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Numerical computation of plastic zone shapes

Numerical computation of plastic zone shapes. V.V.H. Aditya (1327301) Rishi Pahuja (1327303). outline. Introduction to Fracture Mechanics Defining Concepts Framework of the Problem Experimental Data Where we are Stuck? Root finding Introduction Root Finding Basics

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Numerical computation of plastic zone shapes

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  1. Numerical computation of plastic zone shapes V.V.H. Aditya (1327301) Rishi Pahuja(1327303)

  2. outline • Introduction to Fracture Mechanics • Defining Concepts • Framework of the Problem • Experimental Data • Where we are Stuck? • Root finding Introduction • Root Finding Basics • Hybrid Root Finding Methods • Results and Discussion • Relation of Results to Theory • Questions?

  3. Introduction to fracture mechanics • Stress Distribution at the crack tip • Biaxiality proportional to T- Stress,

  4. FRAMEWORK OF the problem • Through-thickness plastic zone in a plate of intermediate thickness

  5. Experimental Data Specimen-3 Single Edge Notched (Tension) Specimen-1 Compact Tension Specimen-2 Single Edge Notched (Bending) Compact Tension (CT): Single-Edge Notch Bend (SEN(B)): Single-Edge Notch Tension (SEN(T)): Material: Aluminum; =20.6 MPa; W=50.4 mm; a/W=0.47;

  6. Where We Are Stuck? • After evaluating the stress distribution in the Von-Mises Stress equation we end up with • Equation (1) for plane stress and Equation (2) for plane Strain which has to be evaluated at [-]

  7. Root FINDING METHODS Introduction • Why do we use the numerical roots finding methods? • Who cares? • Mathematicians • Physicians • All Engineering Disciplines • Lots of people • We often need to find solution(s) of equation f(x) = 0. Courtesy: http://home.ubalt.edu/ntsbarsh/zero/zero.htm

  8. Root Finding BASICS • Given some function, find location where f(x)=0 • Need: • Starting position x0, hopefully close to solution • Ideally, points that bracket the root • The possible modes of failure of root finding Bracketing • Ideal Cases • Well Behaved Function • Initial guess closer to the root

  9. HYBRID ROOT FINDING METHOD’s Dekker’s Method Brent’s Method • Reliability of bisection, speed of secant /interpolation • Super Linear Convergence • Function should be well behaved • Little slower in convergence than Brent’s Method

  10. Error and Computation Time Analysis • Relative Error ranging from 10-3 <err<10-10 • Error Between Brent’s and Dekker’s Method ≈ macheps • Brent’s method is fastest in terms of computing time

  11. Interpreting Results • In the order of best agreement with the Exact Solution Brent>Dekker>Bisection>Secant • Brent converges faster than the remaining three methods • Relative error between Brent’s and Dekker’s Method is in the order of 10-15 ≈ Machine epsilon • Secants Method drawback demonstrated

  12. THANK YOU QUESTIONS ??? -email us @ adityads@uw.edu or rpahuja@uw.edu

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