Dynamic Spherical Cavity Expansion Research Articles

Modeling dynamic spherical cavity expansion in elasto-viscoplastic media

In this paper, we extend the dynamic spherical cavity expansion model for rate-independent materials developed by Durban and Masri (Int J Solids Struct 41(20):5697–5716, 2004), Masri and Durban (J Appl Mech 72(6):887–898, 2005), and Cohen et al. (J Appl Mech 77(4):041009, 2010) to viscoplastic media. For that purpose, we describe the material behavior with an isotropic Perzyna-type overstress formulation (Perzyna in Q Appl Math 20:321–332, 1963; Adv Appl Mech 9:243–377, 1966) in which the material rate dependence is controlled by the viscosity parameter $$\eta $$. The theoretical predictions of the cavity expansion model, which assumes that the cavity expands at constant velocity, are compared with finite element simulations performed in ABAQUS/Explicit (Abaqus Explicit v6.13 User’s Manual, ABAQUS Inc., Richmond). The agreement between theory and numerical simulations is excellent for the whole range of cavitation velocities investigated, and for different values of the parameter $$\eta $$. We show that, as opposed to the steady-state self-similar solutions obtained for rate-independent materials (Durban and Masri 2004; Masri and Durban 2005; Cohen et al. 2010), the material viscosity leads to time-dependent cavitation fields and stress relaxation as the cavity enlarges. In addition, we also show that the material viscosity facilitates to model the shock waves that emerge at the highest cavitation velocities investigated, controlling the amplitude and the width of the shock front.

Depth of penetration of steel-tube-confined concrete targets based on dynamic finite cylindrical cavity-expansion models

Steel-tube-confined concrete (STCC) targets show excellent anti-penetration performance as a result of confinement effect of steel tube on in-filled concrete. A dynamic finite cylindrical cavity-expansion (FCCE) model with elastic radial spring was developed to analyze the confinement effect and predict the depth of penetration (DOP) of STCC targets normally penetrated by rigid sharp-nosed projectiles. Firstly, steady responses of the dynamic FCCE approximation model were obtained on the basis of the assumption of incompressibility of target concrete and failure of comminuted zone with the Heok–Brown criterion. Then, based on the dynamic FCCE approximation model, a DOP model for STCC targets impacted by rigid projectiles was proposed. Lastly, the penetration tests of STCC targets normally penetrated by 12.7 mm armor piecing projectile (APP) were taken as examples to validate the applicability of the dynamic FCCE approximation model and DOP model. The results show that in comparisons with the results based on the dynamic spherical cavity-expansion (FSCE) model, the DOP model based on the dynamic FCCE model in this paper is more applicable to predict the DOP of STCC targets penetrated by rigid conical or other sharp-nosed projectiles with a proper value of empirical constant m in the Heok–Brown criterion.

Dynamic spherical cavity expansion analysis of concrete/rock based on Hoek-Brown criterion

Cavity expansion theory has been widely applied to predict the penetration depth of projectiles into geological materials, in which the yield criterion and the constitutive model are combined to describe the shear and compaction of the material, separately. For a precise description of the material behaviors in plastic for geological materials with varied compressive strength, in this present manuscript, the Hoek-Brown yield criterion and the Dilatant-Kinematic equation were introduced into the plastic region. A response region called elastic-cracking-dilatant-compaction region for concrete and rock was established. Then the effects of various parameters, including brittleness coefficient, material integrity coefficient and dilatant coefficient, on the relationship between cavity pressure and normal expansion velocity were discussed. Finally, a penetration depth formula of rigid projectiles into semi-infinite concrete and rock targets was established. It showed that, compared with the classical Forrestal model, the proposed model had a well-being as well as higher accuracy prediction of the penetration depth of concrete and rock materials with various strengths.

Revising the penetration behavior of concrete-like and metal-like materials against the rigid projectile impact

Many studies indicate that projectile deceleration exhibits a quasi-constant behavior in the tunneling penetration stage of the rigid penetration mode for both metal-like and concrete-like materials. As to the issue, in the present paper the penetration process under the rigid penetration mode was revised, where a two-phase penetration theory is proposed including the ‘Squeeze and Push’ penetration phase and the dynamic spherical cavity expansion penetration phase. When the striking velocity is below the critical striking velocity, the ‘Squeeze and Push’ effect is generated since the cavitation size is less than the projectile diameter, where the quasi-static term overshadows the dynamic term. The interaction from the surface friction between projectile and target and the penetration force due to the main quasi-static term contributes to the occurrence of the quasi-constant phenomenon. Conversely, when the striking velocity is above the critical velocity, the penetration enters into the dynamic spherical cavity expansion penetration phase, the accompanying feature is that the tunnel becomes large beyond the projectile diameter, meanwhile the friction effect between the projectile and target disappears, the deceleration shows a well-defined peak with the rise of the striking velocity. Following the features, semi-empirical formulas are obtained. Finally, the semi-empirical formulae are discussed and validated by comparing with the typical penetration experiments including concrete-like and metal-like materials.

Dynamic spherical cavity expansion in Gurson materials with uniform and non-uniform distributions of porosity

This paper investigates both theoretically and using finite elements the elastoplastic field induced by a pressurized spherical cavity expanding dynamically in an infinite medium modelled using the Gurson–Tvergaard–Needleman porous plasticity approach. The theoretical model, which assumes that the porosity is uniformly distributed in the material and the cavitation fields are self-similar, incorporates artificial viscous stresses into the original formulation of Cohen and Durban (2013b) to capture the shock waves that emerge at high cavitation velocities. The finite element calculations, performed in ABAQUS/Explicit (2013) using the Arbitrary Lagrangian Eulerian adaptive meshing available in the code, simulate the cavity expansion process in materials with uniform and non-uniform distributions of porosity. The finite element results show that the distribution of porosity has small influence on the cavitation velocity, as well as on the location of the shock wave, which are primarily determined by the cavity pressure and the average material properties. In contrast, it is shown that the intensity of the shock wave, evaluated based on the maximum value of the plastic strain rate within the shock, depends on the local material porosity. The ability of the theoretical model to reproduce the numerical results obtained for the various distributions of porosity used in this work is exposed and discussed.

Analysis of dynamic spherical cavity expansion in undrained modified Cam Clay soil

SummaryIn order to capture the influence of the cavity expansion velocity, this paper presents a semianalytical solution for dynamic spherical cavity expansion in modified Cam Clay (MCC) soil. The key problem is solving the six coupled partial differential equations (PDEs) of cavity expansion, in which the dynamic term is considered in the stress equilibrium equation. The similarity transformation technique is used to transform the PDEs into ordinary differential equations (ODEs). Subsequently, the numerical method using the function “ODE45” in MATLAB is selected to solve the ODEs, which allows the stress and excess pore pressure around the expanding spherical cavity wall to be obtained. The proposed semianalytical solution for dynamic spherical cavity expansion was validated by comparting the degenerate solution with the published quasistatic solution for the MCC model. Parametric study was then conducted to capture the influence of the cavity wall velocity on the cavity expansion response. The proposed solution has potential application to geotechnical problems such as dynamic pile driving, the dynamic cone penetration test, and so forth.

Spherical cavity-expansion model for penetration of reinforced-concrete targets

The feature of reinforcing bars is introduced into dynamic cavity-expansion theory. Based on the elastic–plastic response penetration model of plain (i.e., unreinforced) concrete (Forrestal and Tzou, 1997), a dynamic spherical cavity-expansion penetration model for reinforced-concrete targets is developed with consideration of the circumferential restriction effect derived from reinforcing bars in the crushed region. The theoretical solution and simplified calculation formula for the cavity radial stress in incompressible and compressible reinforced concrete are obtained by introducing a reinforcement ratio as the volume fraction of rebars in the concrete target. A damping function is presented to describe the restriction effect of a single layer of reinforcing bars on the surrounding concrete, thus establishing a model to calculate the penetration resistance of multilayer reinforced-concrete targets. Compared with test data for the penetration depth, this model considering the circumferential restriction effect produces better results compared with the existing theory.

Two new penetration models for ballistic clay incorporating strain-hardening, strain-rate and temperature effects

Ballistic clay, such as Roma Plastilina No. 1 (RP # 1) clay, has been used as a standard backing material in back-face signature (BFS) tests of body armor. Hence, it is important to thoroughly understand penetration mechanics of ballistic clay. In the current paper, two analytical solutions for dynamic spherical cavity expansion in an infinite elastic-plastic solid are first derived by including the strain-hardening effect, strain-rate dependence and temperature influence. The elastic region in the solid is treated as either compressible or incompressible and described using Hooke's law, while the plastic region is regarded as incompressible and characterized using two different constitutive models, one of which is the well-known Johnson-Cook model and the other is a modified Johnson-Cook model simplified from a recently proposed model for ballistic clay. Based on these two dynamic spherical cavity expansion solutions, two models for penetration into a ballistic clay by a rigid projectile with a spherical or an ogival nose are then developed, which incorporate the coupled strain-hardening, strain-rate and temperature effects for the first time. The depth of penetration and impact time relations are obtained for non-penetration impacts, and the ballistic limit and residual velocity are determined for penetration impacts by each type of projectile. To quantitatively illustrate the two newly developed penetration models, they are directly applied to simulate drop tests and impact tests of RP # 1 clay. It is found that the predictions by the current new models agree fairly well with existing experimental data and simulation results.

Dynamic spherical cavity expansion analysis of concrete using the Bingham liquid constitutive model

High pressure exists in concrete targets during hypervelocity penetration. An incompressible concrete material under large deformation can be represented by a liquid constitutive model. The present work modifies the cavity expansion theory by characterizing incompressibility using the Bingham liquid constitutive model, in which the viscosity of an incompressible material is considered according to Cleja-Tigoiu's work. An analytic expression is derived for the relationship between the pressure on a cavity boundary and the velocity of cavity expansion. Then, the effects of material parameters in the modified model are studied, which is the basis of the calculation of the drag force in hypervelocity penetration. The viscosity coefficient (μ0) is the most sensitive parameter for the radial stress on the cavity surface. As μ0 increases, the dimensionless stress on the cavity surface (S) increases and the quadratic relation between S and the velocity of cavity expansion (a˙) becomes weaker. Finally, the penetration depths of long rod hypervelocity penetration into a concrete target are predicted based on the modified model, the results are in good agreement with experimental results.

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.

Cavitation analysis of spherical shock wave evolution in concrete medium

This paper investigates the dynamic spherical cavity expansion of shock wave evolution in concrete medium described by compressible Drucker–Prager plasticity with locked hydrostat. Hugoniot jumping conditions across shock waves are fully accounted for and a complete description of the motion and stress distribution evolution is obtained for different response domains. By solving the dimensionless governing differential equations during different periods, numerical illustrations are then developed with application to an explosion scenario inside an infinite concrete medium. The parameter study indicates that the equivalent charge weight of TNT proportionally affects the shock waves evolution, while the standoff distance has an inversely proportional impact on the ultimate locked/compacted boundary area.

Theoretical Analysis of Dynamic Spherical Cavity Expansion in Reinforced Concretes

This paper aims to establish a model that considers the penetration resistance caused by the constraint effects of steel reinforcements on concrete. Firstly, based on the experiment phenomena that reinforcements increase the toughness and tensile strength of concretes, the fitting relational expression between toughness of reinforced concrete and ratio of reinforcement was used to improve the Griffith yield criterion for reinforced concrete. Then, the dynamic spherical cavity expansion analysis was developed using the improved Griffith yield criterion as constitutive model and the dilation equation as equation of state, and the response regions were consisted of six distinct zones: cavity, compaction zone, dilation zone, radially cracked zone, elastic zone and undisturbed zone. This dynamic analysis considered the compression and dilation of concretes at the same time and was applicable to the penetration problem of reinforced concrete target. At last, based on the theoretical model of this paper, the experiments of projectiles with different weights penetrating into reinforced concrete targets with different reinforcement ratios were calculated using penetration analysis method of rigid projectiles. The comparison results showed that the theoretical analysis model of this paper can be used to predict the depth of penetration and other physical parameters such as velocity and deceleration with certain rationality.

Study of steady cavitation assumptions in strain-rate-sensitive solids for rigid projectile penetrations

Spherical cavity expansion is a well-known theory used to develop penetration models and failure analysis on solids. However, some limitations of this theory have been found, especially when complex materials have been tried to model. In this paper, steady cavitation assumptions are studied for strain-rate-sensitive solids, in order to apply cavity-expansion results to the penetration of rigid projectiles. In this way, this paper is the continuation of our previous paper, where an engineering penetration model was formulated using the dynamic spherical cavity-expansion theory for an elasto-plastic, compressible, Cowper–Symonds solid. In the present work, the assumption of self-similarity transformation was studied and verified for rigid projectile penetration in strain-rate-sensitive solids. The cavity-expansion problem was studied by finite-element simulations performed in ANSYS/AUTODYN for semi-infinite 6061-T651 Al plates. Additionally, engineering and computational penetration models were compared for different M300 steel spheres impacting the semi-infinite 6061-T651 Al targets. It was found that self-similarity transformation cannot be applied to strain-rate-sensitive solids; however, if the spherical cavity is constant, as in a rigid projectile penetration event, assumptions of steady stresses are valid, and the self-similarity transformation can be applied at medium and low penetration velocities.

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.

Dynamic spherical cavity expansion analysis of rate-dependent concrete material with scale effect

In this paper, dynamic spherical cavity expansion model is developed for rate-dependent concrete material described by modified Drucker-Prager Cap plasticity model incorporated with size effect. Penetration resistance components are computed accordingly with static resistance dependent on projectile diameter scale. Saturated region is introduced to investigate the response of concrete target under high impacting velocity. The validation of the proposed model is conducted by modelling a rigid projectile with full-scale and 1/10-scale diameter penetrating a semi-infinite concrete target. Depth of penetration numerically obtained is also compared with some widely recognized empirical penetration formulae, and better agreement is achieved by the proposed model for penetration tests with various striking velocities.

An engineering model for the penetration of a rigid-rod into a Cowper–Symonds low-strength material

The impact of a rigid-rod into a low-strength target is an ubiquitous research problem in many scientific and engineering fields. Typically, this impact has been modeled by the expansion of a pressurized spherical cavity in an unbounded elasto-plastic medium. In this paper, an engineering penetration model was developed using the dynamic spherical cavity expansion theory for an elasto-plastic, compressible, strain-dependent and strain-rate-sensitive material. The material plastic flow behavior was modeled using the Cowper–Symonds strength model. The engineering model was numerically solved, and its predictions were compared with results of computational finite-elements simulations performed in ANSYS/AUTODYN. Additionally, engineering and computational models were validated with experimental data using spherical-nosed, M300 steel projectiles impacting a semi-infinite Al 6061-T651 plate. Comparisons showed good agreement among engineering model results, computational model results and experimental data.

An Investigation on the Grooved-Tapered Projectile Penetrating into the Concrete Targets

Based on the dynamic spherical cavity expansion (SCE) theory of the concrete materials and the analysis of the experimental data, both the model of the petaling penetration with low speed and the model of the round hole penetration with high speed are constructed to describe the penetration of the grooved-tapered projectile in this paper. The penetration depth and the mass abrasion are calculated using the models, so are the change of the velocity and the acceleration of the projectile with the time in the stage of the low speed penetration. The results show for the grooved-tapered projectile penetrating the concrete, the error of the penetration depth and the mass abrasion between the theoretical value calculated using the petaling penetration model and the experimental data is less than 11%, when the initial velocity is lower than about 1000m/s. When the initial velocity is higher than about 1000m/s, the error of the penetration depth between the theoretical value calculated using the round hole penetration model and the experimental data is more than 20%, and the mass abrasion calculated is almost coincide with the experimental data. The research shows the models are suitable for the analysis of the grooved-tapered projectile penetrating the concrete target, and the grooved-tapered projectile is more valuable in the high speed penetration.

Projectile Mass Loss Model Using the Coefficient of Friction for High-Speed Penetration into Concrete

In this paper, a model of the friction coefficient in the high-speed penetration process has been used, which considers the micro-asperities of the projectile surface, the adiabatic shearing and the heat conduction on the nose of projectile. It also considers that the coefficient of friction is a function of the sliding velocity. Then, an analytical model of mass loss based on the coefficient of friction and the revised dynamic spherical cavity expansion theory of the concrete material is constructed. An analytical estimate for the work done by friction force in the penetration could be calculated and the evaluation of mass loss of projectile could also be calculated by the heat translated from the work done by friction force. Finally, a comparative analysis between the calculated data and the experimental data of mass loss is done.

An Investigation on the Nose Mass Abrasion of Projectile

Based on the dynamic spherical cavity expansion theory of concrete and the analysis of experimental data, a mass abrasion model of projectile considering the hardness of aggregates, the relative strength of target and projectile and the initial impact velocity is constructed in this paper. The initial impact velocity is the most important factor of mass abrasion. The hardness of aggregates and the strength of projectile are also the significant factor of mass abrasion. But relatively speaking, the sensitivity of strength of projectile to mass abrasion is higher, which indicates that the effect of projectile material on mass abrasion is more dramatic than the hardness of aggregates.

Projectile Nose Mass Abrasion of High-Speed Penetration into Concrete

Based on the dynamic spherical cavity expansion theory of concrete and the analysis of experimental data, a mass abrasion model of projectile considering the hardness of aggregates, the relative strength of target and projectile, and the initial impact velocity is constructed in this paper. Furthermore, the effect of mass abrasion on the penetration depth of projectile and the influence of hardness of aggregates and strength of projectile on penetration depth and mass loss are also analyzed. The results show that, for the ogive-nose projectile with the CRH of 3 and aspect ratio of 7 penetrating the concrete of 35 MPa, the “rigid-body penetration” model is available when the initial impact velocity is lower than 800 m/s. However, when the initial impact velocity is higher than 800 m/s, the “deforming/eroding body penetration” model should be adopted. Through theoretical analysis and numerical calculation, the results indicate that the initial impact velocity is the most important factor of mass abrasion. The hardness of aggregates and the strength of projectile are also significant factors. But relatively speaking, the sensitivity of strength of projectile to mass abrasion is higher, which indicates that the effect of projectile material on mass abrasion is more dramatic than the hardness of aggregates.