Dynamic responses of steel bolts anchored in concrete and subjected to cyclic loads are complex: Qualitative physical attributes include pinching, displacement drift and force-intercept. In fact, these attributes vary with time, involving uncertainties that can lead to different failure patterns. This study explores higher-order elements (HOEs), an emerging theoretical macro-modeling framework originated from electrical engineering and transferred to engineering mechanics by using physical analogies. This study proposes a systematic procedure for a new model formulation – a model assembly that is made up of three distinct modeling elements, a higher-order spring, higher-order damper, and loading-rate independent damper to capture pinching, displacement drift, and force intercepting, respectively. The two higher-order elements play significant roles to model responses of fatigue loading in two opposite directions each involving loading and unloading. To afford more physical insights, the higher-order spring is further formulated into a mem-spring, where new choices of state variables are employed. The proposed model formulation benefits both efficient simulation of time-varying dynamical systems and qualitative failure analysis. In addition, this study offers a case study to reveal a new utility of HOE theory. Two major datasets of anchoring blots are used to develop and validate the proposed modeling approach, while three minor datasets are used for a supplementary investigation. This study shows that, both qualitatively and quantitatively under ordered excitation, HOE can be employed with more modeling power and efficiency compared with existing models for progressive damage, and with the flexibility to interface with some existing models when needed.