Elasticity Part 14 potx

EXAMPLE 14-4: Kelvin State with Unit Loads in CoordinateEXAMPLE 14-5: Force Doublet Consider the case of two concentrated forces acting along a common line of action but in opposite dire

EXAMPLE 14-4: Kelvin State with Unit Loads in Coordinate

for any closed surfaceS enclosing the origin

Using the basic Kelvin problem, manyrelated singular states can be generated Forexample, defineS0(x)¼ S,a¼ {u,a, s,a, e,a}, where a¼ 1, 2, 3 Now, if the state S0

is generated by the Papkovich potentials f(x) and c(x), thenS0is generated by

f0(x)¼ f,aþ ca

c0(x)¼ ci,aei

(14:2:12)

Further, define the Kelvin stateSa(x;j) as that corresponding to a unit load applied in

thexadirection at point j, as shown in Figure 14-7 Note thatSa(x;j)¼ Sa(x j) Also

define the set of nine statesSab(x) by the relation

EXAMPLE 14-4: Kelvin State with Unit Loads in Coordinate

EXAMPLE 14-5: Force Doublet

Consider the case of two concentrated forces acting along a common line of action but

in opposite directions, as shown in Figure 14-8 The magnitude of each force is specified

as1/h, where h is the spacing distance between the two forces We then wish to take thelimit ash! 0, and this type of system is called a force doublet Recall that this problemwas first defined in Chapter 13; see Exercise 13-18

From our previous constructions, the elastic state for this case is given bySaa(x) with nosum over a This form matches the suggested solution scheme presented in Exercise 13-18

EXAMPLE 14-6: Force Doublet with a Moment (About g-Axis)

Consider again the case of a double-force system with equal and opposite forces butacting along different lines of action, as shown in Figure 14-9 For this situation the twoforces produce a moment about an axis perpendicular to the plane of the forces Againthe magnitudes of the forces are taken to be1/h, where h is the spacing between the lines

of action, and the limit is to be taken ash! 0

The elastic state for this case is specified bySab(x), where a6¼ b, and the resultingmoment acts along the g-axis defined by the unit vector eg¼ ea eb It can be observedfrom Figure 14-9 thatSab¼ Sba From the previous equations (14.2.7), (14.2.12), and(14.2.13), the Papkovich potentials for stateSab(x) are given by

xβ-direction

FIGURE 14-9 Double-force system with moment.

EXAMPLE 14-7: Center of Compression/Dilatation

Acenter of compression (or dilatation) is constructed by the superposition of three tually perpendicular force doublets, as shown in Figure 14-10 The problem was intro-duced previously in Exercise (13.19) The elastic state for this force system is given by

mu-So(x)¼ 12(1 2n)CS

1

R, c

o

i ¼ xi2(1 2n)

1

and these yield the displacements and stresses

uoi ¼  xi2mR3

FIGURE 14-10 Center of compression.

EXAMPLE 14-8: Center of Rotation

Using the cross-product representation, acenter of rotation about the a-axis can beexpressed by the state

In order to develop additional singular states that might be used to model distributed ities, consider the following property

singular-Definition: LetS(x;l) ¼ {u(x;l), s(x;l), e(x;l)} be a regular elastic state for each

par-ameter l2 [a, b] with zero body forces Then the state Sdefined by

S(x)¼

ðb a

is also a regular elastic state This statement is just another form of the superposition principle,and it allows the construction of integrated combinations of singular elastic states as shown inthe next example

EXAMPLE 14-9: Half Line of Dilatation

A line of dilatation may be created through the superposition relation (14.2.27) bycombining centers of dilatation Consider the case shown in Figure 14-11 that illustrates

a line of dilatation over the negativex3-axis LetSo(x;l) be a center of compressionlocated at (0, 0,l) for all l 2 [0, 1) From our previous definitions, it follows that

zSo(x)¼ 

ð1 0

So(x;l)dlwhereSo(x;l)¼ So(x1,x2,x3þ l)

ð1 0

dl

^

R3

zuo2¼ x22m

ð1 0

dl

^

R3

zuo3¼ 12m

ð1 0

^R

 

dl¼ 12m

ð1 0

dl

^

R ¼ = log (R þ x3) (14:2:30)The potentials for this state can be written as

EXAMPLE 14-9: Half Line of Dilatation–Cont’d

Notice the singularity atR¼ x3, and of course this behavior is expected along thenegativex3-axis because of the presence of the distributed centers of dilatation

In spherical coordinates the displacement and stress fields become

zuoR¼ 12mR,

zuof¼  sin f2mR(1þ cos f),

Many brittle solids such as rock, glass, ceramics, and concretes contain microcracks It isgenerally accepted that the tensile and compressive strength of these materials is determined

by the coalescence of these flaws into macrocracks, thus leading to overall fracture The need

to appropriately model such behaviors has lead to many studies dealing with the elasticresponse of materials with distributed cracks Some studies have simply developed modulifor elastic solids containing distributed cracks; see, for example, Budiansky and O’Connell(1976), Hoenig (1979), and Horii and Nemat-Nasser (1983) Other work (Kachanov 1994) hasinvestigated the strength of cracked solids by determining local crack interaction and propa-gation behaviors The reviews by Kachanov (1994) and Chau, Wong, and Wang (1995)provide good summaries of work in this field

We now wish to present some brief results of studies that have determined the elastic constants

of microcracked solids as shown in Figure 14-12 It is assumed that a locally isotropic elasticmaterial contains a distribution of planar elliptical cracks as shown Some studies have assumed arandom crack distribution, thus implying an overall isotropic response; other investigators haveconsidered preferred crack orientations, giving rise to anisotropic behaviors Initial researchassumed that the crack density isdilute so that crack interaction effects can be neglected Laterstudies include crack interaction using the well-establishedself-consistent approach In general,the effective moduli are found to depend on acrack density parameter, defined by

(Cracked Elastic Solid)

(Elliptical Shaped Crack)

FIGURE 14-12 Elastic solid containing a distribution of cracks.

EXAMPLE 14-10: Isotropic Dilute Crack Distribution

Consider first the special case of a random dilute distribution of circular cracks of radius

a Note for the circular crack case the crack density parameter defined by (14.3.1)reduces to e¼ Nha3i Results for the effective Young’s modulus
EE, shear modulus
mm, andPoisson’s ratio
nn are given by

E

45(2 n) þ 16(1  n2)(10 3n)e

m

45(2 n) þ 32(1  n)(5  n)e

e¼ 45(n
nn)(2  n)16(1 n2)(10
nn 3n
nn  n)

(14:3:2)

whereE, m, and n are the moduli for the uncracked material

EXAMPLE 14-11: Planar Transverse Isotropic Dilute

Crack Distribution

Next consider the case of a dilute distribution of cracks arranged randomly but with allcrack normals oriented along a common direction, as shown in Figure 14-13 For thiscase results for the effective moduli are

(Transverse Cracked Solid) FIGURE 14-13 Cracked elastic solid with common crack orientation.

EXAMPLE 14-11: Planar Transverse Isotropic Dilute

Crack Distribution–Cont’d

E

3þ 16(1  n2)e

m

EXAMPLE 14-12: Isotropic Crack Distribution Using

Self-Consistent Model

Using the self-consistent method, effective moduli for the random distribution case can

be developed The results for this case are given by

E

E¼ 1 16(1
nn

2)(10 3
nn)e45(2
nn)

m

m¼ 1 32(1
nn)(5 
nn)e

45(2
nn)

e¼ 45(n
nn)(2 
nn)16(1
nn2)(10n 3n
nn 
nn)

(14:3:4)

Continued

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

EXAMPLE 14-12: Isotropic Crack Distribution Using

Self-Consistent Model–Cont’d

It is interesting to note that as e! 9=16, all effective moduli decrease to zero This can

be interpreted as a critical crack density where the material will lose its coherence.Although it would be expected that such a critical crack density would exist, theaccuracy of this particular value is subject to the assumptions of the modeling and isunlikely to match universally with all materials

In the search for appropriate models of brittle microcracking solids, there has been a desire tofind a correlation between failure mechanisms (fracture) and effective elastic moduli However,

it has been pointed out (Kachanov 1990, 1994) that such a correlation appears to be unlikelybecause failure-related properties such as stress intensity factors are correlated tolocal behavior,while the effective elastic moduli are determined by volume average procedures Externalloadings on cracked solids can close some cracks and possibly produce frictional sliding, therebyaffecting the overall moduli This interesting process createsinduced anisotropic behavior as aresult of the applied loading In addition to these studies of cracked solids, there also exists a largevolume of work on determining effective elastic moduli for heterogeneous materials containingparticulate and/or fiber phases; that is, distributed inclusions A review of these studies has beengiven by Hashin (1983) Unfortunately, space does not permit a detailed review of this work

As previously mentioned, the response of many heterogeneous materials has indicated ency on microscale length parameters and on additional microstructural degrees of freedom.Solids exhibiting such behavior include a large variety of cemented particulate materials such

depend-as particulate composites, ceramics, and various concretes This concept can be qualitativelyillustrated by considering a simplelattice model of such materials as shown in Figure 14-15.Using such a scheme, the macro load transfer within the heterogeneous particulate solid ismodeled using the microforces and moments between adjacent particles (see Chang and Ma1991; Sadd, Qui, Boardman, and Shukla 1992; Sadd, Dai, Parmameswaran, and Shukla2004b) Depending on the microstructural packing geometry (sometimes referred to asfabric),

Network of Elastic Elements

(Heterogeneous Elastic Material)

=

?

Inner Degrees

of Freedom (Equivalent Lattice Model)

FIGURE 14-15 Heterogeneous materials with microstructure.

this method establishes a lattice network that can be thought of as an interconnect series ofelastic bar or beam elements interconnected at particle centers Thus, the network represents insome way the material microstructure and brings into the model microstructural dimensionssuch as the grid size Furthermore, the elastic network establishes internal bending momentsand forces, which depend on internal degrees of freedom (e.g., rotations) at each connectingpoint in the microstructure as shown These internal rotations would be, in a sense, independ-ent of the overall macro deformations.

This concept then suggests that an elastic continuum theory including an independentrotation field with concentrated pointwise moments might be suitable for modeling heteroge-neous materials Such approaches have been formulated under the namesCosserat continuum;oriented media; asymmetric elasticity; micropolar, micromorphic, or couple-stress theories.The Cosserat continuum developed in 1909, was historically one of the first models of thiscategory However, over the next 50 years very little activity occurred in this field Renewedinterest began during the 1960s, and numerous articles on theoretical refinements and particu-lar analytical and computational applications were produced The texts and articles by Eringen(1968, 1999) and Kunin (1983) provide detailed background on much of this work, whileNowacki (1986) presents a comprehensive account on dynamic and thermoelastic applications

of such theories

Micropolar theory incorporates an additional internal degree of freedom called therotation and allows for the existence of body and surface couples For this approach, the newkinematic strain-deformation relation is expressed as

sji,jþ Fi¼ 0

mji,jþ eijksjkþ Ci¼ 0 (14:4:3)where sij is the usual stress tensor, Fi is the body force,mij is the surface moment tensornormally referred to as thecouple-stress tensor, and Ci is thebody couple per unit volume.Notice that as a consequence of including couple stresses and body couples,the stress tensor

sijno longer is symmetric For linear elastic isotropic materials, the constitutive relations for amicropolar material are given by

sij¼ lekkdijþ (m þ k)eijþ meji

mij¼ afk,kdijþ bfi,jþ gfj,i (14:4:4)

where l, m, k, a, b, g are the micropolar elastic moduli Note that classical elasticity relationscorrespond to the case where k¼ a ¼ b ¼ g ¼ 0 The requirement of a positive definite strainenergy function puts the following restrictions on these moduli

0
3l þ 2m þ k, 0
2m þ k, 0
k

Relations (14.4.1) and (14.4.4) can be substituted into the equilibrium equations (14.4.3)

to establish two sets of governing field equations in terms of the displacements and rotations Appropriate boundary conditions to accompany these field equations are moreproblematic For example, it is not completely clear how to specify the microrotation fiand/or couple-stress mij on domain boundaries Some developments on this subject havedetermined particular field combinations whose boundary specification guarantees a uniquesolution to the problem

micro-14.4.1 Two-Dimensional Couple-Stress Theory

Rather than continuing on with the general three-dimensional equations, we now move directlyinto two-dimensional problems under plane strain conditions In addition to the usual assump-tion u¼ u(x, y), v ¼ v(x, y), w ¼ 0, we include the restrictions on the microrotation,

fx¼ fy¼ 0, fz¼ fz(x, y) Furthermore, relation (14.4.2) is relaxed and the microrotation

is allowed to coincide with the macrorotation,

fi¼ !i¼1

This particular theory is then a special case of micropolar elasticity and is commonly referred

to couple-stress theory Eringen (1968) refers to this theory as indeterminate because theantisymmetric part of the stress tensor is not determined solely by the constitutive relations.Stresses on a typical in-plane element are shown in Figure 14-16 Notice the similarity ofthis force system to the microstructural system illustrated previously in Figure 14-15 For thetwo-dimensional case with no body forces or body couples, the equilibrium equations (14.4.3)reduce to

Notice that substituting (14.4.9) into (14:4:8)2gives the resultexy¼ eyx.

The constitutive equations (14.4.4) yield the following forms for the stress components:

com-we can solve for the antisymmetric stress term from the moment equilibrium equation(14:4:7)3to get

Proceeding along similar lines as classical elasticity, we introduce a stress function proach (Carlson 1966) to solve (14.4.13) A self-equilibrated form satisfying (14.4.7) identi-cally can be written as

Differentiating the second equation with respect tox and the third with respect to y and addingeliminates F and gives the following result:

Thus, the two stress functions satisfy governing equations (14:4:15)1 and (14.4.16) Wenow consider a specific application of this theory for the following stress concentrationproblem

EXAMPLE 14-13: Stress Concentration Around a Circular Hole: Micropolar Elasticity

We now wish to investigate the effects of couple-stress theory on the two-dimensionalstress distribution around a circular hole in an infinite medium under uniform tension atinfinity Recall that this problem was previously solved for the nonpolar case inExample 8-7 and the problem geometry is shown in Figure 8-12 The hole is to haveradiusa, and the far-field stress is directed along the x direction as shown The solutionfor this case is first developed for the micropolar model and then the additionalsimplification for couple-stress theory is incorporated This solution was first presented

by Kaloni and Ariman (1967) and later by Eringen (1968)

As expected for this problem the plane strain formulation and solution is best done inpolar coordinates (r, y) For this system, the equilibrium equations become

@sr

@r þ1r