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Introduction to Creep Mechanics
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DOI: 10.1007/978-3-662-53605-6_155-1
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Chapter 1
Introduction to Creep Mechanics
Holm Altenbach and Johanna Eisenträger
Synonyms
Mechanics of time-dependent material behavior
Definitions
• Creep mechanics is a branch of continuum mechanics taking into account the
time-dependent material behavior like non-reversible deformations under the influence of constant mechanical stresses, which is named “creep”. Creep can occur
as a result of long-term exposure to certain levels of stresses that are still below
the yield stress of a material. Furthermore, creep is frequently observed in materials during long-term exposure to heat. The tendency to creep increases while
approaching the melting point of the material.
• The phenomenon creep describes the increase in deformation with time under
constant loads. A further time-dependent effect is relaxation, where the continuous decrease of the stress level occurs in a material subjected to constant prescribed strains at elevated temperatures.
• Note that the time-dependent inelastic deformation of structures is also strongly
connected to stress redistribution.
• Furthermore, in contrast to creep in homogeneous bulk materials, where uni-axial
stress states prevail, creep in structures results in multi-axial stress states, which
should be accounted for by a constitutive model.
Holm Altenbach · Johanna Eisenträger
Institut für Mechanik, Fakultät für Maschinenbau, Otto-von-Guericke-Universität Magdeburg,
Universitätsplatz 2, 39106 Magdeburg, Germany e-mail: holm.altenbach@ovgu.de,
johanna.eisentraeger@ovgu.de
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2
Holm Altenbach and Johanna Eisenträger
1.1 Introduction and Historical Remarks
Creep mechanics focuses on modeling the mechanical behavior of engineering components made of metals or alloys at elevated temperatures (0.3–0.7 of the melting
temperature Tm of the given material). In this case, inelastic deformation under sustained loads (below the yield stress σy ) occurs. This phenomenon can change the
mechanical behavior significantly and reduce the lifetime of components.
In addition, creep also occurs in other materials, for example, in polymeric materials (plastics, rubber), ceramics, concrete, composites, etc. In these cases, the
temperature level is usually significantly lower, but sometimes also much higher
compared to metals and alloys. This results also in different time scales. It is worth
noting that the International Union of Theoretical and Applied Mechanics organized a symposium on ”Creep in Structures”, which takes place every ten years
starting from 1960, cf. (Hoff, 1962; Hult, 1972; Ponter and Hayhurst, 1981; Zyczkowski, 1991; Murakami and Ohno, 2001). The establishment of this symposium
is due to the fact that creep tests for metals and alloys are long-term experiments. In
2000, research on the creep behavior of other materials was presented as well during
this symposium (Murakami and Ohno, 2001), whereby also short-term experiments
were taken into account.
A special branch of creep mechanics is devoted to problems of inelastic timedependent deformations of ice. In this case, the temperature level when creep can
be observed is below 0◦ C. Note that in Weertman (1983) experimental results and
general ideas are presented for this particular type of creep. Due to the development of scientific instruments based on the principle of superconductivity in the
past decades, the time-dependent material behavior at extremely low temperatures
is in the focus of researchers, which is discussed for example by Skoczeń (2018).
The starting point of creep mechanics was in the second half of the 19th century after some unexpected failures of steam machines. The first investigations were
made with a focus on metals and alloys. At the beginning of the 20th century, results
were summarized for the first time in da Costa Andrade (1910, 1914). In 1910, da
Costa Andrade established the terminology still used up to the present and introduced the concept of primary, secondary, and tertiary creep for uni-axial creep tests
under constant load or stress.
As a consequence of demands of technology, approximately 20 years later an
increasing interest in better knowledge of creep phenomena arose. Norton (1929)
investigated the power law for secondary creep
ε̇ ≡
dε
= Aσ n
dt
with respect to applications for various metals. The introduced creep law ( where
ε denotes the creep strain, ε̇ the corresponding rate, and σ the uni-axial stress) is
very simple and contains only two material parameters. At the same time (here, we
provide the references to two later publications, Bailey, 1935, 1947), it was shown
that similar to time-independent plasticity, creep deformations of structural metals
1 Introduction to Creep Mechanics
3
take place under constant volume and the superimposed hydrostatic pressure does
not influence creep.
In 1934, multi-axial creep equations were suggested by F. K. G. Odqvist (see
Odqvist, 1974, among others)
ε̇ε = f σeq s
with the strain rate tensor ε̇ε and the equivalent stress in the von Mises sense
3
2
σeq
= s ··ss.
2
Here, s ··ss denotes the double scalar product of the stress deviator s . For the conformity with Norton’s law, the following expression can be used for the scalar function f :
3 n−1
f (σeq ) = Aσeq
.
2
Odqvist’s theory is similar to the von Mises theory for time-independent plasticity.
The approach is straightforward to formulate since one only needs two parameters
obtained through a one-dimensional creep test and it can be extended to primary and
tertiary creep by introducing hardening, softening, and damage parameters (see, for
example Naumenko and Altenbach, 2016). The principal idea of the extension of
the secondary
creep equations to primary and Holm
tertiary
creepandisJohanna
demonstrated
4
Altenbach
Eisenträger in Fig.
1.1. In a first step, the equation for secondary creep will be established for the range
σ = const, T = const
Creep strain
Fracture
ω = ω∗
σ1
σ2 > σ1
Tertiary
ω̇ = gω (σ , H, ω , T)
Secondary
cr = g (σ , T )
ε̇min
ε
Primary
Ḣ = gH (σ , H, T )
Time
 cr
cr = 0 Constitutive equation,
 ε̇ = gε (σ , H, ω, T ), εt=0
Ḣ = gH (σ , H, ω, T ), Ht=0 = 0 Evolution equation (hardening),

ω̇ = gω (σ , H, ω, T ), ωt=0 = 0 Evolution equation (damage)
Fig. 1.1 Classical phenomenological modeling of creep with a hardening variable H and an
cr = 0 Constitutive equation
σ , H,
, T), εT
 ε̇ cr = gωε ((σ
isotropic damage variable
=ωconst,
const)
t=0=

Ḣ = gH (σ , H, ω , T), Ht=0 = 0 Evolution equation (hardening)
ω̇ = gω (σ , H, ω , T), ωt=0 = 0 Evolution equation (damage)
Fig. 1 Classical phenomenological modeling of creep with hardening H and isotropic damage ω
(σ = const, T = const)
• Sintering
They are related to different mechanisms which depends on the temperature and
stress levels. Such mechanisms are bulk diffusion, climb (the strains are accomplished by a climb), climb-assisted glide (dislocations can go around obstacles),
grain boundary diffusion or thermally activated glide. Some mechanisms of creep
have not been verified by direct microstructural examination yet.
In dependence of the mechanism various equations are suggested for the description of creep. Not all are suitable for creep mechanics analyses. Below some of these
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Holm Altenbach and Johanna Eisenträger
of the minimum creep rate. Afterwards, the evolution of the hardening variable can
be established. Last but not least, the damage evolution will be introduced to describe the tertiary creep range. In the end, one obtains a coupled system of ordinary
differential equations with respect to time.
Probably the first approach to describe primary creep was presented in 1938 by
A. Nadai, who suggested an equation of state for strain hardening (Nádai, 1963). In
addition, a different equation was introduced for time hardening by Finnie (1960).
First theories for tertiary creep were developed by Kachanov (1958) and Rabotnov (1959), who introduced the concept of material deterioration. In their simplest
form, the theories are based on two material parameters derived from a standard
creep rupture curve, which illustrates the dependence of the time to rupture on the
applied stress level. Further information can be found for example in Kowalewski
and Ustrzycka (2018).
In addition, Knych et al (2018) report on current trends in creep mechanics. Other
types of time-dependent material behavior, such as viscoelasticity and viscoplasticity, are presented in Milewski (2018); Sumelka and Łodygowski (2018). Special
attention to solving problems is paid in Naumenko and Altenbach (2007, 2016,
2019); Rabotnov (1969); Szuwalski and Ustrzycka (2018) and the references within.
Furthermore, information on the micromechanics of creep is given in KowalczykGajewska (2018).
1.2 Creep of Metallic Materials
1.2.1 Creep Stages
Based on the terminology of da Costa Andrade (1910, 1914), one distinguishes three
stages, while the elastic range is usually ignored. In the initial stage (primary or decelerate creep, sometimes also transient creep), the strain rate is relatively high, but
decreases with increasing time and strain due to a process analogous to work hardening at lower temperatures. For instance, the dislocation density increases and in
many materials a dislocation subgrain structure is formed and the cell size decreases
with strain (Courtney, 2005).
With increasing deformation, the strain rate will attain its minimum and become
nearly constant as the secondary stage begins. This is due to the balance between
work hardening and annealing (thermal softening). The secondary stage is also often referred to as steady-state creep. However, it is worth pointing out that various
microstructural processes take place during this stage. Now, recovery effects are
concurrent with deformation. Note that changes in material strength are only minor
during these first two stages of creep.
The last stage (tertiary or accelerated creep) is again related to an increase of
the strain rate. Here, softening processes are dominant. At the same time distributed
damage occurs and finally fracture can be observed.
1 Introduction to Creep Mechanics
5
1.2.2 Types of Creep
In material science, one distinguishes the following types of creep (Cotrell, 1967):
•
•
•
•
•
•
•
dislocation creep,
Nabarro-Herring creep
Coble creep
solute drag creep
dislocation climb-glide creep
Harper-Dorn creep
sintering
They are related to different microstructural mechanisms, which depend on the temperature and stress levels. Such mechanisms are for example bulk diffusion, climb
(the strains are accomplished by climbing dislocations), climb-assisted glide (dislocations can go around obstacles), grain boundary diffusion or thermally activated
glide. Note that some mechanisms of creep have not been verified by direct microstructural examinations yet. The so-called deformation-mechanism maps summarize the relations between deformation and mechanisms for many materials, cf.
Frost and Ashby (1982); Nabarro and de Villiers (1995); François et al (2012).
Deformation-mechanism maps represent the dominant deformation mechanisms in
diagrams based on the normalized shear stress, which is displayed with respect to
the homologous temperature.
Depending on the microstructural mechanisms, various equations are suggested
for the description of creep. However, not all are suitable for structural analysis in
creep mechanics. Below, some of these equations are presented.
1.2.3 Materials Science-Based Creep Equations
1.2.3.1 General Creep Equation
The general creep equation is formulated for the creep strain rate
ε̇ =
Q
Cσ m −
kT ,
e
db
(1.1)
whereas C is a parameter depending on the material and the particular creep mechanism, m and b are exponents related to the creep mechanism, Q is the activation
energy of the creep mechanism, d denotes the average grain size of the material,
k ≈ 8.31696 J (mol K)−1 is Boltzmann’s constant, and T is the absolute temperature. Note that Equation (1.1) is a power law function.
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Holm Altenbach and Johanna Eisenträger
1.2.3.2 Dislocation Creep
At high stresses (relative to the shear modulus G), creep is controlled by the movement of dislocations. For dislocation creep, the activation energy Q is related to
self-diffusion, the exponent m is in the range 4 . . . 6, and b = 0 holds. It is obvious
that dislocation creep is strongly influenced by the applied stress and the intrinsic
activation energy, while the influence of the grain size can be neglected. Note that
some alloys feature a very large stress exponent (m > 10). In this case, a threshold
stress σth will be introduce: creep cannot be measured below this value. Equation
(1.1) should be modified as follows
ε̇ =
Q
A(σ − σth )m −
kT ,
e
db
(1.2)
where A, Q and m can all be related to conventional mechanisms. The applied stress
tends to drive the dislocations past the barrier. After bypassing an obstacle, dislocations get into a lower energy state. A significant part of the work required to
overcome the energy barrier of passing an obstacle is provided by the applied stress,
while the remaining impulse is provided by thermal energy. As a result, creep increases in these cases.
1.2.3.3 Nabarro-Herring Creep
In contrast to dislocation glide creep, which does not involve atomic diffusion,
Nabarro-Herring creep belongs to the diffusional creep mechanisms. This creep
mechanism dominates the material behavior particularly at high temperatures and
low stresses. The sides of the crystal are subjected both to tensile and compressive
stresses and the atomic volume is changed due to the applied stresses: it increases
under tension and decreases under compression. The activation energy for vacancy
formation is changed by ±σ Ω , where Ω represents the atomic volume, the sign +
refers to compressive regions and a negative sign describes tensile regions. Since
Ω
the fractional vacancy concentration is proportional to exp(− Qf ±σ
kT ), where Qf is
the vacancy-formation energy, the vacancy concentration is higher in tensile regions
than in compressive regions, leading to a net flow of vacancies from the regions
under tension to the regions under compression, which is equivalent to a net atom
diffusion in the opposite direction, which causes the creep deformation: the grain
elongates in the tensile stress axis and contracts in the compressive stress regime.
In the following, we assume the following values for the creep exponents: m = 1
and b = 2, which results in a weak stress dependence and a moderate influence of
the grain size. If the creep rate decreases, the grain size increases. Furthermore,
one should note that Nabarro-Herring creep is strongly temperature dependent. It
dominates at very high temperatures relative to a material’s melting temperature.
1 Introduction to Creep Mechanics
7
1.2.3.4 Coble Creep
Coble creep is another form of diffusional creep. The atoms diffuse along grain
boundaries to elongate the grains in the stress direction. In this case, the influence
of the grain size on creep deformations is significantly higher compared to NabarroHerring creep. Thus, this mechanism is of higher importance for materials made
of very fine grains. Note that the Boltzmann’s constant k is related to the diffusion
coefficient of atoms along the grain boundary, Q describes the grain boundary diffusion, and m = 1 and b = 3 hold for the exponents. Since the activation energy
for grain boundary diffusion is smaller than in the case of self-diffusion, this creep
mechanism occurs at lower temperatures in comparison to Nabarro-Herring creep.
Although Coble creep is temperature-dependent as well, the influence of temperature is not as strong as for Nabarro-Herring creep. It also exhibits the same linear
dependence on stress as Nabarro-Herring creep. In general, the diffusional creep
rate should be obtained as a sum of the Nabarro-Herring creep rate and the Coble
creep rate. Note that diffusion creep results in grain-boundary separation and the
occurrence of voids or cracks between grains. As a result of these mechanisms,
grain-boundary sliding starts. If grain-boundary sliding cannot accommodate the
incompatibility, the voids at the grain boundaries lead to the initiation of creep fracture.
1.3 Creep Equations for Structural Analysis
Creep failure must be prevented, and the lifetime of components should be predicted in an accurate manner. Consequently, there is a high demand for constitutive
models accounting for not only one-dimensional effects, but also three-dimensional
stress and strain states. As additional challenges, non-isothermal, and cyclic operating conditions should be taken into account. Thus, not only long-term creep strains
should be predicted accurately, but also short-term plastic strains as well as the interaction of both deformation modes. Constitutive equations for metallic materials
are well presented in various monographs and textbooks, among them Hult (1966);
Odqvist (1974); Penny and Marriott (1995); Naumenko and Altenbach (2007); Betten (2008); Naumenko and Altenbach (2016).
1.3.1 Basic Idea: Separation Ansatz
An essential part of creep models is the proper description of the stress and temperature influence by so-called response functions. These functions are specific for
each deformation mechanism, and the choice of these functions can be based on
information from deformation mechanism maps (Frost and Ashby, 1982; Nabarro
and de Villiers, 1995; François et al, 2012). In the following, let us assume that
8
Holm Altenbach and Johanna Eisenträger
the influence of the stress and temperature can be separated. We will introduce
some commonly used stress and temperature response functions with respect to onedimensional stress and strain states.
In the simplest case, the creep strain rate is a function of the stress σ and the
temperature T :
ε̇ = f (σ , T ).
(1.3)
This function can be extended to include further influences, but in this case the effort
for the identification of material parameters increases in accordance to the number
of variables. Even in the simplest case, the identification is not straightforward since
all influences are usually coupled.
A separation ansatz can simplify this procedure. To define the dependence of the
creep strain rate ε̇ on the stress σ and the temperature T , the following approximation is frequently introduced:
ε̇ = fσ (σ ) fT (T ),
(1.4)
which is a product of the stress response function fσ and the temperature response
function fT .
The Norton power law represents one of the most commonly used expression for
the stress response function:
fσ (σ ) = Aσ n .
(1.5)
The material parameters A and n (creep exponent) should be estimated from creep
experiments. Diffusion creep is usually described by employing a linear stress function (Herring, 1950; Harper and Dorn, 1957; Coble, 1963; Lifshitz, 1963), i.e. n = 1:
fσ (σ ) = Bσ ,
(1.6)
with the material parameter B. It is obvious that such a behavior is similar to the
behavior of a viscous fluid. Another common choice to describe creep at low temperatures and with respect to wide stress ranges is a hyperbolic function (Dyson and
McLean, 2001):
fσ (σ ) = C sinh (Dσ ) .
(1.7)
Again, the material parameters C and D must be determined based on experimental
data.
To express the dependence of the creep strain rate on temperature, the Arrhenius
function is a popular choice (Dorn, 1955; Ilschner, 1973):
Q
(1.8)
fT (T ) = α exp −
kT
with the material parameter α, the activation energy Q, and the Boltzmann constant
k. As has already been pointed out, the temperature exerts a significant influence on
creep and relaxation phenomena. This is due to the fact that creep is based on microstructural processes, which are highly dependent on the temperature. However,
because of the great variety of materials affected by creep, the classification of creep
1 Introduction to Creep Mechanics
9
with respect to temperature is not uniform in literature, i.e. a clear and widespread
definition of low and high-temperature creep does not exist. Note that temperature
ranges in creep are often indicated with respect to the liquidus temperature TL of
the material under consideration. Other indicators are the solidus TS or melting Tm
temperature.
According to Naumenko and Altenbach (2016), high temperature materials are
used at a temperature range of 0.3–0.7 Tm . In Frost and Ashby (1982), it is stated that
polycrystalline solids start to creep at 0.5 TL , whereas a temperature level of 0.9 TL
is referred to as “very high” temperature level. In addition, according to McLean
(1966), the temperature range 0.3–0.9 TL is most important for engineering applications, while in this main creep range, two separate creep regimes are distinguished,
low temperature (LT) creep at 0.3–0.5 TL and high temperature (HT) creep at 0.5–
0.9 TL . Higher temperatures are related to other mechanisms and should be presented
by the so-called power-break law.
There are also other approaches (Eisenträger and Altenbach, 2020). However,
practical applications involving structural analyses still represent a major challenge.
Only approaches using advanced rheological models, which are highly non+linear,
but reflect the mechanisms separately, simulate real creep behavior with an adequate
accuracy and involving only moderate computational effort, cf. e.g. Eisenträger et al
(2018a,b,c); Eisenträger (2018).
1.3.2 Extension to Three-Dimensional Behavior
The equations suggested in the previous section can be extended to multi-axial stress
and strain states considering the following three main steps:
• substitution of the scalar stress by a tensorial variable,
• substitution of the scalar strain by a tensorial variable, and
• introduction of an equivalent stress.
It seems that the extension is straightforward, but there are several difficulties. It
seems natural to use the stress tensor and the strain tensor. However, with respect to
Bailey’s statements (Bailey, 1935), several questions arise: should one choose the
stress or strain tensor or deviator, respectively. Here, one should also consider the
axioms of rheology (Reiner, 1960) and statements of Palmow (1984).
The last item in above list is much more complicated. Even in the case of
isotropic materials, the choice of an equivalent stress cannot be based on rational
approaches such as balance equations. Instead, the introduction of some hypotheses
and experimental proofs allow the introduction of an adequate expression for the
equivalent stress (for further reading: Altenbach and Öchsner, 2014).
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Holm Altenbach and Johanna Eisenträger
1.3.3 Anisotropic Creep
In the case of anisotropic creep, one should distinguish between:
• initially anisotropic materials, and
• initially isotropic materials which become anisotropic due to creep deformations
(for example in case of anisotropic damage).
In the first case, tensor calculus is helpful since some suggestions for the constitutive equations can be made based on the mathematical analysis (Naumenko and Altenbach, 2016). The second case is more complicated since further evolution equations should be taken into account.
1.4 Cross-References
•
•
•
•
•
•
•
•
•
•
•
Creep at Extremely Low Temperatures
Anisotropy of Linear Creep
Creep Deformation
Creep Fatigue
Creep in Modern Materials
Creep in Structures
Micromechanics of Creep
Viscoelasticity
Viscoplasticity
Nádai, Arpád
von Mises, Richard
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