A steel alloy to be used for a spring application must have a modulus of resilience of at least \(2.07 \mathrm{MPa}(300 \mathrm{psi}) .\) What must be its minimum yield strength?

Yield strength is a fundamental property of materials that marks the onset of plastic deformation. It is the stress at which a material begins to deform permanently. Before a material reaches its yield point, it behaves elastically, meaning it will return to its original shape when the applied stress is removed.

Understanding yield strength is crucial for engineers and designers as it determines how a material will behave under load before it starts to deform irreversibly. It is especially important in the design of structures and mechanical components that must withstand specific load conditions without failing.

In the context of our exercise where we determine the minimum yield strength for a steel alloy used in a spring, the yield strength represents the maximum stress that the spring material can handle without being permanently bent or stretched.

The modulus of elasticity, also known as Young's modulus, is a measure of the stiffness of a material. It defines the relationship between stress (force per unit area) and strain (deformation) in the elastic region of the material’s stress-strain curve.

A higher modulus of elasticity indicates a stiffer material that does not deform much under load. Materials with a high modulus of elasticity, such as steel or diamond, are ideal for applications where rigidity and maintaining shape under stress are important.

In our exercise, knowing the modulus of elasticity, assumed to be 200 GPa for the steel alloy, allows us to calculate the modulus of resilience, which further enables us to derive the minimum yield strength necessary for the spring application. It's a key factor in determining how much energy the material can store elastically.

Strain energy refers to the energy stored in a material due to elastic deformation. When a material is deformed within its elastic limit, the work done on it is stored in the form of strain energy. Once the stress is removed, this energy is released as the material returns to its original shape.

The concept of strain energy is particularly relevant in spring design. Springs are designed to store and release energy. The modulus of resilience, which is the area under the stress-strain curve up to the yield point, represents the maximum strain energy per unit volume that can be absorbed without permanent deformation.

In the steel alloy spring from our problem, the modulus of resilience specified at 2.07 MPa was used to calculate the minimum yield strength required to make sure the spring can perform as needed without undergoing plastic deformation.