## 1 Introduction  
Ships and offshore structures experiencing low-velocity impacts, including collision, grounding, and floating object collisions, are prone to catastrophic disasters. In severe cases, the structures on the side of the ship that are impacted can undergo significant deformations and fractures, potentially leading to environmental pollution and injuries. The serious consequences of these collision incidents have driven researchers to investigate structural crashworthiness. Numerous valuable insights have been gained through analytical studies, numerical simulations, and experiments [1; 2; 3; 4]. Experimental research has been conducted since the last century to evaluate the important parameters, including impact load, stress, and the initiation and propagation of fractures, associated with impact events. Experimental analyses are regarded as an irreplaceable method in the field of impact research and serve as a verification tool [5]. However, owing to their high cost and restricted experimental conditions, small-scale model tests are frequently utilized to study the response of ship structures, enabling the prediction of the damage sustained by the prototype structure during collisions [6]. To accurately capture the behaviors of the prototype, it is crucial to design models based on appropriate similarity laws. Given the significant nonlinear effects involved in impact scenarios, dimensional analysis methods are extensively employed to simplify these problems. The Buckingham-II theorem [7], a widely used approach, establishes scaling laws and asserts that complete similarity is maintained when all dimensionless numbers of the scaled model are identical to those of the prototype, as expressed in Equation (1). \[\left(\Pi_{i}\right)_{m}=\left(\Pi_{i}\right)_{p} \tag{1}\]  
The scaling laws governing impacted structures are well established and extensively employed in the field [8; 9]. These laws, which are based on a dimensional analysis within the classic mass-length-time (MLT) system, facilitate the achievement of complete similarity. The fundamental relationships between the scaled models and the prototypes are summarized as scaling factors in references [6; 10]. These scaling factors establish a connection between the dynamic behavior of the prototype and the perfect similar scaled models. The scaling factor \(\beta\), defined in Equation (2), represents the relationship between the dimensions of the perfect similar scaled model (as subscript \(m\)) and those of the prototype (as subscript \(p\)). Additionally, the relationship defined for variable \(X\) can be denoted using the scaling factor \(\beta_{X}\) and is expressed in Equation (3). \[\beta=L_{m}/L_{p} \tag{2}\]  
\[\beta_{X}=X_{m}/X_{p} \tag{3}\]  
However, in practical problems, particularly those involving impact scenarios, achieving complete similarity as described in the previous statement is often unattainable. This leads to what is known as incomplete similarity, also referred to as similarity for distortion [11]. Previous research has primarily focused on generating solutions and related parameters solely through a dimensional analysis. However, when distortions are present, the aforementioned approach becomes inadequate, emphasizing the need for further investigation and research. Moreover, [PERSON] [6] highlighted that the response of structures under impact loads does not adhere to conventional similarity laws due to several factors, including strain rate sensitivity, structural fracture and thermal responses, and gravity. Consequently, it is imperative to explore the similarity laws that account for distortions and their additional effects. Structural dynamic behaviors, especially those of structures made of materials like steel and alloys, have been extensively studied to understand the impact of strain rate [11; 12; 13; 14; 15]. As the field of similarity applications has evolved, an increasing number of researchers have turned their attention to geometric distortion similarity methods.