# Fracture Mechanics Homework.pdf - Finite Element Analysis...

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Finite Element Analysis II (ENME 6333)Professor H. WaismanSpring 2022Columbia UniversityHomework set 2 (due 03/10/2022)Fracture MechanicsConsider a square plate with an edge crack, as shown in Figure 1. The dimension ofthis square plate is L = 1 and the crack length is a = 0.5.Young’s modulus is E =3e7 andPoisson’sratio isν =0.3. All units are assumed to be consistent and theplate is assumed to be in plane stress conditions.Figure 1: A square plate with an edge crackExact displacements, from Williams asymptotic solution, corresponding to mode Ifracture with the stress intensity factor (SIF) KI= 100, are imposed on the externalboundaries (not on crack surfaces), as follows:??=𝐾𝐼2??2𝜋cos𝜃2[𝜅 − 1 + 2sin2𝜃2]??=𝐾𝐼2??2𝜋sin𝜃2[𝜅 + 1 − 2cos2𝜃2]
where?is the shear modulus and𝜅the Kolosov constant (κ = (3 − ?)/(1 + ?)forplane stress problems). Note that the range of the polar coordinate𝜃is from– 𝜋(lower crack surface) to𝜋(upper crack surface) and KI= 100 is the referencesolution.Assignment:Part 1: Linear Elastic Fracture Mechanics

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