ME 535 Project Report Numerical Computation of Plastic Zone Shapes in Fracture Mechanics V.V.H. Aditya (137301) Rishi Pahuja (137303) June 9, 014 Abstract The purpose of this project is to numerically compute the plastic zone shapes near the crack tip using the Von-Mises criterion, when T-stress is involved. Three standard specimens with different bi-axial ratios were taken to see the influence on the plastic zone shapes. The bi-axial ratio is directly proportional to the T-Stress or constant stress term in the William s stress functions. The important aspect related to this project 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 solution to this kind of equation. Hybrid root finding method are used to solve for this kind of equations and its advantages are discussed. 1
Contents 1 Introduction 3 Theory 4.1 Williams Stress Distribution........................... 4. Von-Mises Stress Criterion............................ 5.3 State of Stress................................... 5.4 Biaxiality..................................... 6 3 Hybrid Root Finding Methods 7 3.1 Dekker s Method (1969)............................. 7 3. Brent s Method (1973).............................. 8 4 Interpretation and Analysis of the Problem 9 5 Results and Discussion 11
1 Introduction Firstly let us understand the concepts of crack tip plastic zones and the plasticity correction to crack length. From the definition of the stress intensity, based on the elastic stress field near a crack tip, linear elastic theory predicts that the stress distribution (σ ij ) near the crack tip, in polar coordinates (r, θ) with origin at the crack tip, has the form.: a σ ij = σ nom r f ij(θ) = K f(θ) (1) πr where, σ ij is the stress distribution with units MP a K is the stress intensity factor (SIF) with units MP a (m) 1/, f ij 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. The function f(θ) when expanded to the first two terms, using Taylor series expansion, yields a constant term known as T stress which is considered to play a very important role in the stress distribution. The relation of T stress to the problem statement will be discussed in later chapters. This implies that a yielded region exists in the material ahead of the crack tip for different stress values. The shape and size of the plastic zone can be determined, to a first order, from the simple models first proposed by Irwin. Consider a material with a simple elastic-perfectly plastic response (i.e. no strain hardening occurs). A first estimate of the plastic zone size ahead of the crack tip (R p ), along the plane of the crack, can be obtained by substituting the yield strength into the above equation : r p = 1 ( ) K () π σ ys (a) Polar coordinates at the 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 plastic zone 3
Plasticity is important in fracture mechanics, as the extent of plasticity, relative to specimen 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 of maximum shear stress and hence the fracture plane. Thus fracture proceeds perpendicularly to the maximum principal stress in plane strain, and at 45 o to this direction in plane stress. An approximate idea of the shape can be obtained by substituting the near-tip stresses into a yield criterion, e.g. the von Mises shear strain energy criterion, and allowing the angle of the stressed element to vary. In our case the equation of plastic zone size is not so straight forward as with the first order approximation. Therefore this project focuses on root finding method which is used to solve the complicated equation. Theory The next few sections in this chapter we deal with Theory and formulating the equations required to solve the problem in order to obtain the desired results. We use the equations related to elasticity theory, LEFM and Failure theories (Von-Mises) to formulate the equation we need to solve in the range [ π π]..1 Williams Stress Distribution A variety of techniques are available for analyzing stresses in cracked bodies. This section focuses on approach developed by Williams and the following equations represent the crack tip stress fields according to Williams Stress Function. We use these equations to estimate the questions asked in this computational project. We can observe that when r tends to 0 the value of n = 1 leads to singularities in stresses the corresponding coefficients A In and A IIn respectively. These coefficients can be defined in terms of SIF K I and K II as A I1 = K I (3) π A II1 = K II (4) π The William s Stress Function given below with λ = n are given by the following equations. ( n σ xx = A In n=1 ( n=1 ) (r) n 1 {( + ( 1) n + n )cos(n 1)θ (n 1)cos(n 3)θ} A IIn n ( n σ yy = A In n=1 ( n=1 ) (r) n 1 {( ( 1) n + n )sin(n 1)θ (n 1)sin(n 3)θ} ) (r) n 1 {( ( 1) n n )cos(n 1)θ + (n 1)cos(n 3)θ} A IIn n ) (r) n 1 {( + ( 1) n + n )sin(n 1)θ (n 1)sin(n 3)θ} 4 (5) (6)
( n τ xy = A In n=1 ( n=1 ) (r) n 1 { (( 1) n n )sin(n 1)θ + (n 1)sin(n 3)θ} A IIn n ) (r) n 1 {( ( 1) n + n )sin(n 1)θ (n 1)sin(n 3)θ} (7) The stress intensity factor defines the amplitude of the crack-tip singularity; all the stress and strain components at points near the crack tip increase in proportion to K, provided the crack is stationary. The precise definition of the stress intensity factor is arbitrary, however; the constants A In and A IIn would serve equally well for characterizing the singularity. We can manipulate the equations 3, 4 and 5 as following by neglecting the higher order terms in this expansion. We consider the 4A 1 term in the equation which is also called the constant T-Stress. σ xx = A I1 r ( 3 cos ( ) θ + 1 ( )) 5θ cos + 4A I + A ( II1 7 r sin ( ) θ + 1 ( )) 5θ sin T = 4A 1 (Second order term in the expansion) (9) (8) σ yy = A ( I1 5 r cos τ xy = A ( I1 1 ( θ r sin ( ) θ 1 ( 5θ cos ) + 1 sin ( 5θ )) + A II1 r )) ( 1 sin A ( II1 3 r cos ( ) θ 1 ( )) 5θ sin ( ) θ + 1 ( )) 5θ sin (10) (11). Von-Mises Stress Criterion According to the Von-Mises theory, a ductile solid will yield when the distortion energy density reaches a critical value for that material and is given by the relation. σys = 1 [ (σ1 σ ) + (σ σ 3 ) + (σ 3 σ 1 ) ] (1) It can also be written in another form as shown below: σ ys = 1 [ (σxx σ yy ) + (σ xx σ zz ) + (σ zz σ xx ) + 6τ xy] (13).3 State of Stress In 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 provides a good approximation. The nature of the plastic zone that is formed ahead of a crack tip plays a very important role in the determination of the type of failure that occurs. Since the plastic region is larger in Plane stress than in Plane strain, plane stress failure will, in general, be ductile, while, on the other hand, plane strain fracture will be brittle, even in a material that is generally ductile. 5
This phenomenon explains the peculiar thickness effect, observed in all fracture experiments, that thin samples exhibit a higher value of fracture toughness than thicker samples made of the same material and operating at the same temperature. From this it can be surmised that the plane stress fracture toughness is related to both metallurgical parameters and specimen geometry while the plane strain fracture toughness depends more on metallurgical factors than on the others. Figure : 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 near the crack tip reduces the maximum shear stress available to drive plastic flow. We can state that the mobility of the material is constrained by the inability to contract laterally. Thus the 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 the side 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..4 Biaxiality In a cracked body subject to Mode I loading, the T stress, like K I, scales with the applied load. The biaxiality ratio relates T to stress intensity: β = T πa K 1 (14) Both dimensionless parameters vary from positive to negative as the T-stress varied from positive to negative. The β ratio results are presented in Figure 5.8 and the general trend is that the as the crack length advances the B ratio increases. Although Linear elastic stress analysis of sharp cracks predicts infinite stresses at the crack tip. In real materials, however, stresses at the crack tip are finite because the crack-tip radius must be finite. Inelastic material deformation, such as plasticity in metals and crazing in polymers, leads to further relaxation of crack-tip stresses. The elastic stress analysis becomes increasingly inaccurate as the inelastic region at the crack tip grows accounting for inaccurate 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) are available when moderate crack-tip yielding occurs. For more extensive yielding, one must apply alternative crack-tip parameters that take nonlinear material behavior into account which is beyond the scope of this project. 3 Hybrid Root Finding Methods The certainty of bisection method, its error bounds and predictable cost and the efficiency of the secant method which has super-linear convergence are very advantageous for a root finding algorithm. Illinois algorithm, Dekker s method and Brent s method are some well known general purpose hybrid methods with super-linear convergence. In this section two of the hybrid root finding methods will be discussed and the above mentioned 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 bisection method, we need to initialize Dekker s method with two points, say a o and b o, such that f(a 0 ) and f(b o ) have opposite signs. If f is continuous on [a o, b o ], the intermediate value theorem guarantees the existence of a solution between a o and b o. Three points are involved in every iteration: b k is the current iterate, i.e., the current guess for the root of f. a k is the "contrapoint," i.e., a point such that f(a k ) and f(b k ) have opposite signs, so the interval [a k, b k ] contains the solution. Furthermore, f(b k ) should be less than or equal to f(a k ), so that b k is a better guess for the unknown solution than a k. b k 1 is the previous iterate 7
(for the first iteration, we set (b k 1 = a o ). Two provisional values for the next iterate are computed. The first one is given by linear interpolation, also known as the secant method: b k b k b k 1 s = f(b f(b k ) f(b k 1 ) k), if f(b k ) f(b k 1 ) m otherwise and the second one is given by the bisection method m = a k + b k If the result of the secant method, s, lies strictly between b k and m, then it becomes the next iterate (b k+1 = s), otherwise the midpoint is used (b k+1 = m). Then, the value of the new contrapoint is chosen such that f(a k+1 ) and f(b k+1 ) have opposite signs. If f(a k ) and f(b k+1 ) have opposite signs, then the contrapoint remains the same: a k+1 = a k. Otherwise, f(bk + 1) and f(b k ) have opposite signs, so the new contrapoint becomes a k+1 = b k. Finally, if f(ak + 1) < f(bk + 1), then a k+1 is probably a better guess for the solution than b k+1, and hence the values of a k+1 and b k+1 are exchanged. This ends the description of a single iteration of Dekker s method. Dekker s method performs well if the function f is reasonably well-behaved. However, there are circumstances in which every iteration employs the secant method, but the iterates b k converge very slowly (in particular, b k b k 1 may be arbitrarily small). 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, whereas Brent 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. Brent s Method (1973) Brent s method for approximately solving f(x) = 0, where f : R R, is a hybrid method that combines aspects of the bisection and secant methods with some additional features that 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 iterations proceed, determines a sequence of nested intervals that share this property and decrease in length. Convergence of the iterates is guaranteed, even in floating-point arithmetic. If f is continuous on the initial interval, then each of the decreasing intervals determined by the method contains a solution, and the limit of the iterates is a solution. Like the bisection and secant 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 builds upon 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 something better. If this step is unsatisfactory, which usually means too long, too short, or too close to an endpoint of the current interval, then the step reverts to a bisection step. There is also a feature that occasionally forces a bisection step to guard against too little progress for too many 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 computed expressions, 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 the better 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 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 ˆb as follows: If a = c, then ˆb is determined by linear (secant) interpolation: ˆb = af(b) bf(a) f(b) f(a) Otherwise a, b, and c are distinct and ˆb is determined by using inverse quadratic interpolation: ˆb = 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) = αy + βy + γ satisfies p(f(a)) = a, p(f(b)) = b, and p(f(c)) = c. set ˆb = γ. if necessary, ˆb is adjusted or replaced with the bisection point. till the set condition is achieved ˆb = a + b Once ˆb has been finalized, a, b, c, and ˆb are used to determine new values of a, b, and c. 4 Interpretation and Analysis of the Problem Defining the criteria for plane stress and plane strain conditions in the equations below. 9
{ 0 Plane Stress σ zz = ν(σ xx + σ yy ) Plane Strain As we are looking into mode-i loading we make the term A 1 equal to 0. We get the following simplified set of stress distributions as: σ xx = A ( ( ) I1 3 θ r cos + 1 ( )) 5θ cos + T (15) σ yy = A ( ( ) I1 5 θ r cos 1 ( )) 5θ cos (16) τ xy = A ( I1 1 ( ) θ r sin + 1 ( )) 5θ sin (17) When we plug the above equations into the Von Mises stress Criterion as mentioned in the eqn(13) we get the following resulting equations which leaves the unknown variable r for the angle ranging from [ pi, pi] σ ys P lanestress = T + 3 A 11 ( 8 r + A 11 cos θ ) r 3 A 11 sin ( θ 8 r ) ( ) sin 5 θ 3 A 11 cos ( θ 8 r + A 11 T cos ( ) θ 4 r ) ( ) cos 5 θ + 3 A11 T cos ( 5 θ 4 r ) 1/ (18) σ ys P lanestrain = T T v + 3 A 11 8 r 3 A 11 cos ( θ 8 r + A 11 T cos ( ) θ 4 r ) ( ) cos 5 θ + 3 A11 T cos ( 5 θ 4 r + T v + A11 cos ( ) θ r 3 A 11 sin ( θ 8 r ) ( ) sin 5 θ 4 A 11 v cos ( ) θ r 4 A 11 v cos ( ) θ r ) 4 A 11 T v cos ( ) θ + 4 A 11 T v cos ( ) θ r r 1/ (19) These are the equations we need to solve using the above hybrid root finding methods to get the desired results. Material and Geometric Properties for various Specimen For this problem we take the material to be Aluminum with yield strength σ ys = 0.6 MP a. We define three different biaxial ratios for Compact Tension (CT) as Specimen 1, Single- Edge Notch Bend (SE(B)) as Specimen 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, and 3 respectively) Specimen 3. All the specimen have a constant crack-width a/w ratio of 0.47, Poisson s ratio ν of 0.3 and width W of 50.4 mm. The Bi-axial stress ratios are given as: ( a β 1 =.9795 W ( a β =.109 W ) + 3.8857 a W ) 0.6154 a W 0.6433... (CT ) 0.0634... (SE(B)) ( ) a a β 3 =.494 0.657 0.3703]... (SE(T )) W W The Biaxiality stress ratios is defined in terms of constant stress term as: Now we calculate the constant T stress using the equation 5 Results and Discussion β i = T πa K 1 Matlab subroutines for Bisection, Secant, Dekker and Brent s method are coded which were used to solve the problem. The equation (18) and (19) are solved by the hybrid root finding methods and their relative advantages are discussed. All the methods have been used at the tolerance of machineepsilon of MATLAB. The bracketing of the root is done in the region of [0 0.1], which is an initial guess. The plastic zone shapes and sizes for these specimen are estimated in the plots shown in figure (5). When the biaxiality ratio is more eventually the constant stress term increases and 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 exhibits a behaviour similar to what we obtain when T Stress 0. This explains the importance of the Constant Stress Term (T) in the equation, irrespective of magnitude effects the plastic zone invariably. Hence the non singular term T is considered to be the most important term of 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 shapes when evaluated by the exact method., is depicted in Figure-5. For the exact method the Matlab command Solve was used to solve the above mentioned equations. Obviously, the Exact method took a lot of time for computing and its CPU time was found to be as high as 15000 ms. The Plastic Zone shapes computed by the numerical root finding methods are shown in Fig-(6) and fig-(7). We observe the typical behaviour of the secant method in fig-(7b). While using secant method for solving this equation the subroutine does not compute the values in the approximate ranges of [0, 60] [10, 80] for plane strain and only [10, 80] for plane stress as the method goes to infinity while computing the root, which is discussed as the major disadvantage of the secant method. The residual portion which we observe in fig-(7b) is the only part Secant method is successful in computing the plastic zone shapes. The other methods predict the plastic zone in a good agreement with the Exact Method as shown in the figures below. 1
(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- (c) Specimen-3 Figure 8: Error Behavior for Brent s Method for all the Specimen (a) Specimen-1 (b) Specimen- (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 exact method is depicted in the figures fig-(8) for the Brent s method and fig-(9) for the Dekker s Method. The average error percentage is about 0.0005% which is considered to be a very good estimate. We find that both the Brent s and Dekker s algorithms predict the roots with a relative error of 10 15 between them. Now to evaluate how good are the hybrid methods over the traditional root finding methods we do a comprehensive time study by noting the computational time taken by MATLAB to solve the extensive equations (18) and (19),with various methods discussed above, using the cputime routine in MATLAB. As discussed above the computing time taken by the exact method is an extensive 15 seconds. The table shown below displays the time taken by various methods. It is observed that by using Brent s method 98.5% of the computing time will be reduced to solve the equations mentioned above. Also Dekker s Method and other traditional methods significantly reduce the computing time to solve for the equation. Brent s Method stands out because of the super-linear rate of convergence and the Inverse quadratic manipulation it incorporates in the algorithm. Although Secant method has super-linear convergence, its inability to find the roots when its denominator is 0 makes it a inefficient method in a larger perspective. Table 1: Computation Time Analysis (All data in Seconds unless specified) Individual Time Analysis Combined Specimen 1 3 Total (using for loop) METHOD Exact 75.8 71.3 71.8 15.4 Brent s Method CPU Time 1.3 1.3 1. 3.1 Time Savings (%) 98.7 98.33 98.44 98.56 Dekker s Method CPU Time 1.7.0 1.7 3.7 Time Savings (%) 97.8 97.39 97.8 98.9 Bisection Method CPU Time 5.6 5.8 5.8 17.3 Time Savings (%) 9.6 9.40 9.38 91.97 Secant Method CPU Time 6.5 6.9 6.5 0.4 Time Savings (%) 91.41 90.94 91.41 90.5 15
References 1. Brent, R. P. (1973), "Chapter 4: An Algorithm with Guaranteed Convergence for Finding a Zero of a Function", Algorithms for Minimization without Derivatives, Englewood Cliffs, NJ: Prentice-Hall, ISBN 0-13-0335-. Dekker, T. J. (1969), "Finding a zero by means of successive linear interpolation", in Dejon, B.; Henrici, P., Constructive Aspects of the Fundamental Theorem of Algebra, London: Wiley-Interscience, ISBN 978-0-471-0300-1 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 4, 005 16