Accurate description of the constitutive relation and the material parameters are necessary for constructing accurate digital twins of structures. Model updating-based parameter identification is a popular approach employing numerical models such as FEM to build and validate the accuracy of the digital twins based on real measurements. However, the real-world structure’s material parameters and constitutive relation might differ from the ideal properties used in the model. Common SHM approaches focus on identifying the material parameters such as Young’s modulus of the structure to detect an anomaly [1,2]. While this information represents the current state of the structure accurately, prediction regarding the remaining life of the structure is challenging as the underlying constitutive behaviour has likely changed, thereby highlighting the necessity of material model-accurate predictive digital twins. In this work, we address the material model identification by first decoupling it into two manageable sub-problems, material parameters identification and constitutive model identification, to gain a deeper understanding of each component’s characteristics. The parameter identification includes identifying the material constants, given a constitutive law. While the constitutive problem discovers the constitutive model influencing the material behaviour given a set of material parameters. Both the sub-problems are formulated as adjoint-based sensitivities driven optimization tasks with the objective to minimize the weighted differences between the model and the deformation measurements obtained from displacement or strain sensors. The parameters identification follows the continuous optimization technique proposed by the authors in [1,2]. For the constitutive model identification, the continuous relaxation approach is used to convert the discrete variable optimization to a continuous problem enabling smoother exploration. Different fidelities for both the sub-identification problems are investigated. To tackle the ill-conditioning of high-fidelity problems, techniques such as Vertex Morphing are explored. The material parameters and constitutive problems are subsequently combined and the comprehensive material model identification problem is formulated. The results and challenges arising from this combined optimization are examined in detail. Various structural problems and material models are analysed to demonstrate the effectiveness of the approach.