Instability of plastic flow and their role in texture transformations

Instability of plastic flow and their role in texture transformations
Authors: Marta Gajewska Grażyna Kulesza Honorata Kazimierczak Katarzyna Stan Jagoda Poplewska Piotr Bobrowski Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Content Plasticity and yield criteria
Fundamental relations between flow stress and strain Mechanical aspects of plastic instability Plastic anisotropy vs. crystallographic texture and sheet metal forming The analysis of some important (microstructural) features of plastic deformation that are important for large strains Roling and formation of 2 types of rolling texture Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Plasticity and yield criteria (uniaxial and multiaxial plastic flow analysis) Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Stress 1D case – normal stress
Measure of the average force per unit area of a surface within the body on which internal forces act (as a reaction to external forces applied). These internal forces are distributed continuously within the volume of the material and may result in deformation of the body's shape. 1D case – normal stress { F II A  shear stress } Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Stress 3D – Cauchy stress tensor
Principal axes – set of coordinates for which all the components of shear stress are zero σij – i-component of force acting on a unit area with normal xj Moment on each axis is equal to zero (stress tensor is symmetric ) six independent variables isotropic part deviatoric part Hydrostatic pressure Stress deviator Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Relative displacement tensor
Strain 1D – linear strain 3D strain The state of strain at a any point of a continuum body is defined as the sum of the changes in distance between this point and every other neighboring point (linear strain) and the changes in the angle between any pair of directions radiating from this point (shear strain) σ1 Relative displacement tensor Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

state of relative displacement
Strain symmetric tensor asymmetric tensor 1 2 B0 Bt state of relative displacement increment dγ21 dγ12 pure strain increment rigid body rotation + = Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Elasticity vs. plasticity
Describes property of a material that returns to its original shape after the applied forces that made it deform is removed Describes the deformation of a material undergoing non-reversible changes of shape in response to applied forces Hook’s law σ = E ε J. Lubliner, 2008, Plasticity theory Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Huber-Mises-Hencky criterion
Yielding occurs when some function of the principal stresses σ1, σ2, σ3 reaches a critical value 2σf2 σf - flow stress Yield surface for this criterion encloses an infinitely long circular cylinder Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Tresca criterion Yielding occurs when the maximum shear stress reaches a critical value 0,5 σf σmax, σmin – largest and smallest principal stress Yield surface corresponding to this criterion encloses an infinitely long hexagonal cylinder Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Yield surface: Tresca vs H-M-H (2D)
Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Yield surface: Tresca vs H-M-H (3D)

σ3=0 σ2 (tension) σ2 σ1 Tresca σ1 von Mises σ1 (tension) σ0
σ1 (compression) σ2 (compression) von Mises Tresca σ3=0 σ1 σ2 4 1 3 σ1 5 4 2 1 σ1 σ2 σ1 σ2 4 5 2 4

Fundamental relations between flow stress and strain (basic descriptions, definitions and constitutive laws used to describe plastic deformations) Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Stress-strain relations
Strengthening consists the increase of flow stress with the increase of permanent plastic deformation. Material with plastic stress in the annealed condition of σy0 after permanent deformation ε has increased plastic stress σp = f(ε). Stress-strain curve obtained from unidirectional tensile, compression or torsion tests is usually described by a simple Ludwik formula: σy, k1 – constans, n – hardening, which for most metals takes values in the range If the initial yield strength for annealed material is relatively small in relation to size of strengthening, the above relationship can be reduced to a good approximation to the model given by Hollomon: k2, n – constans Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Previous dependencies well describe the strengthening behavior of a large part of metals in range of deformation ≤ 0.5, i.e. in the range of deformation typically obtained in a tensile test of plastic metal. In the greater degree of deformation with which we deal in most of the forming operation, the strengthening is strongly reduced and appears more or less apparent linear range of the strengthening curve. Then to analytical description we needs two or more pairs of parameters (k i n) An alternative approach to the analysis of a wide range of deformation is based on the experimentally observed fact that the flow stress of most metals, for very large degrees of deformation aims to achieve saturation. This condition is described characteristic of each metal value of stress σs. Constitutive equation describing the strengthening may be based, in this case on the relationship given by Voce'a: α – a dimensionless constant characteristic of the strengthening behavior This equation was modified by Hocket and Sherby to the form: Given by Lloyd and Kenny constant value p for aluminum was ≈ 0.5 Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Differentiating Voce’s equation we obtain a linear dependence of the strengthening Θ on a decreasing flow stress values given by Kocks-Mecking equation: Θ0 determines the initial value of strengthening Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Considère criterion Referring to Kocks-Mecking equation, in a tension test the sample is getting strained but it is also becoming thinner with elongation. The latter implies that stress, which is load per unit area, is increasing. Thereby, two things are occurring simultaneously: material is strain hardening and also supporting increasing stress, thanks to the reduction in area. Naturally, a situation will arise when the increase in stress, due to decrease in cross section area, will become equal to the increase in load carrying capacity due to strain hardening. Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Geometrical interpretation
Description of the plastic material with strengthening provides a description of the relationship between stress and strain including effect of the strengthening of these relationships. Depending on whether this effect is an isotropic or anisotropic, makes a distinction between isotropic or anisotropic strengthening. Isotropic strengthening of isotropic material is called plastic deformation in which the area of plastic flow grows symmetrically with a maintaining of the form. Yield condition for such material is written by making the condition of plasticity on the parameter of strengthening. < 0 – elastic or stiff material = 0 – plastic material Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Geometrical interpretation
Isotropic strengthening in uni-dimensional case is manifested by the fact that the plastic strain σ is independent on the way of load. This property transfer to the case of (consisted) complex stress. With the growth of plastic deformation the flow area F(σij,κ)=0 is increased symmetrically irrespective of the way of the load. Increase of the plasticity area depends only on the strengthening parameter κ. As a measure of this parameter is accepted substitute plastic deformation ε, or specific work of plastic deformation w. loading loading offloading offloading reloading reloading Geomtrical interpretation of izotropic strengthening of material: a) unidirectional strain, b) in general Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Work hardening Plastic flow of crystalline materials takes place by the movement of dislocations along cystal planes under the influence of an applied stress. Work hardening is a consequence of the fact that the stress required for dislocation movement usually increases during plastic flow as the dislocations become increasingly hindered by microstructural obstacles. In order to increasing size these obstacles are solute atoms, dislocations, precipitates and grain boundaries. The most imporatant variation in obstacle density is usually due to the dislocations themselves. Their behaviour in crystals under stress is therefore of paramount importance for understanding and modelling work hardening. Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Work hardening stages Stage I: The dislocations are usually confined to their slip planes and do not interact with each other so that the work hardening rate is very low (most easily seen in single crystals oriented for single slip). In polycrystals, which begin to deform plastically, Stage I is negligible since the movement of the first few dislocations is restricted by the grain boundaries at which they often pile-up. Stage II: The dislocation interactions on different slip systems give rise to a rapid multiplication of the dislocations and thereby a high, and roughly constatnt, work hardening rate. Stage III: Subsequently, and up to strains of order unity, the flow curve becomes parabolic as the work hardening rate decreases progresively down to values almost an order of magnitude lower than Stage II. In this stage, the dislocation multiplication processes are cunterbalanced by local dislocation annihilations (dynamic recovery due to localized cross-slip, climb and/or annihilation of segment of oppisite sign). These recovery mechanisms and therefore the work hardening rates are strongly temperature dependent. Stage IV: At higher strains ≥ 1 typical of many rolling and extrusion processes, many grains break up into bands of different orientations, separated by transition zones and grain boundaries. Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Microscopic hardening laws
The particular microscopic laws are often complex so the relatively simple analysis given below in terms of the total dislocation density can be considered as an illustration of the principles. The microscopic work hardening rate (dτc/dγ) can be written in terms of the varaition of the dislocation density as: According to the analysis of Kocks and Mecking the rate of creation of dislocations (dρ+) during a small strain increment dγ is inversely proportional to the mean dislocation slip distance λ so that (dρ+ / dγ) = (1 / bλ) and the mean slip distance λ = C1/ρ½, where C1 is the average number of obstacles that the dislocation meets before stopping. Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

During Stage II hardening, where dislocation annihilation is small and can be neglected, the above model gives a constant hardening rate: During Stage III the hardening rate decreases continuously as some of the dislocations are annihilated by dynamic recovery at a rate written (dρ-/dγ). The exact mechanisms of annihilation (cross-slip, climb) are the subject of current research but one can write that the rate of annihilation is proportional to the current density ρ and a probability of elimination P(T), which is strongly temperature dependent: Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Consequently, Stage III work hardening rate becomes linear in τc: At large strains and low temperatures dislocation annihilation is insufficient to completely balance the rate of creation. This result in the low, but non-zero, work hardening rate of Stage IV; there is no general agreement on the basis physical causes of this stage. Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Mechanical aspects of plastic instability (based on stress – strain curves) Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Instability identifies what happens when a state of uniform strain gives way to one in which straining is more or less localized. Instability: necking in tension shear-band development Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Strain-induced hardening
Schematic presentation of instability in tensile test Strain-induced hardening Geometrical softening in an incipient neck Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Rods in tension To initiate instability (a neck in this case):
a/R ratio for characterizing neck sharpness Grafical mean of finding the necking strain Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Rods in tension Mild steel often shows both stable and unstable necks

Rods in tension Mild steel often show both stable and unstable necks

Rods in tension Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Sheets in tension Diffuse necking
Local neck in a strip tension specimen. (Normal strain along X2’ must be zero) Plane strain in the localized neck Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Sheets in tension The criteria for localized and diffuse necking in pure tension. Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Plastic anisotropy vs. crystallographic texture and sheet metal forming Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Crystalline structure
Crystal - periodic arrangement of atoms in space. Different array of atoms on particular planes. Mechanical, physical and chemical properties vary with testing direction An example : graphite – good electircal conductivity in the direction parallel to the layers, and much worse in the perpendicular direction. Graphite crystal structure skywalker.cochise.edu/wellerr/mineral/graphite/graphiteL.htm Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Polycrystalline materials
Polycrystalline materials – many crystallites of different size, shape and orientation. In isotropic materials all orientations occur with the same probability – properties in all directions are the same Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Texture Crystallites orientation in the material is not random,
Preferred orientation is formed Texture is the main structural reason of anisotropy in polycrystalline materials - properties change with testing direction Anisotropic properties: Mechanical properties: plasticity, elasticity, hardness, strength, Physical properties: optical properties, thermal expansion and conductivity, electric conductivity, magnetization, chemical (corrosion resistance) The exact nature of anisotropy depends upon both alloy composition and process history (e.g. casting, rolling, extrusion, annealing, etc). Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Texture of: crystallization, deformation (rolling, extruding), recrystallization
Not only orientation but also the grains size and shape is important. For example during rolling some crystallographic plane arrange parallel to rolling plane and rolling direction. As a result elongated grains in the rolling direction are formed. Aluminium ingot – example of the texture formed during crystallization Texture evolution of an ultrafine grained (UFG) C-Mn steel and its evolution during warm deformation and annealing R. Song, D. Ponge, D. Raabe Plastic anisotropy Rolling direction Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Recrystallization can produce texture completely
different than that of the deformed material. Deformation-related recrystallization processes M. R. Drury and J. L. Urai Globular shape of grains  smaller anisotropy Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Sheet metal forming Anisotropic properties can have a major effect on subsequent process stages, especially sheet metal forming processes such as deep drawing and stretch forming. Sheet metal exibits special anisotropy. It is characterised by occurance of significant differiences in plastic properties in 3 mutually perpendicular directions: 1) rolling direction, 2) direction normal to rolling direction lying in the sheet metal plane 3) direction normal to sheet metal plane These are the main anisotropy directions. Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Types of anisotropy in sheet metal
Planar Normal Plastic properties in the plane of the sheet metal change with direction. This kind of anisotropy is important from the standpoint of sheet metal deep- drawing Plastic properties change in the direction normal to plane of sheet metal and in the direction lying in the plane of sheet metal. Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

How we can determine anisotropy?
Plastic anisotropy coefficients (Langford) – ratio of real deformation width to real deformation thickness of sample from metal sheet subjected to uniaxial tensile. εh – real deformation of thickness εb - real deformation of width b0 – initial sample width b - sample width after deformation h0 – initial sample thickness h – sample thickness after deformation From description of dr inż. Łukasz Cieniek - Anizotropia i tekstura krystalograficzna. Starzenie po odkształceniu, AGH L0 – initial sample length L – sample length after deformation Sample deformation used to determine normal and planar anisotropy of sheet metal Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

If r = 1 sheet metal has isotropic properties
Significant change in r coefficient with the change of α angle  planar anisotropy Constant value of r coefficient but larger from 1  normal anisotropy In practice 3 types of samples are investigated, which form 3 axis with rolling direction: 45°, 90° and 0° . This values are utilized to determine planar anisotropy and mean normal anisotropy Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Geometric interpretation
Biaxial uniform streching Yield criterion – is a mathematical expression of the stress states that will cause yielding or plastic flow. Shape of the yield surface 45° material with planar anisotropy Ellipses are elongated (if r>1)or shorted (if r<1) in relation to ellipse of isotropic material. Retraction of the ellipse main axis from the 45 ° slope indicates a planar anisotropy. Biaxial uniform compression Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Why is that so important?
Sheet metal for deep-drawing Important from the point of view of shaping by cold plastic deformation (it is basic parameter of deep-drawing sheet metal ) aluminium.matter.org.uk Low value of planar anisotropy is favorable High value of normal anisotropy is favorable , in that case material has a greater “thinning resistance” material deforms better in drawing operations Anisotropy in the mechanical properties can cause 'earing' in beverage can bodies. The ears must be cut off, leading to wastage. Photo courtesy of VAW aluminium AG Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

The analysis of some important (microstructural) features of plastic deformation that are important for large strains Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Microstructure evolutution with strain
A microstructure no deformed Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

A microstructure at small strain DDW - dense dislocation wall Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

A microstructure at large strain Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Homogeneous deformation
In cubic metals the two basic methods of deformation are slip and twinning and the most significant material parameter with respect to the choice of method is the value of the stacking fault energy (γSFE). In metals with a low value of γSFE, the difficulty of cross-slip reduces the ability of the material to change its shape during plastic deformation by slip alone, and therefore deformation twinning may occur. Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Stacking fault energy (γSFE)
– is a function of the material, determines the extent to which unit dislocations dissociate into partial dislocations. Copper, α-brass and austenitic stainless steel have a low γSFE and climb and cross slip of dislocations are difficult in these metals (recrystallization is easier). Metals such aluminium and α-iron have a high γSFE and climb is easy (recovery is easier). Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Homogeneous deformation - slip
Slip occurs in directions in which the atoms are most closely packed, since this requires the least amount of energy; Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Homogeneous deformation - twinning
When mechanical deformation is created by twinning, the lattice structure changes; The atoms move only a fraction of an interatomic space and this leads to a rearrangement of the lattice structure. Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Homogeneous deformation - slip vs. twinning

Non-homogeneus deformation
deformation bands transitions bands shear bands Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

deformation bands transitions bands shear bands The resulting deformation bands deform on different slip systems and may develop widely divergent orientations. The narrow regions between the deformation bands, which may be either diffuse or sharp, are termed transition bands. Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

deformation bands transitions bands shear bands - are noncrystallographic in nature, may pass through several grains, and even extend through the specimen. They are a result of plastic instability, and can be thought of as the rolling equivalent of the ‘necking’ which occurs in a tensile test. Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

deformation bands transitions bands shear bands TEM microstructure showing an advanced stage of shear band formation against a background of the compact twin-matrix layer structure after 67% compression Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Influence of alloying elements
Solute Hardening Precipitation Hardening Dispersoids Hardening Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Solid Solution Hardening Precipitation Hardening Dispersoids Hardening Solute atoms make dislocation motion more difficult increase the strength of a material Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Solute Hardening Precipitation Hardening Dispersoids Hardening Precipitations make dislocation motion more difficult increase the strength of a material Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Solute Hardening Precipitation Hardening Dispersoids Hardening A dispersion of particles will exert a retarding force or pressure on a low angle or high angle grain boundary and this may have a profound effect on the processes of recovery, recrystallization and grain growth. Vickers hardness measurements of alloys after cold rolling (z=60%) and annealing for 1h in different temperatures L.Lityńska-Dobrzyńska; Rola cyrkonu i skandu w procesach tworzenia struktur metastabilnych w stopach Al.-Mg-Si-Cu, IMIM PAN, Kraków 2009 Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Solute Hardening Precipitation Hardening Dispersoids Hardening A dispersion of particles will exert a retarding force or pressure on a low angle or high angle grain boundary and this may have a profound effect on the processes of recovery, recrystallization and grain growth. Interaction of particles with grain boundary M.J. Jones, F.J. Humphreys; Interaction of recrystallization and precipitation: The effect of Al3Sc on the recrystallization behaviour of deformed aluminium, Acta Materialia 51 (2003) 2149–2159 Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Roling and formation of 2 types of rolling texture (the influence of the texture on sheet metal forming) Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Rolling and deep drawing
TD RD A B C D E rad cir εRD εND εcir εrad A B C D E A. Rollet „Typical textures” Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Rolling texture Cu rolled to 90%

Brass and copper textures
[111] PF, 90% reduction 0% 2,5% 5% Copper Brass 10% 20% 30% archive.uky.edu Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Negative aspects sheet metal forming deep drawing: Cu cups RD 00 450
900 Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Positive aspects Transformer metal plates (Fe-Si):
good textures for high magnetic coercion and permittivity: <100>{001} – cubic, <001>{110} - Goss permanent magnets (Alnico): <100> direction oriented along the magnet axis Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

References „Deformation Geometry for Materials Scientists”, C. N. Reid, International series on materials science and technology vol. 11, Oxford: New York: Pergamon Press, 1973, 2-30, 83-88 „Mechanika Plastycznego Płynięcia w zarysie”, T. Bednarski, PWN, 1995 „Odkształcenie i pękanie metali”, J. Wyrzykowski, E. Pleszakow, J. Sieniawski, WNT, 1999 „Obróbka plastyczna”, S. Erbel, K. Kuczyński, Z. Marciniak, PWN, 1977 „ Strukturalne aspekty odkształcenia metali”, K. Przybyłowicz „Dislocations and plastic deformation”, I. Kovac, L. Zsoldos „ Deformation processing”, W. A. Backofen, Addison-Wesley Publishing Company, 1972 „Plasticity theory”, J. Lubliner, 2008 „Anizotropia i tekstura krystalograficzna. Starzenie po odkształceniu”, Ł. Cieniek, AGH „Recrystallization nad related annealing phenomena”, F.J. Humphreys, M. Hatherly, Pergamon, 2002 „Pasma ścinania w metalach o sieci reguralnej ściennie centrowanej”, H. Paul, IMIM PAN, 2009 „Engineering Materials 1. An Introduction to Properties, Applications and Design”, M.F .Ashby, D.R.H. Jones, Elsevier Butterworth-Heinemann, 2005 aluminium.matter.org.uk Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

Instability of plastic flow and their role in texture transformations

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