In this work the linear elastic properties of materials containing spherical voids are calculated and compared using finite element simulations. The focus is on homogeneous solid materials with spherical, empty voids of equal size. The voids are arranged on crystalline lattices (SC, BCC, FCC and HCP structure) or randomly, and may overlap in order to produce connected voids. In that way, the entire range of void fraction between 0.00 and 0.95 is covered, including closed-cell and open-cell structures. For each arrangement of voids and for different void fractions the full stiffness tensor is computed. From this, the Young's modulus and Poisson ratios are derived for different orientations. Special care is taken of assessing and reducing the numerical uncertainty of the method. In that way, a reliable quantitative comparison of different void structures is carried out. Among other things, this work shows that the Young's modulus of FCC in the (1 1 1) plane differs from HCP in the (0 0 0 1) plane, even though these structures are very similar. For a given void fraction SC offers the highest and the lowest Young's modulus depending on the direction. For BCC at a critical void fraction a switch of the elastic behaviour is found, as regards the direction in which the Young's modulus is maximised. For certain crystalline void arrangements and certain directions Poisson ratios between 0 and 1 were found, including values that exceed the bounds for isotropic materials. For subsequent investigations the full stiffness tensor for a range of void arrangements and void fractions are provided in the supplemental material.
The shock response of two-dimensional model high explosive crystals with various arrangements of circular voids is explored. We simulate a piston impact using molecular dynamics simulations with a Reactive Empirical Bond Order (REBO) model potential for a sub-micron, sub-ns exothermic reaction in a diatomic molecular solid. In square lattices of voids (of equal size), reducing the size of the voids or increasing the porosity while holding the other parameter fixed causes the hotspots to consume the material more quickly and detonation to occur sooner and at lower piston velocities. The early t…
The elastic properties of a material with spherical voids of equal volume are analysed using a new model, with particular attention paid to the hexagonal close-packed and the face-centred cubic arrangement of voids. Void fractions well above 74 \% are considered, yielding overlapping voids as in an open-cell foam and hence a connected pore structure. The material is represented by a network of beams. The elastic behaviour of each beam is derived analytically from the material structure. By computing the linear elastic properties of the beam network, the Young's moduli and Poisson ratios for di…
The scattering determinant for the scattering of waves from several obstacles is considered in the case of elastic solids with voids. The multi-scattering determinant displays contributions from periodic ray-splitting orbits. A discussion of the weights of such orbits is presented.
A model for the simulation of Kirkendall voiding in metallic materials is presented based on vacancy diffusion, elastic-plastic and rate-dependent deformation of spherical voids. Starting with a phenomenological explanation of the Kirkendall effect we briefly discuss the consequences on the reliability of microelectronics. A constitutive model for void nucleation and growth is presented, which can be used to predict the temporal development of voids in solder joints during thermal cycling. We end with exemplary numerical studies and discuss the potential of the results for the failure analysis…
We consider the influence of elasticity and anisotropic surface energy on the energy-minimizing shape of a two-dimensional void under biaxial loading. In particular, we consider void shapes with corners for which the strain energy density is singular at the corner. The elasticity problem is formulated as a boundary integral equation using complex potentials. By incorporating the asymptotic behavior of the singular elastic fields at corners of the void, we develop a numerical spectral method for determining the stress for a class of arbitrary void shapes and corner angles. We minimize the total…
This paper presents an efficient and quantitative phase-field model for elastically heterogeneous alloys that ensures the two mechanical compatibilities$\unicode{x2014}$static and kinematic, in conjunction with chemical equilibrium within the interfacial region. Our model contrasts with existing phase-field models that either violate static compatibility or interfacial chemical equilibrium or are computationally costly. For computational efficiency, the partial rank-one homogenization (PRH) scheme is employed to enforce both static and kinematic compatibilities at the interface. Moreover, inte…
Voids can be observed at various scales in ductile materials, frequently of sizes lower than the grain size, leading to porous single crystals materials. Two local deformation modes of porous ductile (single crystals) materials have been identified and referred to as void growth and void coalescence, the latter being characterized by strong interactions between neighboring voids. A simple semi-analytical coalescence criterion for porous single crystals with periodic arrangement of voids is proposed using effective isotropic yield stresses associated with a criterion derived for isotropic mater…
Quantitative behaviors of shock-induced dislocation nucleation are investigated by means of molecular dynamics simulations on fcc Lennard-Jones solids: a model Argon. In perfect crystals, it is found that Hugoniot elastic limit (HEL) is a linearly decreasing function of temperature: from near-zero to melting temperatures. In a defective crystal with a void, dislocations are found to nucleate on the void surface. Also HEL drastically decreases to 15 percent of the perfect crystal when a void radius is 3.4 nanometer. The decrease of HEL becomes larger as the void radius increases, but HEL become…
Collapse modes in compressed simple cubic (SC) and body-centered cubic (BCC) periodic arrangements of elastic frictionless beads were studied numerically using the discrete element method. Under pure hydrostatic compression, the SC arrangement tends to transform into a defective hexagonal close-packed or amorphous structure. The BCC assembly exhibits several modes of collapse, one of which, identified as cI16 structure, is consistent with the behavior of BCC metals Li and Na under high pressure. The presence of a deviatoric stress leads to the transformation of the BCC structure into face-cent…
Exact solutions are derived for the problem of a two-dimensional, infinitely anisotropic, linear-elastic medium containing a periodic lattice of voids. The matrix material possesses either one infinitely soft, or one infinitely hard loading direction, which induces localized (singular) field configurations. The effective elastic moduli are computed as functions of the porosity in each case. Their dilute expansions feature half-integer powers of the porosity, which can be correlated to the localized field patterns. Statistical characterizations of the fields, such as their first moments and the…
We use a new variant of Molecular Static method for simulation of the atomic structure near nanovoids. In our model an iterative procedure is employed, in which the atomic structure in the void vicinity and the parameter determining the displacement of atoms embedded into an elastic continuum are obtained in a self-consistent manner. Results show that the atom displacements near nanovoids are significantly different for varies crystallographic directions in bcc metals. Not long ago we have obtained an equation of vacancy diffusion under strain. Now we use this equation and the atom displacemen…
Slow crack propagation in ductile, and in certain brittle materials, appears to take place via the nucleation of voids ahead of the crack tip due to plastic yields, followed by the coalescence of these voids. Post mortem analysis of the resulting fracture surfaces of ductile and brittle materials on the $μ$m-mm and the nm scales respectively, reveals self-affine cracks with anomalous scaling exponent $ζ\approx 0.8$ in 3-dimensions and $ζ\approx 0.65$ in 2-dimensions. In this paper we present an analytic theory based on the method of iterated conformal maps aimed at modelling the void formation…
We simulate structure in the vicinity of different size nanovoids using a new variant of the Molecular Statics, wherein atomic structure in the vicinity of nanovoids and the parameters that define the displacements of atoms placed in elastic continuum around main computation cell are determined in a self-consistent manner. Then we model structure of the surface of different crystallographic planes. Next, the kinetic equations for shifting rate of void surface elements located normal to the void surface are obtained. These equations take into account the dependence of the vacancy flux on strain…
Our goal is to unravel the mechanisms that lead to failure of a ductile two-phase material - that consists of a ductile soft phase and a relatively brittle hard phase. An idealized microstructural model is used to study damage propagation systematically and transparently. The analysis uncovers distinct microstructural features around early voids, whereby regions of the hard phase are aligned with the tensile axis and regions of the soft phase are aligned with the shear directions. These features are consistently found in regions exhibiting damage propagation, whereby the damage remains initiat…
We consider a two phase elastic thin film with soft inclusions subject to bending dominated deformations. The soft (void) phase may comprise asymptotically small droplets within the elastic matrix. We perform a dimension reduction analysis and obtain a novel `Blake-Zisserman-Kirchhoff' model that combines Kirchhoff's classical plate theory with Blake-Zisserman type surface energy contributions at cracks, folds and the boundary of voids.
We study the influence of the atomic structure in the vicinity of voids on their growth rate anisotropy. In the first part, we model the atomic structure in the vicinity of nanovoids in Fe and W using the advanced Molecular Statics method. In the second part, we use the earlier obtained equations that taking the influence of elastic fields into account to calculate the shifting rate of the void surface elements, and to evaluate the components of the strain tensor we use the atomic structure modeling results from the first part. The calculations are performed for voids of several sizes at certa…
In this work, a model based on the Physics - Informed Neural Networks (PINNs) for solving elastic deformation of heterogeneous solids and associated Uncertainty Quantification (UQ) is presented. For the present study, the PINNs framework - Modulus developed by Nvidia is utilized, wherein we implement a module for mechanics of heterogeneous solids. We use PINNs to approximate momentum balance by assuming isotropic linear elastic constitutive behavior against a loss function. Along with governing equations, the associated initial / boundary conditions also softly participate in the loss function…
A systematic computational and scaling analysis of defect textures in polygonal arrangement of cylindrical particles embedded in a cholesteric (Ch) liquid crystal matrix is performed using the Landau de Gennes model for chiral self assembly, with strong anchoring at the particles' surface. The defect textures and LC phases observed are investigated as a function of chirality, elastic anisotropy, particle polygonal arrangement and particle size. The presence of a polygonal network made of N circular inclusions results in defect textures of a net charge of -(N-2)/2 per unit polygonal cell. As th…
We consider the dynamics of pattern formation in a system of point defects under sustained irradiation within the framework of the rate theory. In our study we generalize the standard approach taking into account a production of defects by elastic fields and a stochastic production representing internal multiplicative noise. Using 3D-modelling we have shown that with the damage rate growth, a morphology of clusters composed of vacancies changes. The same effect is observed with variation in the multiplicative noise intensity. Stationary patterns are studied by means of correlation analysis.
The results of electronic structure calculations for a variety of palladium hydrides are presented. The calculations are based on density functional theory and used different local and semilocal approximations. The thermodynamic stability of all structures as well as the electronic and chemical bonding properties are addressed. For the monohydride, taking into account the zero-point energy is important to identify the octahedral Pd-H arrangement with its larger voids and, hence, softer hydrogen vibrational modes as favorable over the tetrahedral arrangement as found in the zincblende and wurtz…
The overall behavior of a 2D lattice of voids embedded in an anisotropic matrix is investigated in the limit of vanishing porosity f. An effective-medium model (of the Hashin-Shtrikman type) which accounts for elastic interactions between neighboring voids, is compared to Fast Fourier Transform numerical solutions and, in the limits of infinite anisotropy, to exact results. A cross-over between regular and singular dilute regimes is found, driven by a characteristic length which depends on f and on the anisotropy strength. The singular regime, where the leading dilute correction to the elastic…
Strong compositional-dependent elastic properties have been observed theoretically and experimentally in Ti$_{1-x}$Al$_x$ N alloys. The elastic constant, C$_{11}$, changes by more than 50% depending on the Al-content. Increasing the Al-content weakens the average bond strength in the local octahedral arrangements resulting in a more compliant material. On the other hand, it enhances the directional (covalent) nature of the nearest neighbor bonds that results in greater elastic anisotropy and higher sound velocities. The strong dependence of the elastic properties on the Al-content offers new i…
The paper describes the first exact results in optimal design of three-phase elastic structures. Two isotropic materials, the "strong" and the "weak" one, are laid out with void in a given two-dimensional domain so that the compliance plus weight of a structure is minimized. As in the classical two-phase problem, the optimal layout of three phases is also determined on two levels: macro- and microscopic. On the macrolevel, the design domain is divided into several subdomains. Some are filled with pure phases, and others with their mixtures (composites). The main aim of the paper is to discuss …
Use of amorphous silicon ($a$-Si) and hydrogenated amorphous silicon ($a$-Si:H) in photovoltaics has been limited by light-induced degradation (the Staebler-Wronski effect) and low hole mobilities, and voids have been implicated in both problems. Accurately modeling the void microstructure is critical to theoretically understanding the cause of these issues. Previous methods of modeling voids have involved removing atoms according to an {\it a priori} idea of void structure and/or using computationally expensive molecular dynamics. We propose a new fast and unbiased approach based on the estab…
Modeling dynamics fracture in materials involves usage of hydrodynamic codes which solve basic conservation laws of mass, energy and momentum in space and time. This requires appropriate models to handle elastic-plastic deformation, equation of state, material strength, and fracture. Nucleation and Growth (NAG) damage model is a micro-physical model which computes amount of damage in the material by accounting for phenomena like nucleation, growth and coalescence of voids or cracks. The NAG model involves several material model parameters, such as nucleation threshold, growth threshold, etc. T…
We develop a new Voronoi protocol, which is a space tessellation method, to generate fully dense (containing no voids) nanocrystalline models of copper (Cu) with precise grain size control; we also perform uniaxial tensile tests using molecular dynamical (MD) simulations to measure the elastic moduli of the grain boundary and the grain interior components at 300 K. We find that the grain boundary deforms more locally compared with the grain core region under thermal vibrations and is elastically less stiff than the core component at finite temperature. The elastic modulus of the grain boundary…
Three dimensional calculations of ductile crack growth under mode I plane strain, small scale yielding conditions are carried out using an elastic-viscoplastic constitutive relation for a progres- sively cavitating plastic solid with two populations of void nucleating second phase particles. Full field solutions are obtained for three dimensional material microstructures characterized by ran- dom distributions of void nucleating particles. Crack growth resistance curves and fracture surface roughness statistics are calculated using standard procedures. The range of void nucleating particle vol…
MAX phases are nanolaminated ternary materials that combine metallic and ceramic properties. Currently, the A-site elements replacement in traditional ones by later transition-metals opens a door to explore new types of MAX phases. In this work, we performed a systematic first-principle study to explore trends in stability, electronic structure and mechanical properties of 288 compositions of M2AX phases (M=Ti, V, Cr, Zr, Nb, Mo, Hf, Ta, W; A=Al, Si, P, S, Ga, Ge, As, Se, In, Sn, Sb, Te, Tl, Pb, Bi, Mn, Fe, Co, Ni, Cu, Zn, Tc, Ru, Rh, Pd, Ag, Cd, Os, Ir, Pt, Au, Hg; X=C). Such a dataset, combi…
Rapid advances in additive manufacturing over the past decade have kindled widespread interest in the rational design of metamaterials with unique properties. However, many applications require multi-physics metamaterials, where multiple properties are simultaneously optimized. This is challenging, since different properties, such as mechanical and mass transport properties, typically impose competing requirements on the nano-/micro-/meso-architecture of metamaterials. Here, we propose a parametric metamaterial design strategy that enables independent tuning of the effective permeability and e…
Theoretical predictions of the ideal strength of materials range from E/30 to E/10 (E is Young's modulus). However, despite intense interest over the last decade, the value of the ideal strength that can be attained experimentally for metals remains a mystery (1-5). In this study, we demonstrated the unprecedented strength of an amorphous Cu-Zr alloy that surpassed the E/10 limit. Laser-induced shock experiments were conducted on Cu50Zr50 to explore its strength and failure mechanisms at ultrahigh strain rates. The material demonstrated a high spall strength of 9.8 GPa, approximately 1/13 of i…
The understanding of dynamic failure in amorphous materials via the propagation of free boundaries like cracks and voids must go beyond elasticity theory, since plasticity intervenes in a crucial and poorly understood manner near the moving free boundary. In this Letter we focus on failure via a cavitation instability in a radially-symmetric stressed material, set up the free boundary dynamics taking both elasticity and visco-plasticity into account, using the recently proposed athermal Shear Transformation Zone theory. We demonstrate the existence (in amorphous systems) of fast cavitation mod…
In this chapter we review our findings on the bonding structure and growth mechanisms of carbon-based thin solid films with fullerene-like (FL) microstructure. The so-called FL arrangements arise from the curvature and cross-linking of basal planes in graphitic-like structures, partially resembling that of molecular fullerenes. This three-dimensional superstructure takes advantage of the strength of planar pi bonds in sp2 hybrids and confers the material interesting mechanical properties, such as high hardness, high elastic recovery, low-friction and wear-resistance. These properties can be ta…
We propose a cloak for coupled shear and pressure waves in solids. Its elastic properties are deduced from a geometric transform that retains the form of Navier equations. The spherical shell is made of an anisotropic and heterogeneous medium described by an elasticity tensor C' (without the minor symmetries) which has 21 non-zero spatially varying coefficients in spherical coordinates. Although some entries of C, e.g. some with a radial subscript, and the density (a scalar radial function) vanish on the inner boundary of the cloak, this metamaterial exhibits less singularities than its cylind…
Very recently, the construction of twist actuators from magnetorheological gels and elastomers has been suggested. These materials consist of magnetizable colloidal particles embedded in a soft elastic polymeric environment. The twist actuation is enabled by a net chirality of the internal particle arrangement. Upon magnetization by a homogeneous external magnetic field, the systems feature an overall torsional deformation around the magnetization direction. Starting from a discrete minimal mesoscopic model set-up we work towards a macroscopic characterization. The two scales are linked by ide…
The macroscopic mechanical properties of colloidal particle gels strongly depend on the local arrangement of the powder particles. Experiments have shown that more heterogeneous microstructures exhibit up to one order of magnitude higher elastic properties than their more homogeneous counterparts at equal volume fraction. In this paper, packings of spherical particles are used as model structures to computationally investigate the elastic properties of coagulated particle gels as a function of their degree of heterogeneity. The discrete element model comprises a linear elastic contact law, par…
We carried out fully-atomistic reactive molecular dynamics simulations to study the elastic properties and fracture patterns of transition metal dichalcogenide (TMD) MoX2 (X=S, Se, Te) membranes, in their 2H and 1T phases, within the framework of the Stillinger-Weber potential. Results showed that the fracture mechanism of these membranes occurs through a fast crack propagation followed by their abrupt rupture into moieties. As a general trend, the translated arrangement of the chalcogen atoms in the 1T phase contributes to diminishing their structural stability when contrasted with the 2H one…
We focused on elastic properties of B- and N-doped graphene in wide range of concentrations up to 20%. The Young's, bulk and shear moduli and Poisson's ratio have been calculated by means of the density functional theory for a representative set of supercells with disordered impurity patterns including aggregates. In contrast to earlier work, it is demonstrated that doping with nitrogen even strengthens the graphene layers, whereas incorporation of boron induces large structural and morphological changes seen in simulated STM images. Young's and shear moduli increase or decrease with the dopin…
We study water adsorption-induced deformation of a monolithic, mesoporous silicon membrane traversed by independent channels of $\sim$8 nm diameter. We focus on the elastic constant associated with the Laplace pressure-induced deformation of the membrane upon capillary condensation, i.e. the pore-load modulus. We perform finite-element method (FEM) simulations of the adsorption-induced deformation of hexagonal and square lattices of cylindrical pores representing the membrane. We find that the pore-load modulus weakly depends on the geometrical arrangement of pores, and can be expressed as a f…
Intermetallic compounds possess unique atomic arrangements that often lead to exceptional material properties, but their extreme brittleness usually causes fracture at a limited strain of less than 1% and prevents their practical use. Therefore, it is critical for them to exhibit either plasticity or some form of structural transition to absorb and release a sufficient amount of mechanical energy before failure occurs. This study reports that the ThCr2Si2-structured intermetallic compound (CaFe2As2) and a hybrid of its structure (CaKFe4As4) with 2 μm in diameter and 6 μm in height can exhibit …
