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Plasticity of materials, Department of materials forming, FMME, VŠB – TU Ostrava

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Jiří Kliber 1

PLASTICITY OF MATERIALS ________________________

Jiří Kliber

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LECTURE NOTES

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Vysoká škola báňská – Technical University of Ostrava Faculty of Metallurgy and Materials Engineering

Department of materials forming Ostrava, 2016


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Jiří Kliber 2

Title: Plasticity of materials

Author: Jiří Kliber

Issue: first, 2016

Number of pages: 52

Study material for the Metallurgical Engineering study program (study branch Modern

metallurgical technologies) of Faculty of Metallurgy and Materials Engineering, Vysoká

škola báňská – Technical University of Ostrava

© Jiří Kliber


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Jiří Kliber 3

Table of contents

1. CRYSTALLINE STRUCTURE OF METALS 6

2. LATTICE DEFECTS 8

3. DEFORMATION OF SINGLE-CRYSTALS 14

4. DEFORMATION OF POLY-CRYSTALS 17

5. STRESS-STRAIN CURVE 23

6. ANALYSIS OF DYNAMIC RECOVERY 28

7. DYNAMIC RECRYSTALLIZATION 37

8. STATIC RECRYSTALLIZATION 41

9. PLASTIC DEFORMATION DIFFUSION MECHANISM 44

TASK 48


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STUDY INSTRUCTIONS

PLASTICITY OF MATERIALS

For the subject PLASTICITY OF MATERIALS of the 3rd

semester of the study branch of

master's study, you have received a study package containing integrated lecture notes for

combined study, including study instructions. The text in this form does not involve the

whole topic of plasticity.

Prerequisites

Knowledge in mathematics (attainments of the introduction chapters of the subject

Materials Science I and the subject Metals forming, which deal with structures of materials

and basic lattice defects).

Aims of the subject and lectures

The main aim of the subject “Plasticity of Materials” is to provide students with theoretical

basics and methodology for solving of challenging phenomena, incidental with plastic

deformation. The task of the subject is to provide the students with knowledge, which is

necessary for them during finding of inventive and complex engineering solutions

concerning subsequent forming technologies. Due to the requirement for limited length of

the text, not all the chapters stated in the program of the subject are herein presented.

After study of this subject, students should be able to:

Further independently develop provided mathematic relations, apply them during

subsequent elaboration of diploma works. Orientate in theoretic literature within the study

branch. Acquire skills and better practice in solving of mathematic tasks.

To whom is the subject addressed

The subject belongs to the consequent master’s study of the Metallurgical Engineering

study program of the Modern Metallurgical Technologies study branch. It focuses on

technologies of forming and modifications of materials, but it can also be studied by other

students from other study branches, if they meet the required prerequisites.

Study of each chapter is recommended to be done according to the following steps:

Primarily, it is necessary to look through each chapter. Subsequently, students should pay

attention to text parts and equations and assign them to figures. After this step, students

should go through equations within the particular subsection and derive them

independently, possibly continue in their derivation. It is not unconditionally necessary to

know in detail the individual steps necessary to derivate the equations, but to know the

input parameters, final relations and verbal definitions of individual parameters is

necessary. It is recommended to independently draw the figures and to complete them step

by step so that students create, in many cases, their own mnemonics.

Communication with lecturer:

Communication with students is performed primarily within the time scheduled for

lectures and, after having made an appointment, within individual consultations. Checking


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Jiří Kliber 5

of given programs is performed in practical/calculative lessons, where other more detailed

instructions are given.

e-mail : jiri [email protected]

tel.: +420/595994463

mailto:[email protected]


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Jiří Kliber 6

1. CRYSTALLINE STRUCTURE OF METALS

Chapter sectioning:

Lattices (ferritic and austenitic).

Number of atoms in lattice and lattice constants.

Slip systems. Time to study: 90 minutes

Aim:

To understand crystalline structure of materials.

To explain differences between substitutional and interstitial solid solutions.

LECTURE

Although this section is the first one, it is be completely different from the other

sections, which contain detailed descriptions of equations, as well as figures. This section

contains references to previously studied subjects and study lectures, which are available

on the FMMI VŠB-TUO webpage. These are Materials Science I, possibly Metals Forming

and Theory of Forming.

Metals are crystalline materials, most often with cubic or hexagonal lattices.

The cubic stereo-centric lattice is in English denoted as BCC (body centered cubic).

The number of atoms belonging to this lattice is 2. This will be taught in a practical lesson

in the form of a program. Metals belonging to this group are Cr, W, V, Mo, Nb, Feα. Feα

lattice has 9 atoms in the lattice including the central one and the lattice constant of aα =

0.3567 nm, approximately 357 pm. In the ferritic lattice, the plane (group of planes) with

the highest atomic density is {110} and the direction with the highest atomic density is

[111], which is a normal to the octahedral plane. For BCC lattice, the coordination number

is 8 – it is the number of near neighboring atoms within a lattice. In a presented lecture, a


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table of slip systems will be shown. Only a note for now – the BCC lattice has 48 slip

systems, although some of them apply only within a certain temperature range.

The cubic planar-centered lattice in in English denoted as FCC (face centered

cubic). Metals belonging to this group are Al, Cu, Ag, Au, Ni, Pb, Feγ with 4 atoms in an

elementary cell and, likewise to BCC lattice, atomic packing factor for this lattice will be a

subject of a practical lesson. The austenitic FCC lattice has 16 atoms and the lattice

constant of α = 0.3567 nm, approximately 287 pm. In the austenitic FCC lattice, the plane

with the highest atomic density is {111}, i.e. the octahedral plane, and the direction is its

basal plane [110]. The coordination number for FCC lattice is 12; FCC lattice has 12 slip

systems.

Hexagonal lattice – its representatives are e.g. Zn and Mg. The height of the lattice,

c, for the lattice constant equal to the distance of atoms in the basal plane, is a/c=1.63.

Hexagonal lattice had 3 slip systems.

Solid solutions are either substitutional, in which the difference between the

diameters of atoms of alloying elements and elements in the matrix are approx. 15 to 20%

and atoms of alloying elements are located in lattices instead of atoms of the matrix.

Interstitial solid solutions are with atoms, the diameters of which are very small and

thus they can be located in free space within lattices, e.g. on the edge up to the distance of

a/2 for FCC lattices. Interstitial elements are (according to their increasing diameters) H B

C N O (the diameter of hydrogen atom is rH = 50 pm).

Summary of terms:

Cubic and hexagonal lattices.

Types of solid solutions.

Questions:

define types of lattices which you know,

draw BCC and FCC lattices,

define the mathematic relation between lattice constant and atomic radius,

characterize theoretical atomic packing factors for both types of the lattices,


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depict significant planes and directions for both lattice types,

how would you determine the number of slip systems.

2. LATTICE DEFECTS

Chapter sectioning:

Vacancies.

Dislocations.

Stacking faults.

Gain boundaries.

Time to study: 120 minutes

Aim:

Get acquainted with lattice defects.

Get acknowledged with numbers of atoms and vacancies in lattices.

Mathematically derive stress and energy in the vicinity of a dislocation.

Assign stacking fault values to structures.

LECTURE

Lattice defect are characterized as:

point defects – vacancies,

line defects – dislocations, which can be edge or screw,

planar – stacking faults

three-dimensional – grain boundaries


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Vacancies are free locations within a lattice, their number is given by the well-

known Arhenius equation of exponential dependence on temperature. The equation for

number of vacancies is the following

U is energy for generation of vacancies, which is given in J/mol, N is number of atoms in

lattice, usually given as their number for 1 mm3, R is gas constant, R=8.314 J/(mol. K),

(sometimes k is given instead of R, which is the Boltzman constant, which is in this area

theoretically used, its value is 1,38. 10-23

J/(mol∙ K)). The value of energy U is

approximately Q/3, where the activation energy for self-diffusion for Feα is Q = 180000

J/mol and for Feγ it is Q = 280000 J/mol. The vacancy generation rate can be given as

where the value of deltaU = 2 Q/3, U + ∆U = Q. The parameter denoted as δ is atomic

spacing and t0 is oscillation period of an atom around its basic position. In a practical

lesson, students will get a task in which values of temperatures will be given in K together

with the number of vacancies in one mm3 of the BCC Feα alpha ferrite lattice, for which

Nv =8,54 ∙ 1019

. The number of vacancies at various temperatures can thus step by step be

calculated.

The rate of moving of vacancies is then given by the following relation; it is thus a

velocity of moving of a volume of metal, thus the self-diffusion rate given by the relation

The self-diffusion rate can also be depicted using a simple linear function with the angular

coefficient of -Q/R.


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The second type of defects is dislocations, which are, likewise to other lattice defects, very

thoroughly described in lecture notes of other authors. Here I would like to mention the

lecture notes Materials Science I and Forming of Metals (faculty of metallurgy and

materials engineering, VŠB – TU Ostrava), in which these basic lattice defects are

described in a great detail. For visualization, examples of screw and edge dislocations are

shown in the following figures.

However, the types of dislocations in materials are not only edge and screw, they

usually occur in the forms of dislocation loops, which, according to the directions of their

Burgers vectors, have a tangential value and a perpendicular value, which is added to the

value of the Burgers vector. Therefore, there are points A, B, C, D selected on a dislocation

loop, which always represent parts consisting primarily either of edge or screw.

Movement of dislocations is performed by slip of edge dislocations in the direction of

Burgers vector in the given slip plane, or by climb, which however requires a presence of

vacancies, thus it occurs more at elevated temperatures. A dislocation has the ability to

escape from its slip plane into another slip plane by climbing and then return into one,

being originally partial. Screw dislocations move by a transversal slip in the direction of all

the slip planes, in which the Burgers vector lies. Several variants can occur during crossing

of dislocations. Crossing of two edge dislocations results in generation of a step with screw

dislocation character, which is no obstacle for another movement. Crossing of two screw

dislocations results in generation of an immobile step, which can develop into a dipole

with vacancies. By the effect of stress, dislocation loops can then generate on pinned

points. Dislocations can also be incomplete, partial, but this is all depicted in a more detail

in other lecture notes.


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Subsequently, we explain and derive some basic equations connected with dislocation

structure. We start with the basic stress around a dislocation, which can be imagined as a

circle with the radius of r and a small Burgers vector b.

From this then ensues a simple conclusion for the value of tangential stress around a

dislocation

Let me mention, that G is shear modulus and is related to Young modulus E

The value of energy around a dislocation can be determined using a simple plane

under a triangle using the following equation

The overall energy is the an integral within the r0 and r1 radii interval and leads to the

following relation, in which a logarithmic dependence ensuing from dr/r integration occurs

Considering a simplification, in which the r1 radius is 100 to 500 times higher than r0

radius, the logarithmic value of this r1/r0 ratio can be determined as 2π


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then, the final value for overall energy has the simple form of

Dislocation density can be imagined as the number of dislocations located within a

lattice with the length l; the density is then determined by a simple relation involving a

fraction containing a square around the considered dislocations

from this then ensues that the stress 𝜏 is given by the value according to equation

Dislocation density 𝜌 is mostly given in cm/cm3, and thus the final result is in cm

-2,

for pure metals (pure – not-annealed), the dislocations density is on the order of 106 to 10

8,

while for metals strengthened by forming the dislocation density increases up to values

around 1012

.

For dislocations crossing, a distance has to be overcome, which requires force.

If a movement by the distance b is required, the energy necessary for this movement is

This value can then be expressed as the energy necessary to overcome dislocations

pinned by forests (what is a forest dislocation density?, explanation is given in the section

deformation of single-crystals). Activation volume V also enters into the equation


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At the temperature of absolute zero, equal signs can be used for two values of energy.

From this then ensues a conclusion in the form of 𝜏 stress, which is limited by the forest

dislocation density

The third type of defects is stacking faults. These are defects in the arrangement of

atomic planes, in which some planes are lacking or redundant, e.g. ABCACABC or

ABCABBCABC. The lower is the stacking fault energy, the greater is the width of a

stacking fault. The stacking fault energy γSF is given in mJ/m2. For ferritic structures, it is

on the order of 100 mJ/m2 and higher, while for austenitic structures it is low, under 70,

and for some types of materials, e.g. the well-known 18CrNi anticorrosion material, the

value is approx. 13. Values for TRIP and TWIP steels are also very low, around 20 mJ/m2.

The fourth type of lattice defects is three-dimensional grain boundaries.

Summary of terms:

Lattice defects.

Questions:

which types of lattice defects you know,

explain various values of energies in the equations for vacancies,

is there any difference between usage of constants R and k,

derive the basic equation for stress around a dislocation,

determine the overall energy of dislocation,

characterize and mathematically explain the units in which dislocation density

is given

what characterizes the stacking fault size,

what is a three-dimensional defect.


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3. DEFORMATION OF SINGLE-CRYSTALS

Chapter sectioning:

Stress-strain dependence for a single-crystal.

Determination of effective stress.


Aim:

Understand values of stresses and dislocation densities.

Perform an outline of stress calculation.

LECTURE

The basic outline is represented by the following

figure, in which strengthening is represented by increasing shear stress 𝜏 in

dependence on shear strain 𝛾, or by increasing stress 𝜎 in dependence on strain 𝜀. It can be

divided in three regions, I, II and III.

Region I. The internal stress is determined as


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It is a region of very easy slip, generation of dislocations is very low and they do

not cross if they do not move. However, the free path for their movement decreases

gradually. Nevertheless, they can easily move due to lack of grain boundaries. The number

of dislocations, nor the value of stress 𝜏, does not increase significantly.

In region II, the angular coefficient of increase is given by an approximate value

determined by shear modulus, which is approximately equal to 1/3 of Young elasticity

modulus (exactly 3/8 of Young elasticity modulus).

In region II, crossing of dislocations, shortening of free paths and strengthening

occur. Strengthening is more or less linear and the strengthening angular coefficient is

G/300 (according to available literature). It is actually a strengthening coefficient and it

results from forest dislocations pinned by dislocation networks, where

τef is stress in the second region and, as will later be shown, it is a parameter the

derivation of the value of which is relatively complicated.

Region III is then a function of stacking fault energy. Other planar slips occur here

and the region is not substantially important from the single-crystals viewpoint.

Let us now return to stress τII, the following mathematic analysis is aimed to

determination of this value. To be able to start the analysis, we have to define the input

parameters, consequently on the following line:

the values of energy around a dislocation during its motion U0, of internal stress τi

determined as a dependence of a radical of dislocation density, of activation volume V,

which is given by coefficient b2, and stacking fault distance recounted to pinned

dislocations density, i.e. forest dislocation density. The first step of the calculation is

determination of τef (the τef which we look for is included in in the relation), and a

subsequent step is determination of the final value τII.


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The equation also includes a dependence of strain rate change on temperature, in

which the R gas constant value, which is 8.314 J/(mol.K), can occur in the denominator in

practical applications. In this step, the k Boltzman constant value enters the calculation.

By modification of the equation via logarithmic calculation we gradually get to the

possibility to express τII. Subsequently we substitute the complex parenthesis in the

equations mentioned in lines 6 and 7 by α. The final equation is then given in the last line.

It is actually a sum of two parts. The first part, determined by the product in front of the

parenthesis and by the first value in the parenthesis, is the so called athermic internal

component. The second value, the thermic component, depends on the size of forest

dislocations. This relation could further be mathematically derived. For a closer approach,

approximate values of the Burgers vector and distance between pinned dislocations are

mentioned. These can be used to further calculate the relation to get a concrete value of

internal stress τef.

Summary of terms:

Stress curve for a single-crystal.

Stress in individual regions.


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Effective stress.

Thermic and athermic components.

Questions:

draw the stress-strain dependence for a single-crystal,

explain various regions of the dependence,

characterize E and G modules,

determine the input equation for subsequent calculation of stress,

characterize thermic and athermic stress components.

First remuneration and relaxation

The time, which was necessary for you to get to this point, cannot be exactly

determined. However, you must be tired by constantly repeating equations.

They cannot be learnt by heart. It is just a quite simple mathematics, it is

sufficient to realize where we come from and where we go to. Nevertheless, I

think that you could not get to this point all at once; you probably already made

several relaxation breaks. But at this point, I recommend you to have a longer

rest. Each student is an individual and there is no general instructions how to

learn such subject. I recommend to individually write down all the equations

after study of each subsection.

4. DEFORMATION OF POLY-CRYSTALS

Chapter sectioning:

Overall stress in a poly-crystal.

Stress increments.



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Aim:

Define gradual stress increments for poly-crystals.

Explain the influence of grain size, alloying elements and precipitates.

LECTURE

Dislocation trajectories in a poly-crystal are shorter, dislocations crossing happens

and there is no region of easy slip. The overall value of tangential stress 𝜏 or also of

stress 𝜎 (flow stress) consists of five or possibly more contributions. The basic figure is

simple, the curve exhibits only strengthening represented as deformation under cold

conditions.

At the beginning, there are two input parameters (derived in the section

Deformation of single-crystals). In the equation then are:

first member – internal stress,

second member – contribution of mutual interaction of dislocation forests,

third member – contribution of deformation strengthening by grain boundaries

fourth member – the influence of alloying elements in solid solution

last member – contribution of precipitates.

The first two values have been derived in the Deformation of single-crystals

section.


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Let us get to the influence of grain boundaries. It is the conventional Hall-Petch

relation, which supposes dislocations to cumulate in front of boundaries, which are

obstacles for their movement. Continuing deformation requires the dislocation density to

increase, which generates stress in adjacent grains, Frank-Read dislocation sources then

generate in distance x from a dislocation edge and slip continues in a greater extent.

also in the form of

Confirmation of this equation ensues from the following analysis

Let us now focus on strain 𝜀, which is equal to a sum of three parameters

Trajectory of a dislocation s is equal to the mentioned product. This relation can be

used to derive the dependence for dislocation density according to another equation and to

derive the contribution of flow stress to strengthening, which is expressed by the constant k

in front of the grain size. In this case, the exponent for grain size d has the value of -0.5.

This procedure can be considered as derivation of the influence of grain boundaries on

material strengthening.

The below shown Figure depicts the graphical dependence of flow stress on grain

size with the exponent of -0.5, where 𝑑−0,5 is a section on the y axis and the angular

coefficient k represents, in the given equation, the contribution to strengthening with

decreasing grain size; the equation is linear.


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The contribution of alloying elements (in this case we consider substitutional

elements) does not have any substantial influence on the angular coefficient of

strengthening in the form of 𝐺

300 (as shown in the Figure in this section). An alloying

element typically has a lower stacking fault energy γsf. A higher energy is thus necessary

for deformation and it depends primarily on the difference between the atomic radii, i.e. on

the increase in lattice distortion.

A dislocation moves through the material and interferes with a substitutional

element, which increases strengthening. A mutual interaction of an alloying element and a

moving dislocation thus generates a stress field, a stress barrier with certain length and

amplitude, which depends on the real difference between the atomic radii of the basic

matrix (let us consider iron, in this case), which can be denoted as 𝑟𝑎 in our equation, and

of the alloying element atom, r0. In the equation, c is the alloying element concentration.

According the above mentioned equation, the contribution is a function of a product of the

three parameters, where the value n is most often stated to be 4/3.

The precipitates and their influence on material strengthening can be categorized

according two independent criteria, by which their influence is determined by mutual ratios

of their sizes and distances.


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The first contribution, denoted as cross, i.e. around or elastical, considers the

precipitates to be relatively small and with long distance spacing. The contribution can

then be determined according to equations

When the diameters of precipitates are close to 0, the final equation is in the form

of the last mentioned.

A more complicated situation is in cases when precipitates are large or with short

distance spacing, then the second case occurs, i.e. crossing through precipitates or trans,

and a loop generates around a precipitate. This contribution can simply be given by the

following equation

I point out that in this case, the shear modulus G is the shear modulus of the matrix

and not of the precipitate. Let us look at an example – this task we can calculate according

to the provided model constants for shear modulus Gm; further we know the supposed

dimensions and spacings, such as as lp – spacing between precipitates; Burgers vector b

After

calculations, this imparts that if the critical size of a precipitate is on the order of up to 10

nm, it is characterized as an incoherent particle, while particles with the size of around 100

nm are coherent. If we now return to the Hall-Petch relation, we can integrate the mutual

interaction of grain boundaries and precipitates. If, in the Hall-Petch relation, k is equal or


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close to 0, then the solid solution contains large incoherent particles and the particles are

certainly larger than 15-20 nm. If k is close or equal to 1, then the solid solution contains

very small incoherent particles, in this case considered to be smaller than 5-10 nm. Other

values between 0 and 1 signify a presence of both the types of particles in the structure and

their contributions are according to the k value.

Summary of terms:

Students will understand the summary equation for strengthening of materials

during forming.

They will be able to explain the content of two basic internal parameters

typical for a given material.

They will be acknowledged with the influence of grain size on strengthening.

They will understand the influence of ratios of atomic sizes during alloying on

stress increase.

They will understand the influence of precipitates.

Questions:

describe and explain the stress-strain curve for deformation of a poly-crystal,

derive the mathematic dependence on deformation density (connect with

derivation for single-crystals),

derive the -0.5 exponent for the influence of grain size and its perform its

graphical depiction,

how the influence of various atomic radii affects the material during alloying,

explain the influence of size and spacing of precipitates on the final equation

of strengthening contributions.


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5. STRESS-STRAIN CURVE

Chapter sectioning:

Shapes of stress-strain curves.

Mathematic notation.

Determination of strengthening coefficient.

Determination of constants in equations.


Aim:

Depict basic equations of strengthening curve.

Explain physical meaning of strengthening coefficient.

LECTURE

The basic figure depicts model theoretic possibilities of flow stress increase,

therefore stress in dependence on increasing strain. We practically neglect elastic

deformation, for which the Hooke’s law applies, and get to the curve denoted as 1, which,

after elastic deformation is over, features increasing dislocation density. Movement of

dislocation is blocked by obstacles created in the material and increasing strain induces

increasing flow stress.


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Curve 2 is a thermally activated phenomenon, which is increasing dislocation

density, but also an opposite phenomenon, thus their mutual annihilation. After a certain

strain is accumulated, an equilibrium state between generation and annihilation of

dislocations is established and the so-called steady state of plastic flow, for which the

constant stress value is denoted as occurshen the stacking fault energy is high, such as for

aluminum, thus when the grain boundary misorientations are low, the values are between

100 and 150 mJ/m2. The ferritic structure has a higher tendency to exhibit such course,

which is also denoted as dynamic recovery).

Curve 3, denoted as dynamic recrystallization, is more typical for alloys with low

stacking fault energy and thus for austenitic structures. After reaching of a certain critical

value of strain, which corresponds to a peak stress To mathematically describe curve 1,

several equations are presented below

Consequently from left to right, the presented equations are according to Hollomon,

Ludvik, Swift and Raisner, respectively. In each equation, stress, which is in practice given

in MPa, is denoted as 𝜎, possibly with an index, e is true (logarithmic) strain, sometimes

the relations consider also relative strain 𝜀.

The differences in the individual equations are caused by the fact that in practical

testing, we can only hardly start from the state in which stress and strain are both equal to

0. Usually, the beginning lies lower or higher, left or right on both sides. From this then

ensue following equations.

On the following figure, where we consider a second vertical axis on the right, the

course of a new parameter 𝑑𝜎

𝑑𝑒is depicted by the dashed line.


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For mathematic analysis of this dependence, we start from the basic Hollomon

equation, from which we consequently calculate the value

we derive the equation and, supposing that the maximum value σm corresponds to

the critical strain er, the analysis results in the relation which is depicted by the final

equation

This value depicts materials plasticity much more than some other parameters. It is

evident that if the value of exponent n in the Hollomon equation is low, then also the value

of critical strain for achievement of maximum stress is low, and contrariwise. From this

ensues a simple hypothesis for any practically obtained equation. As an example we can

consider an equation for Cu, for which, in an annealed state, i.e. when having great

plasticity, the value of exponent n can be higher than 0.5, while identical Cu, which has

been substantially deformed under cold conditions, can have strengthening exponent n

lower than 0.1.

The following determination of constant k in the Hollomon equation ensues

consequently from two relations


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and for determination of true strain

Derivation is not complicated and leads to an equation for mathematical expression

of constant k

This section is finalized with determination of work to stability limit W, for which

is necessary to integrate the stress-strain curve,

and to mathematically adjust the curve gradually

when n is lower or far lower than 1 it comes out that if n is lower than 0.2, we can

substitute exp(𝑛) = 𝑛 + 1 .

These equations ensue from a real physical subject matter, their constants have

certain senses and numerical values imply material properties.It is necessary to point out

that we cannot always use the equations to describe more complex shapes of stress-strain

curves.Therefore, polynomic relations of are used. These, when correctly mathematically

applied, describe the curve exactly. They also have the advantage that, considering using of

contemporary computer software, which simulate the courses of stress-strain, strain rates


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etc. using the finite elements method, they can quite simply be implemented in a software,

which describes strengthening and even recovery of materials during increasing

strain.Their disadvantage is that from constants, which we obtain by this procedure, we are

not able, by the first sight, nor by any other method, to get any physical determination of

material state.

Summary of terms:

Various shapes of stress-strain curves.

Basic types of mathematic equations.

Relation between strain and strengthening coefficient.

Constant k and its determination.

Work to stability limit W.

Questions:

explain physical phenomena in materials during loading,

write some equations for description of strengthening curves,

perform logarithmic calculation of any equation and determine the angular

coefficient,

derive the relation between critical strain and strengthening coefficient,

calculate constant k in Hollomon equation,

determine work to stability limit W,

summarize gained knowledge and perform a brief presentation about

materials strengthening with no softening during deformation

6. ANALYSIS OF DYNAMIC RECOVERY

Chapter sectioning:


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Jiří Kliber 28

Get acquainted with mathematic analyses of dynamic recovery.

Determination of dependence of stress on thermomechanical parameters.


Aim:

Explain equations in the basic analysis according to Sellars.

Get to the sin h relation by derivation.

Describe in detail the procedure for determination of activation energy.

LECTURE

The first analysis, which we perform, is the analysis for dynamic recovery, i.e. for

the curve denoted as 2 in section 5. It is the stress-strain curve, in which we get the stress

σp after reaching the critical er or ep value.This analysis was mathematically performed by

Sellars for a purely ferritic structure 𝐹𝑒𝛼 It is however applicable especially for materials

with ferritic structure.As is finally shown, the final equation can also be used for

recrystallization.We have to start with two basic equations;he equation for strengthening,

which is given on the left, and the equation for dynamic softening, in this case recovery,

which is given on the right.

By gradual adaptation of these equations with the aim to find the 𝑑𝜎 increment for

Adjustment of the equation can be partially shown in presented lectures, especially


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Jiří Kliber 29

adjustment of the left side is well mathematically elaborated, leading to integrating of the

left side via the per partes method. This results in the final 𝜎 value, which is then compared

to the integral on the right side. After quite complex mathematical adjustments, which are

not included in these study notes, we come to the final equation featuring strain rate the left

side, expressed as shear strain rate or relative strain rate, and constant A and the

exponential relation for activation energy Q and stress 𝜎𝑠𝑠𝑘 on the right side.

The second analysis, which considers climb of edge dislocations, starts from the

hypothesis that the effect of stress τ can invoke generation and possibly annihilation of

vacancies between elastic dislocation loops on an edge section with two pinned locations.

The entire analysis consists of 4 parts.The first part consists of determination of the rate of

change in dislocation density during generation of dislocations. The second part consists of

determination of generation of vacancies, which generate on a certain pinned jog, it is

actually an increase in the amount of vacancies during strengthening. The third part is

focused on changes in concentration of vacancies in time, thus decrease in their amount for

kinetics of the given phenomena, and the last fourth part deals with a decrease in the

amount of dislocation by annihilation during climb.The result can then be determined via

comparison of the four parts and leads to determination of dislocation density, It is the

steady state dislocation density.Mathematically, the dependence is then determined by an

equation, it is again a function of stress (in this case, we mathematically consider the value

of shear strain rate �̇�The equation logically corresponds also to the results of the first


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Jiří Kliber 30

analysis. We are now getting to the third

analysis, which is denoted as dynamic recovery via movement of screw dislocations jogs,

which was derived by Barret and starts from the supposition of existence of vacancies on

screw dislocations.A jog, which does not lie in the slip plane, emits or absorbs vacancies,

possibly interstitials.A jog is thus an edge section creating a breaking force, which has to

be overcome.In an equilibrium state, a steady plastic flow develops.The concentration of

vacancies is driven by the rate of their generation and movement from one jog to another.

vicinity of a jog exhibits a certain density of vacancies ensuing from the equation provided

in the section about density of vacancies within materials.

The force necessary to emit vacancies is provided on the left side of the equation

and analogically, the force for their absorption is on the right side. By step by step

adaptation via logarithmic calculations, we can calculate and determine the values of

concentrations of vacancies being emitted and absorbed.

The difference of the concentration of vacancies cess and the equilibrium

concentration value c0 is determined by a relation featuring the velocity of movement of

emitted vacancies ve and other variables – self-diffusion coefficient Dv and Burgers vector

b, both in denominator.


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Jiří Kliber 31

By adjustment of this equation we get two forces for emission and absorption of

vacancies, which can then be mutually compared.

By this, we achieve the state of analysis for steady state plastic flow and we input

the general knowledge for force for movement of vacancies in the form of 𝜏𝑠𝑠𝑏

The subsequent calculation looks rather complicated. We have to include the value

for concentration of vacancies, where N is the number of atoms in a lattice unit, X0 is the

number of vacant locations and, in denominator, the lattice constant a3. Likewise, the value

of diffusion coefficient Dv can be expressed using a velocity, in this case the self-diffusion

rate D, in relation to the number of vacant locations.

The value of velocity of movement of emitted and absorbed vacancies is

determined as


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Jiří Kliber 32

We consider equilibrium state, which means that each of the velocities applies one

half during the subsequent adjustment

determination of shear strain rate and from this ensuing overall velocity vs leads us to the

final result for strain rate in an equilibrium state

This equation looks very complicated with a variety of different constants, which

were however explained before. We thus now come to the equation, which, as is now

evident, was created by a derivation ensuing from a purely physical approach and the

shape of which resemble to a mathematic equation for a hyperbolic sinus relation. We can

use this parameter for substitution of the dislocation density value, which is usually given

by a function of stress 𝜎3 and values for shear strain rate and a relation between normal

and shear stresses

By this, we get the final equation according to Barret analysis.

The difference from the before mentioned two analyses is in the fact that the right

side of the equation features stress in an exponential function, and in sin h function. Such

mathematical analysis leading to regression is very difficult, maybe even impossible.


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Jiří Kliber 33

A group of authors, which can be abbreviated as STG (Sellars, Tegart, Garofalo)

dealt with this issue. They adjusted the current analyses of recovery within a material to the

form of a fourth, let us say universal, equation featuring activation energy of the process,

as well as stress in a hyperbolic sinus relation. The analysis of this equation is presented in

the following text, it has its internal logic and it is often the last applied equation for stress

as a function of a variety of parameters.

The procedure is the following. Firstly, we determine table values of dependences

from experimental values obtained using a plastometer based on torsion (revolutions,

momentum), or nowadays more often used compression-based Gleeble plastometer

(change of sample height with stress) using rolls, or planar tests (PSCT – Plane strain

compression test), and step by step we start to elaborate linear regression analyses.


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Jiří Kliber 34

For high temperatures we consider that the value of product 𝛼𝜎 ≤ 1. For this case,

the relation mathematically (it can be proven) transforms into the form of ln of strain rate

being a function of ln of stress. These individual step by step elaborated linear regression

analyses are depicted in a set of five figures at the end of this section.

For low temperatures, for which 𝛼𝜎 ≥ 1, is then ln of strain rate a direct function of

stress. By analysis of these two equations we step by step get two angular coefficients, n

and 𝛽, between which is a mutual relation, shown in the equation 𝛼 ∙ 𝑛 = 𝛽.

Now let us get back to the original equation, in which we will however exclude the

relation for temperature (temperature is considered to be constant for each case). Again,

during logarithmic calculation of this equation, we get a set of more or less parallel lines in

a dependence on the horizontal axis, on which is ln sin ℎ (𝛼𝜎𝑠𝑠).

From the third figure ensues the necessity to find the values of intersections of these

lines with vertical y axis, which are denoted consequently as ln A1 to ln Ax according to the

temperatures T1 to Tx (ln T3 intersection is shown as an example).

Let us get to the fourth step by returning to the original equation, in which the

hyperbolic sinus alpha sigma is equal to 1 and its natural logarithm is equal to zero. Then,

the equation transforms to the following form and after logarithmic calculation (in the

figure shown as the second dependence in the right bottom corner) and subsequent linear

regression analysis, we get a negative angular coefficient –Q/R, from which the value of

activation energy Q (J/mol) can be calculated if we know the gas constant R, which is

equal to R=8.314 J/(mol∙K).

The whole analysis can be finished by the equation for Zener-Hollomon parameter

(temperature compensated strain rate (s-1

)), which is given on the left side of the equation,

subsequent regression leads to exponent n and constant A (after logarithmic calculation). In

the final equation for strain rate we then apply the found constants A, Q, α and n. (for a

better lucidity, the final constants are highlighted in circles).

The equation can further be adjusted by a recalculation to determine the value of

stress in dependence on thermomechanical parameters, such as strain rate (hidden in Z),

activation energy Q, temperature T and constant A and exponent n, according to the form

of this equation


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Jiří Kliber 35

When applying this dependence, we do not have to strictly use the value of stress in

steady-state; practice has shown that it can be used also for other stresses. This equation is

also generally used to describe softening, which can be in the form of recovery, or

recrystallization. If we perform experimental testing at certain temperatures and strain rates

and we get to a mathematic expression of constants, we can, by substitution of these

constants, calculate the value of stress, i.e. the value of flow stress in other than

experimental conditions. We even can, and it is often performed, extrapolate this value for

higher or lower values of temperatures or strain rates (usually only by a single order),

however we cannot get to another structural phase of the material, for which the conditions

of dynamic recovery are completely different. The various publications and a variety of

performed experiments come out from the last equation also during determination of

activation energy for static and dynamic recrystallization.


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Jiří Kliber 36

Summary of terms:

Determination of dependence of stress during strengthening and softening.

Resulting equation for strain rate, outline of other analyses and their results.

Detail analysis of sin h relation.

Questions:

determine differential increment of stress during strengthening and softening,

which parameters occur in the presented equations,

how would you determine the new, universal, equation,


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Jiří Kliber 37

what do we have to do during linear regression for low (high) temperatures,

describe the whole procedure of achievement of the activation energy value,

which constants occur in the universal equation and where they come from,

can you determine the value of stress as a function of variable parameters and

constants.

7. DYNAMIC RECRYSTALLIZATION

Chapter sectioning:

Generation and growth of nuclei.

Parametric equation.


Aim:

Describe physical approach to the issue of nuclei in a dynamically deformed

structre.

Determine relation for temperature compensated time.

LECTURE

This chapter has a rather different graphical form, figures and equations are

occasionally missing and it includes only commentaries, which will be supplemented by

graphics in presented lectures.

Dynamic recrystallization, as was already mentioned, is typical for metals and

alloys with low stacking fault energy, such as Cu, Ni, austenite for Fe. To achieve peak

stress, its value does not have to necessarily be higher than for dynamic recovery, but it

usually occurs at higher strains. After reaching the critical value of stress, nuclei start to


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Jiří Kliber 38

generate in a deformed material. From the viewpoint of real state, the nuclei have already

been initiated before the critical peak stress value is reached. However, from the viewpoint

of experimental testing on plastometers, dynamic recrystallization is usually considered to

start at the point, in which the strain corresponding to maximum peak stress is reached.

Published literature implies that the critical value of strain for generation of nuclei can be

considered to be 0.8 ep. A nucleus achieves the critical size and grows, while

simultaneously, new nuclei generate, and together they gradually fill the original grain. If

deformation continues, the nuclei start to decrease their growth rate since they are

gradually filling the whole structure. Recrystallization rate is directly proportional to the

rate of moving of new nuclei boundaries. The driving force is the difference in dislocation

densities of the deformed and new grains. Naturaly, the dislocation density in a new grain

is low. A recrystallized grain, which fills the original deformed grain, during repeating

deformation repeatedly deforms, grows and recrystallizes (especially in the direction of

prevailing deformation), the dislocation density increases and decreases and the whole

process is denoted as steady state, during which strengthening and softening are acting

against each other via recrystallization. This can also be depicted as follows – if the

recrystallization rate is significantly higher than strain rate, then a cyclic course resulting in

an oscillating curve with a certain damping occurs. In case the recrystallization rate is

lower than strain rate, a gradual decrease down to the steady state of material flow occurs.

Once again – it is a conflict between annihilation of dislocations, recrystallization, and

increase in the number of dislocations and their density, material strengthening.

This analysis was also elaborated mathematically and the process can be divided

into several parts. The first part consists of generation of nuclei, in a model case at grain

boundaries, where the dislocation density in a nucleus is 𝜌 = 0 and the dislocation density

in a grain is 𝜌0. The surface energy necessary for a nucleus to exist is equal to the overall

energy given by metal strengthening and by a certain value of dislocation density. Starting

from the overall dislocation energy, we thus gradually get results, the first one of which is

the critical size of a nucleus d*. An analysis of movement of a grain boundary (its mobility

and velocity) follows. Subsequent calculation requires finding of the fraction of active

grain boundaries, since not all the sections of grain boundaries are potential locations for

generation of nuclei. This is followed by determination of a volume of metal with a certain

dislocation density, and then kinetics, i.e. dynamic recrystallization rate. Problematic can

also be the fact that before dynamic recrystallization, a certain fraction of structure can


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Jiří Kliber 39

already be softened by recovery. Step by step we thus come to the time necessary to

establish a dislocation density and to very complicated equations ensuing from the

description of the stress-strain curve, which feature a variety of different values of

dislocation densities. This mathematic analysis even leads to a tan h (x) relation. The result

is purely physical, theoretical, for our purposes it would only be suitable in case of a deep

possibility to determine dislocation densities, which is beyond the frame of our plasticity

(does this section include a lot of text? Well, despite this, try to understand it and say in

your own words).

Despite this fact, the kinetics of dynamic recrystallization can be described

mathematically. We ensue from the facts that for static recrystallization we need to know

the influence of strain, strain rate, temperature and activation energy, while for dynamic

recrystallization we moreover need to know the dependence of the critical value of

dislocation density as a function of peak stress

We can thus depict the peak stress σp in relation to the time necessary to achieve the

peak tp.

This value of time to peak can be described by a value for temperature compensated

time W, which is an analogy to the Zenner-Hollomon parameter, which is a temperature

compensated strain rate (s) and can be described by the following equation.

The result is then a parametric equation,

which can either be for peak strain and peak stress, or for steady-state strain and

steady-state plastic flow stress. Adjustment of this equation is relatively simple, the time to

peak can be expressed in relation with Zener-Hollomon parameter and if we know the

relation between strain, strain rate and time, the value for achievement of peak strain can

be depicted by the given relation.


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Jiří Kliber 40

Summary of terms:

Hypothesis of dynamic recrystallization.

Questions:

determine the conditions under which dynamic recrystallization develops,

explain any difference between metalographic and plastometric approaches,

how does a dynamic nucleus generate and grow,

explain temperature compensated time,

can the kinetics of a phenomenon be expressed also by peak strain.

Second remuneration and relaxation

Did you expect a simple method how to learn the topic of plasticity? Do not be

upset, you still have enough time and, also, you now start to understand that

derivation of equations is not the purpose of this subject!! Nevertheless,

knowledge of mathematic procedures is necessary to pass the exam. Each

section includes a verbal introduction, idea, input data (unfortunately usually

mathematical), and it is necessary to rather understand the phenomena occurring

in materials during individual processes. I am convinced – almost – that in real

technological practice (when operating a rolling mill, determining parameters of

press or hammer, or for creating calibration schemes) you will not need to know

how to derivate equations for description of, let us say, dynamic recovery or

recrystallization. However, the information that, for example, the recrystalized

fraction X depends on certain parameters should stay somewhere in the corner of

your mind. (what is X? are you curious? You will get to know in the following

section). When consulting with specialists from other professions, during

specialized activities, when visiting foreign specialists, during designing of new


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Jiří Kliber 41

modern devices, during scientific work and study of modern foreign literature

and other activities, which you will perform during your lives, the knowledge

will be beneficial to you. And when you even remember that you saw it in some

lecture notes – hooray!

8. STATIC RECRYSTALLIZATION

Chapter sectioning:

Mathematic probability of generation of a nucleus.

Kinetics of static recrystallization, Avrami equation.

Influence of temperature and strain rate on Avrami curve.


Aim:

Derive the final equation for fraction recrystallized.

Understand Avrami equation.

LECTURE

The mathematic model is based on the probability of generation of a nucleus and its

growth, which is schematically shown in the following figure


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Jiří Kliber 42

the mathematic analysis finally leads to the Avrami equation. We start from the

probability of generation of nuclei P and we get deep into the probability theory. In the

time equal to 0, no nuclei generate and the probability of their generation is equal to 1. It is

described by an exponential curve, in which the number of nuclei is infinite and the

probability of their generation is close to 0.

Considering the first nucleus – the probability that it does not generate is 0.36,

mathematically we come from an exponential equation.

Nuclei growth; when considering the first nucleus, we have to calculate the volume

of metal containing the supposed recrystallized nucleus, the rate is supposed to be linear

and even the nucleus in the form of a sphere is idealized.

The nucleus VZ thus has the volume of a sphere with a radius r. When more nuclei

generate, we perform integration within the time from 0 to t and the final equation then

features time in the form of t4. Therefore, if probability is described by an equation of the

given type, the fraction of softened structure, in this case fraction recrystallized, depicted

most often as X, is 1-P and we come to an adjusted equation in the following form


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Jiří Kliber 43

Theoretically, we should also consider a certain incubation time t0. Although we

mostly consider the value k as constant, it is minimally a function of temperature.

The Avrami equation is the most often used equation in the field of static

recrystallization. The curve characterizing this dependence is denoted as a sigmoidal curve

and it approximatively approaches zero for minimum times and one for great times. The

horizontal axis features time in the form of ln t. Adjustment of the equation leads to the

possibility of determination of X in dependence on time in a double logarithmic form of

ln ln (1

1−X). A reverse derivation of the double logarithmic relation will be presented in a

lecture. By this reason, there is no figure in this part of the lecture notes.

Summary of terms:

Mathematic probability theory.

Fraction recrystallized.

Avrami equation and its graphical depiction.

Questions:

define the probability of generation of a nucleus.

characterize possible shapes of a nucleus,

define the final relation for fraction recrystallized,

draw the Avrami curve in sigmoidal and linear forms,

how do basic thermomechanical parameters influence the shape and

translation of the curve.


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Jiří Kliber 44

9. PLASTIC DEFORMATION DIFFUSION MECHANISM

Chapter sectioning:

Vacancy diffusion mechanism of plastic deformation.

Viscous and dislocation mechanisms.


Aim

Determine influences of thermomechanical parameters on relaxation

processes.

LECTURE

When temperatures are high and strain rates low, a mechanism of plastic

deformation differing from the dislocation slip mechanism starts to develop. It is the

diffusion mechanism. Whereas the diffusion mechanism is based on the existence of

vacancies, the conventional slip mechanism, which we have dealt with before, needs

dislocations to proceed.

The diffusion mechanism is presented as a theory of relaxation, which is expressed

by the basic equation featuring Tr as a constant. In this case, this has nothing in common

with temperature, it is actually time (s) – it is the so called relaxation time. Viscosity of a

material is then given as a product of the relaxation time and elasticity modulus E.

This mechanism is actually high-temperature diffusion and sometimes also effects

as low-temperature creep. However, ideas about the real phenomena occurring within a


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Jiří Kliber 45

material differ according to different authors. Vacancies can move through the lattice and

grain boundary sliding and moving of atoms from other boundaries through grains occur.

Possibly, only diffusion along grain boundaries can proceed; this occurs at relatively low

temperatures.

By a mathematic analysis we get to the fact that when the time for relaxation is very

high (approaches infinity), the second member in the equation is significantly low

then the whole equation transforms into a simple form of the conventional Hooke’s

law, in which stress is a product of strain and elasticity modulus (strain is expressed either

as shear strain, relative strain or true strain – to determine the value of Young modulus E,

true strain e is always considered).

The dependence of relaxation rate can be in the following forms

A second possibility is when relaxation time is very short, various situations can

then happen. In the first case, stress is constant. By adjustment of the equation, we get a

relation in the form of

where the fraction of stress differential to time differential is almost zero and the

final value of stress is a product of viscosity and strain rate

However, for similarly short relaxation times strain can also be constant. Then the

whole mathematical adjustment is rather difficult and leads to an exponential relation


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Jiří Kliber 46

The influence of forming conditions, especially strain rate and temperature, on the

diffusion mechanism will be presented in lectures as well, since this diagram should

partially be a result of independent students’ thinking. Students should already be able to

design the dependence independently on the basis of known facts about dislocation and

diffusion mechanisms.

Mathematic expression finally leads to the equations

in which the contributions of the individual mechanisms is determined according to

increments from the differential relations in the first part, which is the dislocation part, and

the second part, which corresponds to the diffusion mechanism.

To remember the influence of viscosity, we use a mnemonics – when material

viscosity increases, which happens with decreasing temperature, the fraction, in the

numerator of which is flow stress and in the denominator of which is viscosity, is close to

0. This imparts that plastic deformation can in this case better be described by Hooke’s

law. On the other hand, when viscosity is low, i.e. at high temperatures, the dislocation

mechanism occurs only rarely and a prevailing portion of plastic deformation is executed

by diffusion.


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Jiří Kliber 47

It is also necessary to analyze the influence of strain rate. For high strain rates, the

increment of strengthening is high and consequently the effect of diffusion mechanism

decreases and contrariwise, for very low strain rates, i.e. decreasing strain rate, the effect of

diffusion mechanism increases.

Summary of terms:

Relaxation time, viscosity, Hooke’s law

The influence of differential increments of stress and strain in relation to time.

Questions:

characterize members in the basic equation,

what is relaxation time related to,

describe the dependence for the relation of various mechanisms on

temperature and strain rate,

how can the original equation be adjusted for changed forming conditions,

divide the equation to two parts,

schematically describe the influence of viscosity and strain rate.

P.S. If you find any mistake in these lecture notes, and it is completely improbable that

there is not any, and certainly not only one, notify the author to prepare corrections also

for other students – I have already found a mistake myself, in the figure on page 8.

Thank you.

Author


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Jiří Kliber 48

TASK

2. Activation energy and formability

Keywords

temperature, strain, strain rate, torsion test, linear regression, activation energy, formability

at peak, forming limit

2.1 Theoretic introduction For elaboration of this type of program, use lectures and section 6 ANALYSIS OF

DYNAMIC RECOVERY.

2.2 Task (activation energy) Elaborate a program, i.e. determine activation energy Q (J/mol) of a forming

process. The given values are:

temperature T (K) – range to 10 values; rate of torsion for torsion test o (min -1

) – range to

10 values; number of revolutions at peak Np (-) – range to 100 values; momentum at peak

Mp (N.mm) – range to 100 values.

Calculate:

2.2.1 Strain rate intensity at peak

SN

N

DN

Lep

p

p

argsinh

2

3 2

D = 4 mm; L = 50 mm;

No

60

2.2.2 Stress intensity at peak

SM

Rp

p

3 3

2 3 R = 3 mm

Compile linear regressions:

2.2.3 ln (ln )S f Sep p angular coefficient denote as nT

2.2.4 ln ( )S f Sep p angular coefficient denote as

2.2.5 determine

nT

2.2.6 ln lnsinh( . )S f Sep p for all temperatures, determine

intersections with y axis for sinh( . ) S p 1; denote as A1 to An

2.2.7 A1 to An = fT( )1

angular coefficient is Q

RS

Q = - 8.314 . S


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Jiří Kliber 49

2.2.8 ln lnsinh( . )SQ

RTf Sep p

as a single line, angular

coefficient denote as n, intersection with y axis is ln A

Compose the final relation:

2.2.9 S AQ

RTSep p

n

.exp sinh .

Perform table and graphical elaboration based on constants known from the final

equation (A,Q, ,n), adjust the equation to the form of S p ........ , calculate it, denote the

values as theoretically calculated S p t and compare them in a table with values measured

experimentally from torsion momentum.

2.3 Task (formability at peak) Elaborate a program, i.e. determine formability at peak, Sep

The given values are:

temperature T (K) – range to 10 values; rate of torsion for torsion test o (min -1) – range to

10 values; number of revolutions at peak Np (-) – range to 100 values.

Calculate:

2.3.1 Strain rate intensity at peak

SN

N

DN

Lep

p

p

argsinh

2

3 2

D = 4 mm; L = 50 mm;

No

60

2.3.2 Strain intensity at peak

Sep 2

3 2argsinh


2.3.3 ln ( ln )S f Sep ep angular coefficients for various temperatures

denote as K1 to Kn, calculate average angular coefficient and denote it as k

2.3.4 sinh(ln .ln )S k S fT

ep ep

1

angular coefficient denote as X

and intersection with axis y denote as Y

Compose the final relation

2.3.5 S S YX

Tep ep

k expargsinh( )


equation (k,Y,X), calculate the values of formability at peak, denote the values as


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Jiří Kliber 50

theoretically calculated Sep t and compare them in a table with values measured

experimentally from number of revolutions at peak.

2.4 Task (forming limit) Elaborate a program, i.e. determine forming limit Sef

The given values are:

temperature T (K) – range to 10 values; rate of torsion for torsion test o (min -1) – range to

10 values; number of revolutions to fracture Nf (-) – range to 100 values.

Calculate:

2.4.1 Strain rate intensity to fracture

SN

N

DN

Lef

f

f

argsinh

2

3 2

D = 4 mm; L = 50 mm;

No

60

2.4.2 Strain intensity to fracture

Sef 2

3 2argsinh


2.4.3 ln ( ln )S f Sef ef angular coefficients for various temperatures denote as a1

to an, calculate average angular coefficient and denote it as a, intersections

with y axis for x = 0 and for different temperatures denote as A1 to An

2.4.4 A1 to An = fT

1

angular coefficient is

a Q

R

.,

determine Q, intersection with y axis for x = 0 is ln A, determine A


2.4.5 S A Sa Q

RTef ef

a

. exp

.

Compile linear regression:

2.4.6 sinh( ln .ln )S a S fT

ef ef

1

angular coefficient denote as Xm

and intersection with axis y denote as Ym


2.4.7 S S YX

Tef ef

a

mm expargsinh( )


equation (a,Ym,Xm), calculate the values of limit formability, denote the values as


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Jiří Kliber 51

theoretically calculated Sef t and compare them in a table with values measured

experimentally from number of revolutions to fracture.

P.S. Forming limit only after having input number of revolutions to fracture.

Table 1. Experimental values from torsion tests

Steel

denomination Theating

[°C]

TZK

[°C]

o

[min-1

]

Mp

[Nmm]

Np

[rev]

X5/5 1150 1150 2000 4250 2.73

/30 500 2120 2.08

/25 100 2280 2.26

/10 28 2120 2.20

/5 1.1 1300 2.00

X5/4 1050 1050 2000 6000 2.83

/19 500 5000 2.75

/24 100 4200 2.54

/19 28 3920 2.58

/14 1.1 2500 2.23

X5/3 950 950 2000 10300 3.30

/18 500 9700 3.25

/23 100 8260 2.91

/8 28 6500 2.75

/13 1.1 4150 2.60

X/2 850 850 2000 11800 3.83

/17 500 11200 3.33

/22 100 10600 3.34

/7 28 9000 3.25

/12 1.1 6200 3.83

3 1000 1000 1600 3950 4.0

160 2900 2.4

16 2100 1.5

1.6 1350 0.7

3 900 900 1600 5350 5.9

160 4000 3.6

16 3200 2.3

1.6 2150 1.6

3 850 850 1600 6200 1.6

160 5300 4.7

16 4000 2.9

1.6 2850 2.0

3 825 825 1600 6600 6.9

160 5000 5.6

16 4000 3.5


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Jiří Kliber 52

Steel

denomination Theating

[°C]

TZK

[°C]

o

[min-1

]

Mp

[Nmm]

Np

[rev]

1.6 3450 3.0

3 800 800 1600 6700 8.5

160 6000 6.1

16 4100 4.8

1.6 3450 3.0

15.1 1150 1150 2000 2600 3.0

500 2020 1.17

100 1630 1.04

28 1300 1.00

1.1 590 0.85

15.1 1050 1050 2000 3500 4.00

500 3000 1.99

100 2350 1.20

28 1850 1.28

1.1 1000 1.09

15.1 950 950 2000 4800 4.2

500 3920 1.83

100 3120 1.53

28 2550 1.42

1.1 1550 1.40

15.1 850 850 2000 6300 4.6

500 5580 2.99

100 4700 1.72

28 3460 1.78

1.1 1920 1.70

15.1 750 750 2000 6820 5.33

500 5900 2.99

100 4900 2.8

28 3950 2.3

1.1 2200 1.95

2.5 Instructions

ad 2.2 to 2.4 Tasks of this type can be solved via computer (Excel, Mathcad,

Origin or any other software), by gradual compilation of regressions (in order to

understand and individually elaborate a task, software Energy cannot be used), by

calculation of constants and graphical outputs. A single steel is given, according to the

table (for calculation of forming limit, the values will be given additionally).

plasticity of materials - vsb.czkatedry.fmmi.vsb.cz/opory_fmmi_eng/2_rocnik/mmt/plasticity of...

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