A PD simulation-informed prediction of penetration depth of rigid rods through materials susceptible to microcracking

Abstract

The present investigation relies upon an alternative approach to estimate the penetration depth of rigid projectiles into quasibrittle materials that utilizes simulation-informed modeling of penetration resistance. Penetration at normal incidence of a long rigid rod into massive targets, made of materials with inferior tensile strength predisposed to microcracking, is an event characterized by a high level of aleatory variability and epistemic uncertainty. This inherent stochasticity of the phenomenon is addressed by a model developed based on the particle dynamics (PD) simulations aimed to provide a key modeling ingredient—the functional dependence of the radial traction at the cavity surface on the radial velocity of the cavity expansion. The penetration depth expressions are derived for the ogive nose projectiles. The use of the power law radial traction dependence upon the expansion rate yields the penetration resistance and depth equations defined in terms of hypergeometric functions. These expressions are readily evaluated and offer a reasonably conservative estimate of the penetration depth. This model is validated by using experimental results of the penetration depth of long projectiles into Salem limestone, which is a typical example of quasibrittle materials with random microstructure well known for their pronounced experimental data scatter. This stochasticity is explored in the present paper by a sensitivity analysis of the key input parameters of the model; most notably, uniaxial tensile strength and friction coefficient.

Appendices

Appendix A: Modeling of the radial traction dependence upon the cavity expansion velocity

In the present study, the functional dependence of the radial traction on the CCE radial velocity is assumed in the following form

$$\bar{\sigma }_{r} = \frac{{\sigma_{r} }}{K} = {\mathcal{B}} + {\mathcal{A}} \cdot \left( {\frac{{v_{r} }}{C}} \right)^{\gamma } \,,\quad \gamma \in \Re^{ + } ,\;\gamma > 1$$

(15)

Based on the results of the CCE PD simulations, the ansatz (15) should satisfy the following four boundary conditions illustrated in Fig. 8:

$$\bar{\sigma }_{r} \left| {_{{\bar{v}_{r} = 0}} \, = \left( {\bar{\sigma }_{r} } \right)_{st} \;} \right.$$

(16a)

$$\frac{{d\bar{\sigma }_{r} }}{{d\bar{v}_{r} }}\left| {_{{\bar{v}_{r} = 0}} \, = 0\;} \right.$$

(16b)

$$\bar{\sigma }_{r} \left| {_{{\bar{v}_{r} = \alpha \cdot \bar{v}_{r}^{ * } }} \, = \left( {\frac{1 - \upsilon }{{1 + \upsilon }}} \right)\,\left( {\alpha \cdot \bar{v}_{r}^{ * } } \right).} \right.$$

(16c)

$$\frac{{d\bar{\sigma }_{r} }}{{d\bar{v}_{r} }}\left| {_{{\bar{v}_{r} = \alpha \cdot \bar{v}_{r}^{ * } }} \, = \left( {\frac{1 - \upsilon }{{1 + \upsilon }}} \right).} \right.$$

(16d)

The first boundary condition (16a) yields

$${\mathcal{B}} = \left( {\bar{\sigma }_{r} } \right)_{st} = \sqrt {\frac{{3{\mkern 1mu} \left( {1 - 2\upsilon } \right)}}{{\left( {1 + \upsilon } \right)\left( {3 - 2\upsilon } \right)}}\left( {\frac{{\sigma_{f} }}{K}} \right)}$$

(17)

The subscript st indicates the static solution for the elastic-cracked CCE case, where σ_f designates the indirect tensile strength (the tensile strength corresponding to splitting under the far-field compression rather than tension; also known as, Brazilian tensile strength).

Equation (15) satisfies the second boundary condition (16b) by definition.

The coupled boundary conditions (16c) and (16d) imply that the curve (15) approaches from above the lower bound given by the inclined line

$$\bar{\sigma }_{r}^{lb} = \left( {\frac{1 - \upsilon }{{1 + \upsilon }}} \right)\;\bar{v}_{r}$$

(18)

The parameter α > 1 determines the point of contact ($\alpha \cdot \bar{v}_{r}^{ * }$) as indicated by Fig. 8. The transition velocity ($\bar{v}_{r}^{ * }$) is uniquely defined by the intercept of the horizontal line corresponding to the static radial traction,$\left( {\bar{\sigma }_{r} } \right)_{st}$ (17), and the inclined line corresponding to the lower-bound radial traction (18). This transition velocity is characterized by the balance of kinetic and potential energies [15].

Fig. 8

Schematic representation of the radial traction at the cavity surface vs. CCE velocity. Note that the PD simulation results pinpoint ${\bar{{v}}}_{{{r}}}^{ * }$ to 0.0143, which for the Salem limestone used in the numerical example corresponds to 60 m/s. (Interestingly, it has been observed that the narrow range of radial velocities centered on ${\bar{{v}}}_{{{r}}}^{ * }$ is characterized by the balance of kinetic and potential energies [15].)

The remaining boundary conditions (16c) and (16d) could be, respectively, developed as follows

$$\left( {\bar{\sigma }_{r} } \right)_{st} + \cdot {\mathcal{A}}\left( {\alpha \cdot \bar{v}_{r}^{ * } } \right)^{\gamma } \, = \left( {\frac{1 - \upsilon }{{1 + \upsilon }}} \right)\,\left( {\alpha \cdot \bar{v}_{r}^{ * } } \right)\;$$

(19)

$${\mathcal {A}} \cdot \gamma \cdot \left( {\alpha \cdot \bar{v}_{r}^{ * } } \right)^{\gamma - 1} = \left( {\frac{1 - \upsilon }{{1 + \upsilon }}} \right)\,$$

(20)

If both sides of Eq. (20) are multiplied by $\alpha \cdot \bar{v}_{r}^{ * }$, and (15) substituted into the resulting equality, it follows

$${\mathcal {A}} \cdot \left( {\alpha \cdot \bar{v}_{r}^{ * } } \right)^{\gamma } = \frac{\alpha }{\gamma }\,\left( {\frac{1 - \upsilon }{{1 + \upsilon }}} \right)\,\;\bar{v}_{r}^{ * } \quad \Rightarrow \quad \alpha \;\left( {\frac{1 - \upsilon }{{1 + \upsilon }}} \right)\,\;\bar{v}_{r}^{ * } - \left( {\bar{\sigma }_{r} } \right)_{st} = \frac{\alpha }{\gamma }\,\left( {\frac{1 - \upsilon }{{1 + \upsilon }}} \right)\,\;\bar{v}_{r}^{ * }$$

(21)

Since the transitional velocity is defined by the equality

$$\left( {\bar{\sigma }_{r} } \right)_{st} = \left( {\frac{1 - \upsilon }{{1 + \upsilon }}} \right)\;\bar{v}_{r}^{ * }$$

(22)

the relationship between two parameters

$$\gamma = \frac{\alpha }{\alpha - 1}\;,\quad \alpha > 1$$

(23)

can be readily obtained from Eq. (21)₂.

Furthermore, the parameter A can be expressed from Eq. (21)₁

$${\mathcal {A}} = \frac{1}{\gamma }\,\left( {\frac{1 - \upsilon }{{1 + \upsilon }}} \right)\,\;\left( {\alpha \cdot \bar{v}_{r}^{ * } } \right)^{1 - \gamma } \quad \Rightarrow \quad {\mathcal {A}} = \frac{{\alpha^{1 - \gamma } }}{\gamma }\,\left( {\frac{1 - \upsilon }{{1 + \upsilon }}} \right)\,\;\bar{v}_{r}^{ * } \;\left( {\bar{v}_{r}^{ * } } \right)^{ - \gamma }$$

(24)

By substituting Eq. (23) into (24)₂ the unknown parameter can be obtained straightforwardly

$${\mathcal {A}} = \frac{{{\upalpha }^{{{1} - {\upgamma }}} }}{{\upgamma }}\,\left( {\frac{{{1} - {\upupsilon }}}{{{1} + {\upupsilon }}}} \right)^{{\upgamma }} \,\left( {{\bar{\sigma }}_{{{r}}} } \right)_{{{{st}}}}^{{{1} - {\upgamma }}} = \frac{{1}}{{\upgamma }}\,\left( {{1} - \frac{{1}}{{\upgamma }}} \right)^{{{\upgamma } - {1}}} \,\left( {\frac{{{1} - {\upupsilon }}}{{{1} + {\upupsilon }}}} \right)^{{\upgamma }} \,\left( {{\bar{\sigma }}_{{{r}}} } \right)_{{{{st}}}}^{{{1} - {\upgamma }}}$$

(25)

The expressions (17), (23) and (25) define parameters of the radial traction (15). An example of these model parameters is presented in Table 3.

Table 3 Example of four sets of model parameters based on PD inputs υ = 0.25 and $\left( {\bar{\sigma }_{{{r}}} } \right)_{st} = 0.0086$

Appendix B: Integration of the penetration resistance for μ = f(v _z)

The introduction of the velocity-dependent friction coefficient into the penetration depth calculation presented in Sect. 5 is not only beset with difficulties regarding necessary inputs but also renders the analytical solution intractable. Consequently, in order to simplify the following analysis, it is assumed henceforth that

$$ {\bar{\sigma }}_{{{r}}} = {\bar{\sigma }}_{{{r}}}^{{{{lb}}}} = \left( {\frac{{1 - {\upupsilon }}}{{1 + {\upupsilon }}}} \right)\,{\bar{{v}}}_{{{r}}}$$

(26)

The assumption (26) neglects: (i) the quasistatic resistance, and (ii) the significant deviation from the linear lower bound (18) in the high-velocity range. Based on Fig. 3, the effect of the first assumption may be considered modest, while the effect of the latter assumption disregarding nonlinearity in the high-velocity region should be more pronounced, especially for the higher striking velocities. Nonetheless, the effects of the simplifying linearization (26) should be examined on case-by-case basis. As an example, Wang and coauthors [42], in their calculation of penetration depth of the rigid projectile into rock, use the linear dependence (26) and estimate the quasistatic-resistance abandon to less than 5% for v_s ≥ 400 m/s and the influence of the neglect of nonlinearity in the high velocity range to less than 4% for v_s ≤ 1000 m/s.

Thus, under the assumption (26), the resistance penetration force is obtained as the linear function of the penetration velocity

$$ {{F}}_{{{z}}} = {\uplambda }_{{{o}}} \,\left( {{\upalpha }_{{{o}}} + {\upbeta }_{{{o}}} \,{\upmu }} \right)\;{\bar{{v}}}_{{{z}}} .$$

(27a)

$$\alpha_{o} = \frac{1}{2}\sin 2\theta_{0} + \arctan \left( {\frac{1}{{\tan \theta_{0} }}} \right) - 2\,\sin \theta_{0} \;\arctan {\text {h}}\left( {\cos \theta_{0} } \right)$$

(27b)

$$\beta_{o} = \frac{2}{3} - \sin \theta_{0} \,\cos^{2} \theta_{0} - \frac{2}{3}\,\sin^{3} \theta_{0}$$

(27c)

$$\lambda_{o} = 4\,a^{2} \pi \,\psi^{2} \,K\,\left( {\frac{1 - \upsilon }{{1 + \upsilon }}} \right).$$

(27d)

The corresponding penetration depth is obtained by integration outlined in Sect. 5

$$D = \frac{{m_{p} \,v_{s} }}{{\lambda_{o} \,\left( {\alpha_{o} + \beta_{o} \,\mu } \right)/C}}\;,\quad \mu = const.$$

(28)

In order to take into account the friction coefficient increase with sliding velocity decrease during projectile deceleration, the following ansatz

$$\mu = \mu_{0} \;\exp \,\left( { - \frac{{v_{z} }}{\tau }} \right) = \mu_{0} \;\exp \,\left( { - \frac{{C\,\bar{v}_{z} }}{\tau }} \right).$$

(29)

is introduced and illustrated in Fig. 9a for τ = 100 m/s and 200 m/s and μ₀ = 0.4. The preceding values are selected arbitrarily to facilitate discussion; the evolution of the friction coefficient as a function of loading history is a captivating problem in itself. (Also, note that Eq. (29) neglects, for simplicity, the dependence of the sliding velocity on the penetration velocity: v_t = v_z sinθ.) The constant values of the friction coefficient, used in Sect. 6, are marked in Fig. 9a. It can be seen, that the constant μ values of 0.02 (0.08) corresponds to Eq. (28) value for 600 (320) m/s. In other words, for v_t < 600 (320) m/s, the friction coefficient exceeds 0.02 (0.08) with exponential rate of increase. On the other hand, at v_t > 1000 m/s the friction appears negligible for all practical purposes.

Fig. 9

a Assumed friction-coefficient velocity dependence; Eq. (29). b Schematics of various variants of the friction coefficient evolution during penetrator deceleration at low striking velocities; with the rise in striking velocity, the sliding friction effect weakens to the extent that it can be ignored

The penetration depth that takes into account the velocity dependent friction, as described above, is

$$D = \frac{{m_{p} \,v_{s} \;}}{{\lambda_{o} \,\alpha_{o} \,/\,C}}\;\left\{ {1 + \frac{1}{{\left( {{{v_{s} } \mathord{\left/ {\vphantom {{v_{s} } \tau }} \right. \kern-\nulldelimiterspace} \tau }} \right)}}\ln \,\left( {\frac{{\alpha_{o} \, + \beta_{o} \,\mu_{0} \,\exp \,\left( {{{ - v_{s} } \mathord{\left/ {\vphantom {{ - v_{s} } \tau }} \right. \kern-\nulldelimiterspace} \tau }} \right)}}{{\alpha_{o} + \beta_{o} \,\mu_{0} }}} \right)} \right\}.$$

(30)

where τ and μ₀ are parameters of Eq. (29).

Finally, it seems appropriate to conclude this appendix with reflections on the inherent complexities of the evolution of the friction coefficient depending on the loading history. It is well known that the structure and geometry of the interface between two solid surfaces in contact are of fundamental importance to the phenomenon of friction. The thermomechanisms on the microscale involved with the interfacial failure caused by sliding are extremely complex [43]. The actual contact is established through a myriad of discrete entities (randomly distributed asperities) accounting, cumulatively, to only a small portion of the total nominal area of contact. It cannot be overemphasized that micromechanical and thermomechanical rearrangements of these asperities (and, consequently, the contact interface) during sliding are irreversible. Consequently, it should be noted that although assumed functional dependence defined by Eq. (29) can be, in principle, obtained by fitting the dynamic-friction experimental data, the actual evolution of the friction coefficient in the course of projectile deceleration is more complex and not likely to obey such experimentally determined μ = f (v_z). The reason is that Eq. (29) does not capture the loading history of surfaces involved in the friction contact but is determined by fitting and extrapolating experimental data obtained by a small set of sliding velocities. Namely, if the green dashed line in Fig. 9b represents the experimentally determined Eq. (29), the evolution μ = f (v_z) following projectile deceleration after the impact with the striking velocity v_s would not likely follow the path 1 during the projectile deceleration (again, especially for high velocity impacts). This is because the high-energy impact leads to rapid, irreversible thermomechanical and chemical changes to the contacting surfaces. In the case of the high striking velocity, a thin layer of material at the projectile nose would melt and act as a lubricant that effectively promotes the frictionless penetration (μ = 0). Even if the melting is not fully accomplished, the friction coefficient value defined by the striking velocity μ = f (v_s) in accordance with Eq. (29), may remain largely unchanged for the rest of penetration (the path 3 in Fig. 20). Thus, the friction coefficient evolution paths 1 and 3 represent the upper and the lower bound, respectively, with actual change in reality taking place somewhere in between (the path 2 in Fig. 20). (In that case, the static friction coefficient becomes a function of the entire contact-surface history of change, which is marked in Fig. 9b by μ₀ = μ₀ (v_s).) It seems reasonable to expect that the higher the striking velocity the closer the path 2 gets to the lower bound (the path 3) with the trend to approach the frictionless penetration, which renders the modeling of friction unnecessary [9].