Tài liệu Mechanical Properties of Engineered Materials P2 pptx

The diffusion of point defects such as vacancies may also lead to the growth of grains in a polycrystalline material.. 2.3.2 Line Defects Line defects consist primarily of dislocations,

Defects are imperfections in the structure They may be one-dimensional point defects (Fig 2.6), line defects (Fig 2.7), two-dimensional plane defects (Fig 2.8), or three-dimensional volume defects such as inclusions

or porosity,Fig 1.16(d) The different types of defects are described briefly

in this section

2.3.1 One-Dimensional Point Defects

One-dimensional point defects [Fig 2.6) may include vacancies [Fig 2.6(a)], interstitials [Figs 2.6(a) and 2.6(b)], solid solution elements [Fig 2.6(b)], and pairs or clusters of the foregoing, Fig 2.6(c) Pairs of ions (Frenkel defects)

or vacancies (Schottky defects) are often required to maintain charge neu-trality, Fig 2.6(c) Point defects can diffuse through a lattice, especially at temperatures above approximately 0.3–0.5 of the absolute melting tempera-ture If the movement of point defects produces a net state change, it causes thermally activated stress-induced deformation, such as creep The diffusion

of point defects such as vacancies may also lead to the growth of grains in a polycrystalline material

2.3.2 Line Defects

Line defects consist primarily of dislocations, typically at the edges of patches where part of a crystallographic plane has slipped by one lattice

FIGURE2.6 Examples of point defects: (a)] vacancy and interstitial elements; (b) substitutional element and interstitial impurity element; (c) pairs of ions and vacancies [(a) and (c) are adapted from Shackleford, 1996—reprinted with permission from Prentice-Hall; (b) is adapted from Hull and Bacon,

1984 Reprinted with permission from Pergamon Press.]

spacing (Fig 2.7) The two pure types of dislocations are edge and screw, Figs 2.7(a) and 2.7(b) Edge dislocations have slip (Burgers) vectors perpen-dicular to the dislocation line [Fig 2.7a)], while screw dislocations have translation vectors parallel to the dislocation line, Fig 2.7(b) In general, however, most dislocations are mixed dislocations that consist of both edge and screw dislocation components, Fig 2.7(c) Note that the line segments along the curved dislocation in Fig 2.7(c) have both edge and screw com-ponents However, the deflection segments are either pure edge or pure screw at either end of the curved dislocation, Fig 2.7(c)

FIGURE2.7 Examples of line defects: (a) edge dislocations; (b) screw disloca-tions; (c) mixed dislocations (Adapted from Hull and Bacon, 1980 Reprinted with permission from Pergamon Press.)

2.3.3 Surface Defects

Surface defects are two-dimensional planar defects (Fig 2.8) They may be grain boundaries, stacking faults, or twin boundaries These are surface boundaries across which the perfect stacking of atoms within a crystalline lattice changes High- or low-angle tilt or twist boundaries may involve changes in the crystallographic orientations of adjacent grains, Figs 2.8(a) and 2.8(b) The orientation change across the boundary may be described using the concept of coincident site lattices For example, a
¼ 5 or

FIGURE2.8 Examples of surface defects: (a) low-angle tilt boundary; (b) high-angle tilt boundary; (c) S ¼ 5 boundary; (d) twin boundary; (e) intrinsic stack-ing fault; (f) extrinsic stackstack-ing fault (Adapted from Shackleford, 1996 Reprinted with permission from Prentice-Hall.)

1¼ 1=5 boundary is one in which 1 in 5 of the grain boundary atoms match, as shown inFig 2.8(c)

Twin boundaries may form within crystals Such boundaries lie across deformation twin planes, as shown in Fig 2.8(d) Note that the atoms on either side of the twin planes are mirror images Stacking faults may also be formed when the perfect stacking in the crystalline stacking sequence is disturbed, Figs 2.8(e) and 2.8(f) These may be thought of as the absence

of a plane of atoms (intrinsic stacking faults) or the insertion of rows of atoms that disturb the arrangement of atoms (extrinsic stacking faults) Intrinsic and extrinsic stacking faults are illustrated schematically in Figs 2.8(e) and 2.8(f), respectively Note how the perfect ABCABC stacking of atoms is disturbed by the insertion or absence of rows of atoms

2.3.4 Volume Defects

Volume defects are imperfections such as voids, bubble/gas entrapments, porosity, inclusions, precipitates, and cracks They may be introduced into a solid during processing or fabrication processes An example of volume defects is presented in Fig 2.9 This shows MnS inclusions in an A707 steel Another example of a volume defect is presented in Fig 1.16(d) This shows evidence of 1–2 vol % of porosity in a molybdenum disilicide composite Such pores may concentrate stress during mechanical loading Volume defects can grow or coalesce due to applied stresses or temperature fields The growth of three-dimensional defects may lead ultimately to cat-astrophic failure in engineering components and structures

FIGURE2.9 MnS inclusions in an A707 steel (Courtesy of Jikou Zhou.)

2.4 THERMAL VIBRATIONS AND

MICROSTRUCTURAL EVOLUTION

As discussed earlier, atoms in a crystalline solid are arranged into units that are commonly referred to as grains The grain size may be affected by the control of processing and heat treatment conditions Grains may vary in size from nanoscale (10–100 nm) to microscale (1–100 m), or macroscale (1–10 mm) Some examples of microstructures are presented in Figs 1.13(a–d) Note that the microstructure may consist of single phases [Fig 1.13(a)] or multiple phases [Figs 1.13(b–d)] Microstructures may also change due to diffusion processes that occur at temperatures above the so-called recrystallization temperature, i.e., above approximately 0.3–0.5

of the melting temperature in degrees Kelvin

Since the evolution of microstructure is often controlled by diffusion processes, a brief introduction to elementary aspects of diffusion theory is presented in this section This will be followed by a simple description of phase nucleation and grain growth The kinetics of phase nucleation and growth and growth in selected systems of engineering significance will be illustrated using transformation diagrams Phase diagrams that show the equilibrium proportions of constituent phases will also be introduced along with some common transformation reactions

2.4.1 Statistical Mechanics Background

At temperatures above absolute zero (0 K), the atoms in a lattice vibrate about the equilibrium positions at the so-called Debye frequency, , of

 1013

s1 Since the energy required for the lattice vibrations is supplied thermally, the amplitudes of the vibration increase with increasing tempera-ture For each individual atom, the probability that the vibration energy is greater than q is given by statistical mechanics to be

where k is the Boltzmann constant (1:38
1023Jatom1K1) and T is the absolute temperature in degrees Kelvin The vibrating lattice atoms can only

be excited into particular quantum states, and the energy, q, is given simply

by Planck’s law (q ¼ h ) Also, at any given time, the vibrational energy varies statistically from atom to atom, and the atoms continuously exchange energy as they collide with each other due to atomic vibrations Nevertheless, the average energy of the vibrating atoms in a solid is given

by statistical mechanics to be 3kT at any given time This may be sufficient

to promote the diffusion of atoms within a lattice

2.4.2 Diffusion

Diffusion is the thermally- or stress-activated movement of atoms or vacan-cies from regions of high concentration to regions of low concentration (Shewmon, 1989) It may occur in solids, liquids, or gases However, we will restrict our attention to considerations of diffusion in solids in the current text Consider the interdiffusion of two atomic species A and B shown schematically in Fig 2.10; the probability that nA atoms of A will have energy greater than or equal to the activation barrier, q, is given by

nAeq=kT Similarly, the probability that nB atoms of B will have energy greater than or equal to the activation barrier is given by nBeq=kT Since the atoms may move in any of six possible directions, the actual frequency in any given direction is =6 The net number of diffusing atoms, n, that move from A to B is thus given by

nd

If the diffusion flux, J, is defined as the net number of diffusing atoms, nd, per unit area, i.e., J ¼ nd=ðl1l2Þ, and the concentration gradient, dC=dx,

FIGURE2.10 Schematic illustration of diffusion: activation energy required to cross a barrier (Adapted from Ashby and Jones, 1994 Reprinted with per-mission from Pergamon Press)

which is given simply by ðCA CBÞ=r0, the diffusion flux, J, may then be expressed as

J ¼ D0exp q

kT

dx

ð2:9Þ

If we scale the quantity q by the Avogadro number, then the energy term becomes Q ¼ NAq and R ¼ kNA Equation (2.9) may thus be expressed as

J ¼ D0exp Q

RT

dC dx

ð2:10Þ

If we now substitute D ¼ D0exp Q

RT

into Eq (2.10), we obtain the usual expression for J, i.e., J is given by

J ¼ DdC

The above expression is Fick’s first law of diffusion It was first pro-posed by Adolf Hicks in 1855 It is important to note here that the diffusion coefficient for self-diffusion, D, can have a strong effect on the creep proper-ties, i.e., the time-dependent flow of materials at temperatures greater than

0.3–0.5 of the melting temperature in degrees Kelvin Also, the activation energy, Q, in Eq (2.10) is indicative of the actual mechanism of diffusion, which may involve the movement of interstitial atoms [Fig 2.11(a)] and vacancies [Fig 2.11(b)]

Diffusion may also occur along fast diffusion paths such as dislocation pipes along dislocation cores [Fig 2.12(a)] or grain boundaries [Fig 2.12(b)] This is facilitated in materials with small grain sizes, dg, i.e., a large number

of grain boundaries per unit volume However, diffusion in most crystalline materials occurs typically by vacancy movement since the activation ener-gies required for vacancy diffusion (1 eV) are generally lower than the activation energies required for interstitial diffusion (2–4 eV) The activa-tion energies for self-diffusion will be shown later to be consistent with activation energies from creep experiments

2.4.3 Phase Nucleation and Growth

The random motion of atoms and vacancies in solids, liquids, and gases are associated with atomic collisions that may give rise to the formation of small embryos or clusters of atoms, as shown in Figs 2.13(a) and 2.13(b) Since the initial free-energy change associated with the initial formation and growth

of such clusters is positive (Read-Hill and Abbaschian, 1992), the initial clusters of atoms are metastable The clusters may, therefore, disintegrate due to the effects of atomic vibrations and atomic collisions However, a

FIGURE2.11 Schematic illustration of diffusion mechanisms: (a) movement of interstitial atoms; (b) vacancy/solute diffusion (Adapted from Shewmon,

1989 Reproduced with permission from the Minerals, Metals, and Materials Society.)

FIGURE2.12 Fast diffusion mechanisms: (a) dislocation pipe diffusion along dislocation core; (b) grain boundary diffusion (Adapted from Ashby and Jones, 1980 Reprinted with permission from Pergamon Press.)

statistical number of clusters or embryos may grow to a critical size, beyond which further growth results in a lowering of the free energy Such clusters may be considered stable, although random atomic jumps may result in local transitions in cluster size to dimensions below the critical cluster dimension

Beyond the critical cluster size, the clusters of atoms may be consid-ered as nuclei from which new grains can grow primarily as a result of atomic diffusion processes, Figs 2.13(c) and 2.13(d) The nuclei grow until the emerging grains begin to impinge on each other, Fig 2.13(e) The growth results ultimately in the formation of a polycrystalline structure, Fig 2.13(f) Subsequent grain growth occurs by interdiffusion of atoms and vacan-cies across grain boundaries However, grain growth is mitigated by inter-stitial and solute ‘‘atmospheres’’ that tend to exert a drag on moving grain boundaries Grain growth is also associated with the disappearance of smal-ler grains and the enhanced growth of larger grains Due to the combined effects of these factors, a limiting grain size is soon reached The rate at which this limiting grain size is reached depends on the annealing duration and the amount of prior cold work introduced during deformation proces-sing via forging, rolling, swaging, and/or extrusion

FIGURE2.13 Schematic illustration of nucleation and growth: (a, b) formation

of embryos; (c,d) nuclei growth beyond critical cluster size; (e) impingement

of growing grains; (f) polycrystalline structure (Adapted from Altenpohl, 1998.)

The simple picture of nucleation and growth presented above is gen-erally observed in most crystalline metallic materials However, the rate of nucleation is generally enhanced by the presence of pre-existing nuclei such

as impurities on the mold walls, grain boundaries, or other defects Such defects make it much easier to nucleate new grains heterogeneously, in contrast to the difficult homogeneous nucleation schemes described earlier

In any case, the nuclei may grow by diffusion across grain boundaries to form single-phase or multi-phase microstructures, such as those shown in Fig 1.13

A simple model of grain growth may be developed by using an analogy

of growing soap bubbles We assume that the growth of the soap bubbles (analogous to grains) is driven primarily by the surface energy of the bubble/ grain boundaries We also assume that the rate of grain growth is inversely proportional to the curvature of the grain boundaries, and that the curva-ture itself is inversely proportional to the grain diameter We may then write:

where D is the average grain size, t is time elapsed, and k is a proportionality constant Separating the variables and integrating Eq (2.12) gives the following expression:

where c is a constant of integration For an initial grain size of D0 at time

t ¼ 0, we may deduce that c ¼ D2 Hence, substituting the value of c into

Eq (2.13) gives

D2 D2

Equation (2.14) has been shown to fit experimental data obtained for the growth of soap bubbles under surface tension forces Equation (2.14) has also been shown to fit the growth behavior of metallic materials when grain growth is controlled by surface energy and the diffusion of atoms across the grain boundaries In such cases, the constant k in Eq (2.14) exhibits an exponential dependence which is given by

where k0 is an empirical constant, Q is the activation energy for the grain growth process, T is the absolute temperature, and R is the universal gas constant By substituting Eq (2.15) into Eq (2.14), the grain growth law may be expressed as

D2 D2