Computational Reality

For a new engineering design, we have to perform various analyses. Many of these analyses belong to mechanics. As a consequence of a static or dynamic loading, deformation and stress occur in the continuum body. If the stress lies below the yield limit, the deformation is recoverable upon unloading. This behavior is called an elastic response. This elastic response is instantaneous, i.e., rate of loading does not matter. In order to bring in the effect of the loading rate, we need a viscoelastic response. This behavior is modeled by changing the constitutive (material) equation. The deformation is still recovered upon unloading. In the case of remaining deformation after unloading, we need a constitutive equation modeling a plastic behavior. In all of the aforementioned phenomena, we ignore any change in temperature, thus the process is isothermal. We will discuss mechanical systems and compute the motion of particles belonging to a continuum body. We start with linear elasticity, then solve a problem with geometric nonlinearities, and finally incorporate material nonlinearities. All of these computations belong to elastostatics. By including time rate in the equations, we start off with dynamics; examples of linear and fractional rheology are presented. Moreover, the plastic deformation is addressed, where the material starts flowing beyond the yield stress. We change the understanding of motion from a solid body to fluid. We present the computation of flows of linear and nonlinear fluids. We finalize the chapter with an application of a deformable solid in a viscous fluid, known as a fluid-structure interaction. All applications are discussed theoretically and implemented by using open-source codes.

Thermodynamics includes a theoretical and an applied part. The applied thermodynamics aims for calculating the temperature distribution in a continuum body. We will study this approach for macroscopic and microscopic systems. The difference between macroscopic and microscopic systems relies on the used constitutive equation. The theoretical thermodynamics has the goal of defining constitutive (material) equations that close the balance equations. By using thermodynamics we will derive the constitutive equations necessary in the computational reality. In the preceding chapter we have employed many constitutive equations with an ad-hoc method. In this chapter we will answer the question of how to derive these equations in a thermodynamically consistent manner. We will analyze such an approach and derive the Navier–Stokes–Fourier equations for a viscous fluid. We will employ the same method for viscoelastic materials and then for plastic deformations. Much use of the method will be made in the next chapter, too.

Electromagnetism is the theory of electromagnetic interactions with matter. In this theory there occur various new quantities; and this makes a straight-forward introduction of equations challenging. These new quantities lead to electromagnetic fields, which can be measured. However, they often lack a clear interpretation. For example, a moving electric charge fails to be understood completely, but an electric current is something we use in our daily lives. In order to establish a knowledge of electromagnetism, we will motivate governing equations one-by-one with applications. We consider a conducting wire and investigate how it heats up due to the production term in the balance of internal energy. We introduce electric field and magnetic flux in polarized materials. By using thermodynamical principles we derive the constitutive equations and solve a problem addressing the thermoelectric coupling. Then we include plasticity. Moreover, a piezoelectric sensor is discussed by deriving the constitutive equations in a thermodynamically consistent way. A magnetohydrodynamical problem is presented in the last section. All applications are discussed theoretically and implemented by using open-source codes.