The Basics of Materials Science Ceramics laboratory 2

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1 The Basis of Materials Siene Ceramis laboratory 2 Faulty of Mehanial Engineering and Robotis, 1 st year Mehanial properties of erami materials Introdution Under applied external fore all bodies deform. When the fores are moderate, materials undergo gradual deformation, elongation or ontration, whih is proportional to the load applied. After the load is removed a sample reovers its original shape. Suh reversible strain is alled elasti strain. For moderate strains all materials behave alike, exept that the strain due to a given load is different for various materials. In order to distinguish between the behaviour of individual materials, material onstants are introdued. In the range of strain to stress proportionality a relation between the normal or shear (tangential) stresses and the strains is given by Hook s law: E (1) G (2) where: E Young s modulus, G - shear modulus ( the stiffness onstant), elongation strain, tangential strain, normal stress, tangential stress. Fig.1. Relationship between strain and stress for various solids: I elasti strain, II plasti strain During an uniaxial elongation under tensile loads of ideally elasti and isotropi solid a lateral ontration also takes plae. If the strain along the diretion of the load equals the lateral strain isthe minus sign means that the lateral strain (elongation) and the longitudinal strain (ontration) are of opposite sign. The onstant is known as Poisson s ratio and like E and G depends of a kind and struture of material. Signifiant differenes in the behaviour of bodies our under larger stresses. Fig 1 shows the relationship of stress vs. strain for various materials. The strain in the majority of eramis terminates in a sudden deohesion of the material at point F. On the other hand, most of metals don t undergo a sudden deohesion at ertain stress but, starting from point A, they undergo a deformation whih inreases with time. The deformation arising from stresses higher than A is permanent, i.e. it doesn t disappear after removal of the load and it is alled a plasti strain (a plasti deformation). The point A is defined as a yield point. Bodies whih fail at deformations lower than those orresponding to the yield point are alled brittle bodies, whereas bodies whih are apable for plasti deformations are defined as plasti bodies. 1

2 Deformation in the atomi sale onsists in fored movement of an atom from its equilibrium position (state of minimal potential energy). In ase of rystal, in whih an atom interats with neighbouring atoms suh movement leads to displaement of the whole atomi layers. For small displaements the energy used to move atoms aumulates in the struture (elasti strain energy), and after removal of the load the atoms ome bak to their equilibrium positions. In suh ase we talk about elasti strains (elasti deformations). Suh mehanism may be desribed onsidering potential energy hanges in a bi-atomi model. Potential energy (V) of interation between two atoms vs. the distane between them (r) (Fig. 2) may be represented as a sum of attration energy and repulsion energy: V A B n m r r (3) where: A and B proportionality onstants of attration and repulsion respetively m and n exponents Fig.2. Condon-Morse urves representing potential energy hanges vs. interatomi distane r The distane orresponding to the minimum of potential energy is an equilibrium distane r 0 of both atoms. Movement of the atom in any diretion from position r 0 indues generation of the fores whih ounterat with a displaement. Marosopi deformation of the rystal is thus aused by hange of interatomi distanes in the same diretion wiązanie kowalenyjne (diament) wiązanie jonowe (NaCl) ,25 0,5 1,0 1,5 2,0 2,5 Odległość międzyatomowa r, nm Fig. 3. Energeti models of formation of ioni and ovalent bonds Energies and fores ourring between atoms forming partiular kinds of bonds may be presented using energeti model. Fig. 3 presents energeti model of formation of ioni and ovalent bond. Energeti model of metalli bond is similar to the one of ovalent bond. Charater of variability of potential funtion is related to the bond kind. The stronger is the bond, the deeper and narrower is a well of potential energy. Partiular energy inrease in ase of rystal with stronger bonds auses smaller hange of its dimensions. 2

3 Mehanial strength Mehanial strength in general is an ability of the materials to withstand a load without failing. It is expressed in terms of fore or stress whih auses loss of material ohesion and its rupture into two or more piees. Signifiant feature of all erami materials at ambient, moderate and partially also elevated temperatures is their brittleness. Brittle failure is the one in whih material undergo deohesion without ourrene of signifiant strains before. It is illustrated by the line OF in Fig.1. Brittle failure differs from plasti failure, typial for many metals, where deohesion ours at muh higher strains (OAF urve in Fig.1). In ase of dutile polyrystalline metals with fae-entered ubi struture the plasti failure is a result of redution of the material ross-setion due to its plasti deformation (Fig. 4). An ability of a material to withstand brittle failure is alled dutility or frature toughness. Fig.4. Sheme of elasti strain formation (a) and plasti strain formation (b) Seond typial feature of the erami materials is a onsiderable differene of the strength at different state of stress. It may be generally aepted, that for the erami materials tensile strength is the lowest one. Bending strength is up to three times higher, and ompression strength is even fifteen times higher. Referene point for evaluation of frature toughness of the materials is a tensile strength of interatomi bonds whih is alled maximal or theoretial strength m. It may be stated, that work performed by the load is onsumed by formation of two new surfaes, eah possessing an exessive surfae energy. The bond ruptures when: m E r 0 (4) Equation 4 is often used for alulation of theoretial strength from measurable values of E, and r 0. However, it should be remembered that rystals onsist of many atoms and bonds, whih interations have to be taken into aount. Table 1. Material type Theoretial strength Whiskers strength Strength of bulk polyrystals -Al 2 O SiC graphite C 122 (II to a axis) <0.5 WC-Co

4 Besides generally higher theoretial strength of erami materials omparing to metals it is worth to note, that in ase of urrently fabriated materials real values of the strength lose to the theoretial one are ahieved only in ase of whiskers (monorystalline fibers) and are slightly lower in ase of polyrystalline fibres. The fibers are haraterized by relatively flawless struture and struture of whiskers is almost perfet, without any disontinuities. Mehanial strengths ahieved in ase of typial, bulky polyrystals are two to three times lower than the theoretial values, whih is aused by defets (slits, miro-raks, et.). Fig.5. Diagram showing formation of stresses at pore tip In the situation presented in Fig.5 transfer of an external load along two broken hains of atoms beomes impossible, and it has to be arried out via different path i.e. round the pore. Chain of atoms on the pore tip transfers four times larger load that in ase of a model without the pore, and also stresses in this plae are four times larger than expeted from the average external loads. It means that in this ase a stress onentration ours. Only at higher distane from the pore surfae stresses are lower and equal to the external load, z = P (Fig.5). Aurate values of the oeffiient of stress onentration, i.e. number whih states how many times a loal stress lose to the pore surfae is higher than the average stress far from the pore (the external load) an be alulated using Inglis equation: z z (5) Internal stresses lose to the slit tip (rak tip) are a multiple of the exerted external loads P, and may there ahieve value equal to the theoretial strength of bond m at moderate external loads P (Fig. 5). In suh onditions it may be expeted, that raks whih already exist in the material subjeted will propagate. Combining equations 4 and 5 one will obtain: E E 2 stąd P (6) r z z 0 4r0 4

5 The so far onsiderations may be onluded, that even at low loadings it is diffiult to avoid formation and/or development of ritial raks in the material. That is why a dutility (frature toughness) of the material is so important fator. In fat it is a measure of energy absorbed by the material during development of the raks. The higher the energy is, the higher is a work whih must be performed by the external loads to break the material. Primary parameters desribing dutility of the material are K and. Brittle rupture may our at three different modes of the material deformation, presented in Fig.6, or their ombinations. The highest stress onentration takes plae when frature ours in mode I. Fig.6. Primary modes of brittle rupture (failure) In other words, during flat state of straining in mode I the risk of brittle failure is higher than that in ase of the modes II and III. That s why, aiming the determination of the lower limit of frature toughness, the analysis of the material straining may be restrited to the mode I. Frature toughness. Griffith s theory Fig.7 Griffith s rak and stresses (in the material subjeted to tensile loads Loading a thin plate, ontaining in its entre an infinitely thin elliptial rak (Griffith s rak, Fig.7) perpendiular to the equatorial axis, leads, due to elasti strains, to aumulation of elasti energy in the material, whih magnitude at the unit length is w. At the same time two boundary surfaes of the rak move apart in diretion of the polar axis. Loss of ohesion of the material in a region of the rak enables free hanges of the material volume without obstales from elasti onstraints related to existene of ontinuous system of interatomi bonds. As a result, part of elasti strain energy aumulated in that region (shaded at Fig.7), w R, may be disharged. Elasti strain energy stored in the material, in respet to the unit length, is thus w-w R, where w elasti strain energy of the material without the rak. At the same time, deohesion of the material leads to formation of two new solid-gas ( or vauum) division surfaes at the rak length with an exessive energy. In ase of a rak having length equal to 2 along the equatorial axis it is aompanied by energy absorption, G=4. The rak may inrease its dimensions along the 5

6 equatorial axis, i.e. propagate in the material, only if at every stage of this proess the energy dereases. Fig. 8 presents hanges in the system total energy du vs. inrement of the rak length d. Beause G = f ( ) and w R = f( 2 ), stronger hanges of w R with the raks length than in ase of G may be expeted. Fig.8 Changes of total energy (U), surfae energy, and elasti strain energy stored in the material (w-w R ) vs. length of the Griffith s rak, Neessary ondition of self-propagation of the rak under a given load is formation of rak with ritial (half) length, above whih system energy dereases with inreasing length of the rak. Relationship between the ritial load, above whih the rak starts to propagate may be presented as follows: 2 E P z, (7) E and are both material onstants, so ritial length of the rak depends of the load applied the higher the external load is, the smaller raks may ause atastrophi failure of the material. In ase of appliation where a load is equal to the theoretial strength of bonds the equals approximately three times the equilibrium length r 0. Relation (7) is right for a rak loated inside of the material. In ase of rak plaed on the materials surfae the stress onentration at the rak tip is higher. Beause of that, dimensionless oeffiient Y is introdued, whih value depends of loation of the rak within the material. Moreover, the stress onentration strongly depends of the rak geometry. It dereases when approahing rounded shape. On the basis of those onsiderations dimensionless oeffiient Z is introdued, whih enables to generalize relation (7) for different ases than the flat Griffiths rak. Generalization of equation (7) is: P Z 2 E z, (8) Y Strength of real erami materials is generally higher than the one expeted from relation (8). It implies, that the real materials don t behave like ideally elasti solids, whih were desribed by the Griffiths theory. This fat has many auses. One of them may be loal plasti deformation of the material lose to the rak tip, the other may be branhing of raks, mainly related to presene of weaker intergranular and interphase boundaries. One of the most important fators determining strength and toughness of the materials is state of their surfae. In a perfet form it ours in ase of 6

7 high-quality glasses, whih don t ontain any oarse defets inside. The phenomena listed above are deliberately used for inreasing the frature toughness of erami materials and strength. It should be stated, that the surfae energy in equation (7) should be replaed by an effetive surfae energy, or better an energy of frature ef :... (9) ef pl r m pf i k where: - speifi surfae energy, pl - energy dissipated at loal plasti deformations r energy dissipated due to branhing of raks or formation of miro raks pp energy absorbed due to ourring phase transformations (polymorphi transformations) m, i... k - other auses inreasing ef. In ase of e.g. fiber reinfored omposites the other auses is energy neessary to pull out fibers from the matrix. Coming out from equation (7), in whih = ef, the material onstant whih haraterizes the material resistane toward atastrophi propagation of an unstable rak may be derived. Multipliation of both sides of equation (7) by leads for the flat stress state to: Z z, 2E ef onst (8) Y It means, that in ase of initiation of propagation of unstable rak able to penetrate through the material a produt of z, and is a onstant, whih depends only of E, ef and Z, Y. Besides the effetive frature energy, other riterion of the material resistane to the frature may be derived basing on marosopi approah of Irwin and William, whih defines relations between permissible load and length of the ritial rak. In that respet, Irwin modified Griffith s approah (energy balane) introduing onept of a fore ausing propagation of the rak with a unit length, and he formulated a new riterion of initiation of unontrolled rak propagation. It ours when at the rak tip ritial value of the stress onentration is reahed. Parameter K I (for I mode of raking) is alled a oeffiient of stress onentration. In turn, K I oeffiient is defined as the frature toughness. The essene of the onept of the frature toughness is, that K I oeffiient desribing hange of stress distribution in the elasti material in presene of a rak and it reahes its maximum value K I when atastrophi rak propagation in the material is initiated. K I oeffiient shouldn t be mixed up with the oeffiient of stress onentration, beause, similarly to ef, it is a material onstant. Irwin found following relation of both onstants: K 2 E ef dla PSN E (9) 2 ef K dla PSO 2 1 For the flat stress state (PSN) onsidered by Griffith it may be written as follows: K E (10) z, ef Exemplary K I and ef values for various materials are gathered in Table 2. 7

8 Table 2. Seleted mehanial and elasti properties of various materials Material Hardness HV Yield stress R e Young s modulus Frature toughness [MPa m 0,5 ] -Al 2 O ,4-5,6 Al 2 O 3 +20% ZrO 2 (partiulate omposite) 6,5-8 CZP (ubi) ZrO ,8-5 TZP (tetragonal) ZrO 2 (12% mol CeO 2 ) Na-Ca glass 5,8 1,9 70 0,6 -SiC ,5 SiC(w)-Al 2 O 3 (o) (fibrous omposite) 21 Si 3 N 4 5,5 diamond ,3 WC-Co *) Ti 3 SiC 2 (nano-laminate) 7-8 Cu 0,1-0,8 0, Al 0,1-0,4 0, onstrutional steel 1,6-5,0 0,3-2, PMMA (Plexiglas) 0,16 0,06-0,11 3,4 0,9-1,4 The effetive surfae energy is in fat the effetive frature energy. Ourrene of phenomena, whih ontribute to the effetive frature energy is failitated by properly designed mirostruture of the material. Main types of mirostruture are shown in Fig.9. Fig.9. Shemes of mirostrutures of various erami materials: a), b) polyrystals, ) fiber reinfored omposites, d) partiulate omposites, e) materials with duplex mirostruture, f) materials ontaining nanometri partiles of the other phase, g) materials ontaining short fibers of the other phase, h) laminates, i) materials with elongated grains, j) nanolaminates. Hardness Hardness may be defined as resistane of a material to the permanent (plasti) deformation aused an indenter fored in. Cerami materials suh as orundum (Al 2 O 3 ) arborundum (SiC) and diamond (C) belong to the hardest solids. Almost all other materials may be ut with them, also their powders are used as grinding and polishing media. A hardness of those materials omparing to that of some metals (Table 2) learly indiates predominane of the former ones. This is due to the nature of hemial bonds and its influene of movement of disloations in eramis and metals. During the 8

9 plasti deformation in hardness test disloations existing in the material slip, so suh test prinipally determine ease of moving disloations in the material. In ase of metals that ease is relatively high, beause moving disloation displaes atoms bonded with rather weak metalli bond. Energy of this bond omes from eletrostati interation between positive harged atomi ores and negatively harged eletron gas. Suh bonds are deloalised. Cerami materials exhibit different properties. Due to ovalent bonds (diamond, arborundum) or ioni-ovalent (orundum) their rystalline network put up a strong resistane to the disloations movement. In ase of loalized ovalent bonds it is neessary to break them and subsequent reprodue them. In ase of ioni bonds, disloation slip is easier, but it may take plae mainly in suh rystallographi diretions in whih attration between anions and aptions ours. This redues number of slip systems neessary to move disloations in ioni polyrystals and is a ause of their relatively high hardness. Rightness of this reasoning is onfirmed on the basis of omparison between metals yield stress (R e ) with hypothetial yield stress for eramis (Table 2). Suh estimated R e values for erami materials are even one order of magnitude higher than those of metals. It is worth to emphasise, that similarly to hardness, also Young s modulus of metals and eramis distintly depends of the atomi bond nature (Table 2). Those relationships are more omplex in ase of polymers. If we limit our onsiderations only to examples of seleted onstrutive polymeri materials, whih are brittle at room temperature, suh as hardened epoxy or polyester resins and polymethyl metharylate (PMMA, Plexiglas), it may seem that their Young s modulus and hardness are determined by strong ovalent arbon-arbon bonds forming main polymeri hain. It means, that suh polymers should have mehanial properties similar to those of diamond. But besides the strong C-C bonds there are there also week, seondary bonds between the hains. Their presene determines in fat relatively high deformations under loads, and thus low hardness and Young s modulus. As it was stated above, estimated values of yield stress of erami materials are muh higher than those of metals. However, ontrary to metals, eramis will break under tensile load reahing yield stress. That s why the estimated values of R e for erami polyrystals (Table 2) have only omparative meaning. It is aused by their low frature toughness (K I ). For the same reasons the raks are easily formed inside eramis by a sharp indenter during hardness test. Threshold value of a loading fore P* whih leads to the material raking may be determined. It is a very good measure of the material leavage. P* fore values are useful in a proess of designing of erami materials. If during operational use of a produt ating fores are lower than P* they won t ause raking but only deformations, whih may be lowered by inreasing the material hardness. In opposite situation propagation of raks will take plae, whih an be prevented by inreasing the frature toughness for example by introdution to the material partiles whih may hinder the raks (ide of the omposites). REFERENCES: 1. R. Pampuh, Siedem wykładów o eramie, Wyd. AGH, Kraków R. Pampuh, Materiały Ceramizne. Zarys nauki o materiałah nieorganizno-niemetaliznyh, Wyd. PWN, Warszawa, Skrypt AGH SU 1566, Laboratorium z Nauki o Materiałah, pod redakją J. Lisa, Wyd. AGH, Kraków M. Ashby, D. Jones, Materiały inżynierskie, Wyd. Nauk.-Teh., Warszawa, 1996 By dr G. Grabowski dr inż. A. Gubernat. Edited by dr inż. N. Moskała. 9