How can mathematical models predict the capabilities of a structure from material properties?

Mathematical models predict a structure's capabilities by integrating fundamental principles of solid mechanics with specific material data. To analyze how a structure resists mechanical forces, designers typically rely on three core considerations:1

• Equations of Motion: Applying Newtonian laws to express the conservation of linear and angular momentum for finite bodies.1
• Geometry of Deformation: Expressing strains based on displacement field gradients.1
• Constitutive Relations: Defining the specific relationships between stress and strain, which are characteristic of the material, temperature, and time scale.1

These models are often supplemented by thermodynamics and electromagnetic equations for complex environments. By performing a stress analysis, designers can identify areas of high stress and predict potential failure due to deformation or fracture.13

While constitutive relations are often determined through experimental materials testing, they can sometimes be estimated by applying mechanics at the microscale—such as modeling atomic forces or dislocation motion—to predict overall behavior. Additionally, statistical theories are used to extrapolate performance under random, real-world conditions from cyclic test data.12