### Problem StatementA central through-the-thickness crack, 50 mm long, propagates in a thermoset polymer in an unstable manner under an applied stress of 5 MPa. Find the fracture toughness, $ K_{Ic} $, of the polymer.### SolutionTo determine the fracture toughness $ K_{Ic} $ of the polymer, we can use the following standard equation for fracture mechanics in materials science:\[ K_{Ic} = Y \sigma \sqrt{\pi a} \]Where:- $ K_{Ic} $ is the fracture toughness.- $ Y $ is a dimensionless geometric factor which can often be approximated as 1 for central cracks in large plates.- $ \sigma $ is the applied stress.- $ a $ is the half-length of the crack.Given:- Crack length ($ 2a $) = 50 mm, hence $ a = 25 $ mm = 0.025 m.- Applied stress ($ \sigma $) = 5 MPa = 5 × 10^6 Pa.- Assuming $ Y = 1 $.### Calculations\[ K_{Ic} = 1 \times 5 \times 10^6 \times \sqrt{\pi \times 0.025} \]\[ K_{Ic} = 5 \times 10^6 \times \sqrt{0.07854} \]\[ K_{Ic} = 5 \times 10^6 \times 0.280 \]\[ K_{Ic} \approx 1.4 \times 10^6 \, \text{Pa} \cdot \sqrt{\text{m}} \]### ConclusionThe fracture toughness $ K_{Ic} $ of the thermoset polymer is approximately 1.4 MPa·m\uprhalf.

Given data: Applied stress, σ = 5 MPa Length of the crack, a = 50 mm For a centre through the…

Answered: A central through-the-thickness crack,…

A central through-the-thickness crack, 50 mm long, propagates in a thermoset polymer in an unstable manner under an applied stress of 5 MPa. Find the fracture toughness, K, of the polymer.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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