Fracture+Mechanics+Outline

Fractography of brittle materials can yield both quantitative and qualitative information about a fracture process, which generally entails the (i) initiation and (ii) subsequent propagation of a crack. A number of quantitative relationships have been established. toc

Plastic Deformation
Upon applying a uniaxial tensile stress to a metal, plastic deformation occurs due to the motion of dislocations along slip systems. Impeding dislocation mobility can occur by: (i) increasing grain boundaries by reducing grain size, (ii) introducing lattice strains by solid solution strengthening, (iii) increasing the dislocation density by cold working or strain hardening. In the absence of dislocation motion, the probability of fracture increases. Ceramics have less slip systems than metals and therefore behaves as a brittle material.

Ductile and Brittle Fracture
The applied stress that leads to failure may be tensile, compressive, shear or torsional. A majority of fracture tests are performed under uniaxial tension. For ceramic materials, tensile tests are often conducted under flexure. [\more about flexure, see dissertation section] As mentioned, the fracture process occurs in two stages: (i) crack initiation and (ii) crack propagation. Cracks propagate in a //stable// or //unstable// manner. Stable crack growth is relatively slow and can be observed in ductile fracture, characterized by plastic deformation around the crack as seen by twisting and tearing at the fracture surface. Unstable crack growth occurs fast, or spontaneously "catastrophic," and is observed in ceramic and brittle materials. Since catastrophic failure can occur without warning, it is generally avoid in material design.

Ductile Fracture The ductile fracture process involves the following: Often ductile fracture is associated with a //cup-and-cone// fracture surface on the mating ends of the specimen, where brittle fracture yields relatively planar fracture ends. Under SEM, dimples and parabolic voids are indicative of uniaxial tensile failures.
 * Necking occurs leading to microvoid formation at the crossection.
 * Microvoids enlarge as cavities coalesce, forming an elliptical crack.
 * Rapid crack propagation occurs aorund the internal perimeter of the stressed cross-section by shear deformation occurring at 45 degrees.

Brittle Fracture Brittle fracture often renders telltale signs on the fracture surface that details the fracture process. These markings result from the interaction of a propagating crack with the surrounding microstructure, the stress waves and generated elastic waves. For instance, the fracture origin in brittle materials can usually be discerned by the fanning out of ridge-like or hackle markings radiating from the failure initiation site. This marking behavior suggests a crack was initiated, grew stably to a defined size, then failed catastrophically. Fracture can along crystallographic planes in a //transgranular// manner. Fracture can also occur //intergranularly// along grain boundaries. [\more on fractographic implications of trans vs. intergranular fracture]. Elastic or sonic waves occur during fracture and the intersection of these waves with the propagating crack front generates another feature known as Wallner lines. Wallner lines provide fractographic information regarding the stress distributions in the material during crack propagation. Fatigue failure is indicated on a fracture surface by the presence of beachmarks or striations, the width of which suggests the propagation of crack during a cycle. Such markings are evident on machined surfaces and suggest a slow fracture process.

Griffith
Work by Griffith marked the advent of fracture mechanics. It was observed that the strengths of brittle materials are much lower than theoretical calculations (nearly 10x less) based on atomic bonding theory. Griffith reasoned that the drop in strength is due to the presence of microscopic "flaws" or cracks inside the material. Applied stresses are concentrated at the crack tips as a function of the stress amplitude and crack geometry. These flaws serve as //stress raisers// where the maximum stress, sigma_m, occurs at the crack tips by a relationship that insensifies the applied stress with a geometrical factor of the crack tip radius. sig_m = 2*sig_0(a/rho)^1/2. The stress insensity factor, K a ratio between the maximum stress and the applied stress, sig_m/sig_0.

Griffith established the relationship between the critical failure stress and the crack size. sig_c = ((2*E*gamma_s)/(pi*c))^1/2 [\r.Griffith paper; discuss experiment] When applied stress exceeds the critical stress value at one of these critical flaws, the crack propagates and fracture occurs.

Crack Anatomy
Upon initiation, a crack propagates until it reaches terminal velocity at which point the crack can: [\4 loading modes in glass: impact, bending, torsion, internal pressure] [\ [\Fracture origin, depth (a), width (2b), fracture boundaries (mirror/mist/hackle)] [\Crack deflection (Faber & Evans)] Understand crack geometry is best assessed after understandinf the orientation of the crack with respect to the (i) unfractured topical surface and (ii) fracture surface Common cracks are understood through inelastic and elastic response of the material following indentation loading: [\median] [\radial] [\lateral: can result in chipping]
 * Bifurcate, dividing the energy
 * Change direction beyond a smooth mirror region creating a perturbed surface (mist, hackle regions)

Modes of Failure
[\Mode I] [\Mode II] [\Mode III]

Toughness
[\r.Wiederhorn, Hutchinson, Evans] Fracture toughness is a material's resisitance of brittle fracture due to the presence of a crack. The fracture mechanics equation for fracture toughness is as follows: K_c = Y*sig_c*(pi*c)^1/2. Y is a dimensionless factor that is mathematically determined for various crack geometries. For an interior, a semi-elliptical crack in a plate of infinite thickness, Y = 1.0. For an edge crack of semi-infinite width, Y~1.1. If the specimen thickness is much greater than the flaw size where plane strain occurs, K_c is independent of thickness. The plane stress configuration for toughness is known as the plane strain fracture toughness, K_IC, where the I corresponds to a mode I crack displacement. K_IC is a fundamental material property. K_IC reduces with increased strain rate, solid solution, additives or strain hardening. K_IC increases with reduced grain size.

In fracture mechanics theory, fracture toughness is material property that is independent of test method and geometry. However, experiments demonstrate otherwise, i.e. the K_IC may vary between flexure tests, strength indentation and chevron notch beam tests. For example, K_IC values are reported higher for chevron notch tests. The assumptions sometimes used is that the initiating crack is not atomically sharp or some slow crack growth proceeded which inflated the theoretical K_IC value. Such deviations from theory suggest that the exact K_IC may not always be reported for various testing methods due to experimental error or limitations.

Stress Intensity Factor
The stress intensity factor (K_I) is a "dynamic" value that scales with the applied stress. K_I is strongly responsible for the growth of the crack front. The crack length found on the fracture surface is typically the initial crack (a_i). If the crack front rapidly propagates (> 0.1 m/s), the initial crack length and the critical length are approximately equal. a_i ~ a_cr. There are special cases of //slow or sub-critical crack growth// where the inital and critical crack length are not equivalent.

Relationship between critical fracture toughness and critical crack length
Stress intensity factor can reach maximum (K_IC), a stress at which the critical crack size is made that leads to catastrophic failure.

Eq. K_IC = Y*sig *c_cr^1/2

Mirror Constant (A_j)
The relationship between mirror boundaries and applied stress generates another "dynamic" constant.

Equ. sug*r_j^1/2 = A_j

Kirchner and Kirchner generalized the above equation suggesting that the formation of mirror, mist and crack branching boundaries occur at a constant such that

Eqn. Y(theta) * sig * r^1/2 = K_BJ

Here the crack branching instensity factor K_BJ is related to the mirror constant (A_J) by the geometric factor Y(theta) (typically = 1.24 for a relatively small semi-circular flaw compared to the thickness of the beam)

Residual Stress
One can differentiate far field stress from ancillary stresses by plotting strength vs. r_J^(-1/2), which show up as two lines of different slopes. The typical experiment performed is tempered vs. annealed soda-lime silica.

Marshall performed an experiment with as-indented and annealled, indented soda-lima silica. He determined that mirror radii remained unchanged, thus the removal of residual stress via annealing had no effect on the mirror constant. However a change was observed in the mirror-to-flaw size ration (r/c). r/c is larger for the annealed samples By analysis with stress intensity factors, this suggested that the mirror-to-flaw size ratio is `1.78 times larger for cracks with local stressed compared to cracks that are free of residual stress.



Fracture Energy
There is relationship between fracture energy and elastic modulus.

Sub-critical Crack Growth
The mirror size is independent of crack kinetics. This phenomenon is observed in glasses and polycrystalline ceramics. At the critical crack size, fracture paths switch from intergranular to transgranular. Mirror to initial crack size ratio becomes a function of time as subcritical crack growth occurs.

Fractal Dimension
Fractal dimension is a non-Euclidean measure of the tortuousness of a fracture surface. A higher D* implies a more tortuous surface. Fractal dimension is increasingly interesting as we try to understand its relating bulk fracture properties by atomic separation.

Testing Methods
The following techniques will be discussed:
 * Microhardness Indentation
 * Strength-Indentation
 * Chevron Notch
 * Fractography
 * Weibull Statistics
 * Flexure Testing: 3pt, 4pt, biaxial
 * Diametral Compression
 * Furnace
 * Optical Microscopy

Microhardness Indentation
Indentation is a useful technique for measuring hardness and calculating key fracture properties such as toughness. Typical microhardness indentations include Vickers and Knoop (for loads below 1 kgf). Samples are polished to a smooth finish. Vickers and Knoop are common for brittle hardness testing. Rockwell and Brinell hardness tests are also used, each using a spherical indenter tip. The goal is to make indents large enough to reduce error, but not too large that promotes cracking.

Knoop Indentation
The Knoop technique was developed as an alternative to VIcker's in order to reduce the cracking observed in brittle materials. The Knoop test creates an elongated pyramidal indentation from which the Knoop Hardness Number (KHN) can be calculated. The KHN is proportional the applied force over a given projected area (~stress value).

KHN = F/A = P/ CL^2

F = P = applied load (kgf) L = length of the long diagonal (mm) A = unrecovered projected area (mm^2) C = constant relating A to L^2 = 0.07028



The ratio of the short and long diagonals is 1:7. The depth is approximately 1/30 the length. Knoop is useful with very hard and brittle materials.

Knoop ASTM Standards
 * C 730 //for Glass and Glass Ceramics//, which recommends a load of 0.98 N (100 gf)
 * C 849 //for Whitewares//, with 9.8 N (1 kgf)
 * C 1326 //for Advanced Ceramics//, with 9.8 N (1 kgf)

Vickers Indentation
The Vickers Hardness number (HV) is similarly a ratio of the applied load to the surface area of the indentation (mm^2).

HV = 1.854 F/d^2

F = applied load d = the average of diagonal lengths (d_1 and d_2)



The depth of the indent is 1/7 the diagonal length. The Vickers technique:
 * has deeper penetrations
 * diagonals ~1/3 the Knoop diagonals
 * is less sensitive to surface conditions
 * is more sensitive to measurement error the Knoop

Vickers ASTM Standards


 * ASTM standard E 384, Microhardness of Materials - for Vickers hardness
 * C 1327 - new standard for Vickers hardness of advanced ceramics and recommends a load of 9.8 N (1 kgf)

Knoop and Vickers
Values were once reported as the hardness number over the area in mm^2 (e.g. HV/40). Conversion requires HV conversion from kgf to N ( where 1 kgf = 9.80665N) and mm^2 to m^2 to yield pascals.

Cracks initiated from the corners of a Vickers indent can be used to calculate elastic modulus and fracture toughness K_IC with approximately 30-40% accuracy. It is important to note that the fracture community started another convention defining Vicker's hardness number as normalized by the projected area as opposed to the generally accepted contact area. This has apparently become a source of confusion.

HV = 2 P/d^2

A transition point is observed for hardness indentation. The point at which hardness plateaus is due to cracking beneath the indent. Localized cracking at higher loads may result in crushing.



A ceramic brittleness index can be used to relate the transition point

B = (HE)/(K_IC)^2

Fractography
[\r.Mecholsky] [\r.Quinn NIST Guide]

Weibull Statistics
[\See student lecture on Weibull statistics]

Flexure Testing: 3pt, 4pt, biaxial
Flexure testing can measure failure load-displacement and stress-strain curves, which can be used to calculate failure stress, calculate work of fracture (area under load displacement) and calculate absorbed energy (area under stress strain).

P-d and sig-eps to calc. sig_f, gamma_WOF, K_app [\See dissertation section on flexure]

Furnace
While not a particular testing technique, a furnace is used to prepare samples by modifying microstructural effects. The microstructural change lead to a host of mechanical properties that can be measured as a function of holding time, ramp time.

f(t_holding, t_ramp) = {mechanical properties: sig_f, rho, T_sintering ...}

Optical Microscopy
This is a useful tool for measuring crack size and mirror radius (as will as indent diagonals). The mirror constant can be calculated.

M_c = sig_r/flaw size

Summary

 * ~ If you measure this... ||~ you can find these... ||~ using these techniques... ||
 * mirror radius || mirror constant

mirror to flaw size ratio

residual stress || microscope

microscope

plot strength vs. r^(-0.5) || residual stress and stress free cracks
 * mirror to flaw size ratio || relation for stress intensity factors of

predict time to failure of subcritical growing crack

proportion to 1/D* || indentation and annealing of glass

equation relation crack velocity and stress intensity

math ||
 * fracture energy || Young's modulus || indentation ||
 * fractal dimensional increment || K_IC || K_IC = E * a_0^(0.5) * D*^(0.5) ||
 * initiated cracks || K_IC || Vickers indent ||
 * E/H || K_IC || Knoop ||