# Rapid Reliable Solution of the Parametrized Partial Differential Equations of Continuum Mechanics and Transport

## Pedagogy The CenterCrack Problem Author: DBP Huynh

Q1 Show that the non-dimensional Energy Release Rate (ERR) is given by G (μ) = – ∂ / ∂ μ1 (s (μ)). Find the corresponding non-dimensional SIF [1].

Q2 Write a matlab script which evaluates the ERR and SIF as a function of crack half-length μ1 and specimen half-length μ2.
Hint: Evaluate the derivative by a simple finite-difference approximation — the "virtual crack extension method" [2].

Q3 We next address a fracture application. We consider for a Zr–Ti–Ni–Cu–Be bulk metallic glass material [3] with Young's modulus = 96 GPa (and Poisson ratio ν = 0.36), a specimen of half-width = 5 cm and aspect ratio μ2 = 2, and tensile loading of = 4 kPa.
(a) Plot the dimensional ERR as a function of crack length ã.

(b) If the material Zr–Ti–Ni–Cu–Be is modeled as brittle with Fracture Toughness KIC = 55 MPa/m1/2, and we invoke Griffith's criterion [3], for what crack length ã will the specimen fail?
Q4 Finally, we address a fatigue application. We consider the A514 Martensitic steel material [3] with "material" properties KIC = 150 MPa/m1/2, C = 0.66 × 10-8 and m = 2.25, a specimen of half-width = 5 cm and aspect ratio μ2 = 2, and a constant amplitude load fluctuation of Δ = 1.5 kPa.
(a) Invoke Paris's law to simulate fatigue crack growth for several initial crack length values ã0. How do different initial crack length values affect the rate of crack growth?

(b) Verify your numerical crack growth result by comparing it with the result obtained based on the analytical SIF [1,3] for the case ã0 = 1.5 cm.

References
[1] DBP Huynh and AT Patera, Reduced basis approximation and a posteriori error estimation for stress intensity factors. Int J Numer Meth Engng, 72(10):1219–1259, 2007.

[2] DM Parks, A stiffness derivative finite element technique for determination of crack tip stress intensity factors. Int J Frac, 10(4):487–502, 1974.

[3] JM Barsom and ST Rolfe, Fracture and Fatigue Control in Structures. American Society for Testing and Metals, 1999.