Micro-crack initiation at the tip of a semi-infinite rigid line inhomogeneity in piezoelectric solids

A dislocation emission mechanism for micro-crack initiation at the tip of a semi-infinite rigid line inhomogeneity in a piezoelectric solid is proposed in the present paper. For a rigid line inhomogeneity embedded in a piezoelectric matrix, dislocations of one sign are driven away from the tip due to high stress level, while the stationary dislocations of the opposite sign are left behind near the tip of the inhomogeneity. As a result, a micro-Zener–Stroh crack is initiated ahead of the line inhomogeneity. In the current study, a dislocation pileup mechanism for micro-crack initiation at the inhomogeneity tip is proposed. An interesting result is that the critical stress intensity factors for a line inhomogeneity perpendicular to the poling direction can be related to the fracture toughness of a conventional crack in the same material. Analytical solutions show that the critical plane shear stress intensity factor depends on the plane shear mechanical and displacement loadings, and the critical opening stress and electric displacement intensity factors depend on not only the mechanical and displacement loadings, but the electric field and displacement loadings as well.     

Numerical modeling of projectile penetration into compressible rigid viscoplastic media

We develop computational methods for modeling penetration of a rigid projectile into porous media. Compressible rigid viscoplastic models are used to capture the solid–fluid transition in behavior at high strain rates and account for damage/plasticity couplings and viscous effects that are observed in geological and cementitious materials. A hybrid time discretization is used to model the non‐stationary flow of the target material and the projectile–target interaction, i.e. an explicit Euler method for the projectile equation and a forward (implicit) method for the target boundary value problem. At each time step, a mixed finite element and finite‐volume strategy is used to solve the ‘target’ boundary value problem. Specifically, the non‐linear variational inequality for the velocity field is discretized using the finite element method while a finite‐volume method is used for the hyperbolic mass conservation and damage evolution equations. To solve the velocity problem, a decomposition–coordination formulation coupled with the augmented Lagrangian method is adopted. Numerical simulations of penetration into concrete were performed. By conducting a time step sensitivity study, it was shown that the numerical model is robust and computationally inexpensive. For the constants involved in the model (shear and volumetric viscosities, cut‐off yield limit, and exponential weakening parameter for friction) that cannot be determined from data, a parametric study was performed. It is shown that using the material model and numerical algorithms that developed the evolution of the density changes around the penetration tunnel, the shape and location of the rigid/plastic boundary, the compaction zones, and the extent of damage due to air‐void collapse are described accurately. Copyright © 2007 John Wiley & Sons, Ltd.     

Penetration into semi-infinite reinforced-concrete targets with spherical and ogival nose projectiles

We developed an engineering model for forces on rigid, spherical and ogival nose projectiles that penetrated semi-infinite, reinforced-concrete targets. Post-test target observations and triaxial, material-test data on samples cored from concrete targets guided the model development. The spherical, cavity-expansion approximation simplified the target analysis, and closed-from penetration equations were derived. The model predicted depths of penetration that were in reasonable agreement with penetration data.     

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 of a rigid projectile into an elastic-plastic target of finite thickness

This paper considers the problem of non-steady penetration of a rigid projectile into an elastic-plastic target of finite thickness. A specific blunt projectile shape in the form of an ovoid of Rankine is used because it corresponds to a reasonably simple velocity field which exactly satisfies the continuity equation and the condition of impenetrability of the projectile. The target region is subdivided into an elastic region ahead of the projectile where the strains are assumed to be small, and a rigid-plastic region near the projectile where the strains can be arbitrarily large. Using the above mentioned velocity field, the momentum equation is solved exactly in both the elastic and the rigid-plastic regions to find expressions for the pressure and stress fields. The effects of the free front and rear surfaces of the target (which is presumed not to be too thin) and the separation of the target material from the projectile are modeled approximately, and the force applied to the projectile is calculated analytically. An equation for projectile motion is obtained which is solved numerically. Also, a useful simple analytical solution for the depth of penetration or the residual velocity is developed by making additional engineering approximations. Moreover, the solution procedure presented in this paper permits a straight forward approximate generalization to accommodate a projectile with arbitrary shaped tip. Theoretical predictions are compared with numerous experimental data on normal penetration in metal targets, and the agreement of the theory with experiments is good even though no empirical parameters are used. Also, simulations for conical and hemispherical tip shapes indicate that the exact shape of the projectile tip does not significantly influence the prediction of integral quantities like penetration depth and residual velocity.     

A constitutive model for compressible elastomeric solids

A non-linear thermo-elastic constitutive model for the large deformations of isotropic materials is formulated. This model is specialized to account for the physics and thermodynamics of the elastic deformation of rubber-like materials, and based on these molecular considerations a constitutive model for compressible elastomeric solids is proposed. The new constitutive model generalizes the incompressible and isothermal model of Arruda and Boyce (1993) to include the compressibility and thermal expansion of these materials. The model is fit to existing experimental data on vulcanized natural rubbers to determine the material parameters for the rubbers examined. The fit between the simple model and the data is found to be very good for large stretches and moderate volume changes.List of symbols x\s=f(p) Deformation function - p Material point of a body in a reference configuration - x Place occupied by material point p in the current configuration - F(p)\eq(\t6/\t6p) f(p) Deformation gradient - J\s=det F\s>0 Determinant of F - F\s=RU\s=VR Polar decompositions of F - U, V Right and left stretch tensors; positive definite and symmetric - R Rotation tensor; proper orthogonal - U= _1–1 ³ ₁ ² r₁r₁ Spectral representation of U - V= ₁₌₁ ³ _t ² 1_t1₁ Spectral representation of V - _t > 0 Principal stretches - {r_i} Right principal basis - {l_i} Left principal basis - C\s=F ^T F, B\s=FF ^T Right and left Cauchy-Green strain tensors - \gq\s>0 Absolute temperature - \ge Internal energy density/unit reference volume - \gh Entropy density/unit reference volume - \gy\s=\ge\t-\gq\gh Helmholtz free energy/unit reference volume     

Penetration of a rigid projectile into a multi-layered target: theory and numerical computations

The main objective of the present work is to develop an adequate analytical model for penetration of multi-layered targets by rigid projectiles. The theoretical approach presented here generalizes the single-layer models described in [1] and [2]. As in [1] and [2] an analytical solution is developed for which the momentum equation is satisfied pointwise in the target region, while the boundary and continuity conditions are satisfied only approximately. Also, a single particular velocity field is assumed for all target layers. The predictions of the analytical solution are compared with numerical simulations obtained using the hydrocode Autodyn2D [3]. Attention is focused on two cases of a two-layered target: one consisting of materials which differ only by their hardnesses (yield strengths); and the other consisting of significantly different materials (RHA and Aluminum). The predictions of the analytical model are in reasonably good agreement with those of Autodyn2D for both cases, independently of whether the hard layer is first or second. It should also be mentioned that the computational time is reduced from several hours for Autodyn2D to only a few minutes for the analytical model.     

Steady-state flow of compressible rigid–viscoplastic media

Computational methods for modeling steady-state flow of compressible rigid viscoplastic fluids are proposed. The constitutive equation used captures the combined effects of high-strain rate and high-pressure on the behavior of porous materials.A mixed finite-element and finite-volume strategy is developed. Specifically, the variational inequality for the velocity field is discretized using the finite element method and a finite volume method is adopted for the hyperbolic mass conservation equation. To solve the velocity problem a decomposition–coordination formulation coupled with the augmented lagrangian method is used. This approach is accurate in detecting the viscoplastic regions and permit us to handle the locking medium condition.The proposed numerical method is then applied to model the penetration of a rigid projectile into cementitious targets. The numerical model accurately describes the density changes around the projectile, the stress field, as well as the shape and location of the deformation zone (viscoplastic region) in the target.     

Couette flow of rigid/hardening solids

Steady Couette-type flow of rigid/linear-hardening solids is investigated. Material behaviour is governed by the von-Mises flow rule. The general flow problem is reduced to a semi-coupled system of two partial differential equations. A few possible boundary conditions are suggested. An exact solution for the plane strain problem is derived. That solution is used, in conjunction with average stress boundary data, for simulating a simple Couette-type flow problem through a curved tube with a rectangular cross section. It is shown that wall friction induces plastic boundary layers into the material. The essential features of these boundary layers are discussed.     

Cohesive zone representations of failure between elastic or rigid solids and ductile solids

A cohesive zone model that describes tangential separation as well as normal separation along an interface is reviewed. The model is based on nonlinear traction-separation relations between the normal and tangential components of the interface tractions and relative displacements. To illustrate the application of the cohesive zone model in studies of material failure or crack growth, analyses of matrix-fibre debonding in metal matrix composites are presented, taking into account effects of residual stresses or of nonlocal plasticity for the matrix. Also studies of interface crack growth under mixed mode conditions are discussed.     

Void formation and growth in a class of compressible solids

A new class of compressible elastic solids, which includes the Blatz-Ko material as a special case, is proposed. A closed-form solution is constructed and studied for a bifurcation problem modeling void formation in this class of compressible elastic solids. The relation between the void-formation condition and the material parameters is obtained analytically. An energy comparison of the void-formation deformation and the homogeneous expansion deformation is carried out.     

Stress analysis of a semi-infinite plate with a thin rigid body

Stress analysis of a semi-infinite plate with an oblique thin rigid body is carried out as a mixed boundary value problem. The complex variable method and a rational mapping function of a sum of fractional expressions are used. A closed solution is obtained for the shape, which is represented by a rational mapping function. Stress distribution, stress singularity at the tip of the thin rigid body, the resultant moment over the thin rigid body, and the rotation angle are investigated.     

Tensile instabilities and ellipticity in fiber-reinforced compressible non-linearly elastic solids

In this paper we examine loss of ellipticity and associated failure for fiber-reinforced compressible non-linearly elastic solids under uniaxial plane deformations. We consider first fiber reinforcement that endows the material with additional stiffness only in the fiber direction. It is shown, in particular, that loss of ellipticity under tensile loading in the fiber direction corresponds to a turning point of the nominal stress and may require concavity of the Cauchy stress–stretch curve. Secondly we examine fiber reinforcement that introduces additional stiffness under shear deformations. In this case we find that loss of ellipticity again occurs at a turning point of the nominal stress, in contrast to the situation for incompressible materials.     

Dynamic frictional slip along an interface between plastically compressible solids

Dynamic frictional slip along an interface between plastically compressible solids is analyzed. The plane strain, small deformation initial/boundary value problem formulation and the numerical method are identical to those in Shi et al. (Int J Fract 162:51, 2010) except that here the material constitutive relation allows for plastic compressibility. The interface is characterized by a rate and state dependent friction law. The specimens have an initial compressive stress and are subject to shear loading by edge impact near the interface. Two loading conditions are analyzed, one giving rise to a crack-like mode of slip propagation and the other to a pulse-like mode of slip propagation. In both cases, the initial compressive stress is taken to vary with plastic compressibility such that the associated initial effective stress is the same for all values of plastic compressibility. The volume change for the crack-like slip mode is mainly plastic while the elastic volume change plays a larger role for the pulse-like mode. For the crack-like slip mode, the proportion of plastic dissipation in the material increases with the increasing plastic compressibility, but the effect of plastic compressibility on the energy partitioning for the pulse-like slip mode is much smaller. The predicted propagation speeds approach a speed about the dilational wave speed for both the crack-like and pulse-like slip modes and this speed is not sensitive to the value of the plastic compressibility parameter. Plastic dissipation is found to be mainly associated with the deformation induced by the loading wave rather than with the deformation arising from slip propagation. The amplitude of the slip rate in the slip pulses is found to be largely governed by the value of the initial compressive stress regardless of the value of plastic compressibility.

     

Nonsteady penetration of longs rods into semi-infinite targets

This paper describes a new one-dimensional theory of nonsteady penetration of long rods into semi-infinite targets. The target is viewed as a “finite mass” that resides within the semi-infinite target space. Thus, an equation of motion for the target was constructed so that together with erosion and penetrator deceleration equations, expressions for penetration rates and depths were obtained. Forces acting on the target and penetrator are defined in terms of only ordinary strength levels usually associated with dynamic properties or work-hardened material states. Also, the concept of critical impact velocity was used to establish the onset of penetration in this formulation. This penetration equation corresponds in exact form to hydrodynamic theory in the limits of small strengths and/or high impact velocity. Results for penetration rates agree well with hydrocode calculations, and predicted penetrations agree with experimental data over an impact velocity range of 0–5,000 m/s.     

Constitutive models for ductile solids reinforced by rigid spheroidal inclusions

We study the effective constitutive response of composite materials made of rigid spheroidal inclusions dispersed in a ductile matrix phase. Given a general convex potential characterizing the plastic “in the context of J₂-deformation theory” behavior of the isotropic matrix, we derive expressions for the corresponding effective potentials of the rigidly reinforced composites, under general loading conditions. The derivation of the effective potentials for the nonlinear composites is based on a variational procedure developed recently by Ponte Castaneda (1991a, J. Mech. Phys. Solids 39, 45–71). We consider two classes of composites. In the first class, the spheroidal inclusions are aligned, resulting in overall transversely isotropic symmetry for the composite. In the second class, the inclusions are randomly oriented, and thus the composite is macroscopically isotropic. The effective response of composites with aligned inclusions depends on both the orientation of the loading relative to the inclusions and on the inclusion concentration and shape. Comparing the strengthening effects of rigid oblate and prolate spheroids, we find that prolate spheroids give rise to stiffer effective response under axisymmetric “relative to the axis of transverse isotropy” loading, while oblate spheroids provide greater reinforcement for materials loaded in transverse shear. On the other hand, nearly spherical “slightly prolaterd spheroids are most effective in strengthening the composite under longitudinal shear. Thus, the optimal shape for strengthening composites with aligned inclusions depends strongly on the loading mode. Alternatively, the properties of composites with randomly oriented spheroidal inclusions, being isotropic, depend only on the concentration and shape of the inclusions. We find that both oblate and prolate inclusions lead to significant strengthening for this class of composites.     

Energy bounds for a mixture of two linear elastic solids occupying a semi-infinite cylinder

Summary The aim of this paper is to establish some forms of the Saint-Venant principle for a mixture of two linear elastic solids occupying a semi-infinite prismatic cylinder. We examine the behaviour of the energy for both static and dynamical problems. Under mild assumptions on the asymptotic behaviour of the unknown fields at infinity, we show that in the static case the elastic energy of the portion of the cylinder beyond a distancex ₃ from the loaded region decays exponentially withx ₃. For the dynamical problem we estimate through the data the total energy stored in that part of the cylinder whose minimum distance from the loaded end isx ₃; these estimates, which are based on the assumption that the initial total energy is finite, depend uponx ₃ but do not depend upon the timet.     

The frictionless rolling contact of a rigid circular cylinder on a semi-infinite granular material

The resulting flow and deformation of a semi-infinite granular material under a rolling, smooth rigid circular cylinder is investigated using a perturbation method. Based on the double-shearing theory of granular flow, complete stress and velocity fields, resistance to rolling and the permanent displacement of surface particles are determined to first order; when the internal friction angle is zero, the solutions reduce to those obtained in the corresponding analysis for Tresca or von-Mises materials. The solution scheme and the double-shearing model for granular flow both find their origins in the work of A.J.M. Spencer.     

An Eulerian particle level set method for compressible deforming solids with arbitrary EOS

In this study, an Eulerian, finite‐volume method is developed for the numerical simulation of elastic–plastic response of compressible solid materials with arbitrary equation of state (EOS) under impact loading. The governing equations of mass, momentum, and energy along with evolution equations for deviatoric stresses are solved in Eulerian conservation law form. Since the position of material boundaries is determined implicitly by Eulerian schemes, the solution procedure is split into two separate subproblems, which are solved sequentially at each time step. First, the conserved variables are evolved in time with appropriate boundary conditions at the material interfaces. In the present work a fourth‐order central weighted essentially non‐oscillatory shock‐capturing method that was developed for gas dynamics has been extended to high strain rate solids problems. In this method fluxes are determined on a staggered grid at places where solution is smooth. As a result, the method does not rely on the solution of Riemann problems but enjoys the flexibility of using any type of EOS. Boundary conditions at material interfaces are also treated by a special ghost cell approach. Then in the second subproblem, the position of material interfaces is advanced to the new time using a particle level set method. A fifth‐order Godunov‐type central scheme is used to solve the Hamilton–Jacobi equation of level sets in two space dimensions. The capabilities of the proposed method are evaluated at the end by comparing numerical results with the experimental results and the reported benchmark solutions for the Taylor rod impact, spherical groove jetting, and void collapse problems. Copyright © 2009 John Wiley & Sons, Ltd.