Inverse identification of material defects from limited measurements is a fundamental challenge in nondestructive evaluation (NDE). Such problems are typically ill-posed and sensitive to noise, particularly when only surface responses are available. In this work, we investigate an adjoint-based optimization framework for identifying material defects in solids using measured surface responses. The inverse problem is formulated by minimizing a misfit functional defined in terms of surface displacement measurements, while the forward problem is governed by continuum mechanical models of the solid. As a first step, elastic deformation is considered, and unknown defect-related parameters are inferred using gradient-based optimization with sensitivities obtained through the adjoint method. Numerical results demonstrate that the approach can successfully localize and characterize material defects when surface deformation data are used as the objective function. To improve defect detectability and sensitivity, the framework is extended to elastodynamic wave propagation, motivated by the rich information content of transient wave fields commonly exploited in NDE applications. The use of dynamic measurements is expected to enhance resolution compared to quasi-static data, but they come with increased computational cost and greater sensitivity to modeling errors. Ongoing work includes a systematic comparison between inverse formulations based on elastic and elastodynamic responses, with emphasis on robustness, computational efficiency, and sensitivity to noise. The proposed framework provides a flexible computational tool for studying the strengths and limitations of different inverse strategies for defect identification in NDE.