Elasticity Part 15 ppsx

Methods of numerical stress analysis normally recastthe mathematical elasticity boundary value problem into a direct numerical routine.. The first of these techniques is known as the fin

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Kennedy TC, and Kim JB: Finite element analysis of a crack in a micropolar elastic material, Computers

in Engineering, ASME, ed R Raghavan and TJ CoKonis, vol 3, pp 439-444, 1987.

Kunin IA: Elastic Media with Microstructure II Three-Dimensional Models, Springer-Verlag, Berlin, 1983.

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Lardner RW: Mathematical Theory of Dislocations and Fracture, Univ of Toronto Press, Toronto, 1974.

Mindlin RD: Influence of couple-stress on stress concentrations, Experimental Mech., vol 3, pp 1-7, 1963.

Mindlin RD: Microstructure in linear elasticity, Arch Rat Mech Anal., vol 16, pp 51-78, 1964.

Mura T: Micromechanics of Defects in Solids, Martinus Nijhoff, Dordrecht, 1987.

Nowacki W: Theory of Asymmetric Elasticity, Pergamon Press, Oxford, England, 1986.

Ostoja-Starzewski M, and Wang C: Linear elasticity of planar Delaunay networks: random field terization of effective moduli, Acta Mech., vol 80, pp 61-80, 1989.

charac-Ostoja-Starzewski M, and Wang C: Linear elasticity of planar Delaunay networks part ii: Voigt and Reuss bounds, and modification for centroids, Acta Mech., vol 84, pp 47-61, 1990.

Sadd MH, and Dai Q: A comparison of micromechanical modeling of asphalt materials using finite elements and doublet mechanics, forthcoming, Mech of Materials, 2004a.

Sadd MH, Dai Q, Parmameswaran V, and Shukla A: Microstructural simulation of asphalt materials: modeling and experimental studies, J Materials in Civil Eng, vol 16, pp 107-115, 2004b.

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Schijve J: Note of couple stresses, J Mech Phys Solids, vol 14, pp 113-120, 1966.

Sun CT, and Yang TY: A couple-stress theory for gridwork-reinforced media, J Elasticity, vol 5,

pp 45-58, 1975.

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Weertman J, and Weertman JR: Elementary Dislocation Theory, Macmillan, New York, 1964.

Weitsman Y: Couple-stress effects on stress concentration around a cylindrical inclusion in a field of uniaxial tension, J Appl Mech., vol 32, pp 424-428, 1965.

Exercises

14-1 Show that the general plane strain edge dislocation problem shown in Figure 14-3 can besolved using methods of Chapter 10 with the two complex potentials

g(z)¼ imb4p(1 n)logz, c(z)¼ 

im
b4p(1 n)logzwhereb¼ bxþ iby In particular, verify the cyclic property [uþ iv]C¼ b, where C isany circuit in thex,y-plane around the dislocation line Also determine the general stressand displacement field

14-2 Justify that the edge dislocation solution (14.1.2) provides the required multivaluedbehavior for the displacement field Explicitly develop the resulting stress fields given

by (14.1.3) and (14.1.4)

14-3 Show that the screw dislocation displacement field (14.1.5) gives the stresses (14.1.6)and (14.1.7)

14-4 For the edge dislocation model, consider a cylinder of finite radius with axis along thedislocation line (z-axis) Show that although the stress solution gives rise to tractions

on this cylindrical surface, the resultant forces in thex and y directions will vanish.14-5 The stress field (14.1.7) for the screw dislocation produces no tangential or normalforces on a cylinder of finite radius with axis along the dislocation line (z-axis).However, show that if the cylinder is of finite length, the stress tzyon the ends will notnecessarily be zero and will give rise to a resultant couple

14-6 Show that the strain energy (per unit length) associated with the screw dislocationmodel of Example 14-2 is given by

14-7 Using similar notation as Exercise 14-6, show that the strain energy associated withthe edge dislocation model of Example 14-1 can be expressed by

14-9 Verify that the displacements and stresses for the center of compression are given by(14.2.21) and (14.2.22)

14-10 A fiber discontinuity is to be modeled using a line of centers of dilatation along thex1axis from 0 toa Show that the displacement field for this problem is given by

-u1¼ 12m

1

^

R1R

u2¼ 12m

1R

x1x2

x2þ x21

^R

(x1 a)x2

x2þ x2

u3¼ 12m

1R

x1x3

x2þ x21

^R

14-11* For the isotropic self-consistent crack distribution case in Example 14-12, show that

for the casev¼ 0:5, relation (14:3:4)3reduces to

e¼ 916

1 2
vv

1
vv2

Verify the total loss of moduli at e¼9

⁄16 Using these results, develop plots of theeffective moduli ratios
nn=n,
EE=E,
mm=m versus the crack density Compare theseresults with the corresponding values from the dilute case given in Example 14-10.14-12 Develop the compatibility relations for couple-stress theory given by (14.4.12) Next,using the constitutive relations, eliminate the strains and rotations, and express theserelations in terms of the stresses, thus verifying equations (14.4.13)

14-13 Explicitly justify that the stress-stress function relations (14.4.14) are a

self-equilibrated form

14-14 For the couple-stress theory, show that the two stress functions satisfy

r4F¼ 0, r2C l2r4C¼ 014-15 Using the general stress relations (14.4.25) for the stress concentration problem ofExample 14-13, show that the circumferential stress on the boundary of the hole isgiven by

develop the stress and stress function compatibility forms (14.5.10) and (14.5.11).14-18* Compare the hoop stress sy(r, p=2) predictions from elasticity with voids given byrelation (14.5.18) with the corresponding results from classical theory Choosing

N¼1

⁄2andL¼ 2, for the elastic material with voids, make a comparative plot of

sy(r, p=2)=T versus r/a for these two theories

14-19* For the doublet mechanics Flamant solution in Example 14-15, develop contour plots(similar to Figure 14-22) for the microstressesp1andp2 Are there zones where thesemicrostresses are tensile?

15 Numerical Finite and Boundary

Element Methods

Reviewing the previous chapters would indicate that analytical solutions to elasticity problemsare normally accomplished for regions and loadings with relatively simple geometry Forexample, many solutions can be developed for two-dimensional problems, while only a limitednumber exist for three dimensions Solutions are commonly available for problems with simpleshapes such as those having boundaries coinciding with Cartesian, cylindrical, and sphericalcoordinate surfaces Unfortunately, however, problems with more general boundary shape andloading are commonly intractable or require very extensive mathematical analysis and numer-ical evaluation Because most real-world problems involve structures with complicated shapeand loading, a gap exists between what is needed in applications and what can be solved byanalytical closed-form methods

Over the years, this need to determine deformation and stresses in complex problems haslead to the development of many approximate and numerical solution methods (see briefdiscussion in Section 5.7) Approximate methods based on energy techniques were outlined inSection 6.7, but it was pointed out that these schemes have limited success in developingsolutions for problems of complex shape Methods of numerical stress analysis normally recastthe mathematical elasticity boundary value problem into a direct numerical routine One suchearly scheme is thefinite difference method (FDM) in which derivatives of the governing fieldequations are replaced by algebraic difference equations This method generates a system ofalgebraic equations at various computational grid points in the body, and solution to the systemdetermines the unknown variable at each grid point Although simple in concept, FDM has notbeen able to provide a useful and accurate scheme to handle general problems with geometricand loading complexity Over the past few decades, two methods have emerged that providenecessary accuracy, general applicability, and ease of use This has lead to their acceptance bythe stress analysis community and has resulted in the development of many private andcommercial computer codes implementing each numerical scheme

The first of these techniques is known as the finite element method (FEM) and involvesdividing the body under study into a number of pieces or subdomains called elements Thesolution is then approximated over each element and is quantified in terms of values at speciallocations within the element called the nodes The discretization process establishes an

algebraic system of equations for the unknown nodal values, which approximate the ous solution Because element size, shape, and approximating scheme can be varied to suit theproblem, the method can accurately simulate solutions to problems of complex geometry andloading FEM has thus become a primary tool for practical stress analysis and is also usedextensively in many other fields of engineering and science.

continu-The second numerical scheme, called theboundary element method (BEM), is based on anintegral statement of elasticity (see relation (6.4.7)) This statement may be cast into a formwith unknowns only over the boundary of the domain under study The boundary integralequation is then solved using finite element concepts where the boundary is divided intoelements and the solution is approximated over each element using appropriate interpolationfunctions This method again produces an algebraic system of equations to solve for unknownnodal values that approximate the solution Similar to FEM techniques, BEM also allowsvariation in element size, shape, and approximating scheme to suit the application, and thus themethod can accurately solve a large variety of problems

Generally, an entire course is required to present sufficient finite and boundary elementtheory to prepare properly for their numerical/computational application Thus, the briefpresentation in this chapter provides only an overview of each method, focusing on narrowapplications for two-dimensional elasticity problems The primary goal is to establish a basiclevel of understanding that will allow a quick look at applications and enable connections to bemade between numerical solutions (simulations) and those developed analytically in theprevious chapters This brief introduction provides the groundwork for future and moredetailed study in these important areas of computational solid mechanics

Finite element procedures evolved out of matrix methods used by the structural mechanicscommunity during the 1950s and 1960s Over the years, extensive research has clearlyestablished and tested numerous FEM formulations, and the method has spread to applications

in many fields of engineering and science FEM techniques have been created for discrete andcontinuous problems including static and dynamic behavior with both linear and nonlinearresponse The method can be applied to one-, two-, or three-dimensional problems using alarge variety of standard element types We, however, limit our discussion to only two-dimensional, linear isotropic elastostatic problems Numerous texts have been generated thatare devoted exclusively to this subject; for example, Reddy (1993), Bathe (1995), Zienkiewiczand Taylor (1989), Fung and Tong (2001), and Cook, Malkus, and Plesha (1989)

As mentioned, the method discretizes the domain under study by dividing the region intosubdomains called elements In order to simplify formulation and application procedures,elements are normally chosen to be simple geometric shapes, and for two-dimensionalproblems these would be polygons including triangles and quadrilaterals A two-dimensionalexample of a rectangular plate with a circular hole divided into triangular elements is shown inFigure 15-1 Two different meshes (discretizations) of the same problem are illustrated, andeven at this early stage in our discussion, it is apparent that improvement of the representation

is found using thefiner mesh with a larger number of smaller elements Within each element,

an approximate solution is developed, and this is quantified at particular locations called thenodes Using a linear approximation, these nodes are located at the vertices of the triangularelement as shown in the figure Other higher-order approximations (quadratic, cubic, etc.) canalso be used, resulting in additional nodes located in other positions We present only a finiteelement formulation using linear, two-dimensional triangular elements

Typical basic steps in a linear, static finite element analysis include the following:

1 Discretize the body into a finite number of element subdomains

2 Develop approximate solution over each element in terms of nodal values

3 Based on system connectivity, assemble elements and apply all continuity and boundaryconditions to develop an algebraic system of equations among nodal values

4 Solve assembled system for nodal values; post process solution to determine additionalvariables of interest if necessary

The basic formulation of the method lies in developing the element equation that mately represents the elastic behavior of the element This development is done for the genericcase, thus creating a model applicable to all elements in the mesh As pointed out in Chapter 6,energy methods offer schemes to develop approximate solutions to elasticity problems, andalthough these schemes were not practical for domains of complex shape, they can be easilyapplied over an element domain of simple geometry (i.e., triangle) Therefore, methods ofvirtual work leading to a Ritz approximation prove to be very useful in developing elementequations for FEM elasticity applications Another related scheme to develop the desiredelement equation uses a more mathematical approach known as the method of weightedresiduals This second technique starts with the governing differential equations, and throughappropriate mathematical manipulations, a so-called weak form of the system is developed.Using a Ritz/Galerkin scheme, an approximate solution to the weak form is constructed, andthis result is identical to the method based on energy and virtual work Before developing the

approxi-(Discretization with 228 Elements)

(Discretization with 912 Elements)

(Triangular Element) (Node)

FIGURE 15-1 Finite element discretization using triangular elements.

element equations, we first discuss the necessary procedures to create approximate solutionsover an element in the system.

Linear Triangular Elements

Limiting our discussion to the two-dimensional case with triangular elements, we wish toinvestigate procedures necessary to develop a linear approximation of a scalar variableu(x,y)over an element Figure 15-2 illustrates a typical triangular element denoted by Oein thex,y-plane Looking for a linear approximation, the variable is represented as

u(x, y)¼ c1þ c2xþ c3y (15:2:1)whereciare constants It should be kept in mind that in general the solution variable is expected

to have nonlinear behavior over the entire domain and our linear (planar) approximation is onlyproposed over the element We therefore are using a piecewise linear approximation to representthe general nonlinear solution over the entire body This approach generally gives sufficientaccuracy if a large number of elements are used to represent the solution field Other higher-orderapproximations including quadratic, cubic, and specialized nonlinear forms can also be used toimprove the accuracy of the representation

1

2 3

(Lagrange Interpolation Functions)

FIGURE 15-2 Linear triangular element geometry and interpolation.

It is normally desired to express the representation (15.2.1) in terms of the nodal values ofthe solution variable This can be accomplished by first evaluating the variable at each of thethree nodes

(a1u1þ a2u2þ a3u3)

c2¼ 12Ae

(b1u1þ b2u2þ b3u3)

c3¼ 12Ae

(g1u1þ g2u2þ g3u3)

(15:2:3)

where Ae is the area of the element, and ai¼ xjyk
xkyj, bi¼ yj
yk, gi¼ xk
xj, where

i6¼ j 6¼ k and i,j,k permute in natural order Substituting for ciin (15.2.1) gives

the region using various other interpolation schemes With these representation conceptsestablished, we now pursue a brief development of the plane elasticity element equationsusing the virtual work formulation.

The principle of virtual work developed in Section 6.5 can be stated over a finite elementvolume Ve with boundarySe as

1CAdxdy

he

ð

G e

dudv

3

7 uv

37

7¼

12Ae

b1 0 b2 0 b3 0

0 g1 0 g2 0 g3

g1 b1 g2 b2 g3 b3

24

35

(15:3:7)Hooke’s law then takes the form

Using results (15.3.5), (15.3.6), and (15.3.8) in the virtual work statement (15.3.3) gives

Tn y

[K]¼ heAe[B]T[C][B] (15:3:15)and multiplying out the matrices gives the specific form

(15:3:16)Note that the stiffness matrix is always symmetric, and thus only the top-right (or bottom-left)portion need be explicitly written out If we also choose body forces that are element-wiseconstant, the body force vector {F} can be integrated to give

{F}¼heAe

3 {Fx Fy FxFy Fx Fy}

The {Q} matrix involves integration of the tractions around the element boundary, and itsevaluation depends on whether an element side falls on the boundary of the domain or islocated in the region’s interior The evaluation also requires a modeling decision on theassumed traction variation on the element sides Most problems can be adequately modeledusing constant, linear, or quadratic variation in the element boundary tractions For the typicaltriangular element shown in Figure 15-2, the {Q} matrix may be written as

Tn y

Tn y

Tn y

Tn y

ds(15:3:18)

Wishing to keep our study brief in theory, we take the simplest case of element-wise constantboundary tractions, which allows explicit calculation of the boundary integrals For this case,the integral over element side G12is given by

Tn y

c1Tn y

c2Tn x

c2Tn y

Txn

Tyn

Txn

Tyn00

whereL12is the length of side G12 Note that we have used the fact that along side G12, c1and

c2vary linearly and c3¼ 0 Following similar analysis, the boundary integrals along sides G23

and G31are found to be

Tyn

ds¼heL232

00

Tn x

Tny

ds¼heL312

Tn x

Tn y

00

15.4 FEM Problem Application

Applications using the linear triangular element discretize the domain into a connected set ofsuch elements; see, for example, Figure 15-1 The mesh geometry establishes which elementsare interconnected and identifies those on the boundary of the domain Using computerimplementation, each element in the mesh is mapped or transformed onto a master element

in a local coordinate system where all calculations are done The overall problem is thenmodeled by assembling the entire set of elements through a process of invoking equilibrium ateach node in the mesh This procedure creates a global assembled matrix system equation ofsimilar form as (15.3.13) Boundary conditions are then incorporated into this global system toreduce the problem to a solvable set of algebraic equations for the unknown nodal displace-ments We do not pursue the theoretical and operational details in these procedures, but ratherfocus attention on a particular example to illustrate some of the key steps in the process

EXAMPLE 15-1: Elastic Plate Under Uniform Tension

Consider the plane stress problem of an isotropic elastic plate under uniform tensionwith zero body forces as shown in Figure 15-3 For convenience, the plate is taken withunit dimensions and thickness and is discretized into two triangular elements as shown.This simple problem is chosen in order to demonstrate some of the basic FEM solutionprocedures previously presented More complex examples are discussed in the nextsection to illustrate the general power and utility of the numerical technique

The element mesh is labeled as shown with local node numbers within each elementand global node numbers (1–4) for the entire problem We start by developing theequation for each element and then assemble the two elements to model the entire plate.For element 1, the geometric parameters are b1¼
1, b2¼ 1, b3¼ 0, g1¼ 0, g2¼

1, g3¼ 1, and A1¼ 1=2 For the isotropic plane stress case, the element equationfollows from our previous work:

T

3

21

y

x

4

3 3

2 2

1 1

12

FIGURE 15-3 FEM analysis of elastic plate under uniform tension.

EXAMPLE 15-1: Elastic Plate Under Uniform Tension–Cont’d

In similar fashion for element 2, b1¼ 0, b2¼ 1, b3¼
1, g1¼
1, g2¼ 0, g3¼

1, A1¼ 1=2, and the element equation becomes

T2x(1)

T2y(1)

T3x(1)þ T(2) 2x