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Open Access 2024 | Open Access | Buch

Basic Modeling and Theory of Creep of Metallic Materials

verfasst von: Rolf Sandström

Verlag: Springer Nature Switzerland

Buchreihe : Springer Series in Materials Science

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

Inhaltsverzeichnis

Über dieses Buch

This open access book features an in-depth exploration of the intricate creep behavior exhibited by metallic materials, with a specific focus on elucidating the underlying mechanical properties governing their response at elevated temperatures, particularly in the context of polycrystalline alloys. Traditional approaches to characterizing mechanical properties have historically relied upon empirical models replete with numerous adjustable parameters, painstakingly tuned to match experimental data. While these methods offer practical simplicity, they often yield outcomes that defy meaningful extrapolation and application to novel systems, invariably necessitating the recalibration of parameters afresh.
In stark contrast, this book compiles a compendium of models sourced from the scientific literature, meticulously crafted through ab initio methodologies rooted in fundamental physical principles. Notably, these models stand apart by their conspicuous absence of adjustable parameters. This pioneering effort is envisioned as a groundbreaking initiative, marking the first of its kind in the field. The resulting models, bereft of arbitrary tuning, offer a level of predictability hitherto unattained. Notably, they provide a secure foundation for ascertaining operational mechanisms, contributing significantly to enhancing our understanding of material behavior in high-temperature environments.
This open access book is a valuable resource for researchers and seasoned students engaged in the study of creep phenomena in metallic materials. Readers will find a comprehensive exposition of these novel, parameter-free models, facilitating a deeper comprehension of the intricate mechanics governing material deformation at elevated temperatures.

Inhaltsverzeichnis

Frontmatter

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Chapter 1. The Role of Fundamental Modeling

Abstract

The difference between empirical and basic modeling and its significance is explained. The types of basic models that have been possible to develop and that are describe in the book are summarized. The starting point is a basic model for the dislocation density that is used to derive expression for tensile and creep properties. It is described how the accuracy of the basic models can be verified. For the creep models it is described that they are applicable over a wide range of temperatures and stresses that is of great value to identify operating mechanisms.

Rolf Sandström

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Chapter 2. Stationary Creep

Abstract

An introduction to creep and its main characterstics are given. Stationary creep has been studied extensively in the literature. Stationary creep is a result of a balance between work hardening and recovery processes, which allows for a continues plastic deformation without raising the stress. The starting point for the basic modeling of creep is a differential equation for the dislocation density that describes how it varies with strain or time. The model explains how the dislocation density is influenced by work hardening and recovery. From the dislocation model, a basic equation for the creep rate is derived that is in many respects similar to the classical Bird, Mukherjee and Dorn (BMD) formula but with the values of the parameters given. By taking the role of strain induced vacancies into account, the applicability of the BMD equation is widely expanded because the basic model can also handle low temperatures and high stresses that is usually referred to as the power-law break down regime. It is illustrated that the creep model can represent the creep rate for pure metals such as Al and Ni.

Rolf Sandström

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Chapter 3. Stress Strain Curves

Abstract

Traditionally, stress strain curves for example from tensile testing are described with empirical models with a number of adjustable parameters such as Hollomon, Ludwik, Voce and Swift. With such models it is difficult or impossible to generalize and extrapolate. A model in the form of Voce equation is derived from the same basic dislocation model used for the creep models with the values of constants computed. The derived model is used to describe stress strain curves for Cu including their temperature and strain rate dependence. The dynamic recovery constant ω plays a central to show how the work hardening deviates from a linear behaviour. The temperature dependence of ω is analyzed and shown to be related to that of the shear modulus. In the literature it is frequently assumed that dynamic recovery is controlled by cross-slip. However, the measured activation energy for dynamic recovery is many times smaller than the energy required to make partial dislocations brought together and form a constriction, which is necessary to enable cross-slip, so this is an unlikely possibility.

Rolf Sandström

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Chapter 4. Primary Creep

Abstract

For many materials, primary creep can be described with the phi (ϕ) model and tertiary creep with the Omega (Ω) model (discussed in Chap. 12). According to the phi model, the creep rate is linear in strain and time in a double logarithmic diagram. When using empirical descriptions of the creep curves, these models are recommended. Several basic models for primary creep are derived. They are based on the creep rate in the secondary stage. This means that primary creep can be derived without any new data. The primary creep models are in agreement with the phi model and can describe experimental data. For the martensitic 9–12% Cr steels at least two dislocation densities are needed to represent primary creep because the initial dislocation density is high contrary to the situation for annealed fcc materials.

Rolf Sandström

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Chapter 5. Creep with Low Stress Exponents

Abstract

Primary creep models predict that at low stresses a stress exponent of 1 can be obtained for dislocation creep. Also experimentally this has been observed for an austenitic stainless steel. The time dependence of the primary creep verifies that it is dislocation creep. An other example is for Al at very high temperatures (Harper-Dorn creep), where at sufficiently low stresses, the stress exponent approaches 1. For both materials higher stresses give larger stress exponents as expected for dislocation creep. Obviously, diffusion and dislocation creep can be competing processes. The validity of creep models at low stresses and high temperatures as well as at high stresses and low temperatures demonstrates their wide range of usage. Since this in reality represents an extensive extrapolation, it can be consider as a direct verification of the basic creep models. In cases for Cu and stainless steels, the predicted creep rate by diffusion creep (Coble) exceeds the observed creep rate as well as the predicted one by dislocation creep by an order of magnitude. The likely explanation is that constrained boundary creep is taken place, i.e. the grain boundary creep rate cannot be essentially faster than that of the bulk.

Rolf Sandström

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Chapter 6. Solid Solution Hardening

Abstract

The size and modulus misfit between solute and parent atoms gives rise to strengthening, solid solution hardening (SSH). With the development of Argon’s expression for the interaction energies for solute atoms and dislocations for size and modulus misfit, both effects can now be modeled without the introduction of adjustable or arbitrary parameters. These expressions are used to derive models for SSH during creep. Although the constants for the modulus misfit can be an order larger than those for size misfit, the latter effect is still dominating. The interaction energy gives a direct contribution to the activation energy for creep. The solutes form Cottrell atmospheres around the dislocations. For slowly diffusion elements, these atmospheres give rise to a drag force that slows down the motion of the dislocations. Fast diffusing elements have to break away from the dislocations to enable their motion. This creates a break stress that is the source of SSH in this case.

Rolf Sandström

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Chapter 7. Precipitation Hardening

Abstract

Models for precipitation hardening (PH) at room temperature have been available for a long time. In spite of the importance of PH, it took a long time to establish models for elevated temperatures. In fact, empirically the room temperature models have also been used at higher temperatures. This gives the wrong temperature dependence and overestimates PH. It was for a long time thought that it was an energy barrier for climb across particles that was the controlling mechanism, but it was gradually appearing that this effect was so small that it could be neglected. Instead it is time it takes for dislocations to climb across particles that is the critical factor. Small particles are readily passed and do not contribute to the strengthening. Particles larger than a critical size have to be passed by the Orowan mechanism, because there is not time enough for dislocations to climb across these particles. This mechanism was finally verified for Cu–Co alloys.

Rolf Sandström

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Chapter 8. Cells and Subgrains. The Role of Cold Work

Abstract

In almost all metals and alloys, dislocations are concentrated to narrow regions after plastic deformation that divide the material into cells or subgrains. The cell walls consist of tangles whereas the subgrains are surrounded by thin regular networks of dislocations. The cells are transferred to subgrains with increasing temperature. Although these substructures have been analyzed for many years, basic models of their development have only appeared recently. Models for substructures are presented for plastic deformation at constant stress and at constant strain rate. During straining the dislocations can move in opposite directions creating a polarized structure, where the possibility for recovery of dislocations is reduced. This can be expressed in term of a back stress. Its presence explains why creep curves at near ambient temperatures could have an appearance that is similar to that at elevated temperatures. It is also the basis for the effect of cold work on creep. The models can quantitatively describe why the creep rate can be reduced by up to six orders of magnitude for Cu after cold work.

Rolf Sandström

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Chapter 9. Grain Boundary Sliding

Abstract

During plastic deformation at elevated temperatures, grains move relative to each other which is referred to as grain boundary sliding (GBS). The amount of GBS is proportional to the creep strain with a proportionality constant that is known from finite element analyses, and found to agree with experiments for Cu. The most import effect of GBS is that it gives rise to the initiation of creep cavities, Chap. 10. GBS is also the main mechanism for superplasticity. A basic model for superplasticity is presented.

Rolf Sandström

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Chapter 10. Cavitation

Abstract

Cavitation is of great technical importance. Nucleated cavities grow and link to form cracks that can cause rupture. During creep, cavities are initiated in the grain boundaries. The nucleation takes place at particles or at subboundary—grain boundary junctions. The main mechanism is believed to be grain boundary sliding (GBS), Chap. 9. According to the double ledge model, cavities are formed when the particles or subboundaries meet other subboundaries. With this assumption quantitative models for cavity nucleation can be derived. They show that the nucleated number of cavities is proportional to the creep strain in good accordance with observations. Cavities can grow by diffusion or by straining. It is important to take into account that cavities cannot grow faster than the surrounding creeping matrix, which is referred to as constrained growth. Otherwise the growth rate can be significantly overestimated. Models both for diffusion and strain controlled growth have been available for a long time. A recently developed model for strain controlled growth is presented based on GBS. It has the advantage that is associated with a well-defined initiation size of cavities and that constrained growth is automatically taken into account, features that some previous strain controlled models miss.

Rolf Sandström

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Chapter 11. The Role of Cavitation in Creep-Fatigue Interaction

Abstract

There are many empirical models for the development of creep and fatigue damage. The perhaps most well-known ones are Robison’s and Miner’s damage summation rules. They are based on the mechanical behavior during monotonous and cyclic loading. To improve the accuracy of the damage assessment, it is important to analyze the changes in the microstructure as well, not least the cavitation. To describe cyclic loading, special empirical models have often been used in the past, some with numerous adjustable parameters. Recently, a model for cyclic loading has been formulated that is based on the corresponding expressions for monotonous loading. The main change is that the value of the dynamic recovery constant is increased. In this way, cyclic hysteresis loops can be reproduced without adjustable parameters. Cavitation is believed to be of the same technical importance during cyclic as during static loading. In spite of this, the number of studies of cavitation during cyclic loading is quite limited. One set of data exists for a 1Cr0.5Mo steel. The static cavitation models have been transferred to cyclic conditions. It is demonstrated that these models can describe the cavitation both during low cycle fatigue (LCF) and combined creep and LCF.

Rolf Sandström

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Chapter 12. Tertiary Creep

Abstract

In the tertiary stage, the creep rate is continously increasing eventually leading to rupture. Many mechanisms can contribute to the increasing creep rate such as particle coarsening, substructure coarsening, cavitation, changes in the dislocation density and necking. A large number of empirical models exist for the description of tertiary creep and the development of creep damage not least in the context of continuum damage mechanics (CDM). However, there are also basic models. An equation is presented that can describe the whole creep strain versus time curve. Only parameters that are already defined for secondary creep are needed. During the tertiary stage the true applied stress increases rapidly and faster than the counteracting dislocation strength, which is one main reason for the increase in the creep rate during the tertiary stage. Cavitation is of importance, but the cavitation is often local and therefore gives a modest contribution to the creep rate. According Hart’s criterion, necking starts right at the beginning of the tertiary stage. But the necking is not fully developed until close to rupture. This is demonstrated both by uniaxial and multiaxial models and it is also consistent with available experimental data.

Rolf Sandström

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Chapter 13. Creep Ductility

Abstract

For a number of creep resistant steels, the creep ductiliy decreases with increasing temperature and time. As a function of stress, the ductiity is often describe with an S-shaped curve with an upper and a lower shelf level. As a function of time, the S-shape is inverted. If the ductility is high, the rupture is referred to as ductile, and for low ductility levels as brittle. Ductile rupture is believed to be due to a plastic instability such as necking. Brittle rupture on the other hand is controlled by the nucleation, growth and linkage of creep cavities. With the help of the basic models for creep deformation and cavitation, the rupture stress and ductility can be predicted. Several models exist for the influence of multiaxiality on the creep ductility. Although the models are based on different principles, they predict approximately the same behavior, which is verified by comparison to rupture data for notched bars.

Rolf Sandström

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Chapter 14. Extrapolation

Abstract

The extrapolation of creep data to longer times is technically very important. The traditional way of extrapolating creep rupture data is to use time temperaturer parameters (TTPs). In this way data from several test temperatures are combined to a single master curve that can be used to assess rupture strengths at long times. Recently, there is much focus on machine learning techniques (neural networks, NNs). Both types of procedures can generate accurate results, but a detailed analysis is required. A good way to assess the quality of the results is to use the post assessment tests (PATs) developed by ECCC. Without such tests arbitrary results can be obtained. They are important for both TTPs and NNs. It has been shown that by putting requirements on the derivatives of the creep rupture curves, the PATs can more or less automatically be satisfied. In addition, the error in the extrapolated values should be estimated. Using the basic creep models presented in this book, prediction of rupture strength and ductility can be made in a safer way. It is demonstrated for Cu that accurate extrapolation of many order of magnitude in the creep rate can be made, which is never possible with empirical models.

Rolf Sandström

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Titel: Basic Modeling and Theory of Creep of Metallic Materials
verfasst von: Rolf Sandström
Verlag: Springer Nature Switzerland
Electronic ISBN: 978-3-031-49507-6
Print ISBN: 978-3-031-49506-9
DOI: https://doi.org/10.1007/978-3-031-49507-6

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