Expansion of a cavity in a rubber block under unequal stresses

Elastic expansion of a small spherical void in the interior of a rubber block has been investigated by finite element analysis (FEA). The block was subjected to a far-field tensile or compressive stress while an internal pressure was applied to the cavity. The rubber was assumed to be virtually incompressible in volume and neo-Hookean in elastic behavior. Critical stress states were determined at which the void would become indefinitely large-a form of elastic instability. An applied tensile stress was found to lower the critical inflation pressure for instability, in agreement with experimental observations [1] whereas a compressive stress increased it. The critical mean stress was lowest for an isotropic stress system (when the applied tension was zero). It was higher when either compressive or tensile stresses were applied, i.e. under non-isotropic stresses, in agreement with the analysis of Hou and Abeyaratne [2]. However, the present results show a considerably smaller effect of tensile stress than predicted by Hou and Abeyaratne.     

带形介质内SH型导波对圆柱孔洞的动力分析

弹性板壳的反平面运动中由缺陷引起的应力集中问题可以按照带形介质中圆柱孔洞对SH型导波的散射问题来分析。首先,构造带形介质中相容导波的形式,即其满足上、下边界应力自由条件。之后,由波函数展开法给出圆柱孔洞散射波的级数表示,根据累次镜像叠加的方法构造由上、下边界反射所形成的相容的散射导波。最后,给定入射导波,由圆柱孔洞边界应力自由的条件来定解波函数级数的系数。数值算例求解了特定导波对圆柱孔洞的散射,给出了圆柱孔洞边沿的动应力分布,讨论了导波阶数、频率以及孔洞位置的影响。     

Effect of the static strength of a metal on long-rod penetration at low velocities

The effect of the static strength (yield stress) of a material on major penetration parameters (specific work of cavity formation and pressure at the contact surface) was examined. The analytical equations and the quantitative evaluation of the specific work spent for the formation of a spherical or cylindrical cavity and of the pressure at the surface of such cavities upon their static expansion were obtained for rigid-plastic and elastoplastic materials. The evaluation of the specific work of deformation upon the formation of the spherical cavity in the plate and the cylindrical one upon the penetration of a rigid rod are shown to be adequate. The influence of strain hardening on maximum radial stresses on the cavity surface was evaluated. The resistance to penetration and the expansion of a cavity are shown to be influenced by the elastic compressibility of a material in the inelastic region. The kinetics of the stress-strain state of the material upon the formation of the cavity was analyzed for the case of considerable deformation of a rod at penetration. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev Ukraine. Translated from Problemy Prochnosti, No. 1, pp. 93–110, January–February, 2000.     

Cavitation in elastic and hyperelastic sheets

Plane stress, axially symmetric, cavitation patterns are examined for large sheets with an embedded central circular hole. Both (radially-uniform) remote tension and internal pressure loads are considered. Material behavior is modeled by finite strain Hookean-type elastic and hyperelastic constitutive relations with logarithmic strains.Governing field equations are reduced to a single ordinary differential equation with the principal stress difference as the independent variable. Calculations reveal that under internal pressure the usual definition of cavitation state does not apply as load increases monotonously along the deformation path.Introducing the specific cavitation energy, as a practical measure in cavitation analysis, provides a unified framework for assessment of cavitation phenomena. Comparison with related patterns of spherical and cylindrical cavitation fields supports the main findings of this study.     

Deep penetration analysis with dynamic cylindrical cavitation fields

Dynamic cylindrical cavitation fields are studied for a family of plastic orthotropic solids with arbitrary strain hardening response. Analysis is within the framework of plane-strain, steady state flow theory of associated plasticity. New formulae for cavitation pressure are validated against accurate numerical analysis and contact is made with existing studies. A uniform procedure is presented for estimating penetration depth of rigid axisymmetric projectiles at normal impact. Comparison with available experimental data reveals a very good agreement for both spherical and cylindrical dynamic cavitation models. Quasi-static cavitation pressure formulae can predict penetration depth with an appropriate scaling of the yield stress. The scaling factors appear to be independent of material properties but reflect the shape of head profile.     

Fundamental solutions of cavitation in porous solids: a comparative study

The expansion of internally pressurized cavities, embedded in infinite bodies, in spherical and cylindrical (plane strain and plane stress) configurations, is investigated within the framework of finite plasticity. Material response is modeled by the Gurson theory for porous solids and includes strain hardening. Numerical results, obtained under the assumption of nearly universal loading histories, reveal limit cavitation states for all three deformation patterns. Cavitation is identified with asymptotic levels of the specific cavitation energy, which is highest for the spherical cavity and smallest for plane stress (plate) holes. The influence of material porosity is assessed in context of weight optimization of protective plates. A limited comparison with experimental data for porous titanium plate perforation reveals close prediction of ballistic limit velocity.     

Elastic/plastic indentation hardness and indentation fracture toughness: The inclusion core model

A new model for determining elastic/plastic indentation is presented. This model generalizes Johnson's incompressible core model to a compressible material and allows the indentation pressure to be transmitted via a misfitted inclusion core beneath the indenter which is surrounded by a hemispherical plastic zone. The internal stress field inside the core is obtained by applying Eshelby's spherical inclusion problem together with Hill's spherical-cavity expansion analysis. The plastic deformation considered here exactly ensures compatibility between the volume of a material displaced by the indenter and that accommodated by expansion. The analysis explains the essential relationships between the dimensions of the indentation and plastic zone and the dominant material properties; yield stress, hardness and elastic modulus. The solution is extended to evaluate the indentation fracture toughness by taking into account the reduced half-space constraint by the image force.     

Oscillating two-phase flow in a prestressed thick elastic tube

Summary The pulsating flow of a fluid with dusty particles in a prestressed thick walled elastic tube has been studied. The tube, subjected to a static inner pressure P_i and an axial stretch λ, is taken to be an incompressible, isotropic, elastic material. The fluid with particles is treated as incompressible Newtonian. Employing the theory of small deformation superimposed on large initial deformations, for an axially symmetric perturbed motion the governing equations are obtained in cylindrical polar coordinates. The analytical solutions of the equations of motion for the dust and the fluid have been obtained. Because of the variable character of the coefficients of the resulting equations for the solid body they are solved numerically. The dispersion relation is obtained as a function of the stretch, the thickness ratio and the parameters for dusty particles.     

考虑大变形和排水条件时柱孔扩张问题统一解析

将柱孔周围土体中的应力分布分为弹性区和塑性区,在弹性区中采用小变形理论,在塑性区中采用大变形理论和统一强度准则。根据柱孔扩张问题排水条件时的应力平衡方程、应力和应变连续的边界条件,推导出考虑土体剪胀、大变形、排水条件和中主应力等因素的柱孔扩张问题塑性区半径和极限扩孔压力的理论解答,同时也获得了弹塑性区中的应力和应变场分布规律。理论计算结果与现场实测结果较为接近,初步说明该理论具有一定的工程应用价值。对比分析表明:扩孔压力和塑性区半径受土体剪胀的影响要比b值显著。     

Cavitation Instabilities in Plastics and Rubber-Modified Plastics

Spherical void expansion in plastics and rubber-modified plastics is investigated under radial traction conditions. The plastics are modeled as elastic-plastic pressure-sensitive materials and the rubbers are modeled as nonlinearly elastic materials. First, the growth of a spherical void in an infinite plastic matrix is investigated under remote radial traction conditions. The results show that the cavitation stress of the plastic decreases significantly as the pressure sensitivity increases. Then, the growth of a spherical void located at the center of a spherical rubber particle in an infinite plastic matrix is investigated under remote radial traction conditions. The results indicate that without any failure criteria for the rubber, the cavitation stress does not exist when the void is small and the rubber is characterized by high-order strain energy functions. However, when a failure criterion for the rubber is considered at a finite stretch ratio, the results show that the cavitation stress for the plastic with the rubber particle becomes close to that for the plastic without the rubber particle. This revised version was published online in August 2006 with corrections to the Cover Date.     

不同拉压模量及软化特性材料的球形孔扩张问题的统一解

对于具有不同的拉压模量及软化特性的岩土类材料,提出了不同拉压模量及软化特性的控制参数,采用双剪统一强度理论推导了球形孔扩张问题的应力及位移的统一解。分析了模量、模型和软化等控制参数对球形孔扩张时的扩张压力、塑性区开展规律及应力场的影响。结果表明:圆孔极限扩张压力,塑性区的发展规律,应力场,位移场等均随着模量控制参数、模型参数及软化参数的变化而变化,因此若采用经典的弹性理论、单一的模型参数及传统的不考虑应变软化来对岩土类的工程材料进行设计计算,必会带来较大的误差。     

Concept of Ultimate Fracture-Front Velocity in Cylindrical Cavity Expansion in a Brittle Material

Expansion of a cylindrical cavity in a brittle material is analyzed on the basis of the concept of the ultimate velocity of the fracture front. The boundary problems have been solved for the material regions of various states. The expansion pressure of a cylindrical cavity as a function of the expansion velocity in a ceramic material is given and compared to that of a spherical cavity.     

Porosity-dependence of elastic properties and ultrasonic velocity in polycrystalline alumina — a model based on cylindrical pores

Effective elastic moduli and ultrasonic velocity of materials having aligned cylindrical pores have been derived using a series expansion in terms of the difference between the upper and lower bounds of elastic moduli obtained by the variational method. The theoretical results for polycrystalline alumina agree well with the experimental data, confirming the suggestion of previous researchers that a matrix containing parallel cylindrical pores orientated perpendicular to the applied stress, provide a better model than a spherical one in describing the porosity-dependence of elastic moduli in sintered specimens.     

Evaluating pressure due to the electrohydraulic effect in an infinite elastic medium

Assuming a homogeneous pressure distribution in the volume of water, the pressure developed upon discharge in a spherical water-filled cavity surrounded by an infinite elastic medium is calculated as a function of the action integral S = ∫₀ ^t i ²(t dt, where i is the current in the discharge channel. It is shown that the maximum pressure in discharge significantly decreases for relatively soft chamber wall materials (minerals, concrete) as compared to the approximation of undeformable cavity walls.     

Grain-size dependence of fracture stress in anisotropic brittle solids

The stress concentrations that occur at grain boundaries due to thermal expansion anisotropy and elastic stress concentration are discussed, and the stress intensity factor that results from these stresses is estimated. The procedure for the stress intensity factor calculation is based on the model in which a spherical crystal (grain) is forced into a cavity of equal size possessing annular or radial cracks emanating from the boundary. The stress intensity factor equation thus obtained is extended to include the effect of elastic stress concentration due to the presence of a cavity, and is subsequently used to predict the grain-size dependence of strength in anisotropic brittle ceramics. In assessing the degradation of strength with increasing grain size in non-cubic ceramics, it is shown that, in addition to grain size, the effect of pre-existing crack size must also be considered. Cubic ceramics, on the other hand, are known to exhibit no thermal expansion anisotropy and, based on the present model, their strength is predicted to be governed by the pre-existing flaw size, rather than the grain size.     

不同拉压模量及软化特性材料的柱形孔扩张问题的统一解

对于具有不同的拉压模量及软化特性的岩土类材料,提出了不同拉压模量及软化特性的控制参数,采用双剪统一强度理论推导了柱形孔扩张问题的应力及位移的统一解。分析了模量、模型和软化等控制参数对柱形孔扩张时的扩张压力、塑性区开展规律及应力场的影响。结果表明:圆孔极限扩张压力、塑性区的发展规律、应力场、位移场等均随着模量控制参数、模型参数及软化参数的变化而变化,因此若采用经典的弹性理论、单一的模型参数及传统的不考虑应变软化来对岩土类的工程材料进行设计计算,必会带来较大的误差。     

Analytical solution of the spherical cavity expansion process

The dynamic spherical cavity expansion is treated through a complete analytical solution of the equations of motion for an elasto-plastic solid obeying von-Mises yield criterion. The solutions for various metals, with different elastic and plastic properties, are fitted with a third order polynomial which relates the normalized pressure inside the cavity with the normalized velocity of the cavity wall. An extensive search for material similarities is conducted in order to highlight the roles of the elastic properties of the solid, as well as its strength and equation of state parameters. Using the simple terms we derive, for the coefficients in the third order polynomial, one can easily calculate the relation between pressures inside the cavity and their wall velocities for any solid to within 1%.     

饱和粘土中挤土桩球形孔扩张的弹塑性分析

将桩体的贯入模拟为球形孔扩张过程,基于球形孔扩张理论,在大变形假定情况下,结合修正剑桥模型及水力压裂理论,由最小耗能原理从能量的角度推导出了饱和粘土中挤土按球形孔扩张后,土体整个范围内,包括弹性区、塑性区、破坏区内的应力、位移及超孔隙水压力分布的解析解,并分析了土的超固结比对结果的影响,和其他文献的比较结果证明了理论解析解的准确性。其结论可为饱和土地区的静压桩、打桩等岩土工程问题提供理论依据。     

高压喷嘴空化初生的大涡模拟

考虑瞬时压力脉动对空化初生的影响,采用动态亚格子应力模型对某高压柴油喷嘴内部瞬时流动进行大涡模拟。计算结果表明:喷孔入口分离形成的局部低压区内时均湍流脉动压力可达时均静压的2倍,两相流场时均空泡位置及形态与实验结果十分吻合,并与基于时均静压及湍流脉动压力预测的空化初生区域基本一致。分离剪切层失稳形成柱状涡并激发了转捩过程,在分离转捩区域柱状涡发生扭曲变形并产生强烈的瞬时压力脉动,从而导致了空化。喷孔入口倒角处理使得分离减弱,对分离转捩过程和空化初生具有抑制作用。     

Calculation of penetration resistance of brittle materials using spherical cavity expansion analysis

In this paper, we show that the ‘target resistance’ of brittle materials can be calculated accurately using spherical cavity expansion analysis and a conventional brittle material model. The stress field ahead of the tip of the penetrator is assumed to have spherical symmetry. The brittle material is modeled as an elastic material which cracks under tension. The cracked material is considered to be pulverized when it fails in compression, which is then characterized as a Mohr-Coulomb material with pressure dependent shear strength. The target resistance value found from this analysis compares well with the reported experimental values for AD995 alumina (Al₂O₃) and aluminum nitride (A1N).