

k−1
(x)‖∆tk}¬∆D

upper (6.1)

Discretizing also the spatial coordinates x = [r,z]i,j, by inserting a mesh
∆r = rI − ri−1, ∆z = zj − zj−1, we rewrite equations of heat transfer (3.7)
andmotion (3.6) for the time step tk in terms of FDMwith respect of rI and
zj, respectively. Applying a stage algorithm, first for the damage [D]i,j ≡ 0,
equation of the thermal problem (3.7)with boundaryconditions (5.1) is solved
by making use of the procedure linbcg.for of Conjugate Gradient Method
for sparse system (Press et al., 1993), and the initial ”elastic” temperature



730 A.Ganczarski, P.Foryś

[Te]i,j is found. Then, the system of equations ofmotion (3.6) with the known
temperature field [Te]i,j as the right hand side and the boundary conditions
(cf. Table 1), is numerically solved by the same procedure, and the elastic
displacements are determined [ue]i,j as well. Next, the program enters the
creep-damage loop where the damage and strain rates [Ḋ, ε̇c]i,j are calcula-
ted. Repeating again the stage algorithm a solution to discretized thermo-
creep-damage problem, rates of temperature [Ṫ ]i,j and displacements [u̇]i,j
are computed. In the next time step, the ”new” temperature [T ]i,j and di-
splacements [u]i,j are found, and the process is continued until themaximum
value of damage reaches the critical level max[D]I =Dcrit. The failure crite-
rionmay be derived from the instability condition of the stress-strain relation
det(∂σ/∂ε)= 0.

7. Results

7.1. Data

Numerical resultspresented in thispaperdealwithplatesmadeof copperof
the following thermo-mechanical properties at temperature 523K (Murakami
et al., 1988): Tf0 = 300

◦C, Tref = 0
◦C, E = 60.24GPa, σ0 = 11.0MPa,

ν = 0.3, B = 4.46 · 10−13MPa−kh−1, l = 5.0, k = 5.55, ξ = 1.0,
A1 =2.40·10

−17MPa−n1, n1 =2.60, A2 =3.00·10
−16MPa−n2h−1, n2 =7.10,

α = 0.05h−1, α = 2.5 ·10−51/K, λ0 = 203W/m
◦C, β0 = 75W/m

2◦C. The
loading cycle is definedby the half-amplitude p=±0.1σ0 and the cycle length
t1 =2000h.Reverse, Heaviside-type pressure cycles are applied to the structu-
re, whereas temperature of the heat source is kept constant. The characteristic
parameter of the plate, i.e. thickness-to-diameter ratio is h/R=0.7. For sake
of simplicity, the effect of temperature changes on material constants, and of
the damage on elastic moduli are ignored. The parameter which controls the
type of damage isotropy-to-anisotropy ratio is η0 = 0.5, which means that
mixed isotropic/maximum principal stress mechanism of anisotropic damage
growth is considered.

7.2. Example

Plate response to mechanical reverse cyclic load ±p

Consider a plate subjected to pure mechanical reverse cyclic load
p = ±0.1σ0. A constant, temperature field T = 250

◦C is uniformly applied



Modelling of creep-damage in thick plate... 731

to the structure volume, such that the homogeneous thermal strain field does
not affect the stress field. The plate is subjected to symmetric, reverse uni-
form pressure loading cycles applied to the top surface of the plate (Fig.1).
The external heat source is not active in this case.

Following the symmetric loading cycles, the most damaged zones develop
at the top and bottom plate surfaces around the symmetry axis. The final di-
stribution of the dominant hoopdamage component Dϕ at failure is presented
in Fig.4.

Fig. 4. Zone of hoop damage Dϕ in plate under mechanical cyclic load ±p

The radial damage component Dr grows with a lower rate in this case.
Note, that the damage distributions at the top and bottom surfaces are not
strictly symmetric because the most damaged zone (bottom) corresponds to
the first downward loading cycle. The damage evolution at the bottom surface

centre D
(bottom)
ϕ , compared to the top surface D

(top)
ϕ is sketched in Fig.5.

The damage growth is observed only under the maximum tensile princi-
pal stress σ1, according to the simplified Hayhurst stress function χ(σ)=σ1
used in (4.3) for copper. Damage cumulation at the bottom surface precedes
that at the top one due to the downward first loading cycle. The displace-
ment response of the plate measured at the center point of themiddle surface
w(r = 0,z = 0) is schematically shown in Fig.6. Alternating displacement
amplitudes are nearly symmetric around zero, and no cyclic kinematic effect
is observed.



732 A.Ganczarski, P.Foryś

Fig. 5. Evolution of hoop damage Dϕ in plate under mechanical cyclic load ±p at
point of maximum damage

Fig. 6. Evolution of vertical displacement w(r=0,z=0) at middle surface in plate
under mechanical cyclic load ±p

Plate response to combined mechanical reverse cyclic ±p and damage enhanced ther-
mal T(r,D) loads

In the following considerations, the plate is subjected to simultaneous re-
verse symmetric pressure cycles p = ±0.1σ0 and affected the damage chan-
geable temperature field T(r,D) which is generated by the point heat source



Modelling of creep-damage in thick plate... 733

(Fig.3). The effect of damage on the thermal conductivity tensor λ̃(D) is
given by (3.7), where it is assumed that the elastic constitutive tensor E is
not affected by damage. In contrast to the previous example, the plate simply
heated by the point source is under ”semi-membrane” compressive stress, the
magnitude of which is the highest in the central part of the plate. Hence, the
tensile stress-controlled damage is localized in this case at the plate sidewall
around the bottom corner (cf. Ganczarski, 2001b). However, under a complex
loading, if the plate is simultaneously subjected to both non-uniform tempe-
rature field and mechanical loading cycles, an additional damage localization
zone appears at the top corner of the sidewall (Fig.7). Growth of the hoop
damage component at the top corner takes place when the mechanical and
thermal displacement responses coincide (−p in Fig.8).

Fig. 7. Zone of hoop damage Dϕ in plate under combined cyclic mechanical load
±p and damage enhanced thermal load T(r,D)

Inother case (+p inFig.8), thedamagegrowthat the topbecomespassive,
whereas at the bottom the damage growth due to the thermal stress is enhan-
ced by the additional mechanical effect (Fig.8). The damage accumulation
causes the essential decrease of the hoop component of thermal conductivity
(3.7) however the hoop direction is not active in the modified heat transfer
equation, whereas Dr =Dz = ηDϕ, and consequently, the change of the local
heat flux is rather small (Fig.9).

In the considered case, the displacement response of the plate to its loading
measured at the central point of themiddle surface w(r=0,z =0) is strongly
non-symmetric in away such that theplate curvature drops fromcycle to cycle
(Fig.10). It is mainly due to creep during positive (+p) loading cycles which
is faster than that during negative ones (+p). The process is terminatedwhen
the critical damage is met at the bottom sidewall corner, as seen in Fig.7.



734 A.Ganczarski, P.Foryś

Fig. 8. Evolution of hoop damage Dϕ in plate under mechanical cyclic load ±p and
damage enhanced thermal load T(r,D) at point of maximum damage

Fig. 9. Local heat flux −λ0(1−D) · gradT

8. Conclusions

• When thermo-mechanical loadings are considered, coupled mechanical
creep-damage and heat flux equations result in a damage induced aniso-
tropy. If cyclic thermo-mechanical loadings are considered, an additional
effect of damage deactivation must be included.



Modelling of creep-damage in thick plate... 735

Fig. 10. Evolution of vertical displacement w(r=0,z=0) at middle surface in
plate under combined cyclic mechanical load ±p and damage enhanced thermal

load T(r,D)

• A consistent formulation requires that the damage deactivation effect
influences not only the creep-damage constitutive equations but also the
damage induced anisotropic elasticity law and the anisotropic heat flux
equation of damaged materials (modified Fourier equation).

• The damage deactivation effect on the creep-damage equation is incor-
porated by the extended definition of the effective stress for unilateral
damage conditions (2.3).

• Theelasticity law for theunilateral damage conditions canbe introduced

by applying modified effective elastic moduli Ẽ
+
or Ẽ

−

for active or
passive damage conditions, respectively (Foryś and Ganczarski, 2002).

• Theheat flux equation for the unilateral damage conditions should allow
for not only a drop in the thermal conductivitywith damage growth, but
also for the conductivity recovery in crack closure phenomena.

References

1. Chaboche J.-L., 1992, Damage induced anisotropy: On the difficulties asso-
ciated with the active/passive unilateral condition, Int. J. Damage Mech., 1,
2, 148-171



736 A.Ganczarski, P.Foryś

2. Chaboche J.-L., 1993, Development of continuumdamagemechanics for ela-
stic solids sustaining anisotropic and unilateral damage, Int. J. DamageMech.,
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3. Chaboche J.-L., Lesne P.M., Moire J.F., 1995, Continuum damage me-
chanics, anisotropy and damage deactivation for brittle materials like concrete
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6. Ganczarski A., 2001b, Computer simulation of transient thermo-damage
fields in 3D structures,Proc. ECCM-2001 (on CDROM)

7. Ganczarski A., Skrzypek J., 2001, Application of the modified Muraka-
mi’s anisotropic creep-damage model to 3D rotationally-symmetric problem,
Technische Mechanik, 21, 4, 251-260

8. Halm D., Dragon A., 1996, A model of anisotropic damage by mesocrack
growth; unilateral effect, Int. J. Damage Mechanics, 5, 384-402

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Modelowanie uszkodzenia przy pełzaniu w grubych płytach poddanych

cyklicznym obciążeniom termomechanicznym

Streszczenie

Pracabędąca rozwinięciempoprzednichartykułówautorówjest poświęconasprzę-
żeniu efektów termicznych oraz pełzania z uszkodzeniem występującym w trój-
wymiarowych konstrukcjach obrotowo-symetrycznych poddanych działaniu cyklicz-
nie zmiennych złożonych obciążeń termomechanicznych. Sprzężęnie efektów termicz-
nych i uszkodzenia opisane jest zmodyfikowanym prawem Fouriera, w którym ten-
sor przewodności termicznej jest uzależniony od tensora uszkodzenia. Efekt otwiera-
nia/zamykania mikroszczelin jest uwzględniony w równaniach konstytutywnych po-
przezwprowadzenieoddzielnychdefinicji naprężenia efektywnegodla rozciągania i ści-
skania.Pozwala touwzględnićniezupełnądeaktywacjęuszkodzenia, opisanądiagonal-
nym tensoremzamknięciamikroszczelin, towarzyszącąnaprzemiennymcyklomobcią-
żenia. Ewolucja uszkodzenia podlega mieszanemu izotropowo/anizotropowemu pra-
wu typu Murakami, w którym wprowadzono modyfikację mającą na celu usunięcie



738 A.Ganczarski, P.Foryś

niejednoznaczności rozwiązania, która jest związana z opisem wzrostu uszkodzenia
w konstrukcjach o obrotowej symetrii. Sprzężenie procesów pełzania i uszkodzenia
podlega równaniom typu Murakami przystosowanym do opisu obciążeń cyklicznych.
Jakoprzykładzaprezentowanoanalizęuszkodzeniawtrójwymiarowejpłyciepoddanej
obciążeniom termomechanicznympolegającymna równoczesnymdziałaniu niejedno-
rodnego pola temperatury pochodzącego od punktowego źródła ciepła usytuowanego
ponad górną powierzchnią płyty oraz naprzemiennie zmiennego równomiernego ci-
śnienia.

Manuscript received March 1, 2004; accepted for print April 23, 2004