ME 535 Project Report

Numerical Computation of PlasticZone Shapes in Fracture Mechanics

V.V.H. Aditya (1327301) Rishi Pahuja (1327303)

June 9, 2014

Abstract

The purpose of this project is to numerically compute the plastic zone shapes nearthe crack tip using the Von-Mises criterion, when T-stress is involved. Three standardspecimens with different bi-axial ratios were taken to see the influence on the plasticzone shapes. The bi-axial ratio is directly proportional to the T-Stress or constantstress term in the William’s stress functions . The important aspect related to thisproject is to solve for a non linear equation in one variable with rational exponents.Traditional numerical root-finding methods fail to give a good and converging solutionto this kind of equation. Hybrid root finding method are used to solve for this kind ofequations and its advantages are discussed.

1

Contents1 Introduction 3

2 Theory 42.1 Williams Stress Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Von-Mises Stress Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 State of Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Biaxiality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Hybrid Root Finding Methods 73.1 Dekker’s Method (1969) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Brent’s Method (1973) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Interpretation and Analysis of the Problem 9

5 Results and Discussion 11

2

1 IntroductionFirstly let us understand the concepts of crack tip plastic zones and the plasticity correctionto crack length. From the definition of the stress intensity, based on the elastic stress fieldnear a crack tip, linear elastic theory predicts that the stress distribution (σij) near the cracktip, in polar coordinates (r, θ) with origin at the crack tip, has the form.:

σij = σnom

√a

2rfij(θ) = K

2πrf(θ) (1)

where,σij is the stress distribution with units MPaK is the stress intensity factor (SIF) with units MPa (m)1/2,fij is a dimensionless quantity that varies with the load and geometry.

We can see that, as r tends towards zero, the crack tip stresses become singular. Thefunction f(θ) when expanded to the first two terms , using Taylor series expansion, yieldsa constant term known as T stress which is considered to play a very important role in thestress distribution. The relation of T stress to the problem statement will be discussed inlater chapters. This implies that a yielded region exists in the material ahead of the cracktip for different stress values. The shape and size of the plastic zone can be determined, toa first order, from the simple models first proposed by Irwin. Consider a material with asimple elastic-perfectly plastic response (i.e. no strain hardening occurs). A first estimateof the plastic zone size ahead of the crack tip (Rp), along the plane of the crack, can beobtained by substituting the yield strength into the above equation :

rp = 12π

(K

σys

)2

(2)

(a) Polar coordinates atthe crack tip (b) Plastic Zone

Figure 1: First-order and second-order estimates of plastic zone size (ry and rp, respectively).The crosshatched area represents load that must be redistributed, resulting in a larger plasticzone

3

Plasticity is important in fracture mechanics, as the extent of plasticity, relative to spec-imen dimensions and crack size, determines the state of stress (plane strain or plane stress)and whether LEFM is applicable or not. In turn, stress state affects the direction of planes ofmaximum shear stress and hence the fracture plane. Thus fracture proceeds perpendicularlyto the maximum principal stress in plane strain, and at 45o to this direction in plane stress.An approximate idea of the shape can be obtained by substituting the near-tip stresses intoa yield criterion, e.g. the von Mises shear strain energy criterion, and allowing the angle ofthe stressed element to vary.

In our case the equation of plastic zone size is not so straight forward as with the firstorder approximation. Therefore this project focuses on root finding method which is usedto solve the complicated equation.

2 TheoryThe next few sections in this chapter we deal with Theory and formulating the equationsrequired to solve the problem in order to obtain the desired results. We use the equationsrelated to elasticity theory, LEFM and Failure theories (Von-Mises) to formulate the equationwe need to solve in the range [−π π].

2.1 Williams Stress DistributionA variety of techniques are available for analyzing stresses in cracked bodies. This sectionfocuses on approach developed by Williams and the following equations represent the cracktip stress fields according to Williams Stress Function. We use these equations to estimatethe questions asked in this computational project. We can observe that when r tends to 0the value of n = 1 leads to singularities in stresses the corresponding coefficients AIn andAIIn respectively. These coefficients can be defined in terms of SIF KI and KII as

AI1 = KI√2π

(3)

AII1 = −KII√2π

(4)

The William’s Stress Function given below with λ = n2 are given by the following equa-

tions.

(5)σxx =

∞∑n=1

(AIn

n

2

)(r)n

2−1{(2 + (−1)n + n

2 )cos(n2 − 1)θ − (n2 − 1)cos(n2 − 3)θ}

−∞∑n=1

(AIIn

n

2

)(r)n

2−1{(2− (−1)n + n

2 )sin(n2 − 1)θ − (n2 − 1)sin(n2 − 3)θ}

(6)σyy =

∞∑n=1

(AIn

n

2

)(r)n

2−1{(2− (−1)n − n

2 )cos(n2 − 1)θ + (n2 − 1)cos(n2 − 3)θ}

−∞∑n=1

(AIIn

n

2

)(r)n

2−1{(2 + (−1)n + n

2 )sin(n2 − 1)θ − (n2 − 1)sin(n2 − 3)θ}

4

(7)τxy =

∞∑n=1

(AIn

n

2

)(r)n

2−1{−((−1)n − n

2 )sin(n2 − 1)θ + (n2 − 1)sin(n2 − 3)θ}

−∞∑n=1

(AIIn

n

2

)(r)n

2−1{(2− (−1)n + n

2 )sin(n2 − 1)θ − (n2 − 1)sin(n2 − 3)θ}

The stress intensity factor defines the amplitude of the crack-tip singularity; all the stressand strain components at points near the crack tip increase in proportion to K, provided thecrack is stationary. The precise definition of the stress intensity factor is arbitrary, however;the constants AIn and AIIn would serve equally well for characterizing the singularity.

We can manipulate the equations 3, 4 and 5 as following by neglecting the higher orderterms in this expansion. We consider the 4A12 term in the equation which is also called theconstant T-Stress.

(8)σxx = AI1

2√r

(32cos

(θ

2

)+ 1

2cos(

5θ2

))+ 4AI2 + AII1

2√r

(72sin

(θ

2

)+ 1

2sin(

5θ2

))

T = 4A12 (Second order term in the expansion) (9)

σyy = AI1

2√r

(52cos

(θ

2

)− 1

2cos(

5θ2

))+ AII1

2√r

(12sin

(θ

2

)− 1

2sin(

5θ2

))(10)

τxy = AI1

2√r

(−1

2sin(θ

2

)+ 1

2sin(

5θ2

))− AII1

2√r

(32cos

(θ

2

)+ 1

2sin(

5θ2

))(11)

2.2 Von-Mises Stress CriterionAccording to the Von-Mises theory, a ductile solid will yield when the distortion energydensity reaches a critical value for that material and is given by the relation.

σ2ys = 1

2[(σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2

](12)

It can also be written in another form as shown below:

σ2ys = 1

2[(σxx − σyy)2 + (σxx − σzz)2 + (σzz − σxx)2 + 6τ 2

xy

](13)

2.3 State of StressIn general, the conditions ahead of a crack tip are neither plane stress nor plane strain.There are limiting cases where a two dimensional assumptions are valid, or at least providesa good approximation. The nature of the plastic zone that is formed ahead of a crack tipplays a very important role in the determination of the type of failure that occurs. Sincethe plastic region is larger in Plane stress than in Plane strain, plane stress failure will, ingeneral, be ductile, while, on the other hand, plane strain fracture will be brittle, even in amaterial that is generally ductile.

5

This phenomenon explains the peculiar thickness effect, observed in all fracture experi-ments, that thin samples exhibit a higher value of fracture toughness than thicker samplesmade of the same material and operating at the same temperature. From this it can besurmised that the plane stress fracture toughness is related to both metallurgical param-eters and specimen geometry while the plane strain fracture toughness depends more onmetallurgical factors than on the others.

Figure 2: Through-thickness plastic zone in a plate of intermediate thickness

Plane strain The triaxial stress state set up near the center of a thick specimen nearthe crack tip reduces the maximum shear stress available to drive plastic flow. We can statethat the mobility of the material is constrained by the inability to contract laterally. Thusthe plastic zone size is smaller, corresponding to low fracture toughness.

Plane stress Even in a thick specimen, the z-direction stress must approach zero at theside surfaces. Regions near the surface are therefore free of the triaxial stress constraint,and exhibit greater Shear-driven plastic flow. Thus the plastic zone size is larger,corresponding to high fracture toughness.

2.4 BiaxialityIn a cracked body subject to Mode I loading, the T stress, like KI , scales with the appliedload. The biaxiality ratio relates T to stress intensity:

β = T√πa

K1(14)

Both dimensionless parameters vary from positive to negative as the T-stress varied frompositive to negative. The β ratio results are presented in Figure 5.8 and the general trend isthat the as the crack length advances the B ratio increases.

Although Linear elastic stress analysis of sharp cracks predicts infinite stresses at thecrack tip. In real materials, however, stresses at the crack tip are finite because the crack-tipradius must be finite. Inelastic material deformation, such as plasticity in metals and crazingin polymers, leads to further relaxation of crack-tip stresses. The elastic stress analysisbecomes increasingly inaccurate as the inelastic region at the crack tip grows accounting forinaccurate LEFM solutions. A small region around the crack tip yields, leading to a small

6

Figure 3: Effect of a/W ratio on biaxiality ratio

plastic zone around it. Simple corrections to linear elastic fracture mechanics (LEFM) areavailable when moderate crack-tip yielding occurs. For more extensive yielding, one mustapply alternative crack-tip parameters that take nonlinear material behavior into accountwhich is beyond the scope of this project.

3 Hybrid Root Finding MethodsThe certainty of bisection method, its error bounds and predictable cost and the efficiencyof the secant method which has super-linear convergence are very advantageous for a rootfinding algorithm. Illinois algorithm, Dekker’s method and Brent’s method are some wellknown general purpose hybrid methods with super-linear convergence.

In this section two of the hybrid root finding methods will be discussed and the abovementioned equations will be solved with the two methods.

• Dekker’s Method

• Brent’s Method

3.1 Dekker’s Method (1969)The idea to combine the bisection method with the secant method goes back to T.J. Dekker(1969). Suppose that we want to solve the equation f(x) = 0. As with the bisectionmethod, we need to initialize Dekker’s method with two points, say ao and bo, such thatf(a0) and f(bo) have opposite signs. If f is continuous on [ao, bo], the intermediate valuetheorem guarantees the existence of a solution between ao and bo. Three points are involvedin every iteration: bk is the current iterate, i.e., the current guess for the root of f . ak isthe "contrapoint," i.e., a point such that f(ak) and f(bk) have opposite signs, so the interval[ak, bk] contains the solution. Furthermore, |f(bk)| should be less than or equal to |f(ak)|,so that bk is a better guess for the unknown solution than ak. bk−1 is the previous iterate

7

(for the first iteration, we set (bk−1 = ao). Two provisional values for the next iterate arecomputed. The first one is given by linear interpolation, also known as the secant method:

s =bk −

bk−bk−1f(bk)−f(bk−1)f(bk), if f(bk) 6= f(bk−1)

m otherwise

and the second one is given by the bisection method

m = ak + bk2

If the result of the secant method, s, lies strictly between bk and m, then it becomes thenext iterate (bk+1 = s), otherwise the midpoint is used (bk+1 = m). Then, the value of thenew contrapoint is chosen such that f(ak+1) and f(bk+1) have opposite signs. If f(ak) andf(bk+1) have opposite signs, then the contrapoint remains the same: ak+1 = ak. Otherwise,f(bk+ 1) and f(bk) have opposite signs, so the new contrapoint becomes ak+1 = bk. Finally,if |f(ak + 1)|< |f(bk + 1)|, then ak+1 is probably a better guess for the solution than bk+1,and hence the values of ak+1 and bk+1 are exchanged. This ends the description of a singleiteration of Dekker’s method. Dekker’s method performs well if the function f is reasonablywell-behaved. However, there are circumstances in which every iteration employs the secantmethod, but the iterates bk converge very slowly (in particular, |bk− bk−1| may be arbitrarilysmall). Dekker’s method requires far more iterations than the bisection method in this case.

In summary this method uses only secant line and bisection approximations, whereasBrent’s method adds inverse quadrative interpolation at some steps, for a slight speed gain.The next section describes the methodology of Brent’s Method.

3.2 Brent’s Method (1973)Brent’s method for approximately solving f(x) = 0, where f : R→ R, is a “hybrid” methodthat combines aspects of the bisection and secant methods with some additional featuresthat make it completely robust and usually very efficient. Like bisection, it is an “enclosure”method that begins with an initial interval across which f changes sign and, as the iterationsproceed, determines a sequence of nested intervals that share this property and decrease inlength. Convergence of the iterates is guaranteed, even in floating-point arithmetic. If f iscontinuous on the initial interval, then each of the decreasing intervals determined by themethod contains a solution, and the limit of the iterates is a solution. Like the bisection andsecant methods, the method requires only one evaluation of f at each iteration; in particular,f ′ is not required.

The following provides a rough outline of how the method works. The method buildsupon an earlier method of T.J. Dekker and is the basis of MATLAB’s fzero routine.

At each iteration, Brent’s method first tries a step of the secant method or somethingbetter. If this step is unsatisfactory, which usually means too long, too short, or too close toan endpoint of the current interval, then the step reverts to a bisection step. There is alsoa feature that occasionally forces a bisection step to guard against too little progress for toomany iterations. In the details of the method, a great deal of attention has been paid to

8

considerations of floating-point arithmetic (overflow and underflow, accuracy of computedexpressions, etc.).

An overview of the operation of the method is as follows:• The prerequisites of this method are

– a stopping tolerance δ > 0,– points a and b such that f(a)f(b) < 0– If necessary, a and b are exchanged so that |f(b)|≤ |f(a)| thus b is regarded as thebetter approximate solution. A third point c is initialized by setting c = a.

• At each iteration, the method maintains a, b, and c such that b 6= c and– f(b)f(c) < 0, so that a solution lies between b and c if f is continuous;– |f(b)|≤ |f(c)|, so that b can be regarded as the current approximate solution;– either a is distinct from b and c, or a = c and is the immediate past value of b.

• Each iteration proceeds as follows:– If |b− c|≤ δ|, then the method returns b as the approximate solution.– Otherwise, the method determines a trial point b̂ as follows:◦ If a = c, then b̂ is determined by linear (secant) interpolation:

b̂ = af(b)− bf(a)f(b)− f(a)

◦ Otherwise a, b, and c are distinct and b̂ is determined by using inverse quadraticinterpolation:

b̂ = af(b)f(c)(f(a)− f(b))(f(a)− f(c))+ bf(a)f(c)

(f(b)− f(a))(f(b)− f(c))+ cf(a)f(b)(f(c)− f(a))(f(c)− f(b))

� Determine α, β and γ such that p(y) = αy2 + βy + γ satisfies p(f(a)) =a, p(f(b)) = b, and p(f(c)) = c.

� set b̂ = γ.– if necessary, b̂ is adjusted or replaced with the bisection point.

b̂ = a+ b

2

till the set condition is achieved– Once b̂ has been finalized, a, b, c, and b̂ are used to determine new values of a, b,and c.

4 Interpretation and Analysis of the ProblemDefining the criteria for plane stress and plane strain conditions in the equations below.

9

σzz ={

0 Plane Stressν(σxx + σyy) Plane Strain

As we are looking into mode-I loading we make the term A12 equal to 0. We get thefollowing simplified set of stress distributions as:

σxx = AI1

2√r

(32cos

(θ

2

)+ 1

2cos(

5θ2

))+ T (15)

σyy = AI1

2√r

(52cos

(θ

2

)− 1

2cos(

5θ2

))(16)

τxy = AI1

2√r

(−1

2sin(θ

2

)+ 1

2sin(

5θ2

))(17)

When we plug the above equations into the Von Mises stress Criterion as mentioned inthe eqn(13) we get the following resulting equations which leaves the unknown variable r forthe angle ranging from [−pi, pi]

σys |PlaneStress=

T 2 + 3A211

8 r +A2

11 cos(θ2

)2

r−

3A211 cos

(θ2

)cos

(5 θ2

)8 r

−3A2

11 sin(θ2

)sin(

5 θ2

)8 r +

A11 T cos(θ2

)4√r

+3A11 T cos

(5 θ2

)4√r

1/2

(18)

σys |PlaneStrain=

T 2 − T 2 v + 3A112

8 r + T 2 v2 +A11

2 cos(θ2

)2

r

4A112 v2 cos

(θ2

)2

r

−3A11

2 cos(θ2

)cos(

5 θ2

)8 r −

3A112 sin

(θ2

)sin(

5 θ2

)8 r −

4A112 v cos

(θ2

)2

r

+A11 T cos

(θ2

)4√r

+3A11 T cos

(5 θ2

)4√r

−4A11 T v cos

(θ2

)√r

+4A11 T v

2 cos(θ2

)√r

1/2

(19)

These are the equations we need to solve using the above hybrid root finding methods toget the desired results.

Material and Geometric Properties for various SpecimenFor this problem we take the material to be Aluminum with yield strength σys = 20.6MPa.We define three different biaxial ratios for Compact Tension (CT) as Specimen− 1, Single-Edge Notch Bend (SE(B)) as Specimen − 2 and Single-Edge Notch Tension (SE(T)) as

10

(a) Compact Specimen(b) Single Edge Notched(Bending)

(c) Single Edge Notched(Tension)

Figure 4: Different Specimen Used (Specimen 1, 2 and 3 respectively)

Specimen − 3. All the specimen have a constant crack-width a/W ratio of 0.47, Poisson’sratio ν of 0.3 and width W of 50.4mm.

The Bi-axial stress ratios are given as:

β1 = −2.9795 ∗(a

W

)2+ 3.8857 ∗ a

W− 0.6433 . . . (CT )

β2 = 2.1029 ∗(a

W

)2− 0.6154 ∗ a

W− 0.0634 . . . (SE(B))

β3 = 2.2494 ∗(a

W

)2− 0.6572 ∗ a

W− 0.3703] . . . (SE(T ))

The Biaxiality stress ratios is defined in terms of constant stress term as: Now we calculatethe constant T stress using the equation

βi = T√πa

K1

5 Results and DiscussionMatlab subroutines for Bisection, Secant, Dekker and Brent’s method are coded which wereused to solve the problem. The equation (18) and (19) are solved by the hybrid root findingmethods and their relative advantages are discussed. All the methods have been used at thetolerance of machineepsilon of MATLAB. The bracketing of the root is done in the regionof [0 0.1], which is an initial guess.

The plastic zone shapes and sizes for these specimen are estimated in the plots shown infigure (5). When the biaxiality ratio is more eventually the constant stress term increasesand the curve will show the different behaviour. The second specimen for the given a

W= 0.47

ratio has the least value of B, when compared to all the other specimen so the curve exhibitsa behaviour similar to what we obtain when T − Stress ≈ 0. This explains the importanceof the Constant Stress Term (T) in the equation, irrespective of magnitude effects the plasticzone invariably. Hence the non singular term T is considered to be the most important termof all the higher order terms (H.O.T.).

11

Figure 5: Plastic Zone Shape computed by Exact Method

Now coming to the numerical part of analysis, the behaviour of the plastic zone shapeswhen evaluated by the exact method., is depicted in Figure-5. For the exact method theMatlab command ’Solve’ was used to solve the above mentioned equations. Obviously, theExact method took a lot of time for computing and its CPU time was found to be as highas 215000ms. The Plastic Zone shapes computed by the numerical root finding methodsare shown in Fig-(6) and fig-(7). We observe the typical behaviour of the secant methodin fig-(7b). While using secant method for solving this equation the subroutine does notcompute the values in the approximate ranges of [0, 60] [120, 280] for plane strain and only[120, 280] for plane stress as the method goes to infinity while computing the root, which isdiscussed as the major disadvantage of the secant method. The residual portion which weobserve in fig-(7b) is the only part Secant method is successful in computing the plastic zoneshapes. The other methods predict the plastic zone in a good agreement with the ExactMethod as shown in the figures below.

12

(a) Brent’s Method (b) Dekker’s Method

Figure 6: Plastic Zone Shapes by Hybrid Root finding methods)

(a) Bisection Method (b) Secant Method

Figure 7: Plastic Zone Shapes by Traditional Root finding methods)

13

(a) Specimen-1 (b) Specimen-2

(c) Specimen-3

Figure 8: Error Behavior for Brent’s Method for all the Specimen

(a) Specimen-1 (b) Specimen-2

(c) Specimen-3

Figure 9: Error Behavior for Dekker’s Method for all the Specimen

14

The relative error with respect to the solution obtained by hybrid methods to the exactmethod is depicted in the figures fig-(8) for the Brent’s method and fig-(9) for the Dekker’sMethod. The average error percentage is about ≈ 0.0005% which is considered to be a verygood estimate. We find that both the Brent’s and Dekker’s algorithms predict the rootswith a relative error of ≈ 10−15 between them.

Now to evaluate how good are the hybrid methods over the traditional root findingmethods we do a comprehensive time study by noting the computational time taken byMATLAB to solve the extensive equations (18) and (19) ,with various methods discussedabove, using the ’cputime’ routine in MATLAB. As discussed above the computing timetaken by the exact method is an extensive ≈ 215 seconds. The table shown below displaysthe time taken by various methods. It is observed that by using Brent’s method 98.5% ofthe computing time will be reduced to solve the equations mentioned above. Also Dekker’sMethod and other traditional methods significantly reduce the computing time to solve forthe equation. Brent’s Method stands out because of the super-linear rate of convergenceand the Inverse quadratic manipulation it incorporates in the algorithm. Although Secantmethod has super-linear convergence, its inability to find the roots when its denominator is0 makes it a inefficient method in a larger perspective.

Table 1: Computation Time Analysis (All data in Seconds unless specified)

Individual Time Analysis CombinedSpecimen 1 2 3 Total (using for loop)

METHOD Exact 75.8 71.3 71.8 215.4

Brent’s MethodCPU Time 1.3 1.3 1.2 3.1

Time Savings (%) 98.27 98.33 98.44 98.56

Dekker’s MethodCPU Time 1.7 2.0 1.7 3.7

Time Savings (%) 97.82 97.39 97.82 98.29

Bisection MethodCPU Time 5.6 5.8 5.8 17.3

Time Savings (%) 92.62 92.40 92.38 91.97

Secant MethodCPU Time 6.5 6.9 6.5 20.4

Time Savings (%) 91.41 90.94 91.41 90.52

15

References1. Brent, R. P. (1973), "Chapter 4: An Algorithm with Guaranteed Convergence for Find-

ing a Zero of a Function", Algorithms for Minimization without Derivatives, EnglewoodCliffs, NJ: Prentice-Hall, ISBN 0-13-022335-2

2. Dekker, T. J. (1969), "Finding a zero by means of successive linear interpolation", inDejon, B.; Henrici, P., Constructive Aspects of the Fundamental Theorem of Algebra,London: Wiley-Interscience, ISBN 978-0-471-20300-1

3. Professor Emery’s ME 535 Lecture Notes

4. Professor M.Ramulu’s ME 559 Lecture Notes

5. Fracture Mechanics: Fundamentals and Applications, Third Edition by T.L. Anderson,CRC Press, Jun 24, 2005

16