R. V. Goldstein

Modeling of Bonding at an Interface Crack

A mechanical and mathematical model is suggested for an interface crack with bonding in its end zones. Normal and shear bond tractions occurring under the action of the external loads are searched for by solving a system of two singular integrodifferential equations. The stress intensity factors at the crack tip are calculated taking the compensating action of the bonds into account. Energetic characteristics of the interface crack (the deformation energy release rate and the rate of the energy absorption by the bonds) are analyzed. A sensitivity analysis is performed of the force and energetic characteristics of the interface crack to the end zone size, bond compliance and limit stretching.

Auxetic Mechanics of Crystalline Materials

In the present paper, we analyze uniaxial deformation of crystals of different systems with negative Poisson’s ratios, known as auxetics. The behavior of auxetic crystals is studied on the basis of extensive knowledge on the experimental values of elastic constants of different crystals, gathered in the well-known Landolt-Börnstein tables. The competition between the anisotropy of crystal structures and the orientation of deformable samples results in the dependence of the elastic characteristics of deformation, such as Young’s modulus and Poisson’s ratio, on the orientation angles. In the special case of a single angle, a large number of auxetics were found among crystals of cubic, hexagonal, rhombohedral, tetragonal, and orthorhombic systems and the character of variations in their response due to changes in orientation was determined.

A Compact Analytic Model of the Strain Field Induced by Through Silicon Vias

The thermoelastic strains are induced by through silicon vias due to the difference of thermal expansion coefficients between the copper ( ~ 18 ppm / °C) and silicon ( ~ 2.8 ppm /°C) when the structures are exposed to a thermal ramp in the process flow. A compact analytic model (Bessel function) of the strain field is obtained using Kane-Mindlin theory, and has a good agreement with the finite-element simulations. The elastic strains in the silicon in the radial direction and angular direction are tensile and compressive, respectively. The linear superposition of the analytic model of a single via can be used in the multi-via configuration. Due to the interaction of vias, the slightly larger errors of strain occur between the two close vias when the linear superposition is used.

Negative Poisson’s ratio for cubic crystals and nano/microtubes

The paper systemizes numerous cubic crystals which can have both positive and negative Poisson’s ratios (the so-called partial auxetics) depending on the specimen orientation in tension. Several complete cubic auxetics whose Poisson’s ratio is always negative are indicated. The partial cubic auxetics are classified with the use of two dimensionless elastic parameters. For one of the parameters, a critical value is found at which the orientation behavior of the crystals changes qualitatively. The behavior of mesotubes obtained by rolling up plates of cubic crystals (crystals with rectilinear anisotropy) is considered in detail. Such mesotubes with curvilinear cubic anisotropy can have micron and nanometer lateral dimensions. It is shown that uniform tension of nano/microtubes of cubic crystals is possible only in the particular case of zero chirality angle (the angle between the crystallographic axis and the axis of a stretched tube). It is demonstrated by the semi-inverse Saint-Venant method that solution of the axial tension problem for cylindrically anisotropic nano/microtubes of cubic crystals with a non-zero chirality angle is possible with radially inhomogeneous fields of three normal stresses and one shear stress. In the examples considered, the cylindrically anisotropic nano/microtubes of cubic crystals are auxetics even if they are initially non-auxetics with rectilinear anisotropy.

Modeling electromigration and the void nucleation in thin-film interconnects of integrated circuits

The modern tendency for increasing the productivity of microelectronic devices at the expense of the size shrinkage and the development of densely packed multilevel microelectronic structures stipulates the rising concern for the reliability of integrated circuits. The damage of integrated circuits is mainly caused by electromigration in thin-film interconnects. The current-induced redistribution of vacancies and the action of vacancy sinks/sources lead to heterogeneous volume deformations, which, in turn, cause the rise of mechanical stresses. The interconnect failure is initiated by the nucleation of voids taking place on the crystalline structure heterogeneities like triple points, inclusions, etc. or in the plug region of multilevel metallizations. In the latter case the interconnect damage is also caused by the edge depletion. Mechanical stresses induced by electromigration strongly influence the nucleation process. In the present work we propose a general 3D model for electromigration and the rise of mechanical stresses in a passivated aluminum interconnect. A system of differential equations describing electromigration and induced deformation of an interconnect is derived. We also propose a kinetic model for the void nucleation, elaborated on the basis of the classical theory of the new phase nucleation. Integral equations for the time to the void nucleation are deduced. Based on these models numerical calculations for the void formation in a triple point of the interconnect crystalline structure and for both failure mechanisms in the plug region have been carried out. The times to nucleation and characteristic sizes of voids are calculated as functions of temperature and electric current density. The results obtained agree well with experimental data.

On the Modeling of Surface and Interface Elastic Effects in Case of Eigenstrains

The constitutive equations of interface elasticity in case of eigenstrains are obtained in terms of interface (surface) values defined as integrals of the excesses of the corresponding volumetric values over the normal to the interface. The equations are consistently linearized, which corresponds to the case of both elastic strains and eigenstrains being small. It is shown that the obtained equations possess more general form then Shuttleworth equations. The obtained type of equation was confirmed by considered example: an interface formed by a thin layer of constant properties. It was also shown that the type of energetic restrictions on the surface elastic constants may depend essentially on the definition of the position of the surfaces.

Fracture Structure near a Longitudinal Shear Macrorupture

Fracture evolution the near a main longitudinal shear in the presence of normal stresses is studied. Experiments with model materials (gypsum, cheese) showed that a multiscale echelon structure of cracks feathering the main rupture is formed under the shear domination conditions. A system of small cracks in the initial echelon is replaced by an echelon of larger and sparser cracks. Intensive transverse compression along the normal to the shear plane, which imitates the initial stress concentrator, takes the fracture region away from the shear plane. A model of evolution development of the observed echelon structure along the main rupture front under the shear domination conditions is proposed.

Young’s modulus and Poisson’s ratio for seven-constant tetragonal crystals and nano/microtubes

In the paper, the elasticity theory was applied to consider the mechanical properties of rectilinearly anisotropic seven-constant tetragonal crystals and their cylindrically anisotropic nano/microtubes with and with no chiral angle, being the angle between the crystallographic symmetry axis and elongated tube axis. Pt is found that the number of crystals with negative Poisson’s ratio is the least for rectilinear anisotropy and is much larger for curvilinear anisotropy. With a nonzero chiral angle, all nano/microtubes can have negative Poisson’s ratio. The elastic problem on axial tension of cylindrical nano/microtubes is solved for radially inhomogeneous stresses: three normal stresses and one shear stress.

To determination of the strength of nanodimensional objects

We propose several versions of experiments for determining the strength of nanodimensional objects (nanotubes) contained in the specimens of the form of reinforcing elements of special composites. The experiment methodology foundations were developed and the experiment parameters were chosen on the basis of simulation of the nanotube deformation processes and its interaction with the matrix, which reflects the specific properties of materials in the scale of nanometer dimensions according to the approach proposed earlier. The strength of reinforcing elements is determined from the load at the moment of the change of the fracture mechanism (the transition from the predominate pulling the tubes out of the matrix to their fracture). Different methods for controlling the composite stress state are used: variation in the viscousmatrix strain rate in the reinforced filament testing and variation in the energy of the chemical interaction between the nanotube and the rigidmatrix for nanoobjects of special form such as the nanotube forest grown on a rigid substrate. In the last case, the fracture stresses in nanotubes arise at the moment of separation of an elastic cantilever adjusted to the glue film reinforced by nanotubes. The critical conditions of the fracturemechanism change in the end separation region correspond to the effective specificwork of structural failure.

Distributions of stress and plastic strain in notched tensile bars

The paper deals with a classical problem in fracture mechanics, namely the determination of stresses and plastic strains at the minimum cross-section of a notched tensile specimen. Ideas of Hill and Bridgman are combined to develop a new approximate method for solving this problem. The main aim of the paper is to introduce a damage variable in the analysis without any complicated numerical procedure. The material model proposed by Lemaitre is used to predict the evolution of the damage variable. However, any other model resulting in the incompressibility condition may be adopted with no difficulties.