Mechanical Properties of Engineered Materials 2008 Part 12 pps

Naval Research laboratory were led byGeorge Irwin, who may be considered as the father of fracture mechanics.Using the concepts of linear elasticity, he developed a crack driving forcepa

of plasticity, which were considered in subsequent work by Orowan (1950).Although the work of Griffith (1921) and Orowan (1950) providedsome insights into the role of cracks and plasticity in fracture, robust engi-neering tools for the prediction of fracture were only produced in the late1950s and early 1960s after a number of well-publicized failures of ships andaircraft in the 1940s and early to mid-1950s Some of the failures includedthe fracture of the so-called Liberty ships in World War II(Fig 11.1) and

the Comet aircraft disaster in the 1950s These led to significant research anddevelopment efforts at the U.S Navy and the major aircraft producers such

as Boeing

The research efforts at the U.S Naval Research laboratory were led byGeorge Irwin, who may be considered as the father of fracture mechanics.Using the concepts of linear elasticity, he developed a crack driving forceparameter that he called the stress intensity factor (Irwin, 1957) At aroundthe same time, Williams (1957) also developed mechanics solutions for thecrack-tip fields under linear elastic fracture mechanics conditions Work atthe Boeing Aircraft Company was pioneered by a young graduate student,Paul Paris, who was to make important fundamental contributions to thesubject of fracture mechanics and fatigue (Paris and coworkers, 1960, 1961,1963) that will be discussed inChap 14

Following the early work on linear elastic fracture mechanics, it wasrecognized that further work was needed to develop fracture mechanicsapproaches for elastic–plastic and fully plastic conditions This led to thedevelopment of the crack-tip opening displacement (Wells, 1961) and the Jintegral (Rice, 1968) as a parameter for the characterization of the crackdriving force under elastic–plastic fracture mechanics conditions Three-

FIGURE11.1 Fractured T-2 tanker, the S.S Schenectady, which failed in1941.(Adapted from Parker, 1957—reprinted with permission from the NationalAcademy of Sciences.)

parameter fracture mechanics approaches have also been proposed byMcClintock et al (1995) for the characterization of the crack drivingforce under fully plastic conditions.

The subject of fracture mechanics is introduced in this chapter Thechapter begins with a brief description of Griffith fracture theory, theOrowan plasticity correction, and the concept of the energy release rate.This is followed by a derivation of the stress intensity factor, K, and someillustrations of the applications and limitations of K in linear elastic fracturemechanics Elastic–plastic fracture mechanics concepts are then introducedalong with two-parameter fracture concepts for the assessment of con-straint Finally, the relative new subject of interfacial fracture mechanics

is presented, along with the fundamentals of dynamic fracture mechanics

It is now generally accepted that all engineering structures and componentscontain three-dimensional defects that are known as cracks However, asdiscussed in the introduction, our understanding of the significance ofcracks has only been developed during the past few hundred years, withmost of the basic understanding emerging during the last 50 years of the20th century

Inglis (1913) modeled the stress concentrations around notches with radii ofcurvature,, and notch length, a (Fig 11.2) For elastic deformation, he wasable to show that the notch stress concentration factor, Kt, is given by

Kt¼maximum stress around notch tip

Remote stress away from notch ¼ 1 þ 2

ffiffiffia

of is amplified by a factor of 3 at the notch tip Failure is, therefore, likely

to initiate from the notch tip, when the applied/remote stresses are cantly below the fracture strength of the un-notched material Subsequentwork by Neuber (1945) extended the work of Inglis to include the effects ofnotch plasticity on stress concentration factors This has resulted in thepublication of handbooks of notch concentration factors for variousnotch geometries

signifi-Returning now to Eq (11.1), it is easy to appreciate that the notchconcentration factor will increase dramatically, as the notch-tip radius

approaches the limiting value corresponding to a single lattice spacing, b.Hence, for an atomistically sharp crack, the relatively high levels of stressconcentration are likely to result in damage nucleation and propagationfrom the crack tip.

The problem of crack growth from a sharp notch in a brittle solid was firstmodeled seriously by Griffith (1921) By considering the thermodynamicbalance between the energy required to create fresh new crack faces, andthe change in internal (strain) energy associated with the displacement ofspecimen boundaries (Fig 11.3), he was able to obtain the following energybalance equation:

UT¼   2a2B

where the first half on the right-hand side corresponds to the strain energyand the second half of the right-hand side is the surface energy due to theupper and lower faces of the crack, wihch have a total surface area of 4aB.Also, is the applied stress, a is the crack length, B is the thickness of thespecimen, E0¼ E=ð1  2Þ for plane strain, and E0¼ E for plane stress,

FIGURE 11.2 Stress concentration around a notch (Adapted from Callister,1999—reprinted with permission from John Wiley.)

where E is Young’s modulus, is Poisson’s ratio, and s is the surfaceenergy associated with the creation of the crack faces.

The critical condition at the onset of unstable equilibrium is mined by equating the first derivative of Eq 11.2 to zero, i.e.,

deter-dUT=da ¼ 0 This gives

Equation (11.4) was modified by Orowan (1950) to account for plasticwork in materials that undergo plastic deformation prior to catastrophic

FIGURE11.3 A center crack of length 2a in a large plate subjected to elasticdeformations (Adapted from Suresh, 1999—reprinted with permission fromCambridge University Press.)

failure Orowan proposed the following expression for the critical fracturecondition, c:

c ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðsþ pÞE0

a

r

ð11:5Þwhere p is a plastic energy term, which is generally difficult to measureindependently

Another important parameter is the strain energy release rate, G,which was first proposed by Irwin (1964) This is given by:

This section presents the derivation of energy release rates and complianceconcepts for prescribed loading[Fig 11.4(a)] and prescribed displacement[Fig 11.4(b)] scenarios The possible effects of machine compliance areconsidered at the end of the section

Let us start by considering the basic mechanics behind the definition of theenergy release rate of a crack subjected to remote load, F, Fig 11.4(a) Also,

u is the load point displacement through which load F is applied The energyrelease rate, G, is defined as

is given by (Fig 11.5):

 ¼ PE ¼ SE  WD ¼1

2Fu  Fu ¼ 1

2Fu ð11:9Þwhere SE is the strain energy and WD is the work done By definition, thecompliance, C, of the body is given simply by

FIGURE 11.5 Schematic of load–displacement curve under prescribed load.(Adapted from Suresh, 1999—reprinted with permission from CambridgeUniversity Press.)

FIGURE11.4 Schematic of notched specimens subject to (a) prescribed ing or (b) prescribed displacement (Adapted from Suresh, 1999—reprintedwith permission from Cambridge University Press.)

is zero and hence the change in potential energy is equal to the strain energy.

FIGURE 11.6 Schematic of load–displacement curve under prescribed placement (Adapted from Suresh, 1999—reprinted with permission fromCambridge University Press.)

Let us now consider the influence of machine compliance, CM, on thedeformation of the cracked body shown in Fig 11.7 The total displacement,

FIGURE11.7 Schematic of deformation in a compliant test machine (Adaptedfrom Hutchinson, 1979—with permission from the Technical University ofDenmark.)

T, is now the sum of the machine displacement, M, and the specimendisplacement, If the total displacement is prescribed, then we have

12þ 1

2BC

1

M ðT Þ2 ð11:19Þand the energy release rate is

22dCda

¼ 1

2BC

22dC

da ¼ 12BF

2dCda

ð11:20Þ

Hence, as before, the energy release rate does not depend on the nature of theloading system Also, the measured value of G does not depend on the com-pliance of the loading system However, the experimental determination of G

is frequently done with rigid loading systems that correspond to CM¼ 0

The fundamentals of linear elastic fracture mechanics (LEFM) are presented

in this section Following the derivation of the crack-tip fields, the physicalbasis for the crack driving force parameters is presented along with theconditions required for the application of LEFM The equivalence of Gand the LEFM cracking driving force (denoted by K) is also demonstrated

11.6.1 Derivation of Crack-Tip Fields

Before presenting the derivation of the crack-tip fields, it is important tonote here that there are three modes of crack growth These are illustratedschematically inFig 11.8 Mode I [Fig 11.8(a)] is generally referred to asthe crack opening mode It is often the most damaging of all the loadingmodes Mode II [Fig 11.8(b)] is the in-plane shear mode, while Mode III[Fig 11.8(c)] corresponds to the out-of-plane shear mode Each of the

modes may occur separately or simultaneously However, for simplicity, wewill derive the crack-tip fields for pure Mode I crack growth We will thenextend our attention to Modes II and III.

Now, let us begin by considering the equilibrium conditions for aplane element located at a radial distance, r, from the crack-tip (Fig.11.9) For equilibrium in the polar co-ordinate system, the equilibriumequations are given by

FIGURE11.9 In-plane co-ordinate system and crack-tip stresses

@ r

@r þ

1r

ð11:21eÞFinally, for strain compatibility we must satisfy

@21r

@"rr

@r ¼ 0 ð11:22ÞFor the in-plane problem (Modes I and II), the crack-tip strains areonly functions of r and Also, for the plane stress problem zz¼ 0 Therelationship between stress and strain is given by Hooke’s law:

E"rr ¼ rr ð11:23aÞE" ¼ ys

 2

ð11:61Þ

From the above equations, it is clear that the sizes of the Dugdale and planestress plastic zones are comparable However, the shapes of the plane stressand Dugdale plastic zones are somewhat different (Figs 11.12 and 11.13).Furthermore, it is important to note that the plastic zone size, rp, only gives

an approximate description of the plastic zone since the plastic zone ary varies significantly ahead of the crack tip

bound-Before, closing, it is important to note that the Dugdale model can beused to estimate the crack-tip opening displacement, t, at the points

x ¼ a, and y ¼ 0 The crack-tip opening displacement is given by

t¼ K

2

2 ysE (for plane strain) ð11:64Þ

11.6.3.3 Barenblatt Model

Barenblatt (1962) has developed a model for brittle materials that is gous to the Dugdale model (Fig 11.12) However, in the Barenblatt model,the traction, yy, is equal to the theoretical bond rupture strength of a brittlesolid, th E=10 (Lawn, 1993) The critical condition for fracture may,therefore, be expressed in terms of a critical cohesive zone size, rco, or interms of the critical crack opening displacement,c¼ 2 c(Rice, 1968) Thelatter gives

The stress intensity factor, K is only valid within a small annular region atthe crack tip where the asymptotic singular solutions (K and T terms)characterize the crack-tip fields to within 10% Beyond the annular region,higher order terms must be included to characterize the crack-tip field Ingeneral, however, the concept of K holds when the plastic zone, rp, at thecrack tip is small compared to the crack length ðrp< a=50Þ The concept of

K also applies to blunt notches with small levels of notch-tip plasticity.Furthermore, it applies to scenarios where small-scale deformation occurs

by mechanisms other than plasticity These include stress-induced phasetransformations and microcracking in brittle ceramics Such mechanismsgive rise to the formation of deformation process zones around the cracktip The size of the region of K dominance is affected by the sizes of theseprocess zones

The relationship between G and K is derived in this section Consider thegeneric crack-tip stress profile for Mode I that is shown inFig 11.14(a) Theregion of high stress concentration has strain energy stored over a distanceahead of the crack tip This strain energy is released when a small amount ofcrack advance occurs over a distance,a If G is the energy release rate,

then energy released due to crack extension, Ga, is related to the traction

Ga ¼ð1 þ Þ

8G K ðaÞK ða þ daÞa ð11:68bÞ

FIGURE11.14 Schematic of crack-tip profile and stress state: (a) before mental crack growth; (b) after incremental crack growth (Adapted fromHutchinson, 1983—reprinted with permission from the Technical University

incre-of Denmark.)

is frequently... mode It is often the most damaging of all the loadingmodes Mode II [Fig 11.8(b)] is the in-plane shear mode, while Mode III[Fig 11.8(c)] corresponds to the out -of- plane shear mode Each of the