Table Of Contentvii
Preface
Since the benefit of stress-induced tetragonal to monoclinic phase trans-
formation of confined tetragonal zirconia particles was first recognized in
1975, the phenomenon has been widely studied and exploited in the de-
velopment of a new class of materials known as transformation toughened
ceramics (TTC). In all materials belonging to this class, the microstruc-
ture is so controlled that the tetragonal to monoclinic transformation
is induced as a result of a high applied stress field (e.g. at a crack tip),
rather than as a result of cooling the material below the martensitic start
temperature.
The significance of microstructure to the enhancement of thermome-
chanical properties of TTC is now well understood, as are the mecha-
nisms that contribute beneficially to their fracture toughness. The mi-
cromechanics of these mechanisms has been extensively studied and is
now ripe for introduction to a wide audience in a cogent manner.
The description of the toughening mechanisms responsible for the
high fracture toughness of TTC requires concepts of fracture mechan-
ics, dislocation formalism for the modelling of cracks and of Eshelby's
technique. This has presented us with the opportunity to review these
concepts briefly for the benefit of the reader who is unfamiliar with them.
The advanced readers have our sympathy, if they find this revision su-
perfluous to their needs.
The monograph has its origin in the sets of notes that the first author
wrote on two separate occasions for lectures read to participants from
research and industrial organizations. The preparation of the monograph
has meant that the lecture notes had to be brought up to date and
substantially enlarged to include several topics which have only recently
been fully investigated.
We are indebted to the whole community of researchers who have
contributed to our present understanding of the mechanics of transfor-
mation toughening in TTC. Nothing would have given us greater plea-
sure than to thank all of them individually, but we were bound to miss
some names and to give offence unintentionally. We therefore offer them
a collective thank you and hope reference to their contributions in the
monograph at least partly compensates for this omission on our part.
Chapter 1
Introduction
All ceramics as a rule have very low resistance to crack propagation, i.e.
very low fracture toughness. Cubic zirconia ceramics are no exception
to this rule. They suffer two phase transformations between the melting
point of zirconia at about 2770~ and room temperature. These trans-
formations which result in profuse microcracking can be eliminated by
stabilizing the high temperature cubic phase with calcia (CaO), magne-
sia (MgO), yttria (Y203) or ceria (CeO2). However, the fully stabilized
cubic zirconia still has low toughness and hardness and is not especially
strong for engineering application.
A development in 1975 exploited the phase transformation by using
insufficient amount of stabilizer in order to inhibit the tetragonal (t)
to monoclinic (m) transformation that would normally occur at about
ll00~ on cooling from the cubic phase. This leaves the t-ZrO2 phase in
a metastable state. Substantial toughening is achieved when the retained
metastable t-ZrO2 is induced to transform to the monoclinic phase un-
der high applied stresses, such as those at a crack tip. The t *-- m
phase transformation is martensitic in nature and is accompanied by a
dilatation of about 4% and deviatoric shear strains of about 16%, if the
t-ZrO2 precipitates are unconstrained.
The range of ceramic materials which exploit the controlled t
m transformation has grown extensively. They are collectively called
the zirconia-toughened ceramics (ZTC). Depending upon the matrix in
which the metastable t-ZrO2 precipitates are embedded, the ZTC are
further subdivided into three main groups. These are designated par-
tially stabilized zirconia (PSZ), tetragonal zirconia polycrystals (TZP)
and dispersed zirconia ceramics (DZC).
4 Introduction
The metastable t-phase in PSZ exists as precipitates dispersed within
a cubic stabilized zirconia matrix, the common stabilizing addition being
CaO, MgO and Y203. In TZP the entire polycrystalline body generally
consists of t-phase. This is achieved by alloying with oxides which have
a relatively high solubility in ZrO2 at low temperatures, e.g. Y203 and
CeO2. In DZC materials, the metastable t-phase is dispersed in a non-
ZrO2 matrix which may be either an oxide or a non-oxide, e.g. A1203,
SiC, SiaN4, TiB2, TiN.
The athermal t ~ rn transformation induced as a result of an ap-
plied stress field, e.g. at a crack tip results in a net dilatation of 4%, but
because of shear accommodation processes, e.g. twinning, the net devia-
toric shear strain is much less than that of an unconstrained particle. It
is not surprising therefore that many works dealing with the toughening
induced by the t ~ rn transformation ignore the shear component and
consider only the dilatation. However, as we shall see in this monograph
the shear transformation strains not only contribute to the toughening
but, more importantly, induce some new effects which are not known to
exist under dilatational strains alone. Among these is the phenomenon
of autocatalysis, whereby the stresses induced by the transformation of
a few tetragonal particles are sufficient to induce further transformation
which thus becomes a self-propagating process.
The exact mechanism that triggers stress-induced (athermal) t ~ rn
transformation is still on open question. Some investigators believe that
the transformation occurs spontaneously when the critical mean stress
at the location of a tetragonal precipitate attains a critical value. Others
seem to favour a transformation criterion that includes both hydrostatic
and deviatoric stress components. All known transformation triggering
criteria have been discussed in this monograph to a certain extent, al-
though the greater part of the discussion relies on the critical mean stress
criterion.
The contents of the monograph have been arranged in three parts.
Part I (Chapters 2-6) gives a description of materials, and their con-
stitutive equations and introduces the mathematical tools necessary for
studying the interaction between isolated transformable particles and
cracks. On the basis of these tools, the toughening induced by t ~ rn in
TTC is described in Part II (Chapters 7-10). The interaction of trans-
formation toughening mechanism with other toughening mechanisms is
investigated in Part III (Chapters 11-15) of the monograph, as are the
fatigue and wear characteristics of TTC.
A brief outline of the contents of each chapter follows. Chapter 2
gives a description of the various transformation-toughened ceramics
Introduction 5
(TTC), emphasizing the fabrication processes necessary for the reten-
tion of tetragonal zirconia in a metastable state.
Chapter 3 is devoted to the constitutive description of the whole
class of TTC in the spirit of the classical theory of plasticity. It be-
gins with a description of dilatational transformation plasticity and in-
troduces the concepts of sub- and super-critical transformation. The
influence of shear transformation strains upon the stress-strain relations
of TTC is described next, followed by that of DZC in which the elastic
constants of the matrix and transformable phases are different.
The complex stress potentials for small circular spots of arbitrary
transformation strain are derived in Chapter 4 using Muskhelishvili's
method and Eshelby's formalism. A second method based on the con-
cepts of force doublets and dipoles (strain centres) is also used to develop
Green's functions for infinitesimal transformable spots. It is shown that
in the limit of vanishing transformable spot size, the two methods yield
identical complex stress potentials.
The complex stress potentials for an isolated transformable spot of
arbitrary shape are used in Chapter 5 to derive image potentials for semi-
infinite and finite cracks interacting with such a spot. These potentials
are then used to calculate the stress intensity factors at the tips of the
cracks as a function of the location of spot and the transformation strains
in it.
For many two-dimensional crack problems, especially those involv-
ing single or multiple surface cracks, the Muskhelishvili complex stress
potentials cannot be constructed in a closed form. In such cases it is ex-
pedient to resort to the equivalence between appropriate line dislocations
and cracks. The essential features of this equivalence (i.e. the dislocation
formalism) are briefly outlined in Chapter .6 In particular, the weight
functions necessary for the representation of single and multiple cracks
in a plane or half-plane are derived for use in subsequent chapters.
Chapter 7 is devoted to a continuum or macroscopic description of
the composite zirconia systems in which the discrete transformable spots
are smeared out into a transformation zone. The change in the stress
intensity factor induced by the presence of the transformation zone near
the tip of a semi-infinite crack is calculated when the crack is stationary
or when it is growing under steady-state conditions. The emphasis is on
sub- and super-critical dilatational transformation, although the role of
shear transformation strain is also briefly explored.
The complete analysis of the quasi-static growth of a crack, taking
into account the progressive development of the transformation zone
around the crack, is given in Chapter .8 As in the preceeding Chapter it
6 Introduction
is assumed that super-critical transformation occurs at a critical mean
stress and induces only dilatational strains. The analysis begins with
that of a semi-infinite crack in an infinite medium, followed by that of a
single finite crack and an array of collinear finite cracks, and ends with
that of a single and a periodic array of edge cracks. The computational
difficulties arising from the penetration of a crack into a transformable
inclusion are identified and strategies developed to overcome these.
The plane strain continuum description given in Chapters 7 and 8
to what is essentially a discrete, three-dimensional problem is quite ad-
equate if the number of transformed particles is large. When the trans-
formation zone spans only a few particles and when the remote loading
contains mode II and mode III components, besides that of mode I, a
three-dimensional approach is called for. Chapter 9 is devoted to the
derivation of analytical expressions for stress intensity factors induced
along a half-plane crack front by unconstrained dilatational and shear
transformation strains using three-dimensionM weight functions. The
discrete transformable domain is assumed to be in the shape of a sphere
or a spheroid, and the influence of the orientation of an oblate spheroid
relative to the half-plane crack front upon the transformation toughening
is highlighted.
The role of shear stresses, besides that of the hydrostatic stress,
in triggering the t ~-- m transformation is studied in Chapter ,01 to-
gether with the contribution of transformation-induced shear strains to
the toughening. This study is conducted not in the continuum plane
strain approximation but by assuming that the tip of a semi-infinite
crack is surrounded by a distribution of small transformable spots. It
transpires that shear stresses created by super-critical transformation of
a few spots may be sufficient by themselves to trigger the transformation
of neighbouring spots, thereby creating a self-propagating autocatalytic
reaction.
Chapter 11 is devoted to the study of toughening mechanisms in dis-
persed zirconia ceramics (DZC), such as ZTA. The toughening in these
ceramics can arise from two complementary mechanisms depending on
the volume fraction of tetragonal zirconia. At low volume fractions, there
is practically no stress-induced phase transformation, and the increase
in toughness is primarily due to microcrack-induced dilatation around
thermally formed monoclinic zirconia precipitates. At high volume frac-
tions, on the other hand, the stress-induced transformation toughening
mechanism seems to dominate over the microcrack mechanism. Both
mechanisms are studied with reference to two ZTA compositions. Al-
though the contribution of microcracking mechanism to the toughening
Introduction 7
of PSZ or TZP is believed to be minimal, even in these materials the
slight mismatch in the elastic constants of tetragonal and monoclinic
polymorphs can give a significant effect upon the toughening process.
This question is also addressed in Chapter .11 When the differences in
the elastic moduli of matrix and transformable phases are large, as in all
DZC, the concept of effective transformation strain is introduced.
The toughening of DZC by the shielding of a macrocrack front by
a zone of transformation or microcracks is sensitive to temperature. In
these materials, toughening can also result from the inhibition of prop-
agating cracks by second phase particles, i.e. by crack bridging. This
toughening mechanism is not sensitive to temperature and often acts
in conjunction with the transformation toughening mechanism. The
mechanics of toughening by crack bridging is studied in Chapter ,21
together with its interaction with the transformation toughening mech-
anism.
Another potential mechanism of toughening in DZC is by crack de-
flection in combination with phase transformation. Cracks deviate from
their planes when they encounter second phase particles, the deviation
being all the more noticeable when these particles are non-homogeneous-
ly distributed in the matrix, as is always the case in DZC. The interaction
between crack deflection and phase transformation toughening mecha-
nisms is investigated in Chapter .31
Transformation-toughened ceramics have been found to be suscepti-
ble to mechanical degradation under cyclic loading. As in metals, the
rate of growth of long cracks shows a power-law dependence on the ap-
plied stress intensity range. However, small cracks - the size of naturally
occurring surface flaws - are found to grow at stress intensity levels below
the long-crack fatigue threshold, at which fatigue cracks are presumed
dormant in damage-tolerant designs. Chapter 41 is devoted to the devel-
opment of fatigue crack growth models, which predict the known long-
and short-crack fatigue behaviour of TTC. A detailed study is also made
of the microstructural parameters that ensure crack arrest at a given ap-
plied stress amplitude. It is demonstrated that occasional overloading
can improve the endurance limit of TTC.
TTC are known to exhibit poor wear performance under rolling/-
sliding conditions. The role of surface and subsurface cracks under slid-
ing contact load in this poor performance is investigated in Chapter .51
The tetragonal precipitates are modelled as discrete circular spots. As
in Chapter ,01 the influence of shear tractions under the contact load in
the triggering of transformation is also examined.
Chapter 2
Transformation
Toughening Materials
2.1 General
A ceramic is a combination of one or more metals or semi-metals (such as
Si), with a non-metallic element. Depending on the non-metallic element
present in the composition, a ceramic is classified as being an oxide (if
the non-metallic element is O2) or a non-oxide. The comparatively large
non-metallic ions serve as a matrix with the small metallic ions tucked
into the spaces in between. The basic elements are linked by either ionic
or covalent bonds (or both). The ceramics can be either amorphous or
crystalline.
Ceramics cover a vast field. One of the earliest known materials was
a ceramic, viz. stone. Glass and pottery (and even concrete) are all
ceramics, but the ceramics of most engineering interest today are the
new high-performance ceramics that find application for cutting tools,
dies, internal combustion engine parts, and wear-resistant parts. An
excellent summary may be found in the paper by Morrell (1984).
Diamond is the ultimate engineering ceramic and has been used for
many years for cutting tools, dies, rock drills, and as an abrasive. But it
is expensive. The strength of a ceramic is largely determined by its grain
size, distribution of microcracks and processing technique. A new class
of fully dense, high strength ceramics is emerging that is competitive on
a price basis with metals for cutting tools, dies, human implants and
engine parts. Ceramics are potentially cheap materials.
01 Transformation Toughening Materials
2.2 Modern Zirconia-Based Ceramics
Pure zirconia (ZrO2) suffers two transformations between its melting
point (Tin) at about 2770~ and room temperature. These transforma-
tions which result in profuse microcracking can be eliminated by sta-
bilizing the high temperature cubic form with calcia (CaO), magnesia
(MgO), yttria (Y203) ceria (CeO2). Cubic stabilized zirconia has
or
a low toughness and hardness and is not especially strong. However
an Australian development by Garvie et al. (1975) exploited the phase
transformations by using insufficient amount of stabilizer in order to
inhibit the tetragonal (t) to monoclinic (m) transformationthat would
normally occur at about 1100~ on cooling from the cubic phase.
Toughening is achieved when this retained metastable t-phase is in-
duced to transform to the monoclinic phase under high stresses, such as
those at a crack tip. The t ~ m transformation results in dilatation and
shear strains that impede the progress of a growing crack. This class of
ceramics is now known as transformation-toughened ceramics (TTC).
The t ~ m transformation of confined t-ZrO2 particles under
stress leads to the enhancement of other mechanical properties such as
strength, thermal shock resistance, as well as of fracture toughness. An
excellent account of the dependence of thermomechanical properties of
TTC on microstructure may be found in the article by Hannink (1988)
who is one of the three original developers of TTC. We provide a brief
outline of material from that paper.
The range of ceramics exploiting the athermal (i.e. high stress in-
duced) t ,--- m martensitic transformation of ZrO2 is large and is called
the zirconia-toughened ceramics (ZTC). ZTC may be further classified
into three groups:
.1 Partially stabilized zirconia (PSZ) in which submicron-size
t-ZrO2-precipitates are uniformly dispersed in an essentially
c-ZrO2-matrix. The amount of t-phase can be found by X-
ray diffraction techniques;
2. Tetragonal zirconia polycrystals (TZP) in which the principal
constituent is the very fine-grained t-ZrO2. Usually prepared
with Y203 as the stabilizer (1.75-3.5 tool%), though lately
CeO2 is also being used;
.3 Dispersed zirconia ceramics (DZC) in which t-ZrO2 precipi-
tates are dispersed in a non-ZrO2 matrix, the most common
of which are AI2Oa, SiC, SiaN4, TiB2, TiN. When A12Oa is
2.3. Martensitic Transformation 11
the matrix, the corresponding toughened ceramic is called
ZTA.
The shape of the dispersed t-ZrO2 is determined by the constraint
provided by the matrix. Thus, t-ZrO2 precipitates appear as spheres in
ZTA and as thin oblate spheroids in PSZ (aspect ratio 5:1).
In all ZTC, optimum mechanical properties are intimately connected
with the size of t-ZrO2 precipitates and the metastability of t-phase. To
ensure that the t ~ m transformation does not occur at the martensitic
start temperature Ms requires extreme care and control in sintering, hold
(ageing) and cooling treatments. These will be briefly discussed later
in this introduction. But first a few words on the t ~ m martensitic
transformation.
2.3 Martensitic Transformation
The t ---, m phase transformation in ZrO2 is martensitic in nature. An
important feature of this transformation is the existence of lattice cor-
respondence (LC) between the unit cells of the parent (t-ZrO2) and
product (m-ZrO2) phases. Adoption of a correspondence implies that
the change in polymorphic structure can be approximated by a homoge-
neous lattice deformation in which the principal axes of the parent lattice
remain orthogonal except for a rigid body rotation. This is tantamount
to minimization of the strain energy. The lattice strains (Bain strains)
are given by eqn (2.1).
Table 2.1 and Fig. 2.1 give the lattice parameters of the three poly-
morphic phases of ZrO2, from which it is possible to calculate the Bain
strains due to transformation, e T (i, j - ,1 2, 3 refer to the co-ordinate
axes of c-ZrO2). In view of the close similarity in lattice parameters in
all three polymorphs, the t ~ m transformation in ZrO2 has resulted in
three nearly identical correspondences. These are distinguished on the
basis of which monoclinic axis am, mb or mc is parallel to the tetragonal
ct axis and are designated LC A, B and C, respectively.
eT _ am cos .~ - 1
at
eT = bm
" bt
Cm
33Te = 1
21 Transformation Toughening Materials
1 "17
3Te -- ~T3e -- ~ tan(~- ~)
2Tle --eT1 -- eT3 -- eT2 -- 0 (2.1)
It is clear that the unconstrained t ---, m transformation results in
both dilatational /T/e( -- 1Te 2T2e+ +eT3) and deviatoric (eT;i s5 j) strains
with about 4-6% dilatation and 14-16% deviatoric strains. However,
since the transformation usually takes place in a constrained matrix (a
c-ZrO2 in the case of PSZ and a non-ZrO2-matrix in the case of DZC) a
large proportion of deviatoric strains is cancelled as a result of twinning.
The net deviatoric strains in a constrained matrix are likely to be of the
same order as dilatation.
In all ZTC, the aim is to produce materials whose martensitic start
temperature M, for t ~ m is less than or equal to room temperature in
order to prevent spontaneous transformation taking place during cool-
ing. Ms depends on the particle size of t-ZrO2. The thermal treatment
therefore aims to bring t-ZrO2 to a critical size. The critical size of
metastably retained t-ZrO2 may be roughly calculated from thermo-
dynamic considerations. The transformation of a metastably retained
t-ZrO2 to a lower energy m-ZrO2 state requires that an activation (nu-
cleation) barrier AF* (Fig. 2.2) be overcome. This means a critical
(embryonic) nucleus must be activated before the martensitic t --, m
transformation can occur at a temperature designated Ms.
The actual state of metastability of the t-phase will depend upon
a number of physical and structural factors. This state is depicted
schematically with the use of a free energy diagram in Fig. 2.2 for the
Lattice Parameter (nm)
Material ~ ~ Cubic Tetragonal Monoclinic
~.r ac a t c t a m b m c m J~
Ca-PSZ 8.4 0.5 ! 32 0.5094 0.5180 0.5171 0.5182 0.5296 98.67"
Mg-PSZ 9.4 0.5080 0.5080 0.5183 0.5117 0.5177 0.5303 98.91<'
Y-PSZ 7.4 0.5130 0.5116 0.5157
Ce-TZP 21 - 0.5132 0.5228 0.5193 0.5204 0.5362 98.80"
Table 2.1" Lattice parameters
