CREEP-DAMAGE EQUATIONS FOR LOW AND MODERATE STRESS RANGES
H. Altenbach
Martin-Luther-Universität Halle-Wittenberg, Halle-Saale, Germany
The lifetime of structural components is essentially affected by creep and damage processes. Advanced heat resistant steels (e.g. 9-12% Cr steels) exhibit a stress range dependent creep and damage behavior, which has to be taken into account for the modeling of such materials. Recently published data highlight this behavior. Exemplarily for this behavior is the gradual decrease of the creep exponent in the power law with a decrease of the stress level as well as the change of the damage mode from the pure ductile in the high stress regime to the brittle-ductile or pure brittle in the low stress regime. To describe this behavior with a material model a set of equations is introduced. The effects of hardening, softening and damage are taken into account in a phenomenological sense.
Introduction.
A reliable prediction of the long-term behavior of materials in dependence on the stress regime and the temperature is a very complex challenge. Especially heat resistant materials in responsible structures need to be characterized very exactly. The aim of creep mechanics is to meet this challenge and to develop methods to predict time-dependent changes of stress and strain states in engineering structures up to the critical stage of rupture, e.g. [3] among others. Creep tests for the group of the heat resistant 9%-12% chromium-steels were performed in a big quantity and the experimental data is published for a wide temperature and stress range. Observations at the scale of the microstructure show a significant influence of the stress level on for example the dynamics of dislocations, the coarsening of subgrains and the initiation and growth of microvoids. The summation of the physical effects leads to a dependence of the minimum creep rate on the stress level at the macroscale. Analogue observations for the damage mechanisms at the scale of the microstructure allow a similar conclusion of stress dependence. The physical effects of participate coarsening and initiation and growth of microvoids, which are responsible for observed damage on the macroscale, e.g. necking and/or cracks, are stress dependent. To model this complex behavior of 9%-12% chromium-steels a set of differential equations is derived, one constitutive equation for the creep rate and three evolution equations for the macroscopic effects hardening, softening and damage.
The three evolution equations appear as internal state variables in the constitutive equation of the creep rate. To obtain the parameters for the differential equations the published data is used for their identification.
Uni-axial Creep.
Within the phenomenological approach to creep modeling one usually starts with the description of secondary creep. Here the minimum creep rate is defined as a function of the stress and the temperature. Then the creep constitutive equation is generalized by the introduction of internal state variables. To describe primary creep a set of hardening variables and appropriate evolution equations are utilized. To take into account tertiary creep, damage variables and damage evolution equations are introduced. With the critical values of damage variables the end of the creep process and the creep rupture can be identified. As a result the creep model includes a set of ordinary differential equations, where specific hardening and damage mechanisms are described by the use of independent variables. In this paper we focus on the description of the stress range dependent secondary creep stage. Hardening and damage processes will be ignored. Although, such an approach does not allow estimating the time to creep fracture, it provides a first insight into the stress redistribution in a structure and is frequently used in the early stages of design, e.g. [1].
To describe the creep behavior in the wide stress range, various response functions, which are more or less physically motivated, have been proposed. Overviews are presented in [2; 3], among others. One example is the hyperbolic sine law
 
A sinh( B

) , applied in [4–6] to characterize minimum creep rate of various materials. Another way is to assume that the

minimum creep rate is the sum of the linear and the power law stress functions. As pointed out in
[2] power law and diffusion creep mechanisms involve different defects and may be assumed independent such that the corresponding creep rates add. The constitutive equation can be formulated as follows where
0


0
 
0







0
 n
 
0


0


1








0
 n

1


,
 
0
and n are material parameters. The introduced equation illustrates a typical dependence of the minimum creep rate on the applied stress. Within the „low” stress range the creep rate is a nearly linear function of the stress. Within the „moderate” stress range (transition range) the behavior changes from the linear to the power law creep. The „power law” range is widely established for many metals and heat resistant steels. The creep exponent tales usually the values between 3 and 12 depending on the material, type of alloying and processing conditions.
Within the „power law breakdown” or „high stress” range the creep rate is higher as the power law predicts. Below we assume that the loading is moderate such that the power low breakdown can be neglected. The stress response functions of the abovementioned type which involve the sum of the linear and one or more powers of the stress are well known in the creep mechanics [1;
3] and the rheology, e.g. [19]. An advantage of this equation over the hyperbolic sine law is the possibility to analyze extreme cases of the linear creep for
 
0
1 and the power law creep for 1 . Therefore, in applying this equation to the structural analysis one may examine
0 the influence of the viscous creep and compare the results with the known power law based solutions.
Stress Relaxation. For an accurate description of the stress relaxation a constitutive equation which reflects both the hardening and the steady state creep is required. Below we limit ourselves to a qualitative analysis by neglecting primary creep effects. With the assumed creep law the stress relaxation problem can be formulated as follows

E
 
0 where E is the Young’s modulus and



0
 
0
  ref




0

 n

0 ,
is the reference stress. This equation can be transformed to a Bernoulli-type differential equation and solved for some special cases (for details see [8]).
Creep under Multi-axial Stress States. The structural analysis requires a constitutive model of creep under multi-axial stress states. Following the creep theory proposed by Odqvist
[9] the creep rate tensor
 cr
is defined by the creep potential W and the flow rule
ε cr 

W
 σ
, W

W (
σ
, T ) .
For isotropic materials the creep potential must satisfy some restriction. From this it follows that the potential depends only on the three invariants of the stress tensor. With the principal invariants one can write W
σ
J
1


, tr
σ
, J
2
  
1
2
 
2
,
1 2 3

 tr
σ 2
, J
3
 det
σ
. Any symmetric second rank tensor can be uniquely decomposed into the spherical and the deviatoric part. For the stress tensor the decomposition has the form σ   m
, where s is the deviatoric part of the stress tensor,
 m
is the mean stress and I is the second rank unit tensor. With the principal invariants of the stress deviator
J
2D the potential takes the form W

1
2
σ tr s
2 
1
2 s s ,

J J
1 2D
, J
J
3D
3D


1
3 tr s
3 
1
3
  s
. Applying the chain rule for the derivative of

a scalar valued function with respect to a second rank tensor one can obtain
ε cr 

W

J
1
I 


J
W
2D s 


W

J
3D
 s
2 
1
3 tr s I


.
Discussion. The introduced general isotropic creep law can be specified introducing the incompressibility or the neglecting of tensorial non-linear behaviour. Applying this procedure to the above mentioned uniaxial creep law one obtains
ε cr 
3
2


0
0


1

 vM
0
 n

1

 s .
This law can be extended to hardening, softening and damage behavior; temperature effects can be included. Application examples are presented in [8].
REFERENCES
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Springer, 2007.
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6. Perrin I.J., Hayhurst D.R. Creep constitutive equations for a 0.5Cr-0.5Mo-0.25V ferritic steel in the temperature range 600-675
◦
C // J. Strain Anal. – 1994. – 31(4) . – pp. 299–314.
7. Reiner M. Deformation and Flow. An Elementary Introduction to Rheology, 3rd edn. –
London: H.K. Lewis & Co., 1969.
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University Press, 1974.