INTRODUCTION
2.01 Fundamental Aspects of Hard Ceramics
2.01.5 Mechanical Properties
Although ceramics are hard, wear-resistant materials that retain a reasonable degree of their mechanical properties even at high temperatures, they are very brittle and undergo very little plastic deformation under mechanical stress. Some background to this aspect of the behavior of ceramics is given in the following sections.
2.01.5.1 Strength and Fracture Resistance
Theoretical tensile strength of a material with elastic modulus,E, and interatomic spacing,ro, is given by the following relationship developed byOrowan (1949):
sth ¼ ðEgo=roÞ1=2 (13) Therefore, high strength is associated with high thermodynamic surface energy,go, high elastic modulus or stiffness,E, and small lattice spacing,ro, all usually found in elements, or compounds from elements, in thefirst and second rows of the periodic table exhibiting predominantly covalent bonding (Table 2). Thus, the strongest ceramics are expected to be the oxides, carbides, nitrides, borides of silicon and aluminum and some other metals. Typical values of these parameters for ceramics are
l Thermodynamic surface energy,go¼1 J m2 l Stiffness (elastic modulus),E¼300 GPa l Lattice spacing,ro¼31010m.
Therefore, theoretical strength,sthy30 GPa, i.e.E/10. In practice, actual fracture strengths of ceramics are of the order ofE/1000 although somefibers or whiskers may have strengths closer toE/10. The explanation for the large difference lies in the stress concentrating effects of cracks andflaws in brittle materials such as ceramics, first outlined byGriffith (1921):
sf ¼
2Egf=pc 1=2
(14) wheregfis the fracture surface energy,Eis the elastic modulus andcis the size of a“critical”crack orflaw to cause fracture of the material.
Consideration of the Griffith’s relationship explains why a set of ceramic samples from the same batch will fail over a range of applied stresses because fracture strength will depend on the types and sizes of critical surface flaws and defects present on each component. Thus, it is necessary to test a large number of samples in order to obtain a reliable assessment of the behavior of the ceramic. The failures will usually follow a relationship based on a statistical distribution (Weibull, 1951) where probability of failurePf¼1Ps, andPsis the probability of survival given by
Ps ¼ 1Pf ¼ exp½ fðsscÞ=sogmðV=VoÞ (15) wheres is the applied stress to cause failure,scis the critical value of stress below which the sample is not expected to fail, usually set as zero,mis the Weibull modulus,Vis the volume under consideration, andsoand Voare“normalizing”constants. The Weibull modulus defines the width of the probability distribution. Ifmis large (>10–20), the distribution is narrow showing a small spread of failure strengths and this would be classed as a“reliable”material, whereas ifmis small (<10), the distribution is wide with a large variation in strengths and therefore this would be an“unreliable”material.
Fundamental Aspects of Hard Ceramics 19
Taking logs of both sides of relationship(15)gives ln 1Pf
¼ ðV=VoÞfðsscÞ=sogm (16) Taking logs again gives
ln
ln 1Pf ¼ lnðV=VoÞ þmlnfðsscÞ=sog (17) Thus, assuming thatsc¼0, plotting ln{ln(1Pf)} versus lnsshould give a straight line with gradientm, the Weibull modulus. A more extensive overview of the brittle fracture and stochastic failure of ceramics from a probabilistic fracture mechanics approach is given by Danzer and Bermejo (2014) in this volume (see Chapter 2.09).
The fracture surface energy,gf, and elastic modulus,E, are combined in the expression for the critical stress intensity factorKIc, otherwise known asfracture toughness:
KIc ¼ 2Egf1=2
(18) As brittle materials, such as ceramics, have very little ductility, the values for KIc are generally low (2–10 MPa m1/2). Most advanced ceramics have fracture toughness values <10 MPa m1/2 (e.g. Fantozzi &
Saâdaoui, 2013). The failure of ceramic materials is controlled on the one hand by processing defects (such as porosity, impurities, and microcracking) and on the other by the microstructure (Davidge, 1979), so there are two approaches which can be taken when selecting a material for structural applications:
1. Select a material that is so strong that the fracture strength is never reached in service.
2. Select a material with superior fracture toughness.
Thefirst of these relies on reducingc, the size of the criticalflaw in the Griffith equation, and requires a careful approach to processing, that is, what is termed“flaw-minimal fabrication”. The second approach relies on increasinggfin the Griffith equation and is referred to as“microstructural engineering”(Becher, 1991).
The microstructure depends on the composition and crystal structure of the solid phases as well as the presence of pores, microcracks and impurity phases. Grain size, grain size distribution, and volume fraction of pores are all important parameters to be controlled. The aim of microstructural engineering is to achieve a microstructure that results in high hardness and fracture toughness. Toughening creates microstructures that impart sufficient fracture resistance that the strength becomes much less sensitive toflaw size so that small levels of processing and post-processing damage can be tolerated.
The fracture toughness in ceramics can be improved by various mechanisms (Becher, 1991), such as in situ toughening (growth of platelets or whiskers within the microstructure), transformation toughening (trans- formation of tetragonal zirconia grains into the lower density monoclinic phase at the crack tip, thus generating crack closure forces), crack deflection, crack bridging (filament orfiber toughening), or microcrack toughening.
2.01.5.2 Time Dependence of Strength
Many ceramic materials undergo a progressive weakening with time. Failure occurs under static loads lower than those that cause fracture during“normal”short-term tests. This phenomenon, which is known asstatic fatigue, suggests that the defect population is evolving with time with the occurrence of “subcritical”crack growth (SCCG) in which smallerflaws eventually grow to become“critical”[see Griffith’sEqn (14)]. This SCCG is a function of applied stress intensityKIand environmental conditions. Oxides tend to be more susceptible than nonoxides as water molecules can attack the bonds at the crack tips causing a stress-corrosion effect leading to extension of the cracks. The relationship between crack velocityvand applied stress intensity factorKIis given by (Davidge, 1979; Fett & Munz, 1985):
v ¼ aKIn (19)
It has been shown (Davidge, 1979) that the relationship between the applied stresssand the expected time to failuretfis
tfsn ¼ constant (20)
Data from thesev–Ktests can be combined with Weibull probability data to give strength–probability–time diagrams (Davidge, McLaren, & Tappin, 1974). The Weibull distribution as a function of time taking into 20 Fundamental Aspects of Hard Ceramics
account the effects of SCCG has also been derived (Danzer & Bermejo, 2013; Danzer, Lube, Supancic, &
Damani, 2008; Fantozzi & Saâdaoui, 2013) which allows calculation of times to failure for design purposes.
By proof testing at a higher proof stresssp, it is possible to remove before service any individual samples from a batch that are not up to specification. To do this, it is necessary to determine the ideal proof test conditions.
The proof test will have the effect of growing the cracks in the survivors so effectively weakening them. However, for a given ratio of proof stress to applied stress (sp/sa), the minimum time to failuretfminis proportional to s2a . Thus, a plot of logtfminversus logsa gives a straight line of slope2. When failure probability data is superimposed on these data, it results in a proof test failure diagram from which, knowing the lifetime expected of components under a particular applied stress, the proof stress can be determined and the test carried out (Fuller & Wiederhorn, 1980).
2.01.5.3 Hardness
Hardness is, in effect, the resistance of a material to deformation, scratching or erosion but is defined as resistance to surface indentation or penetration by an applied mechanical load. Therefore, hardness is not necessarily a bulk property and is therefore not necessarily related to the strength, but relates to the ability of the material to withstand penetration of the surface through a combination of brittle fracture and plastic flow.
The hardness of ceramics is measured using the Knoop and Vickers indentation methods that involve the application of a load via a geometrically defined indenter. Vickers uses a square pyramid diamond while in the Knoop technique, an elongated diamond pyramid is used.
Under known loading conditions, the size of the indentation is related to the hardness of the material so the hardness value quoted for any material is a function of the type of test conducted and the loading conditions.
The general relation for hardness numberHis given by (Smith & Sandland, 1922)
H ¼ P=A (21)
wherePis the applied load andAis the representative area of the residual indent.
Generally, the lower the load applied, the higher the hardness values. Typically in a Vickers Hardness test, the notation HV10 or HV20 relates to the applied load in kg (in this case 10 or 20 kgf respectively). It is difficult to convert values from one hardness method to that of another so making comparison difficult. Other factors that need to be taken into account when interpreting hardness data for ceramic materials are the amount of porosity in the surface, the grain size and the effects of GB phases. This is particularly important where the size of the indent is small and the same order of magnitude as microstructural features. Variations in hardness values arise also depending on the particular laboratory and the observer, and errors of up to 15% are common. Some typical hardness values for ceramic materials are given inTable 3.
Whatever the shape of the indenter, the hardness number may increase or decrease with load; it may be independent of load or it may show a complex variation with load changes, depending on the material. This hardness–load dependence is known as the Indentation Size Effect.
Studies on the Vickers hardness of several brittle ceramics show hardness–load curves which exhibit a distinct transition to a plateau of constant hardness level. This involves a relationship between hardness (H), Young’s modulus (E), and fracture toughness (KIc). Using these parameters, a brittleness indexBused in designing with ceramics (Lawn & Marshall, 1979) was proposed that is derived from deformation and fracture energy ratios as follows:
B ¼ HE=KIc2 (22)
Table 3 Comparison of hardness values for“hard”ceramics
Ceramic Knoop hardness (GPa)
Alumina Al2O3 15–18
Silicon nitride, Si3N4 20–22
Tungsten carbide, WC–Co 18–22
Titanium carbide, TiC 28
Silicon carbide, SiC 24–28
Boron carbide, B4C 32
Fundamental Aspects of Hard Ceramics 21
Besides toughness and hardness, there are several mechanical properties of ceramic materials which are relevant to their applications, including wear resistance, hot hardness, creep resistance, tribochemical stability, and thermal expansion coefficient. Wear behavior of ceramic materials is complex and is dependent upon many variables, of which hardness is an important variable but not the only significant one (De Portu & Guicciardi, 2013). For example, in many wear environments, such as the erosive wear behavior of oxide engineering ceramics, it is the ratio of fracture toughness to hardness (related to brittleness index) which is found to be significant in determining wear behavior. Chemical changes, especially corrosion and oxidation, also influence wear behavior, especially at high temperatures which may be operating at the surface of the materials under consideration.