Paper 1 - A dislocation model for the plastic deformation of single-phase alpha iron.

It is assumed that the increase in dislocation density with strain is controlled by the creation, the immobilisation and the remobilisation of dislocations. It is further assumed that the mean free path of dislocation motion is strain independent. Proceeding from these assumptions a theory for the strain dependence of the total dislocation density is derived.

The stress-strain behaviour is obtained by applying Taylor’s equation for the relationship between stress and the total dislocation density. This theory has proved to be very useful for metals with high stacking fault energy and a large number of slip systems like alpha iron.

Paper 2 - The average rest time, t, of immobile dislocations.

It is sometimes of great interest to know the time an immobilised dislocation remains in rest before it remobilises. Proceeding from the theory in Paper 1 it is possible to derive a very simple relationship for, t. It is shown that t is inversely proportional to the applied strain rate and the dislocation remobilisation constant defined in Paper 1. This type of information is valuable when dealing with for instance Cottrell locking of dislocations by nitrogen and carbon atoms. Two examples are static and dynamic strain ageing.

Paper 3 - A theory for the temperature and strain rate dependences of the remobilisation constant, W.

At high enough temperatures the rate of dislocation remobilisation is controlled by vacancy diffusion. The dislocation theory presented in this paper describes this process and it is demonstrated that the model accurately predicts the influence of temperature and strain rate. It is also demonstrated that the activation energy for vacancy diffusion can be estimated by analysing experimental stress-strain curves recorded at different temperatures and strain rates.

In a later paper it will be demonstrated that this theory may also be applied to fcc metals.

Paper 4 - A correlation between the mean free path of dislocation motion and the “grown-in” dislocation density.

Reid et al (1) have drawn the attention to a rather striking and general correlation between the mean free path of dislocation motion, s_{0}, and the “grown-in” dislocation density, r_{0}. In this paper the dislocation theory presented in Paper 1 is used to study this correlation. The results indicate that s_{0} is inversely proportional to the square root of r_{0}, and that s_{0} is 6 to 7 times larger than the mean distance between the “grown-in” dislocations.

Paper 5 - A dislocation model for the plastic deformation of fcc metals - an analysis of pure copper and stainless steel

It is assumed that the increase in dislocation density with strain is controlled by the creation, the immobilisation and the remobilisation of dislocations. It is further assumed that the mean free path, s, of dislocation motion is strain dependent and decreasing with strain from an initial value s_{1} to a final value s_{0} at larger strains. Proceeding from these assumptions a theory for the strain dependence of the total dislocation density is derived.

The stress-strain behaviour is obtained by applying Taylor´s equation for the relationship between stress and the total dislocation density.

The theory has proved to be very useful in describing the true stress-true strain behaviour of metals with low stacking fault energies and a small number of slip systems as fcc metals like cupper and austenitic steels.

Paper 6 - A dislocation based model for the work hardening behaviour of Dual Phase steel.

The dislocation theory discussed in this paper accurately describes the plastic deformation behaviour of DP steel. Results from microstructure investigations as well as results from analysis of the stress-strain behaviour in terms of the present theory, tell us that the plastic deformation process in DP steel is inhomogeneous By introducing the concept of a non-homogeneity parameter

*f*(*e*), that specifies the volume fraction of ferrite that takes active part in the plastic deformation process, it is possible to give a precise physical description of the deformation behaviour until necking.

Paper 7 - Work hardening in single- and dual phase steel - a comparison

In the present paper we have studied the plastic deformation processes in two different steels, namely a single phase Ti-stabilised steel and a dual phase DP800 steel. It is shown that a simple dislocation model, taking the in-homogeneous plastic deformation into account, is capable of accurately describe the true stress-strain curves of the two steels strained at room temperature.

A comparison between the steels reviles that the plastic deformation process is much more in-homogeneous in the DP800 steel than in the Ti-stabilised single phase steel. However, it is also demonstrated that the Ti-stabilised steel exhibits a small in-homogeneous behaviour which leaks out after around 1% of straining. In the DP800 steel the in-homogeneity effect controls the work hardening behaviour up to necking. This also explains why the DP steels show an excellent combination of hardness and formability.

It is also interesting to note that the dual-phase theory is capable of estimating the volume fraction of the plastically non-deforming hard phase. In the present case it predicts that the DP800 steel contains 26.4% martensite and the Ti-stabilised steel 0.2% hard phase.

Paper 8 - The Hollomon n-value, and the strain to necking in steel** **

In this paper the Bergström dislocation theory for the stress-strain behaviour of metals is used to discuss the Hollomon n-value and the strain to necking in mild steel. It is demonstrated, in good agreement with experimental data, that n is not a constant but strongly strain dependent. This explains the double-n and triple-n behaviours often reported. Since n is often used as a measure of the stretch formability of sheet metals it should be stressed that n, in fact, is a bad measure for that purpose. The strain to necking is a much better measure.

In this report the Bergström dislocation theory for the stress-strain of metals is used to derive a simple expression for the strain to necking in mild steel and it is demonstrated, on a physical basis, that the strain to necking and hence stretch formability increases with:

increasing grain diameter, s_{g}

decreasing solution hardening, s_{s}

decreasing precipitation hardening, s_{p }

decreased thermal hardening, s*

increased dislocation creation, U_{0}

decreased dislocation remobilisation, W

The dislocation theory opens for the possibility to calculate exact values for the strain to necking as a function of values on the above mentioned parameters.

Paper 9 - The effect of strain rate in DP steel and the three stages of work

A recently presented dislocation based model for the work hardening behavior of DP steels is used to investigate the strain rate dependence of DP600 steel. Due to the expansion of the active volume fraction, f, with strain the thermal stress s*, and the local strain rate decreases with strain. For this purpose the strain rate dependence of thermal friction stress, s*, is taken into account. It should be observed that also the parameters f(e) and s(e) are strain rate dependent.

An attempt is also made to explain the three different stages of work hardening in this type of steel. Comments are given with respect to the number of parameters involved in the model and various ways to independently estimate the values of some of these parameters.

The Bergstrom dislocation model for homogeneous plastic deformation of pure single-phase metals has been gradually further-developed over the years. We begin with a brief review of the various steps taken in this further-development process.

Around 6 years ago a more thorough further-development of the theory was initiated in order to include also inhomogeneous plastic deformation which, inter alia, occurs in metals composed of hard and soft phases – e.g. modern, advanced high strength metals such as dual phase (DP) steels and LTT Martensitic steels but also high-strength fcc metals containing a mixture of micro-sized hard particles and a soft fcc metal.

Based on this further – development, it has been possible, firstly to improve the physical basis of the theory and, secondly, to improve its ability to describe the uniaxial true stress – true strain behavior of various types of inhomogeneous metals.

Proceeding from this model it is also possible to relate the global strain recorded in a tensile test with the much larger local, strain dependent strain caused by the inhomogeneity of the material.

It is assumed that the increase in dislocation density with strain is controlled by the creation, the immobilisation and the remobilisation of dislocations. It is further assumed that the mean free path of dislocation motion is strain independent. Proceeding from these assumptions a theory for the strain dependence of the total dislocation density is derived.

The stress-strain behaviour is obtained by applying Taylor’s equation for the relationship between stress and the total dislocation density. This theory has proved to be very useful for metals with high stacking fault energy and a large number of slip systems like alpha iron.

Paper 2 - The average rest time, t, of immobile dislocations.

It is sometimes of great interest to know the time an immobilised dislocation remains in rest before it remobilises. Proceeding from the theory in Paper 1 it is possible to derive a very simple relationship for, t. It is shown that t is inversely proportional to the applied strain rate and the dislocation remobilisation constant defined in Paper 1. This type of information is valuable when dealing with for instance Cottrell locking of dislocations by nitrogen and carbon atoms. Two examples are static and dynamic strain ageing.

Paper 3 - A theory for the temperature and strain rate dependences of the remobilisation constant, W.

At high enough temperatures the rate of dislocation remobilisation is controlled by vacancy diffusion. The dislocation theory presented in this paper describes this process and it is demonstrated that the model accurately predicts the influence of temperature and strain rate. It is also demonstrated that the activation energy for vacancy diffusion can be estimated by analysing experimental stress-strain curves recorded at different temperatures and strain rates.

In a later paper it will be demonstrated that this theory may also be applied to fcc metals.

Paper 4 - A correlation between the mean free path of dislocation motion and the “grown-in” dislocation density.

Reid et al (1) have drawn the attention to a rather striking and general correlation between the mean free path of dislocation motion, s

Paper 5 - A dislocation model for the plastic deformation of fcc metals - an analysis of pure copper and stainless steel

It is assumed that the increase in dislocation density with strain is controlled by the creation, the immobilisation and the remobilisation of dislocations. It is further assumed that the mean free path, s, of dislocation motion is strain dependent and decreasing with strain from an initial value s

The stress-strain behaviour is obtained by applying Taylor´s equation for the relationship between stress and the total dislocation density.

The theory has proved to be very useful in describing the true stress-true strain behaviour of metals with low stacking fault energies and a small number of slip systems as fcc metals like cupper and austenitic steels.

Paper 6 - A dislocation based model for the work hardening behaviour of Dual Phase steel.

The dislocation theory discussed in this paper accurately describes the plastic deformation behaviour of DP steel. Results from microstructure investigations as well as results from analysis of the stress-strain behaviour in terms of the present theory, tell us that the plastic deformation process in DP steel is inhomogeneous By introducing the concept of a non-homogeneity parameter

Paper 7 - Work hardening in single- and dual phase steel - a comparison

In the present paper we have studied the plastic deformation processes in two different steels, namely a single phase Ti-stabilised steel and a dual phase DP800 steel. It is shown that a simple dislocation model, taking the in-homogeneous plastic deformation into account, is capable of accurately describe the true stress-strain curves of the two steels strained at room temperature.

A comparison between the steels reviles that the plastic deformation process is much more in-homogeneous in the DP800 steel than in the Ti-stabilised single phase steel. However, it is also demonstrated that the Ti-stabilised steel exhibits a small in-homogeneous behaviour which leaks out after around 1% of straining. In the DP800 steel the in-homogeneity effect controls the work hardening behaviour up to necking. This also explains why the DP steels show an excellent combination of hardness and formability.

It is also interesting to note that the dual-phase theory is capable of estimating the volume fraction of the plastically non-deforming hard phase. In the present case it predicts that the DP800 steel contains 26.4% martensite and the Ti-stabilised steel 0.2% hard phase.

Paper 8 - The Hollomon n-value, and the strain to necking in steel

In this paper the Bergström dislocation theory for the stress-strain behaviour of metals is used to discuss the Hollomon n-value and the strain to necking in mild steel. It is demonstrated, in good agreement with experimental data, that n is not a constant but strongly strain dependent. This explains the double-n and triple-n behaviours often reported. Since n is often used as a measure of the stretch formability of sheet metals it should be stressed that n, in fact, is a bad measure for that purpose. The strain to necking is a much better measure.

In this report the Bergström dislocation theory for the stress-strain of metals is used to derive a simple expression for the strain to necking in mild steel and it is demonstrated, on a physical basis, that the strain to necking and hence stretch formability increases with:

increasing grain diameter, s

decreasing solution hardening, s

decreasing precipitation hardening, s

decreased thermal hardening, s*

increased dislocation creation, U

decreased dislocation remobilisation, W

The dislocation theory opens for the possibility to calculate exact values for the strain to necking as a function of values on the above mentioned parameters.

Paper 9 - The effect of strain rate in DP steel and the three stages of work

A recently presented dislocation based model for the work hardening behavior of DP steels is used to investigate the strain rate dependence of DP600 steel. Due to the expansion of the active volume fraction, f, with strain the thermal stress s*, and the local strain rate decreases with strain. For this purpose the strain rate dependence of thermal friction stress, s*, is taken into account. It should be observed that also the parameters f(e) and s(e) are strain rate dependent.

An attempt is also made to explain the three different stages of work hardening in this type of steel. Comments are given with respect to the number of parameters involved in the model and various ways to independently estimate the values of some of these parameters.

The Bergstrom dislocation model for homogeneous plastic deformation of pure single-phase metals has been gradually further-developed over the years. We begin with a brief review of the various steps taken in this further-development process.

Around 6 years ago a more thorough further-development of the theory was initiated in order to include also inhomogeneous plastic deformation which, inter alia, occurs in metals composed of hard and soft phases – e.g. modern, advanced high strength metals such as dual phase (DP) steels and LTT Martensitic steels but also high-strength fcc metals containing a mixture of micro-sized hard particles and a soft fcc metal.

Based on this further – development, it has been possible, firstly to improve the physical basis of the theory and, secondly, to improve its ability to describe the uniaxial true stress – true strain behavior of various types of inhomogeneous metals.

Proceeding from this model it is also possible to relate the global strain recorded in a tensile test with the much larger local, strain dependent strain caused by the inhomogeneity of the material.

© copyright 2010 Yngve Bergström