Instability of plastic flow and their role in texture transformations

1. 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

2. 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

3. Instability of plastic flow and their role in texture transformations • Plasticity and yield criteria (uniaxial and multiaxial plastic flow analysis) Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

4. 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

5. 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 Stress deviator isotropic part deviatoric part Hydrostatic pressure Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

6. 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) Relative displacement tensor σ1 σ1 Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

7. Strain symmetric tensor asymmetric tensor 2 2 2 2 rigid body rotation state of relative displacement increment = + pure strain increment B0 Bt dγ12 0 0 0 1 1 1 0 1 dγ21 Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

8. Elasticityvs. plasticity Describesproperty 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

9. 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

10. Trescacriterion Yielding occurs when the maximum shear stress reaches a critical value0,5 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

11. Yieldsurface: TrescavsH-M-H (2D) H-M-H: Tresca: Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

12. Yieldsurface: TrescavsH-M-H (3D) http://upload.wikimedia.org/wikipedia/commons/c/cc/Yield_surfaces.svg Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

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

14. Instability of plastic flow and their role in texture transformations • 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

15. Stress-strainrelations 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 0.05-0.5 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

16. Stress-strainrelations 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

17. Stress-strain relations Differentiating Voce’s equation we obtain a linear dependence of the strengthening Θ ona 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

18. Considèrecriterion 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

19. 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 stiffmaterial = 0 – plastic material Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

20. 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

21. 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

22. 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

23. 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

24. Microscopic hardening laws 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

25. Microscopic hardening laws 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 ofStage 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

26. Instability of plastic flow and their role in texture transformations • Mechanical aspects of plastic instability (based on stress – strain curves) Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

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

28. Strain-induced hardening Schematic presentation of instability in tensile test Geometricalsofteningin an incipientneck Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

29. Rodsintension To initiate instability (a neck in this case): a/R ratio for characterizingnecksharpness Graficalmean of findingtheneckingstrain Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

30. Rodsintension Mild steel often shows both stable and unstable necks Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

31. Rodsintension Mild steel often show both stable and unstable necks Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

32. Rodsintension Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

33. Sheetsintension Diffusenecking Localneckin a strip tensionspecimen. (NormalstrainalongX2’ must be zero) Planestraininthelocalizedneck Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

34. Sheetsintension Thecriteria for localized and diffuseneckinginpuretension. Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

35. Instability of plastic flow and their role in texture transformations • Plastic anisotropyvs. crystallographic texture and sheet metal forming Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

36. 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. skywalker.cochise.edu/wellerr/mineral/graphite/graphiteL.htm Graphite crystal structure Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

37. Polycrystalline materials Polycrystalline materials – many crystallites of different size, shape and orientation. www.flickr.com/photos/core-materials/3841194280/ 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

38. 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) Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

39. Texture of: crystallization, deformation (rolling, extruding), recrystallization http://www.spaceflight.esa.int/impress/text/education/Solidification/Crystaline_Growth.html 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

40. 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

41. 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

42. Types of anisotropyinsheet 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

43. 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

44. 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

45. 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. Retractionof 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

46. Why is that so important? Sheet metal for deep-drawing Important from the point of viewof 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

47. Instability of plastic flow and their role in texture transformations • 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

48. A microstructure no deformed • Microstructureevolututionwithstrain Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim

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

50. A microstructure at large strain • Microstructureevolututionwithstrain Interdyscyplinarne studia doktoranckie z zakresu inżynierii materiałowej z wykładowym językiem angielskim