The tensile strength of brittle materials may be determined using a variation of Equation \(8.1 .\) Compute the critical crack tip radius for a glass specimen that experiences tensile fracture at an applied stress of \(70 \mathrm{MPa}\) \((10,000 \mathrm{psi}) .\) Assume a critical surface crack length of \(10^{-2} \mathrm{mm}\) and a theoretical fracture strength of \(E / 10,\) where \(E\) is the modulus of elasticity

The critical crack tip radius is a fundamental concept in the study of material fracture, especially for brittle materials like glass or ceramics. It refers to the size of the tip of a crack at which the material will fail under a given amount of stress.

When the stress at the tip of a crack reaches a critical level, it causes the crack to propagate, leading to fracture. The smaller the crack tip radius, the higher the stress concentration, which increases the likelihood of the material fracturing. Engineers use this concept to predict when a material might fail and prevent catastrophic failures in structures.

In the given exercise, we calculated the critical crack tip radius for a glass specimen, which is crucial for understanding when the glass will break under stress. Remember that accurate assessment of this radius is vital for safety and design in engineering applications.

The fracture strength of a material is the stress at which a material cracks or fails. The calculation involves understanding the relationship between the internal structure's stresses and the external forces applied. For brittle materials, which do not deform significantly before breaking, the fracture strength can be a direct indicator of the material's ability to withstand stress.

Through the solution steps, we relate the applied stress and the fracture strength to find the modulus of elasticity, ultimately being used to calculate the critical crack tip radius. Such calculations are instrumental in predicting the failure point of materials, which informs engineers and designers when selecting appropriate materials for various applications.

The modulus of elasticity, often denoted as E, is a measure of a material's stiffness or rigidity. It is a fundamental property that describes the material's elastic behavior when forces are applied to it. Higher values of E indicate a stiffer material that does not easily deform, whereas lower values suggest a more flexible one.

In the context of the exercise, the modulus of elasticity is used to calculate the fracture strength, highlighting the relationship between the internal resistance to deformation and the external stress. Understanding E is crucial for engineers to make informed decisions about material selection and predict how structures will respond to forces.

The stress-strain relationship is a cornerstone of material science, expressing how a material deforms in response to stress. The relationship is typically linear for elastic materials, meaning that the material returns to its original shape after the stress is removed, up to a certain limit known as the yield strength.

Stress is defined as force per unit area, and strain is the deformation experienced by the material as a result of that stress. Understanding this relationship allows scientists and engineers to anticipate material behavior under load and is used to determine the modulus of elasticity. The exercise we are analyzing does not delve deep into the stress-strain curve, but it does involve the applied stress and properties derived from that relationship, such as the modulus of elasticity and the fracture strength.