## 1. Introduction

Impact-Penetration Dynamics (IPD) is the mechanics of deformation caused by two or more colliding bodies when one of them permanently changes its shape, or its integrity, due to the high forces developed during impact. The study of IPD covers a wide range of problems and applications [

1].

To understand the penetration process, many researchers have grouped various IPD problems into a finite number of categories. For example, Jonas and Zukas [

2] classified them according to impact velocity, where the response of materials at different velocities is a function of the strain-rates from impact. Hopkins and Kolsky [

3] categorized penetration processes by the different physical mechanisms that come into play, which determine the nature of the penetration process. Examples of such mechanisms are plastic, elastic, and thermal behaviors, as well as a decrease in compressibility. Wilbeck [

1] suggested that a pair of variables must be chosen to classify impact regimes: (1) the relative magnitudes of the material’s strengths, both penetrator and target strengths (

${\sigma}_{P}$ and

${\sigma}_{T}$); and (2) the impact pressures

P, where pressure is proportional to the material’s density and to the square of the impact’s velocity (

$P\sim \rho {v}^{2}$). However, adding more variables in the classification makes the analysis more confused. For this reason, Backman and Goldsmith [

4] simplified the study of rod penetration according to the impact velocity of the projectile and the respective penetration depth in the impacted target, as shown in

Figure 1. This schematic only shows a wide concept for the study of IPD, and it could involve different scenarios, from low velocity impacts to high energy projectiles.

In

Figure 1, four different penetration regimes can be analyzed. In the first region, relative velocities are low, and thus, the projectile can remain intact, which produces a deep crater whose diameter is only slightly greater than the projectile’s. In the second region, the projectile deforms while penetrating the target. As the impact velocity increases, a point is reached at which the crater depth increases with the velocity. At this point, the semi-fluid region is reached, where the materials can be modeled as fluids. Over this point, the penetration regime is known as hyper-velocity impact. Finally, the last region involves impacts of relatively ultra-high velocity, the energy of which can generate an explosive reaction in the bodies. Notice that for very low impact velocities (not shown in this figure) impact events are low energy, which could not produce plastic deformation in the bodies. Therefore, this region is not subjected to study in IPD.

Different formulations have been proposed to study each regime in

Figure 1; for example, cavity expansion theory [

6] for studying Regimes 1 and 2, the Tate formulation [

7,

8] for studying Regime 2, or fluid dynamics [

9] for studying Regime 3. Even though it is possible to find some common concepts among those theories, each penetration regime comes with its own limitations, and for this reason, it is not possible to extrapolate one theory for all penetration regimes. In this review, only the first penetration region is covered, which commonly represent events of a deformable target impacted by rigid rods. The study of rigid rod penetration has different applications. The principal fields of application are:

The mechanics of indentation and penetration [

4,

10,

11,

12,

13].

The soil mechanics for modeling various processes (e.g., penetration) in soils and rocks [

14,

15,

16].

Geo-mechanical applications: areas of interpretation of in situ tests (both the cone penetrometer and the pressure meter) and also in the prediction of the behavior of piles [

17,

18,

19].

In mechanical characterization [

20,

21,

22].

Nuclear structural engineering and related areas [

14,

23].

The mechanics of the indentation and penetration of a rigid object into thick targets has been the subject of interest for many decades for both experimentalist and numerical analyst [

13]. The physical processes happening in the rods are separate from those occurring in the target, but they are constrained by the requirement that stress needs to be continuous across the projectile-target boundary and that displacement needs to be compatible at the boundary. For rigid projectile penetration, satisfying results can be obtained with Cavity Expansion Theory (CET) and internal friction theory [

5]. However, in recent years, CET has been more extensively studied than the friction theory [

6,

24,

25,

26], and new analytical models have been created around this CET approach. In general, CET consists of two different formulations: spherical or cylindrical cavity expansion methods. The first one is formulated under spherical symmetry conditions, and the second one is formulated using cylindrical coordinates. Both formulations have shown successful results to model different experimental cavity expansion effects [

27]. It seems that the spherical formulation is easier to understand and simpler to model because the number of variables can be reduced due to the symmetry of the spherical problem. For this reason, many applications can be recently found using this approach [

28,

29,

30,

31].

The main objective of this review is to introduce the Spherical Cavity Expansion (SCE) theory and formulation by explaining the different mathematical features of this approach, its solution, and potential application in penetration problems. For this reason, SCE was utilized to develop an engineering model of penetration for rigid rods penetrating deformable targets. The SCE model is based on Durban et al.’s approach [

32]. The paper is outlined as follows: First, a chronological literature review of the most common models used to study this kind of penetration problem is given, focusing on the cavity expansion theory. After that, the SCE theory is reviewed, using Durban et al.’s approach [

32]. Next, an engineering model of penetration is proposed. This model will be based on the spherical cavity expansion theory for a strain-rate dependent material. The model is then compared and validated with some FE numerical simulations and with previous penetration results.

## 2. Literature Review

During the penetration process, the motion of the rigid rod is opposed by a force

${F}_{x}$ called penetration resistance. Applying Newton’s law, the motion of the projectile is governed by:

and:

where

${m}_{p}$ and

${V}_{x}$ are the mass and instantaneous velocity of the rod and

${x}_{p}$ is the coordinate in the direction of the penetration depth. Initial conditions of this problem are:

${V}_{x}\left(t=0\right)={V}_{0}$ and

${x}_{p}\left(t=0\right)=0$.

Common approaches are well known in penetration mechanics, which are based on making simple assumptions on the force opposing the motion of the projectile; for example, the steady penetration equations developed by Poncelet (Equation (

3)) or by Resal (Equation (

4)), where the penetration resistance is a function of velocity (

${A}_{P}$,

${B}_{R}$, and

${C}_{P}$ are empirical constants) [

33].

Although the previous models showed good agreement with most of the penetration impact events for metallic targets, they are fully empirical, and the plastic behavior of the medium was not well understood. More recently, approaches for calculating ${F}_{x}$ have assumed that the pressure during the penetration process is constant over all the cavity wall.

Forrestal et al. [

34] used this approach to calculate the force for different geometries of rods, as seen in

Figure 2. Due to the lack of detailed data of frictional force, it was assumed that the tangential stress on the nose (

${\sigma}_{t}$) was proportional to the normal or radial stress (

${\sigma}_{r}$):

where

$\mu $ is the sliding-frictional coefficient.

Therefore, assuming a spherical expansion of the cavity wall, the force of the cavity against the movement of the rod, for spherical nose rods, is given as follows:

for ogival nose rods is given as follows:

and for conical-nose rods is given as follows:

where all parameters involved (in the previous three equations) are shown in

Figure 2.

Jones et al. [

35,

36] formulated an equation to calculate

${F}_{x}$ force on the wall of a general nose geometry for axis-symmetric rods, as seen in

Figure 3. In this figure,

$\mathrm{y}=\mathrm{y}\left(x\right)$ is the shape function of an axis-symmetric nose with

$0\le x\le a$; this shape function needs to be nose-pointed

$\mathrm{y}\left(0\right)=0$, and the shape function must satisfy the condition

${\mathrm{y}}^{\prime}\left(x\right)>0$ and has a base radius of

$\mathrm{y}\left(a\right)=a$.

Using those geometrical conditions,

F is given by:

where

${\sigma}_{r}$ is the radial stress and

${F}_{f}$ is the frictional force on the cavity wall. Using Coulomb’s relation:

Equation (

9) is reduced to:

In the previous equations, which are related to calculating

${F}_{x}$ (Equations (

6)–(

8), or Equation (

11)), it is necessary to know the radial stress

${\sigma}_{r}$ on the cavity wall. If it is assumed that the radial stress is constant over all the cavity wall and equal to the pressure on the rod nose (

${p}_{c}$), this stress can be calculated using the cavity expansion theory.

Bishop et al. [

37] were the pioneers of cavity expansion pressure calculations. Bishop et al. [

37] began the study of plasticity in ductile materials for quasi-static conditions. In their study, an approximate formulation was made to calculate the load required to force a cylindrical punch deep into a semi-infinite block of ductile material. Due to the fact that a full theoretical solution of the equations, determining the strains around the head of the punch, is very complex, only two problems were solved:

Spherical Cavity Expansion (SCE): Starting with a small hollow sphere in the body of an infinite block of ductile material to determine the pressure ${p}_{s}$ that will enlarge the spherical hole indefinitely.

Cylindrical Cavity Expansion (CCE): Starting with a cylindrical hole of infinite length to find the pressure ${p}_{c}$ that will enlarge the hole indefinitely.

Bishop et al. [

37] also assumed that the true flow stress in compression

${\sigma}_{i}$, above the yield stress

Y, is given by Equation (

12), where

$\u03f5$ is the equivalent strain.

For this material behavior, pressures

${p}_{s}$ and

${p}_{c}$ were calculated as:

where

E and

$\nu $ are the elastic modulus and the Poisson’s ratio of the material,

a is the radius of the hole, and

c is the radius of the plastic region around the hole. If it is assumed that the material is perfectly plastic (

$f\left(\u03f5\right)=0$ in Equation (

12)), the integral terms in Equations (

13) and (

14) must be omitted.

Hopkins [

3] studied the spherically-symmetric cavity formation under conditions of large elastic-plastic deformations for quasi-static and dynamic motion. The most remarkable of Hopkins’ approaches was the calculation of

${p}_{s}$ for a perfectly plastic (with a constant dynamic yield stress

${Y}_{d}$), incompressible material at dynamic conditions:

where

$\dot{\square}$ represents the time derivative; thus,

$\dot{a}$ and

$\ddot{a}$ are the velocity and acceleration of the spherical radius. Furthermore, Hopkins proposed the differential equation for a perfectly plastic compressible material; however, an explicit relation was not presented.

Knowles and Jakub [

38] studied spherically-symmetric motions of an elastic solid containing a spherical cavity and subjected to large strains (e.g., rubber-like materials). A perfectly plastic medium was described by a general strain energy function. Thus, two special cases were solved: Mooney and non-Hookean materials. For those special cases, values of the static stress and dynamic stress at the cavity wall for various values of the dimensionless equilibrium radius were tabulated.

Durban and Baruch [

14,

39] solved the non-linear problem of a spherical cavity surrounded by an infinite elasto-plastic medium and subjected to uniform and quasi-static radial loads. The governing non-linear equations were solved, in terms of closed integrals, for internal or external pressure conditions.

Using the Hopkins approach, Forrestal and Luk [

40] developed an analytical model for the elastic-plastic response of a compressible material from the uniform expansion of a spherically symmetric cavity. The formulation Forrestal and Luk [

40] was based on the equations of momentum and mass conservation in Eulerian coordinates. Additionally, the medium was described by a perfectly plastic behavior, and the plastic region was defined with a linear pressure-volumetric strain relation:

and the Tresca yield criterion:

where

$\mathrm{P}$ is hydrostatic pressure,

K is the bulk modulus,

${\rho}_{0}$ and

$\rho $ are the densities in undeformed and deformed configurations, and

${\sigma}_{r}$ and

${\sigma}_{\theta}$ are the radial and hoop components of Cauchy stress; they are taken as positive in compression.

Forrestal and Luk [

40] found an approximate solution of the non-linear problem assuming an incompressible material for the elastic region; although, they also proposed a full non-linear solution using a numerical method to solve the differential equations. At the end, the solution for the spherical cavity pressure can be simplified by Equation (

18), where

${A}_{s}$ is expressed by Equation (

19) and

${C}_{s}$ needs to be adjusted to fit cavity expansion results.

Using the same conditions, Forrestal [

41] found a solution for the cylindrical cavity pressure, which can be simplified by Equation (

20), where

${A}_{c}$ is expressed by Equation (

21), and

${C}_{c}$ needs to be adjusted to fit cavity expansion results.

Cylindrical and spherical cavity expansion approximations were used to develop a penetration model for a rigid rod with spherical, ogival, and conical noses that penetrated elastic-perfectly plastic targets. The results of the model were compared with penetration data and results from Lagrangian and Eulerian wave codes [

34,

42]. The predictions were in good agreement with the measured final depths of penetration for aluminum targets at velocities between 400 m/s and 1400 m/s.

Continuing with the Forrestal work, Luk et al. [

43,

44] developed the dynamic expansion of spherical cavities for elastic-plastic, incompressible and compressible materials with a power-law strain hardening relation. The modified Ludik equation was used to describe the power-law of the plastic behavior:

where

n is the strain-hardening exponent.

A closed-form solution was found for the incompressible material, and a numerical solution was proposed for the compressible case. Both solutions can be adjusted to Equation (

18). However, for the incompressible problem,

${A}_{s}$ is given by:

For the compressible case, ${A}_{s}$ and ${C}_{s}$ must be adjusted to fit cavity expansion results. Thus, a penetration model was numerically programmed, and the results were compared with penetration data. Predictions were in good agreement with the final penetration depths for aluminum targets measured at velocities between 300 m/s and 1000 m/s.

After that, Warren and Forrestal [

45] solved the dynamic spherical cavity expansion problem for an elastic-plastic material with a constitutive model that included the effect of strain hardening and strain-rate sensitivity. For a state of uni-axial stress, the stress-strain relation was assumed as:

where

${Y}_{d}$ is the dynamic yield stress,

m is the strain-rate sensitivity exponent,

${\dot{\u03f5}}_{o}$ is a reference strain rate, and

$\alpha $ is a curve fitting parameter.

A closed-form solution was found for an incompressible material, which is expressed by:

The first term of Equation (

26) corresponds to the strain hardening component of the solution; the second term corresponds to the inertial part of the solution; and the last term accounts for the strain-rate effects.

For the compressible problem, a numerical solution was proposed. In this case, a spherical expansion solution can be adjusted to:

where

${A}_{s}$,

${B}_{s}$, and

${C}_{s}$ must be adjusted to fit cavity expansion results.

To compute the structural and component responses of a projectile due to three-dimensional penetration events, Warren and Tabara [

46,

47] implemented the spherical cavity expansion analysis in a transient dynamics finite-element code called PRONTO 3D. The sample demonstrated good agreement between experimental and analytical results. After that, Brown et al. [

48] implemented the spherical cavity expansion problem in a numerical solver as a third-party library, which can be used in explicit codes like ABAQUS.

Chen and Li [

12,

49] used Forrestal et al.’s approach to study the perforation of thick plates by rigid projectiles with various geometrical characteristics. They added plugging formation and movement for blunt-nosed projectiles to their model, obtaining a simple and explicit formula to predict the ballistic limit and residual velocity for the perforation of thick metallic plates, which agreed with available experimental results with satisfactory accuracy.

Teland and Moxnes [

50] compared penetration results among analytical results of the cavity expansion theory and numerical simulation results, using two different nose geometries of projectiles (spherical and conical noses) and two different target materials (steel and concrete). Some agreements were achieved among the results, especially in the first stage of penetration (cratering phase). However, in the second phase (tunneling phase), the results suddenly diverge.

More recently, Buchely et al. [

51,

52,

53] developed an engineering model based on the spherical cavity expansion theory for the penetration of a rigid-rod into an elasto-plastic, J2Mises, compressible, Cowper–Symonds strength model. The computations of Buchely and Maranon confirmed previous theoretical considerations and showed that the cavitation fields for rate-dependent materials are functions of time and cannot be modeled with steady-state self-similar approaches. However, during penetration events involving rigid projectiles, the cavity remains constant because the rigid projectile does not undergo a change, and it only moves into the field due to the movement of the projectile. Therefore, a state of stress is placed ahead of the nose of the projectile, and it moves through the material. It was shown that this state of stress is nearly steady during the penetration process at medium and low penetration velocities; therefore, self-similarity transformation can be used in this specific penetration event, even for strain-rate-sensitive materials.

The previous analyses were developed mainly for ductile targets. However, cavity expansion theory has also been extensively used in brittle materials (e.g., ceramics, glasses, and soils).

Forrestaland Lancope [

54] developed a model of spherical cavity expansion for an elastic-cracked-plastic medium on the quasi-static conditions. In Forestal and Lancope’s work [

54], the plastic region was described by a linear pressure-volumetric strain relation:

where

$\lambda $ defines the pressure-dependent shear strength and is less than one (

$\lambda <1$) and

$\mathrm{P}$ is the hydrostatic pressure defined by Equation (

16). Additionally, it was assumed that

${\sigma}_{\theta}=-T$ at the cracked-elastic interface, where

T is the tensile strength of the material; and the material in the cracked region was taken as linear elastic with

${\sigma}_{\theta}=0$. Thus, the pressure to open a spherical cavity was calculated as follows:

where the parameters

$\alpha $ and

$\beta $ are given as follows:

and the relation

$\left(c/a\right)$ is given as follows:

The solution is simplified when the stress at the cavity surface

${R}_{t}<Y$; thus, the material does not reach its shear strength. For this case, the medium has a cracked region bounded by an elastic one, and the pressure to open a spherical cavity is simplified to the equation:

This work was continued by Satapathy and Bless [

55,

56,

57], who extensively studied the spherical cavity expansion of brittle materials. In their analysis, an elastic-cracked-comminuted medium at quasi-static and dynamic conditions was analyzed, using a two-curve pressure–shear behavior of the material (like the Johnson–Holmquist constitutive relation). At the end, closed-form solutions were proposed, and results were compared with experimental penetration data for AD995 alumina targets.

Durban and Fleck [

58] also studied the spherical cavity expansion on brittle materials at quasi-static conditions, but using a Ducker–Prager constitutive yield criterion. Some special cases of materials were studied, such as the associative flow rule, elastic-perfectly plastic behavior, and Mohr–Coulomb solids. Their model was numerically solved, and different parameters of the models were studied.

Shaw [

13] studied the penetration of rigid objects into semi-infinite compressible solids for penetrators with arbitrary nose geometries. Using smoothed-particle hydrodynamics simulations, he calculated the resistive force of the target through the spherical cavity expansion theory. The effect of friction at the object-target interface was also studied.

Additionally, extra work has been done to model the hyper-velocity expansion cavity, with propagation of shock waves. Cohen and Durban [

59] investigated the shock waves in porous materials based on the dynamic cavity expansion theory. This work was continued by Czarnota et al. [

60] for ductile metallic materials and by Rodriguez-Martinez et al. [

61] for strain-hardening materials.

#### Summary of the Literature Review