FRACTURE MECHANICS II

Graffith’s Theory of Fracture

A.A. Griffith was the one, who introduced in 1920 the scientific study of the effect of cracks on the strength of solids. It is known that the theoretical cohesive strength of solids, based on the inter-atomic forces, is about one hundred to one thousand times higher than the actual values. A.A. Griffith was concerned with the problem that small glass objects generally exhibited higher strengths than larger ones. He proposed that every body contains a distribution of imperfections due to cracks or flaws and that failure occurs at the largest of these. Larger bodies have a greater likelihood of containing bigger or larger number of flaws and will fail at lower stresses. The principle of stress concentration produced in solid continuum at locations, where cracks exist, was found necessary to explain this phenomenon. The concept of imperfections caused by cracks and the propagation of cracks – crack growth - has led to various statistical theories of strength / failure to predict fracture. All of these are based on Griffith’s theory of crack propagation for brittle materials.Griffith’s theory is based on the concept

that certain strain energy is required to break an elastic solid containing an elliptical crack and subjected to tension and

that during the process of crack opening, the increasing area of surface of the crack needs additional surface energy, which is supplied at the expense of strain energy.

In other words, there is no change in the total energy, as it is constant and the increase in surface energy of the crack is therefore equal to the decrease in the potential energy

He started with the familiar stress concentration formula for the maximum stress Smax at extreme end on the major axis of an elliptical hole in an infinite plate under tensile load.

Smax = So [1 + 2a/b]* (1)

where the major axis and minor axis are 2a and 2b and So is the uniform tensile stress applied on the infinite plate. Replacing b by the radius of curvature = b2/a, he derived the basic form of the equation to be developed for further study**.

Smax = So [1 + 2a/] (2)

When a crack is considered, tends to zero and hence the second term to . That means the first term 1 << second term and is neglected leading to

Smax = 2 Soa/ (3)

-------------------------------------------------------------------------------------------------------*Note for circular hole a = b and Smax = 3So, which is familiar in Theory Elasticity. **Students may prove that the radius of curvature = b2/a. using x2/a2 + y2/b2 = 1, and deriving 1/ R = d2y/dx2, noting that dx/dy = 0, at x = a, the extreme end.

Recall that the strain energy due to Smax per unit volume is

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U = (Smax) 2 /2E Substituting from (3)

U = 4 So2 (a/)/2E (4)

In a similar form Griffith derived the difference in the strain energy between that of the plate without and with the hole (crack) as

Ue = a2 So2 B/E (5)

where B is the thickness of the plate. Then he expressed surface energy of the crack Us, as,

Us = As (6)

where is the surface energy per unit surface area of the crack and As is

the crack surface area, which is two-times the crack area A.

As = 2A = 2 (2a) B (7)

Therefore (6) becomesUs = 4 a B (8)

Now the total potential energy is

U = a2 So2 B/E + 4 a B (9)

Since the total potential energy is constant when the crack grows, the rate of change with respect to the change in area of the crack A (=2aB) should be zero: dU /dA = 0.

0 = (2 a So2 B/E + 4 B) /2B (10)

a So2 /E = 2 (11)

So = ( 2 E/ a) (12)

Relation between 2 and R

The surface energy added to the surface is not reversible and (2 ) is therefore a measure of dissipated energy and resistance to crack growth. It has already been derived that this energy is Fracture Toughness R = Stain Energy Release Rate G at equilibrium. Therefore,

R = G = 2 (13)

and from (11), (12) & (13),

a So2 /E = 2 R (14)

So = ( R E/ a) (15)

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Irwin – Orowan Corrections to Griffith’s theory

The surface energy considered by Griffith was to some extent correct for highly brittle materials like glass. But it was not proving for less brittle materials. It has been noticed by Irwin and Orowan independently that even in the so-called brittle materials, there is local plastic flow at the crack tip. This consumes more irreversible energy, which is in the order about 1000 time higher than that was considered by Griffith. Thus they suggested that is replaced by (s + p ), where suffices s and p denote surface and plastic energies. Therefore (11) & (12) become

a So2 /E = 2 ( + p ) (16)

So = [ 2 ( + p ) E/ a] (17)

R = G = 2 ( + p ) (18)

Stress Intensity Factor K

Many useful results may be deduced from the solution for the stresses around an elliptical hole in an infinite plate subjected to uniform tensile stress So, applied normal to its major axis at a remote section. One such solution is related to a crack of total length 2a, where the minor axis tends to zero, in terms of a single parameter K, called Stress Intensity Factor. For elaborate derivation of K, you can read books on the subject. K is deduced from the singular term derived from the complex stress analysis, resulting in a convenient form, easy to use.

K2 = So2 a (!9)

K = So a (20)This is different from the familiar stress concentration factor, which is a non-dimensional factor. But K has dimension of (stress- square root of crack length, Nm /m2). In general cases other than central crack, a is to be correctly chosen. For the present, we will identify three cases:

1) Central Notch - CN a half of crack length

2) Single Edge Notch - SEN a full edge crack length

3) Double Edge Notch - DEN a one side crack length

Correction to K for finite width While the above changes are to take into account of the free edge effect, there

is one more correction that is required to take into account of finite width of the plate.

The general expression for this correction is

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K2 = Y2 (a/D) So2a (21)For CN

Y(a/D} = Y* (22)

For SEN and DENY(a/D}= 1.12 Y* (23)

whereY*2 = [2D/a] tan[a/2D] (approximately) (24)

Since there is lack of constraint for crack opening in the free edge, the stress intensity K is higher and the applied stress So is lower .and hence the factor 1.12.

Note for infinite plate [a/2D]tends to zero and hence tan[] → []

Y* = Y(a/D} = 1 no correction needed.

Relation between R and K

Independently, K has been derived as in (19) &(20) and R as in (14) & (15). Comparing (14) & (15) and (19) & (20)

K2 = a So2 = R E (25)

K = ( R E) (26)

Modes of FractureAll that was discussed so far are relevant to ed Mode-I fracture only. However

with suitable modification , they could be applied to other modes of fracture.

Mode-I: Opening (in-plane)Mode-II: Shearing (in-plane)Mode-III: Tearing (out-of-plane)

Three examples are common for testing and evaluation of materials for their resistance to fracture.

1 Double Cantilever Beam ……………………… Mode-I

2 Peeling of Tape (Scotch, Masking, Cello Tapes) …Mode-I

3 Trousers Test ………………………………… Mode-III

Double Cantilever Beam (Mode-I)A = Length of crack (single edge crack)B = Plate thicknessC = Compliance

D1, D2 = Depth of each limb of beam section on either side of crack

E = Modulus of Elasticity of material of beam under test

Double Cantilever Beam (Mode-I)

A = Length of crack (single edge crack)

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B = Plate thicknessC = Compliance

D1, D2 = Depth of each limb of beam section on either side of crack

E = Modulus of Elasticity of material of beam under test P = Load that corresponds to crack length a and displacements u1 and u2

u1,u2 = Displacements on either side

The displacements at the points of load P on the two sides are

u1 = Pa3 / 3E I1 and u2 = Pa3 / 3E I2 (27)

u = u1 + u2 = [Pa3 / 3E][1/ I1 + 1/ I2] (28)

C = u/P = [a3 / 3E][1/ I1 + 1/ I2] (29)

C/A = C/(aB) = [a2 / BE][1/ I1 + 1/ I2] (30)

G = (1/2) P2C/A = [P2a2 /2 BE][1/ I1 + 1/ I2] (31)

If the crack is symmetrical, D1 = D2 and u1 = u2 and I1 = I2 = BD3/12 and hence

G = 12P2a2 / B2 D3E (32)

Peeling of Tape (Mode-I)

L = Original unstretched peeled tape length L* = Stretched peeled length, L* = L + dL = L[1 + dL/L] = L[1 + ] (33)A = L* B = L[1 + ] B (34)

u = Displacement along the peeled tape = L* (35)ux, uy = x and y components of uP = Load along the peeled tapePx, Py = x and y components of P = Angle of inclination P with x

Now the initial displacement is zero and hence

Px =P cos; Py = P sin (36)

uy = L* sin; ux = - L*[1 – cos] (note minus sign) (37)

W = Work done = Px ux + Py uy

= - Pcos L*[1 – cos] + Psin L* sin

= P L[1 + ][1 – cos] (38)

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U = Energy stored = (1/2) PdL = (1/2) PL dL/L = (1/2) PL (39) Energy released U = W – U = PL[(1 + )(1 – cos) - /2]

= PL[1 + /2 – (1 + ) cos] (40)

G = U/A = PL[1 + /2 – (1 + ) cos]/[1 + ]LB

= (P/B)[1 + /2 – (1 + ) cos]/[1 + ] (41)

Now if = 90o G = [P/B][ 1 + /2]/[1 + ] (42)

If the small stretch is further neglected, then =0 and

G = P/B (43)

Trousers Test (Mode-III)

As in the case of tape with = 90o, the work done W follows similarly, except that the displacement is the total for the two limbs,

u = 2L* (44)

W = 2PL* = 2PL[1 + ] (45)

The strain energy stored U is due to dL in each of the two limbs = 2dL and

U = (1/2) P (2dL) = P L (46)

The computation of G follows, noting that the crack area isA = aB = L* B

G = (2P/A) [1 + - /2] = (2P/B) [1 + /2] / [1 + ] (47)

If the small elastic recovery is neglected, then G becomes

G = 2P/B (48)

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