Institute of Fundamental Technological Research

In metal single crystals, the observed formation of deformation banding pattern has been explained by greater latent hardening of slip systems than their self-hardening, which promotes spatial segregation of plastic slips and lamination towards single-slip domains. Numerical studies focusing on the formation of deformation bands usually involved initial imperfections, boundary-induced heterogeneity, or the postulate of minimal global energy expenditure which additionally promoted non-uniformity of deformation. This article analyses the case when no such mechanism enforcing locally non-uniform deformation is implemented in the finite element (FE) method, while the global system of equations of incremental equilibrium is solved in a standard way. The new finding in this paper is that the deformation banding pattern can appear spontaneously in FE simulations of homogeneous single crystals even in the absence of any mechanism favouring deformation banding in the numerical code. This has been demonstrated in several examples in the small strain formalism using a plane-strain model in which the twelve fcc slip systems are reduced to three effective plastic slip mechanisms. Incremental slips are determined at the Gauss-point level either by incremental work minimization in the rate-independent case or by rate-dependent regularization. In the rate-independent approach, the trust-region algorithm is developed for the selection of active slip systems with the help of the augmented Lagrangian method. Conditions under which a banding pattern appears spontaneously or is suppressed are discussed. In particular, a critical rate sensitivity exponent is identified.

Crystal plasticity, Small strain, Slip-system selection, Path instability, Microstructure formation, Finite element method

The recently introduced quasi-extremal energy principle for incremental non-potential problems in rate-independent plasticity is applied to select the deformation pattern and active slip systems in single crystals. The standard crystal plasticity framework with a non-symmetric slip-system interaction matrix at finite deformation is used. The incremental work criterion for the formation of deformation bands is combined with the quasi-extremal energy principle for determining the active slip systems and slip increments in the bands. In this way, the incremental energy minimization approach has been extended to the non-potential problem of deformation banding in metal single crystals. It is shown that fulfilment of the mathematical criterion for incipient deformation banding in a homogeneous crystal in the multiple-slip case under certain conditions requires non-positive determinant of the hardening moduli matrix. Numerical examples of energetically preferable patterns of deformation bands are presented for Cu and Ni single crystals.

Solids, Plasticity, Crystal plasticity, Material stability, Slip-system selection, Energy minimization, Quasi-minimization, Time integration, Implicit, Channel-die, Deformation band

The direct incremental energy minimization in rate-independent plasticity does not account for the skew-symmetric part of the tangent stiffness matrix. In crystal plasticity, this corresponds to neglecting the asymmetry of the matrix of interaction moduli for active slip-systems. This limitation has been overcome in the recently proposed quasi-extremal energy principle (QEP) applicable to nonpotential problems. In the present article it is shown how to extend QEP to finite increments in the backward-Euler computational scheme. A related constitutive algorithm is proposed which enables automatic selection of active slip systems using an energetic criterion, along any path of large deformation of a rate-independent single crystal with a nonsymmetric slip-system interaction matrix. Numerical examples have been calculated for a fcc single crystal subjected to simple shear or uniaxial tension. The slip system activity predicted by using the QEP algorithm has been found to be more reliable in describing the actual plastic response of metal crystals than conventional rate-dependent modeling in cases where the selection of active slip-systems is essential.

solids, plasticity, material stability, Lagrangian, slip-system selection, time integration, implicit

The issue of applicability of the Morphologically Representative Pattern (MRP) approach to elastic-plastic composites is addressed. The extension to the regime of non-linear material behaviour is performed by employing the concept of incremental linearization of the material response in two basic variants: tangent and secant. The obtained predictions are evaluated through comparison with the outcomes of numerical analyses. Finite Element simulations are carried out using periodic unit cells with cubic arrangements of spherical particles and representative volume elements (RVE) with 50 randomly placed inclusions. In addition to the analysis of the packing effect in two-phase composites, the size effect is also studied by assuming an interphase between the matrix and inclusions. It is concluded that the MRP approach can be used as an effective predictive alternative to computational homogenization, not only in the case of linear elasticity but also in the case of elastic-plastic composites.

particulate composites, elastoplasticity, micromechanics, size effect, packing effect, morphologically representative pattern

Nonlinear properties of metal matrix composites (MMCs) are studied. The research combines results of loading-unloading tensile tests, microstructural observations and numerical predictions by means of micromechanical mean-field models. AA2124/SiC metal matrix composites with SiC particles, produced by the Aerospace Metal Composites Ltd. (AMC) are investigated. The aluminum matrix is reinforced with 17% and 25% of SiC particles. The best conditions to evaluate the current elastic stiffness modulus have been assessed. Tensile tests were carried out with consecutive unloading loops to obtain actual tensile modulus and study degradation of elastic properties of the composites. The microstructure examination by scanning electron microscopy (SEM) showed a variety of phenomena occurring during composite deformation and possible sources of elastic stiffness reduction and damage evolution have been indicated. Two micromechanical approaches, the incremental Mori–Tanaka (MT) and self-consistent (SC) schemes, are applied to estimate effective properties of the composites. The standard formulations are extended to take into account elasto-plasticity and damage development in the metal phase. The method of direct linearization performed for the tangent or secant stiffness moduli is formulated. Predictions of both approaches are compared with experimental results of tensile tests in the elastic–plastic regime. The question is addressed how to perform the micromechanical modelling if the actual stress–strain curve of metal matrix is unknown.

metal matrix composites, tension with unloadings, damage, microstructure, non-linear effective properties

Several micromechanical and numerical approaches to estimating the effective properties of heterogeneous media are analyzed. First, micromechanical mean-field estimates of elastic moduli for selected metal matrix composite systems are compared with the results of finite element calculations performed for two simplified unit cells: spherical and cylindrical. Advantages and deficiencies of such numerical verification of analytical homogenization schemes are indicated. Next, predictions of both approaches are compared with available experimental data for two composite systems for tension and compression tests in the elastic-plastic regime using tangent and secant linearization procedures. In the examined range of strain and ceramic volume content, both the Mori-Tanaka averaging scheme and the generalized self-consistent scheme lead to reliable predictions when combined with the tangent linearization, while the use of secant moduli results in a too stiff response. It is also found that the mean-field predictions for a small ceramic volume content are very close to the results obtained from the finite-element analysis of a spherical unit cell.

Metal-matrix composites, Effective properties, Analytical estimates, Numerical homogenization, Nonlinear analysis

Effects of particle packing and size on the overall elastic properties of particulate random composites are analyzed. In order to account for the two effects the mean-field Morphologically Representative Pattern (MRP) approach is employed and an additional interphase surrounding inclusions (coating) is introduced. The analytical mean-field estimates are compared with the results of computational homogenization performed using the finite element (FE) method. Periodic unit cells with cubic crystal-type arrangements and representative volume elements with random distributions of particles are used for verification purposes. The validity of the MRP estimates with respect to the FE results is assessed.

Composite materials, Elasticity, Micro-mechanics, Packing and size effects

We propose a new class of modular-robotic structures, intended to produce forces which scale with the number of modules. We adopt the concept of a spherical catom and extend it by a new connection type which is relatively strong but static. We examine analytically and numerically the mechanical properties of two collective-actuator designs. The simulations are based on the discrete element method (DEM), with friction and elastic deformations taken into account. One of the actuators is shown to generate forces proportional to its volume. This property seems necessary for building modular structures of useful strength and dimensions.

modular robots, self-reconfiguration, programmable matter, actuators, mechanical strength

A new constitutive algorithm for the rate-independent crystal plasticity is presented. It is based on asymptotically exact formulation of the set of constitutive equations and inequalities as a minimum problem for the incremental work expressed by a quadratic function of non-negative crystallographic slips. This approach requires selective symmetrization of the slip-system interaction matrix restricted to active slip-systems, while the latent hardening rule for inactive slip-systems is arbitrary. The active slip-system set and incremental slips are determined by finding a constrained minimum point of the incremental work. The solutions not associated with a local minimum of the incremental work are automatically eliminated in accord with the energy criterion of path stability. The augmented Lagrangian method is applied to convert the constrained minimization problem to a smooth unconstrained one. Effectiveness of the algorithm is demonstrated by the large deformation examples of simple shear of a face-centered cubic (fcc) crystal and rolling texture in a polycrystal. The algorithm is extended to partial kinematic constraints and applied to a uniaxial tension test in a high-symmetry direction, showing the ability of the algorithm to cope with the non-uniqueness problem and to generate experimentally observable solutions with a reduced number of active slip-systems.

solids, crystal plasticity, rate-independent constitutive equations, material stability, variational methods, incremental energy minimization, augmented Lagrangian method

The optimization procedure is worked out for finding an optimal content of phases in metal–ceramic composites in case of conflicting objectives regarding thermo-mechanical properties of the material for a specific target application. Relationships between the material composition and effective properties of the composite are calculated by employing several methods of continuum micromechanics. A constrained minimization problem is solved for a single objective function based on the weighted squared distances from the best available thermo-mechanical properties for the material system selected. A compound block diagram is proposed for quick assessment of the consequences of deviating from the optimal composition. The developed procedure is applied to practical examples of Al2O3–Cu composites for brake disks and Al2O3–NiAl composites for valves of potential use in automotive industry.

Metal–matrix composites (MMCs), Thermomechanical, Plastic deformation, Micro-mechanics, Multi-criteria optimization

The phenomenon of spontaneous formation of deformation bands in metal single crystals deformed plastically by crystallographic multislip is investigated theoretically by using the energy criterion of instability of a uniform deformation path. The second-order energy criterion for incipient deformation banding is derived in a time-continuous setting for a rate-independent elastic–plastic crystal. The need for selective symmetrization of the local interaction matrix for active slip-systems is demonstrated. A computational approach to deformation banding is developed by using non-convex constrained minimization of the incremental work with respect to increments in crystallographic shears and kinematical degrees of freedom. Calculated examples of deformation banding patterns in fcc single crystals are compared with experimental observations.

Metal crystal, Plasticity, Finite deformation, Laminate microstructure, Incremental energy minimization

The symmetry issue of the interaction matrix between multiple slip-systems in the theory of crystal plasticity at finite deformation is revisited. By appealing to possibly non-uniform distribution of slip-system activity in a representative space-time element of a crystal, symmetry of the slip-system interaction matrix for the representative element is derived under assumptions that have a physical meaning. This conclusion refers to active slip-systems only. Accordingly, for any given hardening law, a new symmetrization rule is proposed that is restricted to active slip-systems and leaves the latent hardening of inactive slip-systems unchanged. Advantages of the proposal in comparison with full symmetrization are illustrated by a simple example of uniaxial tension.

finite deformation, metal crystal, plasticity, hardening, symmetry

The subject of the paper is an optimal choice of material parameters characterizing the core layer of sandwich plates within the framework of the conventional plate theory in which the core layer is treated as soft in the in-plane direction. The mathematical description is similar to the Hencky–Reissner model of plates with transverse shear deformation. Here, however, the bending stiffnesses and the transverse shear stiffnesses can be designed independently. The present paper deals only with optimal design of the core layer to make the plate compliance minimal. Two core materials are at our disposal, which leads to the ill-posed problem. To consider it one should relax this problem by admitting composite domains and characterize their overall properties by the homogenization formulae. The numerical approach is based on this relaxed formulation thus making it mesh-independent. The equilibrium problem is solved by the DSG3 finite element method. The optimization results are found with using the convergent updating schemes of the COC method.

Minimum compliance problem, Sandwich plates, Topology optimization

A collective actuator is a self-reconfigurable modular-robotic structure which produces useful mechanical work through simultaneous reconfiguration of its constituent units. An actuator is additionally called scalable if its force-to-weight ratio does not depend on the number of its member modules. In this work, we consider scalable collective actuators built from spherical catoms with two connection types: strong but fixed and weak but mobile. We investigate how to construct these actuators in such a way, as to maximize their force-to-weight ratio. We present a number of designs of high strength, whose force capacities significantly exceed those of similar actuators reported previously.

The term Programmable Matter (PM) describes the class of future meta-materials of programmable and controllable properties and behavior, e.g., able to autonomously transform into an arbitrary shape. The robotic approaches towards PM are based on the concept of cooperation of millions of micro-robots (modules), acting at a very fine length-scale and collectively imitating deformation of a macroscopically continuous material. Recent ideas about reconfiguration of a collective of modules to obtain a desired overall mechanical response are promising. However, they are limited by the strength of individual connections between modules. In the present work, we propose a way of arranging spherical modules into microstructures, in which some connections are fixed and mechanically stronger, and the rest are active (reconfigurable) but weaker. If the fixed connections are sufficiently strong, the proposed microstructures perform the function of collective actuation by exerting forces proportional to their volumes. Two variants of a linear-actuator microstructure are presented and studied in more detail. A rotary-actuator microstructure is also introduced.

Micromechanical modelling of metal-ceramic composites has been carried out to obtain a material of required thermo-mechanical properties. Quantitative transition from phase properties and morphology to macroscopic properties of a composite has been modelled by mean-field approaches, including the self-consistent scheme. An optimization method has been developed for the objective function that expresses a distance between the required values of macro-variables and those determined for a given set of microstructural parameters. The presented example concerns application of Al2O3-Cu composite to brake disks.

multi-objective optimization, composite selection, metal matrix composites

Crystal plasticity, Material instability, Automatic selection of active slip-systems

mean-field modelling, numerical homogenization, elasto-plasticity

Crystal Plasticity, Slip-systems Selection, Incremental Energy Minimization

Different approaches to model packing and size effects are studied to model overall properties of particulate composites of different morphological features of phase distribution. The micromechanical schemes originating in the composite sphere model and its extension by morphologically-based pattern approach are taken as a basis. Analytical predictions are compared with results of computational homogenization performed on the generated representative volume elements of prescribed statistical characteristics.

micromechanics, morphologically representative pattern, computational homogenization, size and scale effect

In the modelling of metal single crystals in the framework of rate-independent plasticity, there are known difficulties caused by non-uniqueness in selection of active slip-systems. A related challenge is to predict emergence of non-uniform deformation patterns and the formation and evolution of experimentally observed microstructures. A new constitutive algorithm is presented that tackles those problems using the energy approach. It is based on asymptotically exact formulation of the set of constitutive equations and inequalities as a minimum problem for the incremental work expressed by a quadratic function of non-negative crystallographic slips. The calculated examples of deformation banding patterns and of reduction of multiplicity of active slip-systems in fcc single crystals are compared with the experimental observations.

crystal plasticity, slip-systems selection, incremental energy minimization

We examine, analytically and numerically, forces produced by collective actuators–possible future building blocks of Programmable Matter. The actuators are composed of tiny spherical robotic modules which can strongly attach to their neighbors, and move by rolling over one another using electric or magnetic local propulsion mechanisms. An actuator works through collective reconfiguration–a coordinated motion of its constituent modules–which results in a global deformation of the structure. The simulations are performed using specially adapted discrete element method software, and account for friction and elastic deformations of modules.

programmable matter, active materials, actuators, mechanical strength, modular robots