Brittle Fracture and Toughening Mechanisms in ...

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Ann. Rev. Mater. Sci. 1984. 14: 373-403

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BRITTLE FRACTURE AND

Annu. Rev. Mater. Sci. 1984.14:373-403. Downloaded from www.annualreviews.org by UNIVERSITY OF FLORIDA - Smathers Library on 09/22/10. For personal use only.

TOUGHENING MECHANISMS IN CERAMICS

1

S. M. Wiederhorn

National Bureau of Standards, Washington, DC 2 02 34 1.

INTRODUCTION

In many applications, brittle fracture limits the use of ceramic materials. In the electronics industry for example, ceramics are used as substrates and dielectrics because of their electrical properties, and yet failure in these applications is often caused by brittle fracture, which results from thermal expansion mismatch between ceramic and metallic parts of electronic packages. Similarly, new developments in heat engines require ceramic parts in order to achieve the high temperatures that result in greater engine efficiency, and yet here, too, failure has been shown to occur by brittle fracture, caused by thermal shock as components are heated to and cooled from their operating temperatures. In order to understand the fracture behavior of ceramic materials, it is necessary to understand the mechanisms of fracture of materials that are entirely brittle. In these materials plastic deformation by dislocation motion does not occur, or occurs to such a limited extent that cracks are sharp to the atomic level of the solid. Resistance to fracture is provided by the lattice itself, and not by the movement of dislocations. Ceramics can be made tougher by modifying the microstructure of the solid in such a way as to reduce stresses near crack tips. Our ability to make tougher ceramics has increased gradually with our deepening understanding of brittle fracture. The purpose of this review is to outline the evolution of this understanding and to show how the knowledge gained is currently being applied to the development and manufacture of tougher ceramics. 1

The US Government has the right to retain a nonexclusive royalty-free license in and to

any copyright covering this paper.

37 3


374

WIEUERHORN

To set the stage, the earlier works of Inglis, Griffith, and Barenblatt on crack stability in homogeneous materials are reviewed. The importance of the lattice to fracture, as embodied in concepts of lattice trapping, is then discussed. It is shown that lattice trapping provides a natural description of subcritical crack growth in brittle materials. Following these discussions, the importance of microstructure to the fracture process is reviewed. Grain boundaries and crystal anisotropy are shown to be especially important in establishing the fracture resistance of ceramic materials. The review ends with a discussion of toughening methods for ceramic materials. These methods are leading to practical advances that may eventually result in applications in which ceramics will be used as primary structural elements. 2.

THEORIES OF BRITTLE FRACTURE

The earliest explanations of fracture in brittle materials were those de veloped by Inglis (1) and by Griffith (2), who demonstrated the impor tance of microscopic flaws to the strength of brittle materials. By ana lyzing the stresses in the vicinity of a notch, Inglis showed that notches serve as stress concentrators, which locally magnify the stresses applied to the solid. Failure of the solid occurs when the stresses at the root of the notch exceed the theoretical strength, O"th' of the material. As an example of a stress concentrator (Figure 1) the stress, 0", at the tip of a two-dimensional elliptical notch with a major axis, a, a minor axis, b, and major axis radius, p b2/a is related to the applied stress, S, by the following equation: =

0" =

2SjaiP,

1.

where a is assumed to be much larger than p. Fracture occurs when 0" = O"th' The extent of stress magnification in ceramics can be estimated from the fact that crack lengths in brittle materials are normally of the order of 10 to 100 flm ; whereas the "effective" radius of the crack tip is of the order of the atomic spacing of the solid, 0 .3 to 1 nm, if the crack is assumed to be atomically sharp. Using these values, stress concentrations ranging from 100 to 1000 times the applied stress are expected ( 3). Indeed, silica glass with a theoretical strength of approximately2 0 GPa has a practical strength of2 to 20 MPa as a consequence of microscopic damage due to manufacturing and handling. By contrast, silica glass fibers that are made under conditions designed to minimize surface damage have measured strengths in vacuum or in water-free environments of 10 GPa (4, 5), which is close to the theoretical value of the strcngth. Similar decrcases from thc theoretical strengths have been observed for other glasses and ceramic materials. A somewhat different approach to the fracture of brittle materials was used by Griffith (2). Crack stability was determined from a mechanical .�

BRITTLE FRACTURE IN CERAMICS

375

energy balance in which the energy required to form a differential area of new surface was equated to the mechanical energy lost as the crack advanced. By the principle of virtual work, Griffith was able to show that, for a two-dimensional elliptical crack (Figure 1), the far-field tensile stress, S, required for fracture was related to the Young's modulus, E, of the solid, the crack length, a, and the surface energy, }" of the solid:


S

=

(2EYlna)1/2.

2.

As with the Inglis approach to fracture, the strength of the solid was found to be inversely proportional to the square root of the crack length. For a theoretical surface energy (6) of approximately 2J1m2, silica glass has been shown to have strengths of the same order as that given by the Inglis calculation: 30 MPa for a 1 00-l1m crack. The Griffith and Inglis analyses are alternative ways of describing the fracture of brittle solids. In the Inglis analysis, the material parameters that control the fracture are the theoretical strength and the radius of curvature of the crack tip; in the Griffith approach, the controlling material parameter is the surface energy. From simple atomic considerations it has been argued that the two criteria are equivalent, when the radius of �

Figure 1

Elliptically shaped crack in an infinite two-dimensional plane. Stresses near the end

of the maj or axis approximate those of a crack when

a

� b.


376

WIEDERHORN

curvature of the crack tip in the Inglis model is of the order of the three times the lattice spacing of the solid (7). In this interpretation, either equation provides both the necessary and sufficient condition for fracture. The Griffith approach to fracture employs an energy balance, and does not make specific reference to the exact details of the structure of the crack tip. Resistance of the material to fracture originates with the surface energy of the material, which can be evaluated experimentally by measuring the strength of specimens that contain well-defined cracks. These experimental techniques (8, 9) lie within the realm of fracture mechanics and have been used to determine the fracture surface energy of a large number of glasses and ceramics. A compendium offracture energy values is being collected by Freiman, Baker, and Wachtman and will be published in the near future. Part of the compendium on glasses is given in ( 10). Some fracture energy values determined on single crystals and on silica glass are given in Table 1 where they are compared with theoretical estimates of the surface tension. The theoretical surface tension values were determined either by considering energy requirements for bond annihi lation on the cleavage plane (6, 15), or by using simple atomic force laws to describe the rupture process ( 13, 14). Considering the approximate nature of the theoretical values (Table 1), the factor-of-two agreement between experiment and theory suggests that the fracture resistance of these materials arises primarily from the atomic forces that bind the atoms of the solid together. This finding is consistent with the view that crack tips in brittle materials are essentially atomistic in their structure. Their shape is controlled by the lattice structure of the solid and the atomic forces that hold the solid together. The works of Griffith and Inglis have been extended and now form the basis for the discipline of linear elastic fracture mechanics. In this discipline, a crack is modeled as a slit in a continuum ( 16, 17) (Figure 2). The stresses, Table 1

Fracture energies, strengths, and elastic constants of brittle solids

Material(Ref.) Si (11) Ge(l1) SiC(12) AI203

(13)

Si02(14)

E

y(Jm - 2)

(nm)

1.7

1.5

1.2

1.1 10

0.07 0.07 0.16 0.10 0 .39

(GPa)

(110) (111) (0001) (1010)

168 140 470

425

5 5.8

70

1.8

4.3

glass

AI203 (49)

polycrystalline

SiC(49)

polycrystalline

d

Experimental

Crystal surface

Theoretical

7.3

10--50 15-30


377

(Jij' at some point near the crack tip are related to the distance, r, ofthe point

from the crack tip, the angle, 0, of the point from the crack plane, and a parameter called the stress intensity factor, K.:


3.

where k(O) is a function of the angle O. The stress intensity factor depends on the geometry of the crack and the applied stresses that are the driving forces for fracture. The stress intensity factor also establishes the magnitude of the stress field near the crack tip, and as such, is the local determinant for the fracture driving forces. It is for this reason that fracture in brittle materials is usually assumed to occur when the stress intensity factor in a solid reaches a critical value, K, KIC' Because of the inverse square root dependence of stress on distance from the crack tip (Equation 3), the stress at the crack tip is singular regardless of the level of the applied load. As discussed below, this crack tip singularity was the impetus behind =

Barenblatt's work, which introduced the idea of crack tip cohesive forces.

Linear elastic fracture mechanics also treats fracture and crack stability from an energy viewpoint. In this approach, fracture occurs when the crack

I

-

-�.�r

X

20 Figure

2

Fracture mechanics model of a slit in a continuum. Note that the stress on the

projected crack plane goes to infinity as the distance to the crack tip approaches zero, regardless of the magnitude of the far field stresses applied to the crack ( 1 6).

378

WIEDERHORN

extension force, G, also known as the energy release rate, is equal to a critical value, Gc: G Gc. The crack extension force is defined as the total mechanical energy release per unit area,A, of crack formed: =

G

=

4.

aU/aA,

where U is the total mechanical energy of the system (the sum of the work done by the external forces minus the elastic energy of the solid). In brittle ceramic materials Gc 2y. It can be shown that the stress and the energy approaches to fracture are equivalent, and that in fact Kf EG for opening mode failure, plane stress condition, where E is Young's modulus. The works of Inglis and Griffith are important because they focused the attention of ceramic science on the flaws and surface cracks that determine the strength of brittle materials. Subsequent investigations have shown that the removal of flaws in brittle materials resulted in a considerable improvement in the strength of these materials. In fact, by careful surface preparation, strengths of the order of the theoretical strength of solids can be obtained on fibers and thin rods of glasses and other ceramics (4,5,18). From a more practical viewpoint, these early theories of fracture led directly to the development of thermally and chemically tempered glasses (19,20). These glasses are modified by chemical or thermal treatments that place their surfaces into a state of compressive stress, thus inhibiting the motion of cracks that are normally present. In these glasses, equilibrium cracks cannot move until the applied stress exceeds the surface stress by a sufficient amount that the Griffith criterion for fracture is satisfied for the most critical flaw in the glass surface. Another practical result of the work by Griffith and Inglis has been the improvement of the microstructure of ceramic materials through proces sing (21). The strength of ceramics is often determined by surface cracks introduced during machining and finishing operations (22). Strength can be increased by modifying machining procedures (22), or by post-finishing thermal or chemical treatments as has been done on nonoxide ceramic parts for heat engines (23, 24). Shrinkage voids, a source of crack-like cavities,can be avoided by removing the sources of void formation during processing. Although they might not be as severe as cracks, inclusions are also a source of stress concentration. The severity of the stress near an inclusion, depends on the type of inclusion, and the thermal and elastic properties of the inclusion vis-a-vis that of the matrix (21). A considerable amount of research effort is currently being spent on developing methods of improving strength by modifying the microstructure of ceramics. In a sense, this effort had its origin in the early recognition by Griffith and Inglis that small cracks or notches are the source of fracture in brittle materials. =


=



3.

379

CRACK TIP FORCES

Since Inglis and Griffith published their work, a number of other authors have presented theories for fracture that were based on more "realistic" models of the crack tip. The theory presented by Barenblatt introduced the idea of a crack-tip cohesive force as the primary resistance to fracture in brittle solids (25, 26). Barenblatt's model is based on the theory of fracture mechanics in which the crack is assumed to be a planar discontinuity in a solid. For this geometry, the stresses, (Jii' at the crack tip are singular, regardless of the forces applied to the material (see Equation 3). Barenblatt recognized that the stresses cannot be infinite at the crack tip (r = 0) as implied from Equation 3 and argued that cohesive forces acting between the crack surfaces near the crack tip had to be of sufficient magnitude to remove the stress singularity. For a cohesive stress, (J(t), acting on the crack surface over a distance, d, from the crack tip (Figure 3), the stress intensity factor, K", resulting from the cohesive forces is given by: KG

= _

(�)1/2 fd (JJt(t) dt. TC

5.

0

The stress at the crack tip can now be calculated by substituting K( + KG for y

o

a(x.O)

I

!

!

C

C'

'---- -�, . .,�--

C

----+-.I

D

x

-j

-a(x)

Figure 3

Dugdale-Barenblatt model of the crack tip. Cohesive forces acting between the faces

of the crack, near the crack tip, remove the stress singularity so that the stresses are finite at the crack tip (17).

380

WIEDERHORN

K( in Equation 3. When K( = -Ku the stress singularity is removed. It is worth noting that Ku is an integrated form of the force equation. As K( is increased, the forces at the crack tip increase until they reach a maximum, at which point K( Ku = K(C and crack motion occurs. Thus, crack motion occurs at a critical value of the applied stress intensity factor corresponding to achievement of a maximum stress at the crack tip. This condition is a well-known fracture mechanics criterion for failure. Baren blatt's theory led to the conclusion that the crack contour closes smoothly so there is no strain singularity at the crack tip. Annu. Rev. Mater. Sci. 1984.14:373-403. Downloaded from www.annualreviews.org by UNIVERSITY OF FLORIDA - Smathers Library on 09/22/10. For personal use only.

=

4.

-

CRACK TIP STRUCTURE

The theory developed by Barenblatt can be used to estimate the width of the cohesive zone at the crack tip. Assume that the stress, u(t), within the cohesive zone is equal to the theoretical strength; then Equation 5 is easily integrated to yield the following equation for the width of the cohesive zone: 6. This is the famous Dugdale-Barenblatt equation, suggested by Dugdale (27) for a plastic zone at the crack tip and by Barenblatt (25, 26) for the cohesive zone in brittle materials. The zone size can be estimated from Equation 6 using the data in Table 1. In this table, theoretical estimates of the cohesive strength were obtained from a simple sinusoidal force law approximation originally suggested by Orowan (28): Uth = E/n. In the absence of nonlinear deformation at the crack tip, the cohesive zone is of the order of the spacing between atoms of the solid. This finding is consistent with a similar conclusion reached by Cribb & Tompkins (29) using the same sort of analysis. To obtain a more realistic estimate of the forces and displacements at the crack tip, Sinclair & Lawn (30) modelled atomic bonding in diamond structure crystals by a modified Morse potential. In their model the crack tip region consisted of512 atoms, each of which was bonded to its neighbors by atomic forces. Outside of the crack tip region, the solid was treated as a continuum. Estimates of the forces between atoms bordering the crack plane (Figure 4) suggest that complete bond rupture occurs within one interatomic spacing. Calculations of displacements along the crack plane give a cohesive zone width, d, of approximately three nearest neighbors, i.e. 0.7 nm for silicon. Although this estimate of the cohesive zone is larger than that given in Table 1, the zone is still of atomic dimensions, which indicates the importance of atomic forces at crack tips in brittle materials. The importance of atomic bonding to crack stability has been discussed "'"


381

extensively by Thomson, Fuller, and others using one-, two-, and three dimensional atomic models of the crack tip (31-38). Because of the narrow width of the cohesive zone, the atomic structure of the crack tip gives rise to energy barriers to crack motion. As a consequence of these energy barriers, the crack can be "trapped" by the lattice so that the crack is stable for a range of applied forces (Figure 5). The crack can move forward ifthe applied crack tip force exceeds the maximum value G +, for trapping. Conversely, crack healing can occur when the applied force is less than the mini mum critical value, G for trapping. The Griffith condition for crack stability is found to be between these two values of the crack tip force, G < G < G + (39). An important implication of the theory of lattice trapping is the occurrence of "environment-free" subcritical crack growth in brittle materials. Even though the crack is trapped by the discrete nature of the solid, available thermal energy may be sufficient to drive the crack over the energy barrier and result in subcritical crack growth. Based on the concept of lattice trapping, models of subcritical crack growth were developed by Thomson & Fuller (38) and by Lawn (40). These theories predict that the growth of cracks in brittle materials satisfies an Arrhenius function in which


_,

_

3

1.5

t Z 2i

1.0 t

....

0

c:

';;j f...

�

..., Ul