Elasticity Part 10 ppsx

10.4.3 Infinite DomainsFor the region shown in Figure 10-7c, the general form of the potentials is determined in ananalogous manner as done in the previous case.. Theseproblems are discu

10.4.3 Infinite Domains

For the region shown in Figure 10-7(c), the general form of the potentials is determined in ananalogous manner as done in the previous case The logarithmic terms in (10.4.6) may beexpanded in the regionexterior to a circle enclosing all m contours Ckto get

log (z
zk)¼ log z þ log 1
zk

z

¼ log z
zk

z þ12

zkz

g(z)¼

Pm k¼1

Fk2p(1þ k)logzþ

s1

x þ s1 y

4 zþ g(z)

c(z)¼

kPm k¼1

Fk2p(1þ k)logzþ

s1y
s1

x þ 2it1 xy

2 zþ c(z)

(10:4:7)

where s1x , s1y , t1xy are the stresses at infinity and (z) and P(z) are arbitrary analyticfunctions outside the region enclosing allm contours Using power series theory, these analyticfunctions can be expressed as

We now develop some solutions of particular plane elastic problems involving regions of acircular domain The process starts by developing a general solution to a circular region witharbitrary edge loading as shown in Figure 10-8 The region 0 r  R is to have arbitraryboundary loadings at r¼ R specified by sr¼ f1(y) try¼
f2(y), which can be written incomplex form as

f ¼ f1(y)þ if2(y)¼ sr
itryjr¼R (10:5:1)The fundamental stress combinations and displacements in polar coordinates were given inrelations (10.2.12) The tractions given by (10.2.14) may be expressed in polar form as

Txrþ iTr

y¼
id

dsg(z)þ zg0(z)þ c(z)

Integrating this result around the boundaryr¼ R (ds ¼ Rdy) gives

i

ð(Trþ iTr)Rdy¼ g(z) þ zg 0(z)þ c(z)

annzn
1þ
ann
zzn
1
e2iy[
zzann(n
1)zn
2þ bnnzn
1]

¼ a1þ
a1þX1

k¼1(
[akþ1(k2
1)rkþ bk
1(k
1)rk
2]eiky

þ
akþ1(kþ 1)rke
ikyÞ

(10:5:4)

This relation can be recognized as thecomplex Fourier series expansion for sr
itry On theboundaryr¼ R, the complex boundary-loading function f can also be expanded in a similarFourier series as

f (y)¼ X1 k¼
1

Ckeiky

Ck ¼ 12p

ð2p 0

This solution scheme then only duplicates previous methods based on Fourier analysis Amore powerful use of complex variable techniques involves the application of Cauchy integralformulae In order to discuss this method, consider again the circular region with unit boundaryradius Relation (10.5.3) becomes

þC

g(z)

z
zdzþ

12pi

þC

zg

0(z)

z
zdz

þ 12pi

þC

c(z)

z
zdz ¼

12pi

þC

g(z)þ
a1zþ 2
a2þ c(0) ¼ 1

2pi

þC

g(z)

We also find thatan¼ 0 for n > 2, and so g(z) ¼ aoþ a1zþ a2z2 These results can be used tosolve for the remaining terms in order to determine the final form for the potential g(z) Using asimilar scheme but starting with the complex conjugate of (10.5.7), the potential c(z) may befound Dropping the constant terms that do not contribute to the stresses, the final results aresummarized as

g(z)¼ 1

2pi

þC

g(z)

z
zdz

a1z, a1þ
a1¼

12pi

þC

g(z)

z2 dzc(z)¼ 1

2pi

þC

g(z)

z
zdz

g0(z)

z þa1z

(10:5:10)

Note that the preceding solution is valid only for the unit disk For the case of a disk of radius

a, the last two terms for c(z) should be multiplied by a2 We now consider a couple of specificexamples of using this general solution

EXAMPLE 10-2: Disk Under Uniform Compression

Consider the case of uniform compression loading of the circular disk, as shown inFigure 10-9

The boundary tractions for this case become

Trxþ iTr

y ¼ (srþ itry)eiy¼
peiyand thus the boundary-loading function defined by (10.5.3) reduces to

g¼ iþ

C

pz(z
z)dzþ

EXAMPLE 10-2: Disk Under Uniform Compression–Cont’d

srþ sy¼ 2(
p

a1
p
a1)¼
2p

sy
srþ 2itry¼ 0Separating the real and imaginary parts gives individual stresses

df (x)d(x
x)dx ¼ f (x) for any parameter d and continuousfunction f Using this representation, the resultant boundary-loading function can beexpressed as

g¼ iþ

C(Trþ iTr)ady¼

1z

2

(10:5:16)Continued

EXAMPLE 10-3: Circular Plate With Concentrated

(
zz2
a2)þ 1

a2

(10:5:18)

which was the problem previously solved in Example 8-10, giving the stresses specified

in relations (8.4.69) Solutions to many other problems of circular domain can be found

in Muskhelishvili (1963), Milne-Thomson (1960), and England (1971)

Complex variable methods prove to be very useful for the solution of a large variety of space and half-space problems Full-space examples commonly include problems with varioustypes of internal concentrated force systems and internal cavities carrying different loadingconditions Typical half-space examples include concentrated force and moment systemsapplied to the free surface and indentation contact mechanics problems where the boundaryconditions may be in terms of the stresses, displacements, or of mixed type over a portion ofthe free surface This general class of problems involves infinite domains and requires thegeneral solution form given by (10.4.7)

EXAMPLE 10-4: Concentrated Force-Moment System

iM2pz

sy
sxþ 2itxy¼ 2
ðzzg00(z)þ c0(z)Þ ¼ Xþ iY

p(1þ k)

zz

z2þk(X
iY)p(1þ k)

1

z
iM

pz2(10:6:2)

while the resulting displacements are

2mU¼ kg(z)
zg0(z)
c(z) ¼

k(Xþ iY)2p(1þ k)( logzþ log
zz)þ

X
iY2p(1þ k)

z

zzþiM2p
zz

FIGURE 10-11 Concentrated force system in an infinite medium.

EXAMPLE 10-4: Concentrated Force-Moment System

For the special case ofX¼ P and Y ¼ M ¼ 0, the stresses reduce to

sx¼
Px2p(1þ k)r2[4x

2

r2þ k
1]

sy¼ Px2p(1þ k)r2[4x

2

r2þ k
5]

txy¼ Py2p(1þ k)r2[4y

Following the solution pattern from Example 10-4, the complex potentials can bewritten as

g(z)¼
Xþ iY

2p logzc(z)¼(X
iY)

sy
srþ 2itry¼ 2e2iy[
00(z)þ c0(z)]¼ 2e2iy Xþ iY

2p

zz

z2þX
iY2p

1z

EXAMPLE 10-5: Concentrated Force System on the Surface

The boundary condition related to the concentrated force involves integrating thetractions around the contourC (a semicircle of arbitrary radius centered at the origin) asshown in Figure 10-12 Thus, using (10.4.4)

þ

C

(Txnþ iTn

y)ds¼ i[g(z) þ zg0(z)þ c(z)]C¼ X þ iY

which verifies the appropriate boundary condition By using the moment relation (10.3.2),

it can also be shown that the resultant tractions on the contourC give zero moment

For the special caseX¼ 0 and Y ¼ P, the individual stresses can be extracted fromresult (10.6.8) to give

sr¼
2P

prsin y, sy¼ try¼ 0 (10:6:9)Again this case was previously presented in Example 8-8 by relation (8.4.35)

By employing analytic continuation theory and Cauchy integral representations, other morecomplicated surface boundary conditions can be handled Such cases typically arise from contactmechanics problems involving the indentation of an elastic half space by another body Such a

problem is illustrated in Figure 10-13 and the boundary conditions under the indenter couldinvolve stresses and/or displacements depending on the contact conditions specified Theseproblems are discussed in Muskhelishvili (1963), Milne-Thomson (1960), and England (1971).

EXAMPLE 10-6: Stressed Infinite Plane with a Circular Hole

The final example in this section is a full plane containing a stress-free circular hole,and the problem is loaded with a general system of uniform stresses at infinity, asshown in Figure 10-14 A special case of this problem was originally investigated inExample 8-7

The general solution form (10.4.7) is again used; however, for this problem the termswith stresses at infinity are retained while the logarithmic terms are dropped because thehole is stress free The complex potentials may then be written as

g(z)¼s

1

x þ s1 y

s1

x þ s1 y

1

y
s1

x þ 2it1 xy

2iy

¼X1 n¼1

EXAMPLE 10-6: Stressed Infinite Plane with a Circular

The method of conformal mapping discussed in Section 10.1 provides a very powerful tool tosolve plane problems with complex geometry The general concept is to establish a mappingfunction, which will transform a complex region in thez-plane (actual domain) into a simpleregion in the z-plane If the elasticity solution is known for the geometry in the z-plane, thenthrough appropriate transformation formulae the solution for the actual problem can be easilydetermined Because we have established the general solution for the interior unit disk problem

in Section 10.5, mapping functions that transform regions onto the unit disk (see Figure 10-5)will be most useful Specific mapping examples are discussed later

In order to establish the appropriate transformation relations, we start with the generalmapping function

g(z)¼ g(w(z) ), c(z) ¼ c(w(z) ) (10:7:1)where w is an analytic single-valued function Using this result, the derivatives arerelated by

dg

dz¼dg1dz

y)ds¼ g1(z)þw(z)

w0(z)g

0

1(z)þ c1(z) (10:7:6)The complex displacement transforms to

a scheme that maps the region onto the exterior of the unit disk in the z-plane Either mappingscheme can be used for problem solution by incorporating the appropriate interior or exterior

solution for the unit disk problem Mappings for the special cases of circular and ellipticalholes are shown in Figure 10-16, and additional examples can be found in Milne-Thomson(1960) and Little (1973).

For the exterior problem, the potential functions are given by relations (10.4.7) and (10.4.8),and when applied to the case under study gives

g(z)¼
F

2p(1þ k)log [w(z)]þ

s1x þ s1 y

4 w(z)þ g(z)c(z)¼ k
F

2p(1þ k)log [w(z)]þ

s1y
s1

x þ 2it1 xy

w (ζ) = Rζ− 1

+ mz w(ζ)=R 1

z-plane: Circular Case

whereF is the resultant force on the internal boundary C, and the functions g(z) and c(z) areanalytic in the interior of the unit circle For the geometry under investigation, the mappingfunction will always have the general formw(z)¼ Cz
1þ (analytic function), and thus thelogarithmic term in (10.7.8) can be written as logw¼
log zþ (analytic function) Thisallows the potentials to be expressed as

g1(z)¼ F2p(1þ k)log zþ

s1

x þ s1 y4

C

zþ c(z)

(10:7:9)

We now investigate a specific case of an elliptical hole in a stressed plane

EXAMPLE 10-7: Stressed Infinite Plane with an

EXAMPLE 10-7: Stressed Infinite Plane with an Elliptical Hole– Cont’d

For this case, relations (10.7.9) give the potentials

g1(z)¼S4

where g(z) and c(z) are analytic in the unit circle These functions may be determined

by using either Fourier or Cauchy integral methods as outlined in Section 10.5 Details

on this procedure may be found in Little (1973), Muskhelishvilli (1963), or Thomson (1960) The result is

of the importance of this topic, we further investigate the nature of the stress distributionaround cracks in the next section A plot of the stress concentration factor (sj)max=Sversus the aspect ratiob/a is shown in Figure 10-18 It is interesting to observe that thisrelationship is actually linear For aspect ratios less than 1, the concentration is smallerthan that of the circular case, while very high concentrations exist forb=a > 1 Furtherdetails on such stress concentration problems for holes of different shape can be found

Continued

EXAMPLE 10-7: Stressed Infinite Plane with an

Elliptical Hole–Cont’d

in Savin (1961) Numerical techniques employing the finite element method are applied tothis stress concentration problem in Chapter 15; see Example 15-2 and Figure 15-5

As shown in the previous example and in Example 8-8, the elastic stress field around crack tips canbecome unbounded For brittle solids, this behavior can lead to rapid crack propagation resulting

in catastrophic failure of the structure Therefore, the nature of such elevated stress distributionsaround cracks is important in engineering applications, and the general study of such problemsforms the basis of linear elastic fracture mechanics Complex variable methods provide aconvenient and powerful means to determine stress and displacement fields for a large variety

of crack problems We therefore wish to investigate some of the basic procedures for suchapplications

Several decades ago Westergaard (1937) presented a specialized complex variable method

to determine the stresses around cracks The method used a single complex potential nowrespectfully called theWestergaard stress function Although this scheme is not a completerepresentation for all plane elasticity problems, it was widely used to solve many practicalproblems of engineering interest More recently Sih (1966) and Sanford (1979) have reex-amined the Westergaard method and established appropriate modifications to extend thevalidity of this specialized technique More detailed information on the general method can

be found in Sneddon and Lowengrub (1969) and Sih (1973), and an extensive collection ofsolutions to crack problems have been given by Tada, Paris, and Irwin (2000)

Crack problems in elasticity introduce singularities and discontinuities with two importantand distinguishing characteristics The first is involved with the unbounded nature of thestresses at the crack tip, especially in thetype of singularity of the field The second feature

is that the displacements along the crack surface are commonly multivalued For open cracks,the crack surface will be stress free However, some problems may have loadings that can

0 5 10 15 20 25

produce crack closure leading to complicated interface conditions In order to demonstrate thebasic complex variable application for such problems, we now consider a simple example of acrack in an infinite plane under uniform tension loading.

EXAMPLE 10-8: Infinite Plane with a Central Crack

Consider the problem of an infinite plane containing a stress-free crack of length 2a lyingalong thex-axis as shown in Figure 10-19 The plane is subjected to uniform tension S inthey direction, and thus the problem has symmetries about the coordinate axes

The solution to this problem follows the general procedures of the previous sectionusing the mapping function

z2
a2p

Continued

x y

y = S

a a

s∞

FIGURE 10-19 Central crack in an infinite plane.

EXAMPLE 10-8: Infinite Plane with a Central Crack–Cont’d

For this case the stress combinations become

z¼ rbeib, z
a ¼ reiy, zþ a ¼ raeia, andr sin y¼ rasin a¼ rbsin b

Using these new geometric variables, the stress combinations and displacements can

Evaluating these relations for smallr gives

EXAMPLE 10-8: Infinite Plane with a Central Crack–Cont’d

and solving for the individual stresses produces the following:

sx¼ KffiffiffiffiffiffiffiffiI2pr

p cosy

2 1
siny

2sin

3y2

sy¼ KffiffiffiffiffiffiffiffiI2pr

p cosy

2 1þ siny

2sin

3y2

txy¼ KffiffiffiffiffiffiffiffiI2pr

(10:8:7)

where the parameterKI¼ Spffiffiffiffiffiffipa

and is referred to as thestress intensity factor Usingrelation (10.2.9), the corresponding crack-tip displacements can be expressed by

u¼KIm

ffiffiffiffiffiffir2p

rcosy2

ffiffiffiffiffiffir2p

rsiny2

kþ 1

2
cos2y

2

As observed in Section 8.4.10, these results indicate that the crack-tip stresses have a

r
1=2singularity, while the displacements behave asr1=2 Thestress intensity factor KIis

a measure of the intensity of the stress field near the crack tip under theopening mode(mode I) deformation Two additional shearing modes also exist for such crack problems,and the crack-tip stress and displacement fields for these cases have the samer depend-ence but different angular distributions For the central crack problem considered in thisexample, the stress intensity factor was proportional topffiffiffia

; however, for other crackgeometries, this factor will be related to problem geometry in a more complex fashion.Comparing the vertical displacements on the top and bottom crack surfaces indicates thatv(r, p)¼
v(r,
p) This result illustrates the expected multivalued discontinuousbehavior on each side of the crack surface under opening mode deformation

As mentioned, Westergaard (1937) developed a specialized complex variable technique tohandle a restricted class of plane problems The method uses a single complex potential, andthus the scheme is not a complete representation for all plane elasticity problems Neverthe-less, the technique has been extensively applied to many practical problems in fracturemechanics dealing with the determination of stress fields around cracks Sih (1966) andSanford (1979) have reexamined the Westergaard method and established appropriate modifi-cations to extend the validity of the specialized technique

In order to develop the procedure, consider again the central crack problem shown in Figure10-19 Because this is a symmetric problem, the shear stresses must vanish ony¼ 0, and thusfrom relation (10.2.11)

Im[
zzg00(z)þ c0(z)]¼ 0 on y ¼ 0 (10:9:1)