An Alternative Ballistic Limit Equation for the Whipple Shield in the Shatter Regime, Based on Characteristics of the Large Central Fragment

In the shatter regime of a Whipple shield, a large central fragment makes a significant contribution to the damage-causing capacity of the debris cloud. Herein we present a feasible scheme for the identification and measurement of this large central fragment and propose an alternative approach to the ballistic limit equation (BLE) for the Whipple shield, deducing an alternative ballistic limit in the shatter regime based on the large central fragment’s characteristics. This alternative BLE is compared with the phenomenological Whipple BLE, the JSC Whipple BLE and the Ryan curve. Our alternative BLE, modified at the incipient fragmentation and completed fragmentation point, is shown to agree well with experimental results.

A fracture equation of mixed mode cracks

On the basis of the multiaxial fatigue limit equation by Liu and Yan, a fracture equation of mixed mode crack is proposed in this note. By means of the test data on the mixed crack fracture reported in literature, the fracture equation has been verified to be not only simple in computation but also high in accuracy.

A vanishing dynamic capillarity limit equation with discontinuous flux

We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the dynamics capillarity equation $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u_{\varepsilon ,\delta } +\mathrm {div} {\mathfrak f}_{\varepsilon ,\delta }(\mathbf{x}, u_{\varepsilon ,\delta })=\varepsilon \Delta u_{\varepsilon ,\delta }+\delta (\varepsilon ) \partial _t \Delta u_{\varepsilon ,\delta }, \ \ \mathbf{x} \in M, \ \ t\ge 0\\ u|_{t=0}=u_0(\mathbf{x}). \end{array}\right. } \end{aligned}$$ ∂ t u ε , δ + div f ε , δ ( x , u ε , δ ) = ε Δ u ε , δ + δ ( ε ) ∂ t Δ u ε , δ , x ∈ M , t ≥ 0 u | t = 0 = u 0 ( x ) . Here, $${{\mathfrak {f}}}_{\varepsilon ,\delta }$$ f ε , δ and $$u_0$$ u 0 are smooth functions while $$\varepsilon $$ ε and $$\delta =\delta (\varepsilon )$$ δ = δ ( ε ) are fixed constants. Assuming $${{\mathfrak {f}}}_{\varepsilon ,\delta } \rightarrow {{\mathfrak {f}}}\in L^p( {\mathbb {R}}^d\times {\mathbb {R}};{\mathbb {R}}^d)$$ f ε , δ → f ∈ L p ( R d × R ; R d ) for some $$1<p<\infty $$ 1 < p < ∞ , strongly as $$\varepsilon \rightarrow 0$$ ε → 0 , we prove that, under an appropriate relationship between $$\varepsilon $$ ε and $$\delta (\varepsilon )$$ δ ( ε ) depending on the regularity of the flux $${{\mathfrak {f}}}$$ f , the sequence of solutions $$(u_{\varepsilon ,\delta })$$ ( u ε , δ ) strongly converges in $$L^1_{loc}({\mathbb {R}}^+\times {\mathbb {R}}^d)$$ L loc 1 ( R + × R d ) toward a solution to the conservation law $$\begin{aligned} \partial _t u +\mathrm {div} {{\mathfrak {f}}}(\mathbf{x}, u)=0. \end{aligned}$$ ∂ t u + div f ( x , u ) = 0 . The main tools employed in the proof are the Leray–Schauder fixed point theorem for the first part and reduction to the kinetic formulation combined with recent results in the velocity averaging theory for the second. These results have the potential to generate a stable semigroup of solutions to the underlying scalar conservation laws different from the Kruzhkov entropy solutions concept.

A Predictive Non-Dimensional Scaling Law for the Plate Perforation of Several Aluminum Alloys by Fragment-Simulating Projectiles

An equation was previously-presented to predict the ballistic-limit velocity for the perforation of aluminum armor plates by fragment-simulating projectiles (FSP). The ballistic-limit equation was presented in terms of dimensionless parameters so that the geometric and material problem scales are identified. Previously published predictions and data for two different FSP projectile calibers (12.7 mm and 20 mm) and two different strength aluminum alloys show the scaling law to be accurate. In this paper we extend the same concept to several other alloys and show that this scaling law is predictive.

A rupture limit equation for COPVs following a perforating MMOD impact

Most spacecraft have at least one pressurized vessel on board. One of the primary design considerations for earth-orbiting spacecraft is the anticipation and mitigation of the possible damage that might occur in the event of a micrometeoroid or orbital debris (MMOD) particle impact. To prevent mission failure and possibly loss of life, protection against perforation by such high-speed impacts must be included. In addition to a hole, it is possible that, for certain pressure vessel designs, materials, impact parameters, and operating conditions, a pressure vessel may experience catastrophic failure (i.e. rupture) as a result of a hypervelocity impact. If such a tank rupture were to occur on-orbit following an MMOD impact, not only could it lead to loss of spacecraft, but quite possibly, for human missions, it could also result in loss of life. In this paper we present an update to a Rupture Limit Equation, or RLE, for composite overwrapped pressure vessels (COPVs) that was presented previously. The update consists of modified RLE parameters and coefficients that were obtained after the RLE was re-derived using new / additional data. The updated RLE functions in a manner similar to that of a ballistic limit equation, or BLE, that is, it differentiates between regions of operating and impact conditions that, given a tank wall perforation, would result in either tank rupture or only a relatively small hole or crack. This is an important consideration in the design of a COPV pressurized tank – if possible, design parameters and operating conditions should be chosen such that additional sizable debris (such as that which would be created in the event of tank rupture or catastrophic failure) is not created as a result of an on-orbit MMOD particle impact.