## 1 Introduction  
Although a great majority of experimentally ballistic data is gathered under ideal impact conditions of normal incidence, the probable fact is that almost all ballistic impacts occur at some non-ideal level of obliquity [1, 2, 3, 4, 5, 6]. [PERSON] [7] comprehensively summarized analytical, numerical and experimental investigations of targets subjected to the nonstandard collisions, penetration and perforation of projectiles. Depending on projectile materials, shapes and speeds, as well as a host of different target substances, there will be a critical angle of obliquity beyond which the projectile will ricochet from the target surface. Ricochet is a special phenomenon in the oblique penetration of projectiles, where the projectile deflects from its course while maintaining its integrity after its impacting on the target [8, 9]. The ricochet projectile proves worthless to its launcher, as its warhead fails to penetrate the interior of target and strike the facilities and personnel of the enemy effectively. Additionally, it may bring about the secondary impacting on an undesirable place [10]. This is highly manifested in the case of low-altitude or high-speed delivery modes against large steel targets like armoured fighting vehicles [11, 12]. The ricochet phenomenon has been studied worldwide for a long time considering its significance in ballistics. [PERSON] [13] developed an early model for the ricochet of rods based on the impacting of flat-nosed projectiles, which allows the local erosive deformation of the rod in the immediate vicinity of the impacting, where the asymmetric forces acting on this deforming rod tip are evaluated to ascertain their capacity to induce a rotation sufficient to bring about ricochet during the limited time before the rod tip becomes fully engaged in the target. [PERSON] [14] proposed a ricochet model based on the mechanics of materials-based approach, which focused on the interaction stresses and fluxes in the rod and target that produce the forces and moments required to continuously sustain a plastic hinge at the rod-target interface. However, [PERSON]'s model failed to predict the ricochet of rigid rods because of the method used to calculate the interaction force, and its requirement to ricochet by way of rigid rod rotation, and [PERSON]'s model is represented by a handful of algebraic equations that must be simultaneously solved subject to various constraints. Currently, the analytical models concerning oblique incidence and ricochet are pitfully few and tend to be either extremely simple or limited to the initial stages of the impacting process. Therefore, it is essential to develop a relatively simplified model to not only meet the practical applications in engineering but also take into account the different stages of the impacting process. Moreover, the ricochet problem can be investigated experimentally, and substantial studies concentrate on the governing factor in the generation of the ricochet and the penetration ability of the projectile. [PERSON] and [PERSON] [15] studied the effects of the impacting angle, impacting velocity and dynamic hardness on the crater length and crater volume by impacting a hard ball against a semi-infinite target. [PERSON] [1,2] conducted ballistic experiments to investigate the damage behaviour of a thick steel armour plate at different obliquities impacted by an ogive-shaped steel projectile. Extensive work has been carried out involving penetration and ricochet studies on thin metallic and composite plates impacted by hard balls, long rod penetrators and small arms ammunitions [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. On the other hand, there exist several potential studies concerning the threat from ricochet projectiles or fragments, such as using shields to deflect space debris and meteorites of exploded projectiles or protecting critical equipment from intentional and accidental impacts [7,30].