1. The Role of Fundamental Modeling

Modeling and prediction of materials properties have had a rapid development in recent years. Ab initio methods are used to compute the electronic structure of crystals based on quantum mechanical methods. The full multi particle problem is not possible to solve but a number of first principle procedures such as Density Functional Theory (DFT) are available to handle the problem. By minimizing the total energy of the system, lattice parameters and the most stable crystal and surface structures can be established. A range of physical parameters such as thermal expansion coefficient, heat capacity, electric and thermal conductivity can be computed. Interface energies and elastic constants can be derived. This type of modeling is referred to as fundamental because it is based on physically well founded algorithms and no parameters are fitted to experimental data.

Computational thermodynamics (CTD) is another area where great progress has taken place in recent decades. Its base is unique. Expressions for the free energy are fitted to a range of thermophysical properties as a function of alloy content, which is referred to as the Calphad approach. The functions determined in this way can then be used to find the equilibrium phases for specific amounts of alloying elements. Phase diagrams can be generated. By using data also for interface energies and diffusion constants the development of microstructure can be predicted, which is the basis of much research in materials science. The variation of the interface energies and diffusion coefficients with alloying elements can also be derived with CTD, which is usually referred to as fundamental in spite of the fitting of the free energy functions to experimental data, because once this process is completed no further fitting is involved.

Mechanical properties are of great technical and scientific interest. In spite of this, the amount of fundamental modeling that has been performed for mechanical properties is much more limited than for ab initio methods and CTD. Mechanical properties are primarily controlled by the motion of dislocations and this has created a barrier for more fundamental modeling.

Probably the most common way of measuring mechanical properties is the tensile test. It gives a stress strain curve that can be used to assess the yield and tensile strengths as well as the ductility in the form of elongation or reduction of area. To describe stress strain curves almost exclusively empirical models are used. Some of the most well-known models are Hollomon, Ludwik, Voce and Swift. These models do not have any physical background. They are simply mathematical expressions with a number of adjustable parameters that can be fitted to the data. Only the Voce model can readily be derived from basic physical principles. This will be described in Chap. 3.

Also for creep, mainly empirical models are used. The development of them has taken place over many decades. A major event was when Norton in his book from 1929 gave an equation for the stress dependence of the creep rate [1]. This relation is now often referred to as the Norton equation. It was the only equation in the book. The stress dependence of the creep rate has played a profound role in the development. An important result by Bird, Mukherjee and Dorn (BMD) in the 1960ties gave an explicit expression for the temperature dependence in the Norton equation [2]. The creep rate was assumed to be proportional to an Arrhenius equation of the self-diffusion coefficient which is natural when climb is the controlling dislocation mechanism. This implies that the activation energy for creep is the same as that for self-diffusion, a relation that has been experimentally confirmed for a number of pure metals. Some temperature dependence was also incorporated by explicitly taking into account the temperature dependence of the shear modulus. This generalization of the Norton equation is referred to as the BMD equation.

The BMD equation can be considered as a semi-empirical equation. The inclusion of the self-diffusion constant was based on physical thinking, but the equation still has at least two adjustable parameters: a proportionality constant and the stress exponent n_N. In spite of these limitations the equation is frequently used till this day. The value of n_N was assumed to be related to the operating mechanisms. Weertman suggested that climb control would give n_N ≈ 5 and glide control n_N ≈ 3 [3, 4]. Together with the knowledge that diffusion control gives n_N ≈ 1, it was thought that the value of n_N could be used to identify the operating mechanism. For this reason much focus in creep research in the coming decades was on measuring the secondary creep rate and determining the stress exponent. This seemed logical at the time but has turned out to be unfortunate. It gradually became apparent that the stress exponent was not automatically related to the rate controlling creep mechanism [5]. With the fundamental models that are presented in this book, it is shown that climb controlled creep can be associated with stress exponents from 1 to 50, demonstrating that a specific range of stress exponents does not necessarily make it possible to identify the creep mechanism.

For a more detailed modeling of creep deformation, it must be possible to describe how the dislocation density is evolving with time and strain. Initially models were semi-empirical in nature, but later it has been possible to derive and verify some of these models. There are three contributions to the changes in the dislocation density. Work hardening involves the generation of dislocation that raises the strength. In the 50ties and 60ties numerous scientists were engaged in developing models for work hardening. For creep of polycrystalline materials, a simple expression derived from the Orowan equation has turned out to be useful. This was first utilized by Lagneborg [6]. Two recovery processes that reduce the dislocation density are central aspects of creep: dynamic recovery that is strain controlled and static recovery that is time controlled. The expression for dynamic recovery was first presented by Bergström [7]. It was also given in a well-known paper by Kocks [8]. The role of static recovery was initially emphasized by Lagneborg [6].

What has been described so far is the empirical and semi-empirical modeling of creep. Before entering fundamental modeling, it is worth-while to specify what we mean by the different types of modeling. These definitions are not supposed to be general, but it is essential to clarify what we mean in this book.

When fundamental models exist they are more valuable tools because they can be used to make predictions and generalize results. However, empirical models can be quite useful as well. A classic example is the empirical Bohr model for the hydrogen atom, where an electron circles around the nucleus as a particle. The Bohr model inspired a large number of scientists to perform experiments and to develop models. The Bohr model is now superseded by the quantum mechanical description of the hydrogen atom.

Empirical models are usually the first ones to be established for a specific phenomenon. The Norton equation is an example of that. It created the understanding about some basic facts about creep. It took about three decades before semi-empirical improvements started to appear and even further decades before fundamental versions were available. There is always a risk to focus too much on empirical or semi-empirical models. With the help of adjustable parameters, it is usually possible to get a good fit to experimental data and that can easily create the impression that good understanding of the phenomenon has been established. But it is important to recall that it is very difficult to make predictions, generalizations and to identify operating mechanisms with empirical and semi-empirical models. The use of the BMD equation is unfortunately an example of that in a number of cases.

The starting point for development of basic creep models concerns diffusion creep. In these models the deformation is assumed to take place by diffusion of atoms inside the grains [9] or along the grain boundaries [10]. These models are quite simple and easy to understand intuitively. They both predict a stress exponent of 1. It was thought that these models would be easy to verify experimentally. But this has not turned out to be the case. The interpretation of many earlier tests has also been questioned [11‐13]. One of the main reasons for the difficulty is the assumption that only diffusion creep is associated with a stress exponent of 1. It has more recently been shown that dislocation creep can give a stress exponent of 1 at low stresses. This has been found for aluminum at very high temperatures [14, 15]. In the past a stress exponent of 1 was observed. But by carrying out the testing until sufficiently high strains were reached, a stress exponent of 3 was obtained. Low stress exponents for dislocation creep have also been observed for the martensitic 9Cr1Mo steel P91 and for the austenitic stainless 17Cr12Ni2Mo steel 316H [16, 17]. Although the tests for these steels were performed till 1000 h, the secondary stage was far from reached. If the stress exponent is determined in the primary rather than in the secondary stage it can give a low value of about unity.

It is evident from these experimental results and also from modeling findings in the present book that dislocations can be of importance at high temperatures and low stresses and that contribution from them can even be much larger than that from diffusion. When studying diffusion creep it is consequently important to check the amount of dislocation creep that is present. A simple way to do that is to observe if primary creep occurs, which would be a clear sign that dislocation creep is present. In addition, the creep exponent must be unity and the measured creep rate should agree with the formulae for diffusion creep. Unfortunately, it is not easy to find studies fulfilling these requirements in the literature. This does not mean that diffusion creep is not a real effect. However, it seems to be masked by other mechanisms in many cases.

Basic modeling of dislocation creep can be said to be started by the formulation of the climb mobility by Hirth and Lothe [18]. Climb is by far the most important mechanism for controlling the creep rate. This climb mobility has been implemented in the expression for static recovery proposed by Lagneborg [6]. Also the parameters in his expression have been derived and the expression validated. The first to give a basic modeling of dynamic recovery was Roters et al. [19]. This item is not fully settled yet because the modeled dynamic recovery constant ω is not always in agreement with experiments. With these achievements a basic differential equation for the development of the dislocation density could be established. The equations are derived in Chap. 2.

From the differential equation for the dislocation density, the stationary creep rate can be derived. In this way the Norton equation is obtained where all the parameters are well defined with given values. In the past many attempts have resulted in expressions with poorly defined back stresses, activation energies or activation areas. This is now avoided. The new formulation gives a stress exponent of three or four in its simplest form. This is sometimes referred to as the “natural creep law” because it is a direct consequence of the balance between work hardening and recovery during stationary creep [20, 21].

It is well established that the stress exponent is raised when higher stresses and lower temperatures are considered. A dramatic increase from the values of n_N = 4–7 in power-law creep is observed, which is often referred to as power-law break down. The increase in the stress exponent has been possible to be fully explained by taking strain enhanced formation of vacancies into account [22]. This effect is now integrated in the expression for the creep mobility and it is taken into account in the Norton equation. This will be described in Chap. 2.

It is natural to assume that stress controlled, load controlled as well as rate controlled plastic deformation are governed by the same mechanisms and equations. To follow this principle, stress strain curves are described with the same dislocation models as for creep. For fcc alloys, the stress strain curves obey the Voce model in the most direct derivation. This can give an as accurate representation as empirical models for stress strain curves. The dynamic recovery constant ω plays a special role because it controls the work hardening behavior of an alloy. The value of this parameter as well as stress strain curves are covered in Chap. 3.

Basic expressions for primary creep have not been derived until recently. In [23] a formulation was presented for 9–12%Cr steels and in [24, 25] for copper. In these papers, the primary creep rate is derived from dislocation equations without introducing new quantities. For copper the observed exponential decrease in creep rate with increasing strain can be reproduced. With a satisfactory description of primary creep, the behavior at very low stresses can be modeled. As discussed below this is of importance for analyzing data for diffusion creep. In addition, an accurate representation is important in many cases in design at high temperatures. Empirical equations for primary creep are typically difficult to generalize and transfer to suitable expressions for stress analysis. However, from basic equations this is readily possible. Primary creep will be covered in Chap. 4.

Creep at low stresses and with low stress exponents has always created a special interest amongst scientists due to the simple expressions for diffusion creep. With the event of basic formulae for primary dislocation creep, it is possible to analyze its role at low stresses. Since stationary conditions are rarely reached at very low creep stresses, it is essential to take primary creep into account. It is shown in the book that dislocation creep can give stress exponents of 1 and that situation is thus not restricted to diffusion creep. Examples are given for an austenitic stainless steel, for aluminum and for copper. Both for aluminum and copper, the basic creep model can accurately represent creep measurements at high temperatures and low stresses as well as at low temperatures and high stresses. Thus, the model can handle a wide range of conditions in temperature and stress. These findings can be considered as a direct verification of the basic creep model. The results are presented in Chap. 5.

For solid solution hardening (SSH) already Hirth and Lothe [18] derived expressions for slowly diffusion elements. Surprisingly enough, the expressions have not been used extensively in the literature. The author has shown that the expressions can reproduce experimental results quite well. For fast diffusing elements the mechanism is different [26]. The dislocations have to break away from their Cottrell atmospheres of elements to move. The most well characterized case is phosphorus in copper where ppm quantities have a pronounced effect on the creep strength. SSH will be discussed in Chap. 6.

Precipitation hardening (PH) is a potent method to increase the creep strength of alloys. This was realized early on and many scientists were engaged to try to model the magnitude of the contributions. They worked from the assumption that there is a barrier against climb when the dislocations pass the particles. A number of estimates of the size of the barrier were made. However, eventually the values became so low that they had no technical interest anymore [27]. With the lack of a proper model for a long time, in many papers PH contribution to the creep strength was estimated with the Orowan strength, which strongly overestimates the PH contribution and in addition gives the wrong temperature dependence. Later, it was assumed that the controlling factor was the time it takes for a dislocation to climb across a particle [28, 29]. This mechanism was used to describe the creep strength of austenitic stainless steels [30‐32]. These studies had unfortunately the situation that the PH was only a smaller part of creep strength. To verify the model, Co particles in Cu were studied [33]. The Cu–Co alloys had the advantage that PH was a major part of the creep strength and the validity of the model could be verified. The influence of composition and heat treatment could be reproduced. PH is analyzed in Chap. 7.

Cells or subgrains are formed in virtually all materials during plastic deformation and are collectively referred to as substructure. If the substructure can be locked with the help of particles, it can give a significant contribution to the creep strength. A well-known example is the martensitic steel P91 where M₂₃C₆ particles can be used to stabilize the substructure [23]. Models for the formation of substructure during creep and during plastic deformation at near ambient temperatures are presented. Unbalanced dislocations can be formed where the presence of opposite Burgers vector is absent. This has the consequence that static recovery does not occur and the substructure can build up a significant contribution to the creep strength. This is an important mechanism for how the creep strength can be raised after cold work [25]. The model can accurately describe how cold work can raise the rupture time by several orders of magnitude. This can be considered as an additional verification of the basic creep models. Substructures are analyzed in Chap. 8.

Grain boundary sliding (GBS) is assumed to be the main mechanism for initiation of creep cavities. The grain boundary displacements have been quantified with the help of finite element analysis (FEM) [34, 35]. The displacement is proportional to the creep strain with a proportionality constant C_s that can be assessed from the FEM results [36, 37]. The amount of data on GBS in the literature is limited. However, for copper three types of measurements have been performed, and these measured values for C_s are in agreement with the theoretical value. GBS is also believed to be the main mechanism for superplasticity. The results for GBS are used to derive a basic model for superplasticity. The model can reproduce literature data for Al22Zn. GBS is discussed in Chap. 9.

Nucleation, growth and linkage of creep cavities are the source of crack initiation in many cases and are therefore of considerable technical interest. Nucleation of cavities can be well described by assuming that it is controlled by GBS. In particular, it gives the number of cavities that is proportional to the creep strain in a way that is in quantitative agreement with observations [38]. Cavities are mainly nucleated at particles or subboundary junctions in the grain boundaries. In the past, attempts have been made to base cavity nucleation on classical nucleation theory. However, it suggests a strong stress and temperature dependence that is at variance with observations.

Models for diffusion growth of creep cavities have been available for a long time, but the original expressions overestimated the observed growth. This was solved by requiring that the growth rate should not be faster than the creep rate (constrained growth). However, these modeled growth rates are still higher than the experimental values. By analyzing the balance between the cavity growth rate and the creep rate with the help of finite element methods (FEM), further improvements have been achieved and now the data can be described in a satisfactory way [39]. Strain controlled growth of cavities is also analyzed. A number of models can be found in the literature. However, several of these models give a very low growth rate if the normal size of cavity nuclei is assumed. That makes it difficult to use them for prediction of growth rates. In addition, some models do not take constrained growth into account. On the other hand for larger initial cavity sizes, the predicted growth rates can exceed the observed ones in a pronounced way. One approach that relates the growth rate directly to the amount of GBS avoids these problems [40]. Cavitation is discussed in Chap. 10.

Cavitation during cyclic loading is expected to play the same important role for rupture prediction as during static creep. To describe the nucleation of cavities, the amount of creep during creep-fatigue interaction must be possible to predict. The basic models for static creep can be taken over when describing the stress strain loops during cycling with one important change. The dynamic recovery parameter ω must be increased. The reason is that dislocations encounter each other much more frequently during cycling than during static loading, leading to an enhanced annihilation of dislocations. The principles for nucleation and growth of cavities can essentially be taken over from static loading. This is verified by comparison to experiments for 1Cr0.5Mo steel, which is handled in Chap. 11.

Numerous empirical models for tertiary creep can be found in the literature. They are used to describe the creep damage for example during the analysis of residual lifetime of components with the help of the continuum damage mechanics (CDM). There are many mechanisms that can contribute to tertiary creep such as cavitation as well as particle and substructure coarsening. However, recent investigations suggest that another mechanism is often the dominating one [41]. The true stress during a constant load test increases rapidly with strain. During primary and secondary creep, the increase rate in the dislocation stress matches that of the true stress. However during tertiary creep this is no longer the case and the creep rate increases. For a more complete picture, the role of the substructure must be taken into account as well. The model results suggest that necking starts to form right at the beginning of the tertiary stage, but the neck is not fully developed until very close to rupture. These findings are consistent with available observations [41]. Findings on tertiary creep are presented in Chap. 12.

Modeling of creep ductility is natural to divide between brittle and ductile rupture because the failure mechanisms are different. For common creep resistant alloys brittle rupture is initiated by cavitation. When the cavitated area fraction in the grain boundaries reaches a critical factor, cracking and rupture are initiated. The general behavior of the ductility of austenitic steels has been modeled, where the ductility decreases with increasing temperature and rupture time. For ductile rupture, the modeling suggests that necking activates the failure. So far this has been demonstrated for copper and for steels that obey the Omega model, where the logarithm of the strain rate is linear in the strain in the tertiary stage. These materials include low alloy steels, martensitic 9–12%Cr steels and some austenitic stainless steels. Creep ductility is covered in Chap. 13.

Extrapolation of creep rupture data to longer times is technologically most important due to the extended design life of modern high temperature plants of 30 years or more. Extrapolation is in most cases performed with empirical statistical methods. In particular, time-temperature parameters (TTP) are commonly used. To obtain accurate results a large number of data points must be available and careful post assessment tests must be performed. It is shown that the results are found in a safer way if requirements are placed on the derivatives of the creep rupture curves in the analysis. A method for the assessment of the errors in the extrapolated values is presented. An example is also given of the use of neural networks (NNs) in the assessment of creep rupture data. NNs are straightforward to use but stringent requirements on the analyses must be fulfilled to get meaningful results. Fundamental models have reached a sufficient degree of development that they can be used to predict creep rupture data. This is demonstrated for austenitic stainless steels. The results of fundamental models can be generalized and extrapolated. In conventional empirical extrapolation with statistical methods, recently safe extrapolation can reach a factor of 3–5 in time [42]. This should be contrasted with the use of the basic model for primary and secondary creep of copper. It has been demonstrated that the model can describe experimental data at low stresses even after extrapolating the creep rate by many orders of magnitude [24]. For copper canisters for spent nuclear fuel, the canisters should stay intact for 100000 years. In such a situation the use of fundamental models is absolutely essential. Even such a large time scale can be covered. Extrapolation is discussed in Chap. 14.

The starting point for the study of creep can be one of the excellent text books by Ilschner [43], Evans and Wilshire [44], Kassner [45] or Zhang [46]. There are also many high quality review articles about creep, for example: Sherby and Burke [47], Lagneborg [6], Nix and Ilschner [48], Orlova and Cadek [49], Kassner and Pérez-Prado [50], and Blum [51].

It is not the aim to review the complete literature on creep modeling. It would neither be possible nor meaningful. Instead the book is concentrated to models that can be derived from physical principles and can give results in quantitative agreement with observations. Such models have mainly been presented in recent years.

The purpose of this book is fourfold

The author has taken a number of steps to make it easier for the reader to understand the models that are presented:

It is planned to provide supplementary material to the book text. The supplementary material will contain values of material constants and other information that would simplify repeating some of the computations in the book. This material will be placed at the author’s home page.

Or as an SKB report at

The title of the book will be included in the name of the supplementary material.