Multi-scale defect interactions in high-rate failure of brittle materials,Part II: Application to design of protection materials

Micromechanics based damage models, such as the model presented in Part I of this 2 part series (Tonge and Ramesh, 2015), have the potential to suggest promising directions for materials design. However, to reach their full potential these models must demonstrate that they capture the relevant physical processes. In this work, we apply the multiscale material model described in Tonge and Ramesh (2015) to ballistic impacts on the advanced ceramic boron carbide and suggest possible directions for improving the performance of boron carbide under impact conditions. We simulate both dynamic uniaxial compression and simplified ballistic loading geometries to demonstrate that the material model captures the relevant physics in these problems and to interrogate the sensitivity of the simulation results to some of the model input parameters. Under dynamic compression, we show that the simulated peak strength is sensitive to the maximum crack growth velocity and the flaw distribution, while the stress collapse portion of the test is partially influenced by the granular flow behavior of the fully damaged material. From simulations of simplified ballistic impact, we suggest that the total amount of granular flow (a possible performance metric) can be reduced by either a larger granular flow slope (more angular fragments) or a larger granular flow timescale (larger fragments). We then discuss the implications for materials design.     

A thermomechanical constitutive model for cemented granular materials with quantifiable internal variables. Part II – Validation and localization analysis

We study the mechanical failure of cemented granular materials (e.g., sandstones) using a constitutive model based on breakage mechanics for grain crushing and damage mechanics for cement fracture. The theoretical aspects of this model are presented in Part I: Tengattini et al. (2014), A thermomechanical constitutive model for cemented granular materials with quantifiable internal variables, Part I – Theory (Journal of the Mechanics and Physics of Solids, 10.1016/j.jmps.2014.05.021). In this Part II we investigate the constitutive and structural responses of cemented granular materials through analyses of Boundary Value Problems (BVPs).The multiple failure mechanisms captured by the proposed model enable the behavior of cemented granular rocks to be well reproduced for a wide range of confining pressures. Furthermore, through comparison of the model predictions and experimental data, the micromechanical basis of the model provides improved understanding of failure mechanisms of cemented granular materials. In particular, we show that grain crushing is the predominant inelastic deformation mechanism under high pressures while cement failure is the relevant mechanism at low pressures. Over an intermediate pressure regime a mixed mode of failure mechanisms is observed. Furthermore, the micromechanical roots of the model allow the effects on localized deformation modes of various initial microstructures to be studied. The results obtained from both the constitutive responses and BVP solutions indicate that the proposed approach and model provide a promising basis for future theoretical studies on cemented granular materials.     

Solution existence conditions for elastoplastic constitutive models of granular materials

In this paper, the conditions of solution existence for stress rates under given strain rates are investigated. The focus of the solution existence investigation is on the non-associated flow rule and elastic stress–strain relationship. Granular materials characterized with strong non-associated plastic flows are used as a particular example for analysis. Various flow rules for granular materials are analyzed, including Rowe’s, Roscoe’s flow rules and their modified versions. In the elastic stress–strain relationships of materials, the effects of Poisson’s ratio on solution existence are investigated. Both isotropic and anisotropic elasticity are considered. Given a granular material and its states, it is found that there exists a critical Poisson’s ratio for a particular non-associated flow rule. When the Poisson’s ratio of a material is above this critical Poisson’s ratio, its constitutive model is susceptible to solution non-existence. It is suggested that special attentions should be paid to the selection of material Poisson’s ratio and non-associated flow rule to ensure the existence of elastoplastic solutions.     

Mechanics of robust and releasable adhesion in biology: Bottom-up designed hierarchical structures of gecko

Gecko and many insects have evolved specialized adhesive tissues with bottom-up designed (from nanoscale and up) hierarchical structures that allow them to maneuver on vertical walls and ceilings. The adhesion mechanisms of gecko must be robust enough to function on unknown rough surfaces and also easily releasable upon animal movement. How does nature design such macroscopic sized robust and releasable adhesion devices? How can an adhesion system designed for robust attachment simultaneously allow easy detachment? These questions have motivated the present investigation on mechanics of robust and releasable adhesion in biology. On the question of robust adhesion, we introduce a fractal gecko hairs model, which assumes self-similar fibrillar structures at multiple hierarchical levels mimicking gecko's spatula ultrastructure, to show that structural hierarchy plays a key role in robust adhesion: it allows the work of adhesion to be exponentially enhanced with each added level of hierarchy. We demonstrate that, barring fiber fracture, the fractal gecko hairs can be designed from nanoscale and up to achieve flaw tolerant adhesion at any length scales. However, consideration of crack-like flaws in the hairs themselves results in an upper size limit for flaw tolerant design. On the question of releasable adhesion, we hypothesize that the asymmetrically aligned seta hairs of gecko form a strongly anisotropic material with adhesion strength strongly varying with the direction of pulling. We use analytical solutions to show that a strongly anisotropic elastic solid indeed exhibits a strongly anisotropic adhesion strength when sticking on a rough surface. Furthermore, we perform finite element calculations to show that the adhesion strength of a strongly anisotropic attachment pad exhibits essentially two levels of adhesion strength depending on the direction of pulling, resulting in an orientation-controlled switch between attachment and detachment. These findings not only provide a theoretical foundation to understand adhesion mechanisms in biology but also suggest possible strategies to develop novel adhesive materials for engineering applications.     

Effective pressure-sensitive elastoplastic behavior of particle-reinforced composites and porous media under isotropic loading

The composite under investigation consists of an elastoplastic matrix reinforced by elastic particles or weakened by pores. The material forming the matrix is pressure-sensitive. The Drucker–Prager yield criterion and a one-parameter non-associated flow rule are employed to formulate the yield behavior of the matrix. The objective of this work is to estimate the effective elastoplastic behavior of the composite under isotropic tensile and compressive loadings. To achieve this objective, the composite sphere assemblage model of Hashin [Z. Hashin, The elastic moduli of heterogeneous materials, ASME J. Appl. Mech. 29 (1962) 143–150] is used. Exact solutions are thus derived as estimations for the effective secant and tangent bulk moduli of the composite. The effects of the loading modes and phase properties on the effective elastoplastic behavior of the composite are analytically and numerically evaluated.     

A model for plasticity kinetics and its role in simulating the dynamic behavior of Fe at high strain rates

The recent diagnostic capability of the Omega laser to study solid-solid phase transitions at pressures greater than 10 GPa and at strain rates exceeding 10⁷ s⁻¹ has also provided valuable information on the dynamic elastic-plastic behavior of materials. We have found, for example, that plasticity kinetics modifies the effective loading and thermodynamic paths of the material. In this paper we derive a kinetics equation for the time-dependent plastic response of the material to dynamic loading, and describe the model’s implementation in a radiation-hydrodynamics computer code. This model for plasticity kinetics incorporates the Gilman model for dislocation multiplication and saturation. We discuss the application of this model to the simulation of experimental velocity interferometry data for experiments on Omega in which Fe was shock compressed to pressures beyond the α-to-ε phase transition pressure. The kinetics model is shown to fit the data reasonably well in this high strain rate regime and further allows quantification of the relative contributions of dislocation multiplication and drag. The sensitivity of the observed signatures to the kinetics model parameters is presented.     

Dual-phase steel sheets under cyclic tension–compression to large strains: Experiments and crystal plasticity modeling

In this work, we develop a physically-based crystal plasticity model for the prediction of cyclic tension–compression deformation of multi-phase materials, specifically dual-phase (DP) steels. The model is elasto–plastic in nature and integrates a hardening law based on statistically stored dislocation density, localized hardening due to geometrically necessary dislocations (GNDs), slip-system-level kinematic backstresses, and annihilation of dislocations. The model further features a two level homogenization scheme where the first level is the overall response of a two-phase polycrystalline aggregate and the second level is the homogenized response of the martensite polycrystalline regions. The model is applied to simulate a cyclic tension–compression–tension deformation behavior of DP590 steel sheets. From experiments, we observe that the material exhibits a typical decreasing hardening rate during forward loading, followed by a linear and then a non-linear unloading upon the load reversal, the Bauschinger effect, and changes in hardening rate during strain reversals. To predict these effects, we identify the model parameters using a portion of the measured data and validate and verify them using the remaining data. The developed model is capable of predicting all the particular features of the cyclic deformation of DP590 steel, with great accuracy. From the predictions, we infer and discuss the effects of GNDs, the backstresses, dislocation annihilation, and the two-level homogenization scheme on capturing the cyclic deformation behavior of the material.     

Dynamic stability analysis of an elastic composite material having a negative-stiffness phase

The rigorous classical bounds of elastic composite materials theory provide limits on the achievable composite stiffnesses in terms of the properties and arrangements of the composite's constituents. These bounds result from the assumption, presumably made for stability reasons, that each constituent material must have positive-definite elastic moduli. If this assumption is relaxed, recently published elasticity analyses and experimental measurements show these bounds can be greatly exceeded, resulting in new materials of enormous potential.The key question is whether a composite material having a non-positive-definite constituent can be stable overall in the practically useful situation of applied traction boundary conditions. Drugan [2007. Elastic composite materials having a negative-stiffness phase can be stable. Phys. Rev. Lett. 98 (5), article no. 055502] first proved the answer is yes, by applying the energy criterion of elastic stability to the basic two- and three-dimensional composites consisting of a cylinder or sphere having non-positive-definite (but strongly elliptic) moduli with a thin positive-definite coating and proving overall stability provided the coating is sufficiently stiff.Here, we perform a complete and direct dynamic stability analysis of the plane strain fundamental elastic composite consisting of a circular cylinder of non-positive-definite material firmly bonded to a positive-definite concentric coating, for the full range of coating thicknesses (i.e., volume fractions). We determine quantitatively the full permissible range of inclusion and coating moduli, as a function of coating thickness, for which the overall composite is stable under dead traction boundary conditions. Among the results, we show that in the thin-coating case, the present dynamic stability analysis leads to precisely the same analytical stability requirements as those derived via the energy criterion by Drugan [2007. Elastic composite materials having a negative-stiffness phase can be stable. Phys. Rev. Lett. 98 (5), article no. 055502], and we derive new analytical stability requirements that are valid for a wider range of coating thickness. At the other extreme, we show that in the case of very thick coatings (corresponding to the dilute case of a matrix-inclusion composite), even an inclusion with merely strongly elliptic moduli can be stabilized by a positive-definite matrix satisfying weak requirements, for which we derive analytical expressions. Overall, our results show that surprisingly weak restrictions on the moduli and thickness of the positive-definite coating are sufficient to stabilize a non-positive-definite inclusion, even one whose moduli are merely strongly elliptic. These results legitimize expanding the search for novel materials with extreme properties to those incorporating a non-positive-definite constituent, and they provide quantitative restrictions on the constituent materials’ moduli and volume fractions, for the geometry examined here, that ensure overall stability of such composite materials.     

Multiaxial constitutive model accounting for the strength-differential in Inconel 718

The nickel-base alloy Inconel 718 exhibits a strength-differential, that is, a different plastic flow behavior in uniaxial tension and uniaxial compression. A phenomenological viscoplastic model founded on thermodynamics has been extended for material behavior that deviates from classical metal plasticity by including all three stress invariants in the threshold function. The model can predict plastic flow in isotropic materials with or without a flow stress asymmetry as well as with or without pressure dependence. Viscoplastic material parameters have been fit to pure shear, uniaxial tension, and uniaxial compression experimental results at 650°°C. Threshold function material parameters have been fit to the strength-differential. Four classes of threshold functions have been considered and nonproportional loading of hollow tubes, such as shear strain followed by axial strain, has been used to select the most applicable class of threshold function for the multiaxial model as applied to Inconel 718 at 650 °C. These nonproportional load paths containing corners provide a rigorous test of a plasticity model, whether it is time-dependent or not. A J₂J₃ class model, where J₂ and J₃ are the second and third effective deviatoric stress invariants, was found to agree the best with the experimental results.     

Grain boundary versus transgranular ductile failure

The competition between intergranular and intragranular fracture is investigated using a bilayer damage model, which incorporates the relevant microstructural features of aluminium alloys with precipitate free zones (PFZ) nearby the grain boundary. One layer represents the grain behaviour: due to precipitation, it presents a high yield stress and low hardening exponent. The other layer represents the PFZ which has the behaviour of a solid solution: it is much softer but with a much higher strain hardening capacity. In both layers, void growth and coalescence is modelled using an enhanced Gurson-type model incorporating the effects of the void aspect ratio and of the relative void spacing. The effects on the ductility (i) of the flow properties of each zone, (ii) of the relative thickness of the PFZ, and (iii) of the particles spacing and volume fraction in the PFZ are elucidated. Comparisons are made with experimental data. Based on the previous analysis, qualitative understanding of trends in the fracture toughness of aluminium alloys can be gained in order to provide a link with the thermal treatment process.     

Couette flow of granular materials

The flow of granular materials between rotating cylinders is studied using a continuum model proposed by Rajagopal and Massoudi (A method for measuring material moduli for granular materials: flow in an orthogonal rheometer, DOE/PETC/TR90/3, 1990). For a steady, fully developed condition, the governing equations are reduced to a system of coupled non-linear ordinary differential equations. The resulting boundary value problem is non-dimensionalized and is then solved numerically. The effect of material parameters, i.e., dimensionless numbers on the volume fraction and the velocity fields are studied.     

Slip-line modeling of machining with a rounded-edge tool—Part II: analysis of the size effect and the shear strain-rate

Part II of the present study quantitatively analyzes orthogonal metal cutting processes based on the new slip-line model proposed in Part I. The applicable range of the model is illustrated, followed by an explanation of the non-unique nature of the model. It is suggested that the tool edge roundness be comprehensively defined by four variables. Namely: tool edge radius, position of the stagnation point on the tool edge, tool-chip frictional shear stress above the stagnation point on the tool edge, and tool-chip frictional shear stress below the stagnation point on the tool edge. The effects of these four variables on eight groups of machining parameters are investigated. These include (1) cutting force, thrust force, resultant force, and the ratio of cutting force to thrust force; (2) ploughing force; (3) chip up-curl radius; (4) chip thickness; (5) tool-chip contact length; (6) thickness of the primary shear zone; (7) average shear strain in the primary shear zone; and (8) average shear strain-rate in the primary shear zone. The importance of tool edge roundness is further reinforced by a series of new research findings made in this paper. It is revealed that the size effect highly depends on the material constitutive behavior in machining. The dependence of the thickness of the primary shear zone and the dependence of the magnitude of shear strain-rate in the primary shear zone on the tool edge radius are well demonstrated. A surprisingly good agreement between theory and experiments is reached.     

Three-dimensional micromechanical modeling of voided polymeric materials

A three-dimensional micromechanical unit cell model for particle-filled materials is presented. The cell model is based on a Voronoi tessellation of particles arranged on a body-centered cubic (BCC) array. The three-dimensionality of the present cell model enables the study of several deformation modes, including uniaxial, plane strain and simple shear deformations, as well as arbitrary principal stress states.The unit cell model is applied to studies on the micromechanical and macromechanical behavior of rubber-toughened polycarbonate. Different load cases are examined, including plane strain deformation, simple shear deformation and principal stress states. For a constant macroscopic strain rate, the different load cases show that the macroscopic flow strength of the blend decreases with an increase in void volume fraction, as expected. The main mechanism for plastic deformation is broad shear banding across inter-particle ligaments. The distributed nature of plastic straining acts to reduce the amount of macroscopic strain softening in the blend as the initial void volume fraction is increased. In the case of plane strain deformation, the plastic flow is observed to initiate across inter-particle ligaments in the direction of constraint. This particular mode of deformation could not have been captured using a two-dimensional, plane strain idealization of cylindrical voids in a matrix.The potential for localized crazing and/or cavitation in the matrix is addressed. It is observed that the introduction of voids acts to relieve hydrostatic stress in the matrix material, compared to the homopolymer. It is also seen that the predicted peak hydrostatic stress in the matrix is higher under plane strain deformation than under triaxial tension (with equal lateral stresses), for the same macroscopic stress triaxiality.The effect of void volume fraction on the macroscopic uniaxial tension behavior of the different blends is examined using a Considère construction for dilatant materials. The natural draw ratio was predicted to decrease with an increase in void volume fraction.     

散料在料仓内流动特性的实验研究

用振动筛测量多种散料的筛分尺寸，测量散料的容积密度、安息角，用应变直剪仪进行散料内摩擦角、壁面摩擦角的测量，分析了散料颗料的形状、尺寸对散料的安息角、内摩擦角、壁面摩擦角的影响。用示踪法进行了圆筒内散料重力场中颗粒流线和等时线的试验，以及出口直径与流量的关系、筒仓内散料高度对流量的影响的试验，流量与筒仓内散料高度关系的试验。分析了散料各物性参数对散料流动性的影响，为散料存贮和运输设备的结构设计提供实验依据。     

Effects of texture on shear band formation in plane strain tension/compression and bending

In this study, effects of typical texture components observed in rolled aluminum alloy sheets on shear band formation in plane strain tension/compression and bending are systematically studied. The material response is described by a generalized Taylor-type polycrystal model, in which each grain is characterized in terms of an elastic–viscoplastic continuum slip constitutive relation. First, a simple model analysis in which the shear band is assumed to occur in a weaker thin slice of material is performed. From this simple model analysis, two important quantities regarding shear band formation are obtained: i.e. the critical strain at the onset of shear banding and the corresponding orientation of shear band. Second, the shear band development in plane strain tension/compression is analyzed by the finite element method. Predictability of the finite element analysis is compared to that of the simple model analysis. Third, shear band developments in plane strain pure bending of a sheet specimen with the typical textures are studied. Regions near the surfaces in a bent sheet specimen are approximately subjected to plane strain tension or compression. From this viewpoint, the bendability of a sheet specimen may be evaluated, using the knowledge regarding shear band formation in plane strain tension/compression. To confirm this and to encompass overall deformation of a bent sheet specimen, including shear bands, finite element analyses of plane strain pure bending are carried out, and the predicted shear band formation in bent specimens is compared to that in the tension/compression problem. Finally, the present results are compared to previous related studies, and the efficiency of the present method for materials design in future is discussed.     

On a geometrically nonlinear damage model based on a multiplicative decomposition of the deformation gradient and the propagation of microcracks

We aim to derive a damage model for materials damaged by microcracks. The evolution of the cracks shall be governed by the maximum energy release rate, which was recently shown to be a direct consequence of the variational principle of a body with a crack (Arch. Appl. Mech. 69 (5) (1999) 337). From this, we get the path of the growing crack by introducing a series of thermodynamically equivalent straight cracks. The equivalence of the energy dissipated by microcrack growth and the damage dissipation leads to our damage evolution law. This evolution law will be embedded in a finite deformation framework based on a multiplicative decomposition into elastic and damage parts. As a consequence of this, we can present the anisotropic damaged elasticity tensor with the help of push and pull operations. The connection of this approach to other well known damage theories will be shown and the advantages of a finite element framework will be worked out. Numerical examples show the possibilities of the proposed model.     

Constitutive response of idealized granular media under the principal stress axes rotation

This paper is dedicated to the understanding of the phenomena, which give rise to anisotropy and non-coaxiality in granular materials. In achieving three-dimensional numerical simulation under static condition of granular media, granular element method (GEM) is adopted in this study. The method has been incorporated into the so-called mathematical homogenization theory for quasi-static equilibrium problems, which enables us to obtain the macroscopic/phenomenological inelastic deformation response of a representative volume element (RVE). To examine the anisotropic macroscopic deformation properties of the assumed RVE, which is solved by granular element method (GEM), a series of numerical experiments involving the pure rotation of the principal stress axes are carried out, and its results are discussed in relation to induced anisotropy and non-coaxiality.     

A three-invariant cap model with isotropic–kinematic hardening rule and associated plasticity for granular materials

In this paper, a three-invariant cap model is developed for the isotropic–kinematic hardening and associated plasticity of granular materials. The model is based on the concepts of elasticity and plasticity theories together with an associated flow rule and a work hardening law for plastic deformations of granulars. The hardening rule is defined by its decomposition into the isotropic and kinematic material functions. The constitutive elasto-plastic matrix and its components are derived by using the definition of yield surface, material functions and non-linear elastic behavior, as function of hardening parameters. The model assessment and procedure for determination of material parameters are described. Finally, the applicability of proposed plasticity model is demonstrated in numerical simulation of several triaxial and confining pressure tests on different granular materials, including: wheat, rape, synthetic granulate and sand.