Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 4, pp. 949-971, Warsaw 2008

CONSTITUTIVE MODEL OF PLASTIC STRAIN INDUCED
PHENOMENA AT CRYOGENIC TEMPERATURES

Błażej Skoczeń

Cracow University of Technology, Institute of Applied Mechanics, Cracow, Poland

e-mail: blazej.skoczen@pk.edu.pl

FCCmetals and alloys are frequently used in cryogenic applications, ne-
arly down to the temperature of absolute zero, because of their excellent
physical andmechanical properties, including ductility. Thesematerials,
often characterized by the Low Stacking Fault Energy (LSFE), undergo
at low temperatures three distinct phenomena: Dynamic Strain Ageing
(DSA), plastic strain induced transformation from the parent phase γ
to the secondary phase α′ and evolution of micro-damage. As all three
phenomena lead to irreversible degradation of lattice and can accelerate
the process of material failure, a combined constitutive description ap-
pears to be fundamental for the correct analysis of structures applied
at very low temperatures. The constitutive model presented in the pa-
per takes into account all the three phenomena as well as the relevant
thermodynamic background.

Key words: constitutive behaviour, yield condition, phase transforma-
tion, voids and inclusions, cryogenic temperature

1. Introduction

FCC metals and alloys (such as copper, copper alloys or stainless steel) are
often applied in cryogenic conditions, down to the temperature in the proximi-
ty of absolute zero, because of their remarkable properties including ductility.
A broad class of these materials is characterized by the Low Stacking Fault
Energy (LSFE). Therefore, they undergo bothDynamic StrainAgeing (DSA)
and transformation from the parent phase γ to the secondary phase α′ at
extremely low temperatures. Both phenomena can be classified as material
instabilities associated on one hand with an oscillatory mode of plastic flow
(DSA effect) and, on the other hand, with a particular sensitivity to ine-
lastic strain (γ → α′ phase transformation). Thermodynamic conditions of



950 B. Skoczeń

DSAandplastic strain induced phase transformation are strictly linked to the
so-called thermodynamic instability related to vanishing specific heat when
the temperature approaches absolute zero. The DSA effect manifests itself
at the macroscopic level by the so-called discontinuous plastic flow (serra-
ted yielding), whereas the phase transformation converts the material from a
homogeneous to heterogeneous two-phase continuum. Both types of material
instability alternate in the vicinity of temperature T1 (material parameter)
that characterizes transition from screw dislocations to the edge dislocations
mode. Finally, both phenomena are accompanied by nucleation and evolution
of micro-damage fields, driven by inelastic strains that develop at very low
temperatures.

The present paper is focused on constitutive description of FCCmaterials
applied at very low temperatures. As an example, Fe-Cr-Ni austenitic stain-
less steels are commonly used tomanufacture components of superconducting
magnets and cryogenic transfer lines since they preserve ductility practically
down to 0K. The constitutive description addresses all the three above listed
phenomena driven by plastic strains at low temperatures: discontinuous (ser-
rated) yielding, strain induced γ→α′ phase transformation and evolution of
micro-damage. All of them are of dissipative nature and lead to irreversible
processes in the material lattice. Even if the metastable stainless steels have
been chosen in the present paper as a field of application of the constitutive
description, themodels presented in the course of the paper can be easily ad-
opted to describe othermaterials used at cryogenic temperatures (like copper,
copper and aluminium alloys, etc.). Their ductile behaviour down to 0K im-
plies evolution of plastic strain fields as soon as the stresses exceed the yield
point characteristic of a given temperature.

Stainless steels (typically: grades 304L, 316L, 316LN) applied at low tem-
peratures prove unstable both with respect to plastic flow and to γ → α′

phase transformation. Three distinct domains of the response of LSFEmate-
rials are identified for one of the most frequently applied materials: stainless
steel 316LN (cf. Obst and Nyilas, 1991). Domain I corresponds to the tempe-
rature range below T1 and to plastic flow instability called the discontinuous
or serrated yielding.Domain II stretches between T1 and Md, the latter being
the temperature above which the process of plastic strain induced γ → α′

phase transformation does not take place. Inside this domain the plastic flow
is smooth and accompanied by transformation from the parent γ phase to the
secondary α′ phase. The phase transformation leads to a significant increase
of the yield stress. Finally, domain III above the temperature Md is characte-
rized by smooth plastic flow and rather stable behaviour with respect to the



Constitutive model of plastic strain... 951

phase transformation. It is worth pointing out that the evolution of micro-
damage occurs in all three domains and is driven by stable or unstable plastic
flow. As all the above mentioned phenomena lead to irreversible degradation
of lattice and can accelerate the process of material failure, a combined con-
stitutive description is fundamental for correct prediction of the critical state
of the material.

2. Kinetics of plastic strain induced phenomena at cryogenic
temperatures

2.1. Kinetics of discontinuous (serrated) yielding (domain I)

At very low temperatures (below T0 or T1) and for sufficiently high stra-
in rate, discontinuous plastic flow (serrated yielding) is observed. The main
feature of serrated yielding consists in frequent abrupt drops of stress as a
function of strain duringmonotonic loading. Themechanism of discontinuous
yielding is related to formation of dislocation pile-ups at strong obstacles such
as the Lomer-Cottrell locks during the strain hardening process. The back
stresses of the piled-up groups block motion of newly created dislocations.
The local shear stress at the head of dislocation pile-up, proportional to the
number of dislocations in the pile-up,may reach the level of cohesive strength
and the Lomer-Cottrell lock may collapse by becoming a glissile dislocation.
This process takes place below the temperature T0 or T1, where the dislo-
cations have a predominant edge character and cannot leave the pile-up by
cross-slip.
Such a local catastrophic event can trigger similar effects in other groups of

dislocations. Thus, the final result ismassive andhas a collective character. At
low temperatures, where very high stresses are expected, this avalanche-like
process is followed by spontaneous generation of dislocations by the rapidly
increasing number of sources. This – in turn – leads to load drops observed in
the stress-strain curve.Duringa tensile test at low temperature, theavalanche-
like barrier crossing by dislocation pile-ups ismanifested by acoustic effects of
”dry” sounds emitted by the specimen. Periodic vibrations induced by a series
of catastrophic events can be investigated by an analytical method related to
impacting mechanical systems (cf. Palej and Nizioł, 1986). Each ”serration”
(sudden drop of stress as a function of time) is accompanied by a considerable
increase of temperature, related to dissipation of plastic power and ”thermo-
dynamic instability” described in the previous section. A typical stress-strain



952 B. Skoczeń

curve for thematerial that exhibits discontinuous yielding (stainless steel 316L
at 4.2K) is illustrated in Fig.1a. Every ”spike” in the stress-strain diagram
shows a similar pattern: after the initial elastic process (stage 1), smooth pla-
stic flow occurs (stage 2) until the abrupt drop of stress (stage 3) and further
relaxation (stage 4) take place (cf. Obst and Nyilas, 1998).

Fig. 1. (a) Serrated yielding in FCCmetals (316L at 4.2K). (b) RVE and dislocation
groups localized at Lomer-Cottrell locks

No significant increase of temperature is observed during the smooth pla-
stic flow. The temperature increases dramatically when the abrupt relaxation
of stress begins. During the process of stress relaxation, the temperature ri-
se induces a significant decrease of the yield point as the properties of FCC
materials are highly temperature dependent. With increasing temperature,
the dislocations become again more mobile and the further straining process
occurs at much lower stress levels.
The main function that reflects the readiness of lattice with respect to

discontinuous plastic flow is the cumulative volume of dislocation groups (lo-
cated at the Lomer-Cottrell locks) per unit volume of lattice. This para-
meter will be called the volume fraction of dislocation groups and denoted
by B

B=
dVB
dV

0¬B¬ 1 (2.1)

where dVB denotes the volume fraction occupied by dislocation groups and
dV stands for the total volume. TheRVE containing the dislocation groups is
shown in Fig.1b. It is assumed that the incremet of B strictly related to the
increment of accumulated plastic strain (Odqvist parameter)

dp=

√
2
3
dεp : dεp (2.2)



Constitutive model of plastic strain... 953

Increasing intensity of theplastic flowgeneratesmorebarriers formotionof the
dislocations. Therefore, the following kinetic law of evolution for the volume
fraction of the dislocation groups is postulated

Ḃ=FLC(T,σ)ṗH(p−pLC) (2.3)

where FLC is a function of temperature and the level of stress whereas pLC
represents a threshold above which the Lomer-Cottrell barriers massively de-
velop. For an isothermal process and small variation of theflow stress, a simple
linear representation is obtained

dB=FLCdp p­ pLC (2.4)

2.2. Kinetics ofphase transformationand relatedphenomena(domain II)

Theplastic strain induced γ →α′ phase transformation inmetastablema-
terials like stainless steels occurs in a wide range of temperatures below Md.
For instance, it can be easily activated at 77K in liquid nitrogen. The process
is controlled via the transformation kinetics, represented by the phase trans-
formation curve. Kinetics of the γ → α′ phase transformation, developed by
Olson and Cohen (1975) is reflected by a typical sigmoidal curve defining the
evolution of the martensite content ξ as a function of the plastic strain. Un-
der isothermal conditions and for a given strain rate, the classical sigmoidal
curve has the form shown in Fig.2b. At very low temperatures, the phase
transformation process can be subdivided into three stages: low rate trans-
formation below the threshold pξ (stage I), fast transformation with a high
and nearly constant transformation rate (stage II) and asymptotically vani-
shing transformation with the rate decreasing to 0 and the volume fraction of
martensite reaching a maximum ξL (stage III). If the plastic strain induced
phase transformation occurs at very low temperatures (typically liquid helium
4.2K or liquid nitrogen 77K), then the steep part of the transformation curve
(stage II) remains in the domain of relatively small strains (below 0.2). In
this case, the constitutive modelling can be considerably simplified and stays
within the scope of the classical rate-independent theory of plasticity.
A simplified evolution law for the volume fraction of martensite has been

introduced for the linearpart (II)of the sigmoidal curvebyGarionandSkoczeń
(2002)

ξ̇=A(T, ε̇p,σ)ṗH((p−pξ)(ξL− ξ)) (2.5)

where ṗdenotes the rate of the accumulated plastic strain.Here, ξ denotes the
volume fraction ofmartensite, A(. . .) is a function of temperature, stress state



954 B. Skoczeń

Fig. 2. (a) RVEwith α′-martensite inclusions andmicro-damage fields. (b) Volume
fraction of α′-martensite versus accumulated plastic strain p

and strain rate, pξ denotes the accumulated plastic strain threshold (to trigger
the formation of martensite), ξL stands for the martensite content limit and
H represents the Heaviside function. The phase transformation from FCC to
BCC lattice is drivenby the accumulated plastic strain obtained bymonotonic
or cyclic straining at low temperatures. For an isothermal process and small
variation of stress, again a simple linear representation is obtained

dξ=Adp p­ pξ ξ¬ ξL (2.6)

2.3. Kinetics of micro-damage evolution (domains I, II and III)

The classical kinetic law of micro-damage evolution (Chaboche, 1988; Le-
maitre, 1992) for isotropic andductile damagepostulates the following relation
between the damage rate and the accumulated plastic strain rate

Ḋ=
(Y
S

)s
ṗH(p−pD) (2.7)

where Y stands for the strain energy density release rate, S denotes the
strength energy of damage and pD stands for the damage threshold. Here, D
and Y form a pair of dual state variables, and S is a material modulus. The
driving force of damage evolution is the accumulated plastic strain. In the case
of anisotropic ductile damage, the damage parameter D (Fig.2) is replaced
by the damage tensor D (cf. Murakami, 1990) and the scalar function of the
strain energy density release rate Y is replaced by the relevant tensor Y (cf.
Lemaitre, 1992). Again, both D and Y form a pair of dual state variables.



Constitutive model of plastic strain... 955

In thepresentpaper, it is assumed that thedriving force of evolution of ani-
sotropic ductile damage remains the accumulated plastic strain. The strength
energy of damage S is replaced by the tensor C, which defines material pro-
perties in the principal directions of damage. Furthermore, it is assumed that
as soon as the damage threshold pD has been reached, damage starts deve-
loping driven by the increase of the accumulated plastic strain p. However,
the damage evolution is different along the principal directions described by
the eigenvectors of the tensor D. Thus, the kinetic law of damage evolution is
postulated in the following form (cf. Garion and Skoczeń, 2003)

Ḋ=CYC⊤ṗH(p−pD) (2.8)

which assures that the damage tensor is symmetric. In the direct notation,
Eq. (2.8) is equivalent to

Ḋij =CikYklCjlṗH(p−pD) (2.9)

Here, the tensor C has been imposed on the tensor Y with respect to the
index k, and the product has been again imposed on the tensor C⊤ with
respect to the index l. The tensor C is defined as follows

C=
3∑

i=1

Cini⊗ni (2.10)

and can be classified as a symmetric tensor containing the material moduli.
The kinetic law of damage evolution has been built as a direct extrapola-
tion of the isotropic reference case. The isotropic conjugate damage variables,
D and Y , can be obtained from the anisotropic state variables D and Y by
the following operations

Y = tr[Y] (2.11)

and, assuming that C=CI for an isotropic material, the first invariant of the
damage rate tensor reads

Ḋ= tr[Ḋ] =C2Y ṗ (2.12)

which corresponds to the isotropic damage evolution law with S = 1/C2. In
the simple isotropic case and for s=1, one obtains:

dD=
Y

S
dp p­ pD (2.13)



956 B. Skoczeń

3. RVE-based constitutive description of discontinuous yielding,
γ→α′ phase transformation and micro-damage evolution

3.1. Discontinuous (serrated) yielding (domain I)

The discontinuous plastic flow reflects the dynamic strain ageing effect
that occurs below the temperature T0 or T1. It has been explained (cf. Obst
and Nyilas, 1991) by the mechanism of rapid formation of dislocation pile-
ups at strong obstacles such as Lomer-Cottrell locks. The shear stress applied
to the head of dislocation pile-up is multiplied by a factor proportional to
the number of dislocations in the group. Simultaneously, the back stresses of
dislocation groups block movement of new dislocations. As soon as the stress
concentration at the leading edge is high enough, local catastrophic failure
of the Lomer-Cottrell (LC) locks occurs and triggers the avalanche process
that spreads over a larger portion of the material. Failure of LC locks leads
to massive motion of released dislocations as well as spontaneous generation
of dislocations by new sources, accompanied by a step-wise increase of the
strain rate. The initially microscopic process becomes macroscopic and leads
to load drops observed in the stress-strain curve. The local shear stress τ
is accompanied by the amount of crystallographic slip, denoted by γ. Let
us assume that the local dislocation density at a temperature T is denoted
by ρ. The evolution of dislocation density with deformation is described by
the following equation

dρ

dγ
=
dρ

dγ

∣∣∣
+
+
dρ

dγ

∣∣∣
−

(3.1)

where the component denoted by ”+” represents the rate of production of
dislocations and the component denoted by ”−” stands for the rate of an-
nihilation of dislocations (cf. Bouquerel et al., 2006). The production part is
expressed by the formula

dρ

dγ

∣∣∣
+
=
1
λb

(3.2)

where λ is the mean free path of dislocation and b denotes length of the
Burgers vector. The annihilation part is given by the following relation

dρ

dγ

∣∣∣
−

=−kaρ (3.3)

where ka represents the dislocation annihilation constant. Themean free path
of dislocation obeys the following rule

λ=
1

∑
iλ
−1
i

(3.4)



Constitutive model of plastic strain... 957

with λi denoting the mean free path related to a specific type of obstacle. In
the present paper, three types of obstacles will be taken into account: grain
boundaries, dislocations and LC locks

1
λ
=
1
d
+k1
√
ρ+k2

√
B (3.5)

where d is the average grain size and k1, k2 are constants. Combining Eqs.
(3.1) through (3.5), one obtains

dρ

dγ
=
1
db
+
k1
b

√
ρ+
k2
b

√
B−kaρ (3.6)

Assuming the following relations for the macroscopic stress and strain

σ=Mτ γ=Mε (3.7)

where M is the Taylor factor, and accepting that

ε≈ εp (3.8)

the following formula can be derived

dρ

dεp
=M

( 1
db
+
k1
b

√
ρ+
k2
b

√
B−kaρ

)
(3.9)

Making reference to Eq. (2.3) and assuming that

FLC = const ε
p ­ εpLC (3.10)

the volume fraction of dislocation groups reads

B=FLC(ε
p−εpLC) (3.11)

Thus, the evolution of dislocation density can be expressed as

dρ

dεp
=M

( 1
db
+
k1
b

√
ρ+
k2
b

√
FLC(εp−ε

p
LC)−kaρ

)
(3.12)

The average shear stress in the lattice is composed of lattice friction and
interaction between the dislocations

τ = τ0+µαb
√
ρ (3.13)

where µ is the shear modulus and α is the coefficient of dislocations interac-
tion. The shear stress at the head of dislocation pile-up (stress concentration
point) amounts to

τe =
π(1−ν)
µb

λτ2 (3.14)



958 B. Skoczeń

and is a quadratic function of the average shear stress in the lattice. Here, the
mean free path of dislocation can be interpreted as a distance between the
source and the barrier.
The following criterion of avalanche-like failure of LC locks is postulated

B=Bcr (3.15)

Thus, as soon as the volume fraction of dislocation groups reaches its critical
value, the avalanche-like process is triggered and – at least theoretically – all
the LC barriers are broken. The process of massive failure of LC locks results
in an instantaneous increase of the strain rate

ε̇= ε̇0+∆ε̇H[(ε−ε1)(ε2−ε)] (3.16)

where
ε2 >ε1 ∆εs = ε2−ε1 (3.17)

denotes the amount of slip during the catastrophic failure of dislocation bar-
riers. Assuming that the process is kinematically controlled, the following equ-
ation for relaxation of strains in the portions of material outside the slip zone
can be derived

ε= ε̃−∆εs (3.18)

where ε̃ is the value of macroscopic strain just before the catastrophic event.
Strain relaxation results in the proportional elastic drop of stress (Fig.3b)

∆σ=E∆εs (3.19)

with the residual stress after unloading equal to

σr = σ̃−E∆εs (3.20)

where σ̃ is the value of macroscopic stress just before the catastrophic slip.
During the massive failure of LC locks followed by fast motion of glissile

dislocations in the lattice, a quantity of heat is produced. The heat quantity
is a function of the plastic work and internal friction in the lattice when the
abrupt slip occurs

∆Q= η(∆Wp+∆Wf) (3.21)

Here, η denotes the relevant conversion factor. Following the curve of specific
heat under constant volume and assuming that temperature remains above
absolute zero, the following temperature increment occurs

∆T =
∆Q

mCV (T)
(3.22)



Constitutive model of plastic strain... 959

Fig. 3. (a) Initiation of discontinuous plastic flow. (b) Unloading outside the slip
zone

This increase of temperaturemay even beof the order of 40-50K.The tem-
perature becomes a driving force in the process of stress relaxation (stage 4)
down to a basic level determined by equilibrium conditions (cf. Zaiser and
Haehner, 1997). The process of stress relaxation is described by the following
set of equations

σ=σr+∆σT
d

dt
(∆σT)=

1
tT
[(σ∞−σr)−∆σT ] (3.23)

where σr,σ∞ are the stress level after the catastrophic slip and the asymptotic
strain rate sensitivity, respectively (stress level after the transients have died
out), tT denotes the characteristic time. The temperature relaxes towards a
new steady state value inducing an additional evolution of the stress ∆σT .
After the process of stress relaxation has been completed, another dynamic
strain ageing cycle begins.
Thefirst two stages of theprocess reflect elastic-plastic loadingundernear-

ly isothermal conditions, corresponding to low excitation of the lattice. Under
these circumstances (no thermal activation), the rate-independent plasticity
can be applied, at least until the catastrophic failure of LC locks. Thus, the
yield surface has the form

fy(σ,X,R)= J2(σ−X)−σy −R (3.24)

where

J2(σ−X)=
√
3
2
(s−X) : (s−X) (3.25)

is the second invariant of the stress tensor. Here, s, X denote the deviatoric
stress and the back stress tensors, whereas σy,R are the yield stress and the



960 B. Skoczeń

isotropic hardening variable, respectively. Furthermore, it is assumed that the
continuum containing LC locks obeys the associated flow rule

dεp =
∂fy
∂σ
dλ (3.26)

with the yield function postulated as the potential of plasticity. Thehardening
model is represented by the following equations

dX=
2
3
CXdε

p dR=CRdp (3.27)

where CX, CR denote the kinematic and the isotropic hardening moduli, re-
spectively. The evolution of parameter B can be computed from Eq. (2.3)

dB=FLC(T,σ)dp (3.28)

It is assumed that in every loading/unloading ”cycle” (single tooth in the
stress-strain curve) the parameter B is accumulated from 0 to Bcr. As soon
as condition (3.15) is fulfilled, the plastic flow instability (drop of stress) takes
place.

3.2. Plastic strain induced γ→α′ phase transformation (domain II)

The RVE-based constitutive model presented in the paper describes be-
haviour of ductile materials in which the phase transformation occurs. The
model is based on the following assumptions:

• two-phase continuum is composed of the austenitic matrix andmarten-
site platelets represented by small Eshelby type ellipsoidal inclusions,
randomly distributed and randomly oriented in the matrix,

• theausteniticmatrix is elasto-plastic, whereas the inclusions showpurely
elastic response (the yield stress of martensite fraction is much higher
than the yield stress of austenite),

• rate-independent plasticity is applied: it is assumed that the influence of
the strain rate ε̇p is small in the range of temperatures 2-77K, and the
function A(. . .) depends on the temperature and stress state only,

• small strains are assumed: the accumulated plastic strain p does not
exceed 0.2,

• mixed isotropic/kinematic hardening affected by the presence of mar-
tensite fraction is included,

• the two-phase material obeys the classical associated flow rule.



Constitutive model of plastic strain... 961

The constitutivemodel has been developed for a two-phase (γ+α′) isotro-
pic and ductile material (cf. Garion and Skoczeń, 2002). The kinetics of mar-
tensitic transformation has been already described in Section 2.2, Eq. (2.5).
Thegeneral constitutive law includesplastic, thermal and transformation stra-
ins

σ=E : (ε−εp−εth− ξεbs) (3.29)

where εp is the plastic strain tensor, εbs = (∆υI)/3 denotes the free de-
formation called the bain strain expressed in terms of the relative volume
change ∆υ, εth stands for the thermal strain tensor and E is the fourth-rank
elasticity tensor. It is assumed that the mesoscopic strain tensor εbs is obta-
ined by integrating the microscopic eigen-strain tensor εbsµ over the RVE (cf.
Garion et al., 2006)

ε
bs = ξ

1
3
∆υI (3.30)

For convenient description, it is assumed that the elastic stiffness tensor is
expressed by

E=3kJ+2µK (3.31)

where the tensors J, K are volumetric and deviatoric 4th rank projectors,
respectively

J=
1
3
I⊗ I K= I−J (3.32)

In the standard notation, the above equations are equivalent to

Jijkl =
1
3
δijδkl Iijkl =

1
2
(δikδjl+ δilδjk) (3.33)

Here, symbol ⊗ denotes the dyadic product, δij is the Kronecker delta and
k, µ denote the bulk and the shear moduli, respectively. Themodel of plastic
flow is again based on the rate-independent plasticity, as indicated in Section
3.1, Eqs. (3.24)-(3.27). The hardening model is represented by the following
equations

Ẋ=
2
3
CXε̇

p =
2
3
g(ξ)ε̇p Ṙ=CRṗ= f(ξ)ṗ (3.34)

Since the hardening variables R and X are affected by the presence of mar-
tensite, the corresponding evolution laws are postulated in the following in-
cremental form

dX= dXa+dXa+m =
2
3
C(ξ)dεp+G(ξ)dεp =

2
3
g(ξ)dεp

(3.35)

dR= f(ξ)dp



962 B. Skoczeń

It is assumed that the back stress increment is composed of the classical term
which corresponds to the behaviour of the austenitic phase dXa in the presen-
ce of localized small inclusions, uniformly distributed and randomly oriented
in the RVE and a term related to the combination of austenite and marten-
site via the homogenization algorithm (dXa+m). Furthermore, it is assumed
that themechanism of plastic flow at low temperatures is based onmotion of
dislocations in the lattice. If massive motion of dislocations occurs, they are
stopped by themartensite inclusions and the corresponding local stress fields
(Fig.4a).

Fig. 4. (a) Interaction between dislocation and inclusions – the Orowanmechanism.
(b) The principle of homogenization based on the local tangent stiffness moduli

Theprincipal components that constitute the two-phasematerialmodelare
the elasto-plastic matrix (austenite) and the highly localized elastic inclusions
(martensite platelets). The linear kinematic hardening law is applied tomodel
the plastic behaviour of pure austenite

dXa0 =
2
3
C0dε

p (3.36)

where C0 is the hardening modulus of the original non-transformed pure-
ly austenitic phase. For the two-phase transformed material, the hardening
modulus C0 is replaced by the modulus C that is higher than C0 because
of the interactions between the dislocations in the austenite and martensite
inclusions

C =C0ϕ(ξ) for 0¬ ξ¬ ξL
(3.37)

ϕ(0)=1



Constitutive model of plastic strain... 963

It can be easily shown (cf. Skoczeń, 2007) that the shear stress necessary for
dislocation to pass across two inclusions of the average size d, separated by the
distance l, depends roughly linearly on the volume fraction of martensite ξ

τp =
µb

d
3

√
6ξ0
π

(
1+
ξ− ξ0
3ξ0

)
(3.38)

where µdenotes the shearmodulus of austenite, b is the length of theBurgers
vector and ξ0 stands for the initial volume fraction of inclusions. Here, for
the sake of simplicity, an assumption has been made that the average size
of inclusions d is constant and much smaller than the distance between two
inclusions d≪ l. In the light of Eq. (3.38), the function ϕ(ξ) takes the linear
form

ϕ(ξ) =hξ+1 (3.39)

where h is a material-dependent parameter. The function ϕ(ξ) can be inter-
preted as that part of the hardeningprocesswhich corresponds to the increase
in volume fraction of martensite, and an enhanced probability that the dislo-
cation will stack on the inclusion. The back stress increment corresponding
to the behaviour of austenite in the presence of highly localized martensite
inclusions can be decomposed in the following way

dXa = dXa0+dXaξ =
2
3
C0dε

p+
2
3
C0hξdε

p =
2
3
C(ξ)dεp (3.40)

where dXaξ corresponds to the interactions between dislocations in the auste-
nitic matrix andmartensite inclusions.
The second contribution to the hardeningmodel is based on the principle

of homogenization applied on the step-by-step basis to the current linearized
tangent stiffnessmoduli of thematrix and the inclusions (Fig.4b). As thema-
trix (γ-phase) is elastic-plastic, the relevant local linearized tangent stiffness
tensor is derived. The inclusions are assumed to be ellipsoidal in shape and
elastic, therefore, the elastic tangent stiffness is applied. The process of step-
by-step homogenization based on the local tangent stiffnessmoduli follows the
concept introduced by Hill (1965).
For the pure austenitic phase, a linearization of the stress/strain relations

in the vicinity of the current state is expressed by

∆σa =Et :∆ε (3.41)

where Et is the tangent stiffness tensor. A similar principle can be applied to
a two-phase continuum. However, the tangent stiffness tensor is obtained by



964 B. Skoczeń

the homogenization process

∆σa+m =EH :∆ε (3.42)

The additional hardening increment induced by the presence of martensite is
given by

∆σ=∆σa+m−∆σa =(EH −Et) :∆ε (3.43)

Here, the same ”trial” strain increment has been assumed for pure austenite
and for the homogenized two-phase continuum in order to compute themacro-
scopic stress response in the case of a strain controlled process. In the present
paper, mainly the kinematically controlled γ →α′ phase transformation pro-
cesses is analysed. The local linearized stiffness of austenite can be described
by the following tangent stiffness tensor

Eta =3ktaJ+2µtaK (3.44)

where

µta =
Et

2(1+ν)
kta =

Et
3(1−2ν)

Et =
EC

E+C
(3.45)

As the tangent operator for plastically active processes contains a dyadic squ-
are product of the vector normal to the yield surface in the stress space which
makes the operator anisotropic, a linearization based on the quasi-isotropic
operator is particularly justified in the case when the absolute values of the
principal stresses are close to each other. When compared to the matrix, the
inclusions are isotropic and elastic (their yield point is much higher than for
pure austenite) and the corresponding elastic stiffness tensor is given by

Em =3kmJ+2µmK (3.46)

where
µm =

E

2(1+ν)
km =

E

3(1−2ν)
(3.47)

The inclusionsare assumedellipsoidal anduniformlydistributed in thematrix.
Application of the Mori Tanaka homogenization scheme (cf. Garion et al.,
2006) yields

EH =EMT =3kMTJ+2µMTK (3.48)

with EMT obtained from the following relation

[EMT +E
∗]−1 =

∑

i=a,m

fi[Ei+E
∗]−1 (3.49)



Constitutive model of plastic strain... 965

where fi is the volume fraction of the component i and E
∗ stands for theHill

influence tensor. It is assumed that the strain increment is mainly due to the
plastic strains: ∆ε∼=∆εp. Thus, Eq. (3.46) becomes

∆σ= [EMT −Et] :∆ε
p (3.50)

The plastic strains are represented by a deviatoric tensor, therefore

J :∆εp = 0 K :∆εp =∆εp (3.51)

Finally, the additional stress increment due to the presence of martensite in
the austenitic matrix is equal to

∆σ=2(µMT −µta)∆εp (3.52)

If pure kinematic hardening is considered, the evolution of the back stress for
the two-phase continuum obeys the following equation

∆Xa+m =∆σ (3.53)

or in the incremental form

dXa+m =2(µMT −µta)dεp (3.54)

On the other hand, if pure isotropic hardening is considered, the evolution of
the hardeningparameter is obtained by imposing a suitable normon the stress
tensor

∆R=∆Ra+m =
2
3
J2(∆σ)=

2
3

√
3
2
∆σ :∆σ=2(µMT −µta)∆p (3.55)

where ∆p=
√
2(∆εp :∆εp)/3. In the incremental form, one obtains

dR= dRa+m =2(µMT −µta)dp (3.56)

Thus, for a unidirectional process of monotonic loading, the stress increments
corresponding to the same increment of plastic strain are identical both for the
kinematic hardening and for the isotropic hardening models. This linearized
approach to the evolution of isotropic hardening is replaced by amore general
nonlinear formulation, suitable for a greater martensite content

dR=(R∞(ξ)−R)dp (3.57)

where R∞ is the parameter that defines the ultimate size of the yield surface

R∞(ξ)= 2(µMT −µta) (3.58)



966 B. Skoczeń

Eq. (3.57) can be reduced to Eq. (3.56) in the vicinity of the initial state.
In order to establish a proportion between the kinematic and the isotropic

hardening, the Baushinger parameter β has been introduced (after Życzkow-
ski, 1981)

β=
σ′+σ

′
−

2(σ′−σ0)
0¬β¬ 1 (3.59)

where σ′ denotes the stress level at unloading and σ
′
− is the stress level

corresponding to the reverse active process. The parameter varies between 0
for the isotropic hardening (no Bauschinger effect) and 1 for the kinematic
hardening (perfect Bauschinger effect).
Finally, the mixed hardening is described by the following model

dX=
2
3
CXdε

p =
2
3
[C(ξ)+3βb(ξ)(µMT −µta)]dεp

(3.60)

dR=CRdp= b(ξ)(1−β)(R∞(ξ)−R)dp

The relaxation term b(ξ) = 1− ξ has been added in order to compensate for
the assumption that themartensite inclusions are elastic, whereas – in reality
– they behave in an elastic-plastic way. The process of phase transformation
induced strain hardening stops when ξ=1.

3.3. Evolution of micro-damage (domains I, II and III)

Let us assume that at a given point in the RVE (Fig.2) a local set of unit
base vectors ni, tangent to the principal directions, has been defined. The
damage tensor is postulated in the following form (cf. Murakami, 1990)

D=
3∑

i=1

Dini⊗ni (3.61)

where ni stands for the base vector associated with the principal direction i,
and Di denotes the componentof thedamage tensor related to thedirection i.
It is defined by

Di =
dSDn

i

dSni
(3.62)

where SDn
i
is the area of damage in the section Sni represented by the nor-

mal ni. The effective stress σ̃ is supposed to obey the strain equivalence
principle (cf. Lemaitre, 1992)

σ̃=E : εe (3.63)



Constitutive model of plastic strain... 967

For ductile materials, with a limited production of transverse damage, the
strain equivalence principle turns out to be a good approximation of the real
behaviour and does not lead to major errors. The relation between the stress
and the effective stress tensors is postulated in the following form

σ=
1
2
[(I−D)σ̃+ σ̃(I−D)] (3.64)

with I being the identity tensor. The general relationship between the stress
and the effective stress tensors reads

σ̃=M−1 :σ σ=M : σ̃ (3.65)

where M stands for the symmetric damage effect tensor that depends on the
damage state. and fulfils the conditions

Mijkl =Mjikl =Mklij (3.66)

In its general form, the damage effect tensor is expressed by

M=
1
4
[(I−D)⊗I+(I−D)⊗I+ I⊗(I−D)+ I⊗(I−D)] =

(3.67)

=
1
2
I−
1
4
[D⊗I+D⊗I+ I⊗D+ I⊗D]

where a⊗b= aijbkl, a⊗b= aikbjl, a⊗b= ailbjk or in direct notation

Mijkl =
1
2
(δikδjl+ δilδjk)−

1
4
(Dikδjl+Dilδjk+ δikDjl+δilDjk) (3.68)

The strain energy density release rate tensor is defined by

Y=−ρ
∂Ψ

∂D
(3.69)

where the Helmholtz free energy state potential coupled to damage can be
written in the following form

Ψ =
1
ρ

(1
2
ε
e :M :E : εe

)
+Ψp (3.70)

Here, Ψp is the plastic part that does not explicitly depend upon D. Eqs.
(3.69) and (3.70), combined together, lead to the following representation

Y=
1
4
[εe(E : εe)+(E : εe)εe] (3.71)



968 B. Skoczeń

The potential of dissipation is postulated in the following form (cf. Garion,
Skoczeń, 2003)

Φ=
1
2
(CYC⊤) :Y

√√√√ (̃s− X̃) :L : (̃s− X̃)
(̃s− X̃) : (̃s− X̃)

(3.72)

where s̃ denotes the effective deviatoric stress, C is the tensor of material
properties and

L(D)=M−1(D)M−1(D) (3.73)

or in the direct notation

Lijkl =M
−1
ijmnM

−1
mnkl

(3.74)

Here, Φ is a quadratic function of the conjugate force Y, and is related to the
damage rate by

Ḋ= λ̇
∂Φ

∂Y
(3.75)

The kinetic law of damage evolution can be derived directly from the above
equation

Ḋ=CYC⊤ṗH(p−pD) (3.76)

where pD denotes the damage threshold and C represents the texture-induced
anisotropy of damage. Finally, the anisotropic damage evolution law becomes
a generalization of the classical isotropic damage theory.
The general constitutive law includes again plastic, thermal and transfor-

mation strains
σ̃=E : (ε−εp−εth− ξεbs) (3.77)

As the model is based on the rate-independent plasticity, the yield surface
takes the form

fy(σ̃,X̃, R̃)= J2(σ̃− X̃)−σy − R̃ (3.78)

where

X̃=M−1 :X R̃=
R

1−
√
D :D

(3.79)

are the effective kinematic and isotropic hardening variables, respectively. As
already stated, it is assumed that – in the case of phase transformation – the
two-phase continuum obeys the associated flow rule

dεp =
∂fy(σ̃,X̃, R̃)
∂σ̃

dλ (3.80)



Constitutive model of plastic strain... 969

Finally, the hardeningmodel is represented by the following equations

dX̃=
2
3
CXdε

p =
2
3
g(ξ)dεp

(3.81)

dR̃=CRdp= f(ξ)dp

where f(ξ), g(ξ) were defined in the previous section.

4. Conclusions

The constitutive model presented in the paper results from identification of
three fundamental phenomena that occur at very low temperatures in struc-
tural materials characterized by low stacking fault energy:

• dynamic strain ageing reflected by discontinuous (serrated) yielding,

• plastic strain induced transformation from the parent phase γ to the
secondary phase α′, characteristic of meta-stable materials,

• evolution of micro-damage (micro-voids and micro-cracks) reflected by
the decreasing elasticity modulus in the course of deformation.

All the three phenomena lead to irreversible degradation of lattice and acce-
lerate the process of material failure. Therefore, a combined constitutive de-
scription is fundamental for correct dimensioning of structures applied at very
low temperatures like superconductingmagnets or cryogenic systems (Skoczeń
andWróblewski, 2007;Egner andSkoczeń, 2007).Theconstitutivemodel takes
into account the relevant thermodynamic background related to mechanisms
of heat transport in weakly excited lattice at very low temperatures.
It is worth pointing out that the combined model is attractive in view

of its simplicity and a relatively small number of parameters to be identified
at cryogenic temperatures. The experiments carried out in liquid helium or
liquid nitrogen are laborious, expensive and usually require complex cryogenic
installations tomaintain stable conditions (constant or variable temperature).
Therefore, any justified simplification leading to reduction of the number of
parameters to be determined is of great importance.

Acknowledgements

Grant of Polish Ministry of Science and Higher Education: PB4T07A02730
(2006-2008) is gratefully acknowledged.



970 B. Skoczeń

References

1. Bouquerel J., Verbeken K., de Cooman B.C., 2006, Microstructure-
based model for the static mechanical behaviour of multiphase steels, Acta
Materialia, 54, 1443-1456

2. Chaboche J.L., 1988a, Continuum damagemechanics: Part I – General con-
cepts, Jour. Applied Mechanics, 55, 59-64

3. Chaboche J.L., 1988b, Continuum damage mechanics: Part II – Dama-
ge growth, crack initiation and crack growth, Jour. Applied Mechanics, 55,
64-72

4. GarionC., SkoczeńB., 2002,Modeling of plastic strain inducedmartensitic
transformation for cryogenic applications, Journ. of Applied Mechanics, 69, 6,
755-762

5. GarionC., SkoczeńB., 2003,Combinedmodel of strain inducedphase trans-
formation and orthotropic damage in ductile materials at cryogenic tempera-
tures, Int. Journ. Damage Mech., 12, 4, 331-356

6. GarionC., SkoczeńB., Sgobba S., 2006,Constitutivemodelling and iden-
tification of parameters of the plastic strain inducedmartensitic transformation
in 316L stainless steel at cryogenic temperatures, Int. Journ. of Plasticity, 22,
7, 1234-1264

7. Egner H., Skoczeń B., 2007,Model of anisotropic damage in ductile mate-
rials at cryogenic temperatures,Proc. Conf. Thermal Stresses TS’07, Taiwan

8. Hill R., 1965, A self consistent mechanics of composite materials, J. Mech.
Phys. Solids , 13, 213-222

9. Lemaitre J., 1992,A Course on Damage Mechanics, Springer-Verlag, Berlin
and NewYork

10. Mori T., Tanaka K., 1973, Average stress in matrix and average elastic
energy ofmaterials withmisfitting inclusions,ActaMetall. Mater., 21, 571-574

11. Murakami S., 1990, A continuum mechanics theory of anisotropic damage,
In: Yielding, Damage, and Failure of Anisotropic Solids, Proceedings of the
IUTAM/ICM Symposium, EGF Publication 5, 465-482

12. Obst B., Nyilas A., 1991, Experimental evidence on the dislocation me-
chanism of serrated yielding in f.c.c. metals and alloys at low temperatures,
Materials Science and Engineering,A137, 141-150

13. Obst B., Nyilas A., 1998, Time-resolved flow stress behavior of structural
materials at low temperatures,Advances in Cryogenic Engineering (Materials),
44, PlenumPress, NewYork, 331-338

14. Olson G.B., Cohen M., 1975, Kinetics of strain-induced martensitic nucle-
ation,Metallurgical Transactions, 6A, 791-795



Constitutive model of plastic strain... 971

15. Palej R., Nizioł J., 1986, On a direct method of analyzing impacting me-
chanical systems, J. of Sound and Vibration, 108, 2, 191-198

16. Skoczeń B., 2004,Compensation Systems for Low Temperature Applications,
Springer-Verlag, Berlin, Heidelberg, NewYork

17. Skoczeń B., 2007, Functionally graded structural members obtained via the
low temperatures strain induced phase transformation, Int. Journ. Sol. Struct.,
44, 5182-5207

18. Skoczeń B., Wróblewski A., 2007, Thermo-mechanical stress and strain
fields resulting from the plastic strain induced phase transformation at cryoge-
nic temperatures,Proc. Conf. Thermal Stresses TS’07, Taiwan

19. ZaiserM., Haehner P., 1997,Oscillatorymodes of plastic deformation: the-
oretical concepts,Phys. Stat. Sol. B, 199, 267-330

20. Życzkowski M., 1981,Combined Loadings in the Theory of Plasticity, PWN
– Polish Scientific Publishers,Warsaw, Poland

Opis konstytutywny zjawisk wywołanych odkształceniami plastycznymi
w temperaturach kriogenicznych

Streszczenie

Materiały konstrukcyjne (metale lub ich stopy) o sieci krystalicznej regularnej
ściennie centrowanej są często stosowane w temperaturach kriogenicznych (także
w temperaturach bliskich absolutnego zera) z uwagi na bardzo dobre własności fi-
zyczne i mechaniczne, a w szczególności wysoką plastyczność. Materiały te, charak-
teryzujące się z reguły niską energią błędu ułożenia, podlegają w niskich tempera-
turach trzem procesom: dynamicznemu starzeniu odkształceniowemu (które generuje
nieciągłe płynięcie plastyczne), przemianie fazowej od struktury regularnej ściennie
centrowanej do struktury regularnej przestrzennie centrowanej oraz ewolucji mikro-
uszkodzeń. Wszystkie te zjawiska prowadzą do nieodwracalnej degradacji sieci kry-
stalicznej i szybszego zniszczenia materiału, dlatego budowamodelu konstytutywne-
go staje się nieodzownym elementem projektowania konstrukcji pracujących w tem-
peraturach kriogenicznych. Model konstytutywny przedstawiony w artykule ujmuje
wszystkie wyżej wymienione zjawiska, jak również termodynamikę procesów zacho-
dzących w sieci krystalicznej w niskich temperaturach.

Manuscript received February 7, 2008; accepted for print July 10, 2008