Lecture #19 Failure & Fracture

Strength Theories Failure Theories Fracture Mechanics

Failure = no longer able to perform design function FRACTURE in brittle materials YIELDING / excessive deformation in ductile materials

Stages of Cracking Failure Behavior of concrete in compression. Discuss the development of cracking as a function of stress level.

Static Fatigue If a constant load is maintained between 75% and 100% of the strength, failure will eventually occur, because the unstable cracks are given sufficient time to propagate catastrophically.

Bond and Microcracking There is more linearity stress-strain between the paste and aggregates in high strength concrete rather than normal strength concrete because of the reduced micro-cracking. The same is true of the light weight concrete.

Stress Conditions Mechanical testing under simple stress conditions Design requires prediction of failure for complex stress conditions principal stresses (s1 > s2 > s3) biaxial stress state (s3=0)

Strength Envelope For Concrete

Simple Failure Theories Rankine s1=sft St. Venant e1= eft neither agree w/ experimental data either are rarely used

Complex Failure Theories Max Shear Stress (Tresca) ductile materials tmax= ty s1-s3= sy s2-s3= sy s1-s2= sy sy/2 = max shear stress at yield 2 = y If 2 > 1 > 0 1- 2 = -y If 1< 0 and 2 > 0 1 = y If 1 > 2 > 0 2 = -y If 1 < 2 < 0 1- 2 = y If 1> 0 and 2 < 0 1 = -y If 2 < 1 < 0

Complex Failure Theories Max Distortional Strain Energy (octahedral shear stress, von Mises) best agreement with experimental data hydrostatic + distortional principal stresses

Failure Theories Mohr’s Strength both yielding & fracture sft  sfc OR

Failure Theories Mohr’s Strength

Failure Envelope Mohr’s Strength failure envelope

Effect of Confinement

Comparison of Failure Theories equivalent to Max Shear Stress if sft=sfc ductile and modified if sft  sfc (brittle)

concentrated stress at crack tip (see Fig. 6.7) Fracture Mechanics max stress criterion not sufficient relationships between applied stress, crack size, and fracture toughness probability of failure, critical crack size (size effect, variability of material properties) focus on linear fracture mechanics, tensile loading, brittle materials all materials contain flaws, defects, cracks concentrated stress at crack tip (see Fig. 6.7)

Crack Growth

Fracture Mechanics Theoretical cohesive strength Griffith Theory fracture work resisted by energy to create two new crack surfaces Griffith Theory flaw / crack size sensitivity

Fracture Mechanics stress concentration at crack tip (see Fig 6.9) for C>>

Stress Intensity Factor x Crack Tip Stress Distribution

Fracture Mechanics Three modes of crack opening Focus on Mode I for brittle materials

Fracture Mechanics

Fracture Mechanics

Fracture Mechanics KI = stress intensity factor = Fs(pC)1/2 F is a geometry factor for specimens of finite size KI = KIC OR GI=GIC unstable fracture KIC= Critical Stress Intensity Factor = Fracture Toughness GI=strain energy release rate (GIC=critical)

2 d KI c c 2 a Alpha = a/d F Alpha

Flexure (Bending) Fracture Yielding similar as in tension brittle materials nonlinear s distribution initiates as tensile failure flexural strength > tensile strength Yielding similar as in tension ductile materials first @ extreme fiber progresses inward gradual change masks proportional limit In a brittle material, nonlinearity of the stress distribution contributes to the flexure strength exceeding the tensile strength by approximately 50%.

Failure Criterion

Linear Fracture Mechanics Non-Linear Fracture Mechanics

a cf Crack Process Zone KI d Alpha = a/d

Fracture specimens

Specimen Apparatus

Specimen Preparation

Test Specimens

Determination of Fracture Parameters sN = cn KIf / [g’(a0)cf + g(a0)d]1/2 sN = cn P/(sr) - split tensile (eq. 5.12) sN = cn P/(bd) - beam (eq. 5.13) Linear Regression Y = AX + B Y = cn2 / [g’(a0) sN2] X = g(a0) d / g’(a0) KIf = 1 / A1/2 cf = B / A

Application of Fracture Method Strength Determination g(a ) = c2nF2(a) Basic Geometry - split tensile cn = 2/p ;  = (1) 0.0, (2) 0.1667, or (3) 0.6667 (1) F(a) = 0.964; g(a ) = 0.0; g’(a ) = 2.9195 (2) F(a) = 0.964 - 0.026a + 1.472a2 - 0.256a3 F() = 0.9994, g(a ) = 0.5230; g’(a ) = 3.6023 (3) F(a) = 2.849 - 10.451a + 22.938a2 - 14.940a3 F() = 1.6497, g(a ) = 5.6997; g’(a ) = 10.0214 Basic Geometry - beam cn = 1.5 s/d ;  = a/d F(a) = 1.122 - 1.40a + 7.33a2 - 13.08a3 + 14.0a4

Failure Criterion

Applications of Fracture Parameters Strength Determination sN = cn KIf / [g’(a0)cf + g(a0)d]1/2

Applications of Fracture Parameters Strength Determination Size effect on strength ( a0 = 0.2; Bfu = 3.9 MPa = 566 psi; da = 25.4 mm = 1 in) log (d/da) Specimen or structure size log (sN / Bfu) sN d (mm or inch) (MPa or psi) 0.70 127 or 5 - 0.18 2.57 or 373 1.00 305 or 12 - 0.26 2.15 or 312 1.30 507 or 20 - 0.35 1.75 or 254