Spherical Cavity-expansion Approximation Research Articles

The embedment of a high velocity rigid ogive nose projectile into a concrete target

The early stage of penetration of a rigid projectile into a target is denoted as the entrance stage. It begins when the nose tip impacts the target surface and the projectile starts its penetration into the target. The entrance stage is characterized by an increasing contact area between the projectile and the target which yields an increasing resistance and deceleration. This is the more difficult and less studied part of the penetration process. The present paper is focused on the entrance stage aiming at clarifying the interaction behavior during this early stage penetration and contributing to this less investigated problem.The literature survey indicates that the depth of the entrance stage is rather unclear and different researchers define the entrance stage depth differently. All definitions relate this depth to the projectile diameter. The shape of the deceleration curve is examined, and the linear deceleration-depth relationship assumed by the widely used spherical cavity expansion (SCE) approximation is criticized. This paper examines these definitions and postulates a new rational definition which agrees well with test data. A new analytical solution that is based on the SCE approximation is developed for the initial stage of penetration of a rigid projectile into a concrete medium. This modified spherical cavity expansion solution (denoted as SCEM) considers the variation of the “projectile-target” contact area with time, which is typical to the projectile entrance stage. Comparing the deceleration time histories of the new solution with the experimental data and with the DISCS model analysis results shows very good agreement, and the difference between the proposed analytical model and the commonly assumed deceleration-depth linear relationship is demonstrated. Characteristics of the entrance stage are analyzed and features like the embedment duration and the effect on the final penetration depth are examined.

Approximate solutions of finite dynamic spherical cavity-expansion models for penetration into elastically confined concrete targets

Confined concrete has superior anti-penetration performance over unconfined concrete. A finite dynamic spherical cavity-expansion approximation model with radially elastic confinement is proposed to analyze the confinement on concrete targets and predict the depth of penetration (DOP), taking steel-tube-confined concrete (STCC) targets normally penetrated by rigid projectiles as an example. Firstly, the validity of a nonlinear failure criterion to describe strength feature of concrete in the comminuted region is demonstrated by triaxial compressible tested data and the reference range of dimensionless parameter m in the modified Griffith criterion is recommended. Secondly, the relationship between the stresses in concrete and cavity-expansion velocity is developed. Furthermore, the confinement effects on response modes of confined concrete, radial stress at cavity wall and stresses distribution in concrete are analyzed. Lastly, an engineering model is also established to predict the DOP of STCC targets normally penetrated by rigid projectiles. The results show that the radial stress at cavity wall is not a constant during the cavity-expansion process in finite concrete with radially elastic confinement, which is different from the steady spherical cavity-expansion of infinite material in which the radial stress at cavity wall is a constant with constant cavity-expansion velocity. The possible response phases of confined concrete targets normally penetrated by rigid projectiles include “elastic-cracked-comminuted”, “cracked-comminuted” and “full-comminuted”, depending on the relationship among cavity-expansion velocity, radial confining stiffness and radius ratio of cavity to target. The DOP data from the engineering model established in this paper agree well with those of experiments in the published references.

Penetration of common ordinary strength water saturated concrete targets by rigid ogive-nosed steel projectiles

In this paper, we employ a dynamic spherical cavity-expansion solution for use with the spherical cavity-expansion approximation to analyze the penetration of common ordinary strength water saturated concrete targets by small scale rigid ogive-nosed projectiles with normal impact. To do this, we first obtain a quasi-static spherical cavity-expansion model for the radial stress at the cavity surface of a plastic–cracked–elastic material. Next, we add on a target inertia based term to the quasi-static radial stress at the cavity surface to obtain an approximate expression for the dynamic radial stress acting at the surface of the spherical cavity. This spherical cavity-expansion solution is employed with spherical cavity-expansion approximation based penetration models that previously required prior depth of penetration data to obtain the quasi-static target resistance function. With the newly proposed penetration models, a description of the common ordinary strength water saturated concrete material is based on a linear pressure–volumetric strain relation and a pressure dependent shear strength plasticity envelope with a tensile cutoff and is obtained from laboratory scale material tests; therefore, no prior depth of penetration data are required. Analytical model predictions obtained with the newly proposed model for final depth of penetration as a function of striking velocity, along with analytical models for acceleration, velocity and displacement as a function of time, are shown to be in good agreement with the corresponding experimental penetration data.

Thrombolysis compared with heparin for the initial treatment of pulmonary embolism: A meta-analysis of the randomized controlled trials

We developed penetration equations for ogival-nose projectiles that penetrated soil targets after normal impact. The spherical cavity-expansion approximation simplified the target analysis, so we obtained closed-form penetration equations. To verify our model, we conducted field tests into soil targets with 23.1 kg, 95.25 mm diameter projectiles and obtained rigid-body decelerations and final penetration depths. Model predictions show good agreement with measurements for five tests conducted at an impact velocity of 280 ms−1.

Effects of strain hardening and strain-rate sensitivity on the penetration of aluminum targets with spherical-nosed rods

We present penetration equations for rigid, spherical-nosed rods that penetrate 6061-T651 aluminum targets. The penetration models use the spherical cavity-expansion approximation and constitutive equations for the target that include strain hardening and strain-rate sensitivity. We obtained closed-form penetration equations for an incompressible target material; however, predictions for a compressible target require the numerical solution of a coupled set of nonlinear ordinary differential equations. Numerical results show the effects of compressibility, strain hardening, and strain-rate sensitivity. We also show that our penetration model requires compressibility, strain hardening, and strain-rate sensitivity to obtain good agreement with previously published depth of penetration data for striking velocities between 300 and 1200 m/s.

Penetration into ductile metal targets with rigid spherical-nose rods

We developed penetration equations for rigid spherical-nose rods that penetrate ductile metal targets. The spherical cavity-expansion approximation and incompressible and compressible elastic-perfectly plastic constitutive idealizations simplified the target analyses, so we obtained closed-form penetration equations. We compared predictions from our models with previously published penetration data and results from Lagrangian and Eulerian wavecodes.

Penetration into soil targets

We developed penetration equations for ogival-nose projectiles that penetrated soil targets after normal impact. The spherical cavity-expansion approximation simplified the target analysis, so we obtained closed-form penetration equations. To verify our model, we conducted field tests into soil targets with 23.1 kg, 95.25 mm diameter projectiles and obtained rigid-body decelerations and final penetration depths. Model predictions show good agreement with measurements for five tests conducted at an impact velocity of 280 ms−1.