The elasticityElasticity theory introduced in Chap. 2 provides a theoretical basis for analyzing the reversible deformationDeformation of solids. However, when the applied stress is sufficiently high, irreversible deformationDeformation may occur, resulting in permanent changes even after the load on the solid is removed. PlasticityPlasticity theory extends the principles of elasticityElasticity theory to address these irreversible deformationsDeformation. Classical plasticityPlasticity theory has developed from studies on the deformationDeformation of metals in the late nineteenth century. Over time, its applications have broadened to include geomaterials like rocks and sand. Incorporating plasticityPlasticity theory is critical to modeling earthquake rupture because the regions where earthquake rupture is nucleated are typically under high stress. This chapter presents fundamental concepts of plasticityPlasticity theory, and the theory is applied to study the properties of the plastic zone formed at the tip of two-dimensional cracks. For the sake of clarity, we assumeRate-independent plasticity rate-independent plasticityPlasticity in this chapter.