In the present work, a spatial analysis of the stress state and the strain state in a piezoelectric material at points near the crack tip for the case of tensile, transverse shear, and longitudinal shear deformation is made using the theory of linear elastic fracture mechanics and linear elasticity theory. The method of analytic functions with a complex variable in polar coordinates is used. The functions of mechanical stress and electrical potential are introduced. Our analysis shows the characteristic behavior of the coordinates of the mechanical stress tensor, piezoelectric stress, and the relative strain tensor and electric field. These components of the electromechanical state of the material have singular points at the crack tip. This vicinity of the crack tip is a stress concentration zone and the most sensitive area to the later occurrence of material failure. The paper also made a computational analysis of the obtained analytical results for all three cases of crack deformation: Mode I, mode II and mode III and the corresponding graphical representations of the distribution of parameters of the electromechanical state in the vicinity of the crack tip of the piezoelectric material.