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CHAPTER 11CHAPTER 11

Mechanical PropertiesMechanical Properties ：：Failure and Fracture MechanicsFailure and Fracture Mechanics

Mechanical Properties of CeramicsMechanical Properties of Ceramics

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◎ Simple fracture ( 斷裂 ): separation of a body into two or more pieces in response to an imposed stress (tensile, compressive, shear, or torsional; the present discussion : uniaxial( 單軸 ) tensile loads.)

◎ two fracture modes ( 模式 ) : ductile and brittle. (based on the ability of a material to experience plastic deformation.)

11-I. Failure and Fracture Mechanics11-I. Failure and Fracture Mechanics

A. Fundamentals of FractureA. Fundamentals of Fracture

F 6-13

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◎Any fracture process involves two steps---crack

formation ( 裂痕生成 ) and propagation ( 裂痕擴張 )

A-1. Ductile fracture

(1) extensive( 大幅度地 ) plastic deformation with high energy absorption before fracture;

(2) proceeds relatively slowly; such a crack is often said to be

stable:

(3) evidence( 證據 ) of appreciable( 明顯地 ) gross deformation at

the fracture surfaces( 斷裂面 )

F 6-11

F8.3

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A-2. Brittle fracture

(1) may proceed extremely rapidly;

(2) very little or no plastic deformation with low energy absorption;

(3) such cracks may be said to be unstable : once started, will continue spontaneously without an increase in magnitude of the applied stress.

◎ Ductile fracture is almost always preferred :

First, presence of plastic deformation gives warning ( 警告 )

Second, more strain energy is required to induce( 引發 ) fracture.

◎ most metal alloys : ductile

ceramics : brittle

polymers: may exhibit both types of fracture.

F 6-13

T 8-1

F8.3

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◎ extremely soft metals, e.g., pure gold and lead at room

temperature; other metals, polymers, and inorganic glasses

at elevated temperatures.( 高溫 )

˙neck down to a point

˙showing virtually100% reduction in area.

◎ common ductile metals

˙only a moderate amount of necking

˙fracture process (several stages) :

(1) after necking begins, small cavities( 孔洞 ), or

microvoids ( 微小之孔隙 ), form;

B-1. Ductile FractureB-1. Ductile Fracture F 8-1

F 8-2

B. Fracture PhenomenaB. Fracture Phenomena

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(2) microvoids enlarge( 變大 ), come together, and coalesce( 聚合 )

to form an elliptical ( 橢圓形 ) crack,

(3) fracture by shear deformation at an angle of about 45°

(the shear stress is a maximum.)

˙cup-and-cone fracture

˙interior region : irregular( 不規則的 ) and fibrous ( 纖維狀的 )

appearance ( 外觀 )

˙studies of this type : fractographic (using scanning electron

microscopy)

˙“dimples”( 半圓凹洞 ), each dimple : one half of a microvoid,

spherical or elongated

(C-shaped) : indicative ( 顯示 ) of shear failure.

F 8-3

F 8-4

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B-2. Brittle FractureB-2. Brittle Fracture

◎ in some steel pieces : a series of V-shaped “chevron”

markings ( 痕跡 )

◎ Other brittle fracture surfaces : lines or ridges( 線痕 ) that

radiate from the origin of the crack in a fanlike ( 風扇型的 )

pattern

◎ amorphous materials (e.g. ceramic glasses) : shiny and

smooth surface.

◎ most brittle crystalline materials : breaking bonds along

specific crystallographic planes (cleavage).

F 8-5

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˙transgranular (or transcrystalline( 穿晶斷裂 )) :

the fracture cracks pass through the grains (a grainy or faceted texture)

˙ Intergranular ( 沿晶斷裂 ): crack propagation is along grain boundaries. three-dimensional nature of the grains may be seen.

C. Principals of fracture mechanics C. Principals of fracture mechanics (( 破斷力學破斷力學 )) ：：

Stress ConcentrationStress Concentration(( 應力集中應力集中 ) ) ：：

Experimentally measured fracture strengths are significantly lower

than theoretical calculations (based on bonding energies): existance

of very small, microscopic flaws or cracks

an applied stress is amplified( 放大 ) or concentrated at the trip( 端點 ) ： Stress ConcentrationStress Concentration

F 8-6

F 8-7

◎ Fracture of polycrystalline material

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◎ The maximum stress, :

(8.1)

: magnitude of the nominal applied tensile stress,

: radius of curvature of the crack tip

a : length of a surface crack, or half of the length of an

internal crack.

m2/1

02

tm

a

0

t

F 8-8

◎ flaws or cracks ： stress raisers

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(8.2)

˙effect of a stress raiser is more significant in brittle than in ductile materials. (a ductile material, plastic deformation causes more uniform distribution of stress in the vicinity of the stress raiser)

2/1

0

2

t

mt

aK

◎ stress concentration factor Kt :

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A. Brittle Fracture Of Ceramics

◎ The brittle fracture process consists of the formation and propagation of cracks: through the grains (i.e., transgranular ) and along specific crystallographic (or cleavage) planes, planes of high atomic density.

11-II. Mechanical Properties of Ceramics

The principal drawback: fracture in a brittle manner with very little energy absorption.

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◎ The measured fracture strengths are substantially lower than predicted by theory from interatomic bonding forces: explained by very small and omnipresent flaws that serve as stress raisers according to Equation 8.1, minute surface or interior cracks (microcracks), internal pores, and grain corners. For example, even moisture and contaminants in the atomosphere can introduce surface crracks in freshly drawn glass fibers.

◎ The measure of a ceramic material’s ability to resist fracture when a crack is present is specified in terms of fracture toughness. The plane strain fracture toughness KIC

aYK IC (12.2)

Y: dimensionless that depends on both specimen and crack geometries,: applied stress, a: length of a surface crack or half of the length of an internal crack.

Crack propagation will not occur as long as the right-hand side of Equation 12.2 is less than the plane strain fracture toughness of the material.

KIC for ceramic materials are smaller than for metals. F12-28

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◎ Considerable variation and scatter in the fracture strength for many specimens of a specific brittle ceramic material:

dependence of fracture strength on the probability of the existence of a flaw that is capable of initiating a crack, depending on fabrication technique, subsequent treatment and specimen aize◎ For compressive stresses, there is no stress amplification associated with any existent flaws: brittle ceramics display

much higher strengths in compression than in tension (on the order of a factor of 10)

◎ Under some circumstances, specifically when moisture is present in the atmosphere, fracture of ceramic materials will occur by the slow propagation of cracks: static fatigue, or delayed fracture.

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B. Stress-Strain Behavior ： FLEXURAL STRENGTH

◎ The stress-strain behavior of brittle ceramics is not usually ascertained by a tensile test as outlined in section 6.2, for three reasons:

˙It is difficult to preapare and test specimens having the required geometry.

˙ it is difficult to grip brittle materials without fracturing them;

˙Ceramics fail after only about 0.1% strain, specimens must be perfectly aligned to avoid the presence of bending stresses.

◎ A more suitable transverse bending test is most frequently employed: a rod specimen is bent until fracture using a three or four- point loading technique: flexure test.

F12-29F6-3 F6-2 F6-3

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◎ The stress at fracture using this flexure test is known as

the flexural strength, modulus of rupture, fracture

strength, or bend strength, an important mechanical

paramenter for brittle ceramics.

◎ For a rectangular cross section, the flexural strength fs

is equal to

22

3

bd

LFffs (12.3a)

Ff: load at fracture, L: distance between support points, when the cross section is circular

3R

LFffs

(12.3b)T12.5

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◎ During bending, a specimen is subjected to both compressive and

tensile stresses, fs will depend on specimen size; with increasing

specimen volume (under stress ) there is an increase in the

probability of the existence of a crack-producing flaw and,

consequently, a decrease in flexural strength

C. ELASTIC BEHAVIOR◎ Elastic stress-strain behavior: using flexure tests a linear

relationship exists between stress and strain; the slope:

modulus of elasticity.

◎ Most ceramis: do not experience plastic deformation prior to fracture

D. Mechanisms of Plastic Deformation

At room temperature, most ceramic materials suffer fracture before the onset of plastic deformation. Plastic deformation is different for crystalline and noncrystalline ceramics

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E. CRYSTALLINE CERAMICS

◎ Plastic deformation occurs, as with metals, by the motion of

dislocations(Chapter7). One reason for the hardness and brittleness:

difficulty of slip (or dislocation motion).

◎ For bonding predominantly ionic, there are very few slip systems.

This is a consequence of the electrically charged nature of the ions,

ions of like charge are brought into close proximity to one another.

For bonding highly covalent, slip is also difficult and they are brittle

(1) the covalent bonds are relatively strong, (2) there are also limited

numbers of slip systems, and (3) dislocation structures are complex.

F7-1 F7-2 F7-3 f2-4-5

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F. NONCRYSTALLINE CERAMICS

Does not occur by dislocation motion: noncrystalline ceramics have

no regular atomic structure , these materials deform by viscous flow,

same manner in which liquids deform.

Atoms or ions slide past one another by the breaking and reforming of

interatomic bonds . There is no prescribed manner or direction .

Viscoity: a measure of a noncrystalline material’s resistance to

deformation.

The viscosity shear stress and change in velocity

dv with distance dy:

dydv

AF

dydv /

/

/

(12.4)

As the temperature is raised, an attendant decrease in viscosity.

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G. Miscellaneous Mechanical Considerations

◎ INFLUENCE OF POROSITY

˙Subsequent to compaction or forming of ceramic powders into the

desired shape, heat treatment, and sintering: some resiural porosity

will remain (Figure 13.15), any residual porosity will have a

deleterious influence on mechanical properties (and also on

thermal, electrical and optical properties.)

(G-1) Effects on elastic properties and strength

Modulus of elasticity E decreases with volume fraciton porosity P:

)2.09.11( 20 PPEE (12.5)

F13-13 F13-14 F13-15

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Eo: modulus of elaticity of the nonporous material.

Porosity is deleterious to the flexural strength for two reasons;

(1) pores reduce the cross-sectional area (2) they also act as stress concentrators – for an isolated spherical pore, an applied tensile stress is amplified by a factor of 2 . For example, 10 vol% porosity will decrease the flexural strength by 50%

Flexural strength decreases exponentially with volume fraction

porosity (p):

)exp(0 npfs (12.6)

0 and n are experimental constants

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(G-2) Effects on flexural strength

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H. HARDNESS

One beneficial mechanical property of ceramics is their hardness, the hardest known materials are ceramics

I. CREEP

A result of exposure to stresses (usually compressive) at elevated temperatures. Similar to that of metals (Section 8.14)

T12.6