Mathematics and Mechanics of Solids

A short review is presented of progress in the theory of localized eigenwaves in anisotropic solids. The main attention is paid to the problem of existence of surface wave solutions in semi-infinite elastic bodies. Here the contribution of David Barnett is most important. The general theorems of existence and uniqueness are formulated and their modifications for media with piezoelectric and/or piezomagnetic couplings are considered. The conditions for existence of supersonic surface waves in purely elastic half-spaces are displayed. The situations on both sides of the vicinity of exceptional transonic states are discussed. The criterion for the existence of quasi-bulk subsonic surface waves in elastic media of unrestricted anisotropy is obtained in terms of the general Barnett and Lothe theorem. The conditions for the existence of exceptional bulk waves and their main properties are considered. The interface wave theory of David Barnett et al is reviewed for rigid and slip contacts between elastic and piezoelectric sub-media.

An accelerating dislocation exhibits a logarithmic singularity in the near field (associated with the acceleration), which had been earlier derived from the full solution. Here, the existence and evaluation of the logarithmic singularity are obtained solely from the leading terms (1/r) of the near-field expansions (which are the same as those of the steady-state motion with the instantaneous velocity of the accelerating motion as the uniform velocity) by means of a conservation law involving the dynamic energy—momentum tensor. It is also shown that logarithmic terms of the near-field expansions are independent of the angular coordinate, a question that was posed in the past by David Barnett, and which is critical in this analysis. The self-force and effective mass for an accelerating dislocation depends essentially on that logarithmic singularity as shown by Eshelby and more generally by Ni and Markenscoff.

Steady waves propagating in an anisotropic elastic layer that is attached to an anisotropic elastic half-space is studied. By attached we mean that the interface between the layer and the half-space can be perfectly bonded (b) or in sliding contact (s). The other surface of the layer can be traction-free (F), a rigid surface (R) or a slippery surface (S). We also study steady waves in an anisotropic elastic layer that is attached between two different anisotropic elastic half-spaces. The two interfaces between the layer and the half-spaces can be both perfectly bonded (b/b), one of the interfaces is perfectly bonded while the other is a slippery surface (b/s or s/b) or both interfaces are slippery surfaces (s/s). In the derivation the thickness h of the layer is assumed to be small, and the solution is in the form of an infinite series in the power of h from which an approximate solution can be obtained by keeping the terms up to O(h ⁿ ) for any n. However, the infinite series has a closed-form expression so that the thickness h of the layer need not be small.

The explanation of transition between ductile and brittle modes of fracture based on the “incubation time” concept is proposed. This approach allows us to establish relation between the influences of different loading parameters (strain-rate, temperature) on the mode of fracture. Proposed criteria give explanation to anomalous behavior of yield limit and possible high-temperature embrittlement.

Motivated by recently reported chirality-dependent mechanical phenomena of small-radius carbon nanotubes, an anisotropic elastic shell model is developed in the present paper for small-radius single-walled carbon nanotubes. Due to curvature-derived elastic anisotropy, small-radius single-walled carbon nanotubes are better described by anisotropic plane-stress relations rather than graphite sheets of hexagonal symmetry which are governed by an isotropic plane-stress relation. Based on an orthotropic plane-stress elastic relation for zigzag and armchair single-walled carbon nanotubes, the suggested model is formulated for chiral single-walled carbon nanotubes of arbitrary chiral angle through a small-angle (less than π/12) rotation of the coordinate system. The results obtained show that the suggested anisotropic shell model can explain the chirality-dependent mechanical phenomena reported in the recent literature, and could be used to study chirality-dependent mechanical behavior of multi-walled carbon nanotubes of smaller innermost radii.

An arbitrarily curved three-dimensional piezoelectric thin interphase between two piezoelectric solids is considered. In this study the thin interphase is modeled by a zero thickness interface which separates the two media that are adjacent to the interphase. The model is characterized by jump conditions for the mechanical and electrical fields across the interface. The derivation makes use of Taylor expansions of the fields, and is correct to O(h) where h is the constant thickness of the interphase which has been replaced.

This paper examines the axisymmetric problem of the uniform circular surface loading of an isotropic elastic halfspace, which is internally reinforced with an inextensible membrane of finite extent.

The variation of the Lamé compliance with direction is explored for elastic crystals associated with the trigonal class possessing six independent compliances (associated with point groups 32, 3m, 3m). A new decomposition of the compliance tensor is presented that simplifies the Lamé compliance by reducing the number of elastic parameters required to analyze how it varies with direction. The common form of quartz referred to as α-quartz falls within this class and is known to possess such interesting elastic properties as negative Poisson ratios. The Lamé compliance is a convenient indicator of this negativity and furnishes a sharp connection between its extreme directions and the underlying lattice orientation of the material.

The biomembrane force probe is an experimental apparatus that has been developed to measure the response of a single molecular bond to applied force. A key element in the probe is a small vesicle that is held and manipulated by means of a pipette tip. The interest in this element is due to the fact that it is positioned in series with the bond being interrogated. As a result, the interpretation of data hinges on a knowledge of the mechanical properties, principally the load versus deflection behavior, of the vesicle. Under the assumptions that the vesicle has fixed membrane surface area and fixed internal volume, its load—deflection behavior is considered here. The analysis is exact but the final result does not appear in closed form1 consequently, it must be evaluated by numerical means. For a given geometry, the initial stiffness is found to scale with the pipette pressure and pipette channel radius.

The problem of spatial correlation within an array of dislocations in a two-dimensional crystalline solid is addressed. A system of equations for joint probability densities is derived based on the assumption that the force on each dislocation remains finite. For arrays of screw dislocations moving on several slip planes the equations are consistent with balanced positive and negative dislocations forming dipoles or mutually cancelling, leaving geometrically necessary dislocations to interact and correlate. The resulting pair distribution function for the geometrically necessary screw dislocations is found, and used to demonstrate the strain gradient correction emerging in the case of micro-scale plasticity.

Coble creep in thin films with heterogeneous grain boundary (GB) diffusivity and non-uniform grain size is investigated in the model problem of a thin foil with a high diffusivity GB joined with a low diffusivity GB of unequal length and subjected to a far field uniaxial tensile stress. It is found that a transient stress peak emerges near the GB junction on the characteristic time scale of the high diffusivity boundary and then disappears when the system reaches steady-state on the characteristic time scale of the low diffusivity boundary. We show that for a given GB size ratio, this stress depends logarithmically on the GB diffusivity ratio. Based on the GB diffusivity and size ratios, we present a diagram demarcating the regime in which peak stress is determined by the steady state solution and that determined by the transient stress solution.

The energy of a prismatic dislocation loop in an isotropic elastic cylinder is derived, resulting in semi-analytic expressions. Analytic expressions are obtained in two limits: when the radius of the dislocation loop approaches the cylinder radius and when the radius of the dislocation loop is much smaller than the cylinder radius. These expressions can be used to predict the critical condition for misfit dislocation formation in semiconducting nanowires.

Yielding and strain hardening in metallic thin films on substrates are studied using a simple edge dislocation climb model, modified to mimic dislocation processes in passivated, single crystal FCC metal films with a (111) texture. The aim of the modeling is to produce closed-form solutions for the yield strength and rate of strain hardening that can be compared with experiment. The models give a good account of the dependence of the yield strength of passivated gold films on silicon substrates on the film thickness and they are in broad agreement with the experimental observation that plastic flow in passivated metal films is characterized by very high rates of strain hardening. However, these simple models fail to predict the observed decrease in the rate of strain hardening with increasing film thickness, a result that requires more computationally intensive discrete dislocation modeling.

Computational simulations based on time-dependent Ginzburg—Landau equations are used to model martensitic phase transformations induced in pressure-shear plate impact experiments. Symmetric impact experiments on polycrystalline NiTi plates, and so-called “sandwich impact” experiments on thin polycrystalline NiTi foils sandwiched between two hard plates, are simulated by characterizing each grain as a three-dimensional finite element that can transform to twinned martensite involving twenty-four habit plane variants. The threshold stress at which the transformation occurs is obtained from the amplitude of the leading shear wave in the symmetric impact experiments. The kinetic coefficient in the Ginzburg—Landau equation is obtained by matching the risetime of the transverse particle velocity in the sandwich impact experiments. These simulations highlight the importance of including the effects of self-accommodation in which clusters of several habit plane variants nucleate and grow simultaneously to reduce the constraining effect of the surrounding material.

A desirable curvilinear coordinate system for problems in antiplane strain is one in which the shear stress acting across orthogonal trajectories is either zero or has the maximum value. In elastic isotropic solid problems any coordinate system given by z ≡ x+iy = f ( w ) ≡ f (u + iv) satisfies this condition. This paper examines some requirements for curvilinear coordinate systems with a power law plastic solid (of power m). In particular, if the curvilinear trajectories are given by x = x (u, v) and y = y (u, v) the trajectories must satisfy the equation

where 7 represents either x or y.

Nano-scale defect clusters, such as voids, dislocation loops, stacking-fault tetrahedra and irradiation-induced precipitates, are produced in metals by irradiation with high-energy atomic particles. They are obstacles to dislocation glide and can give rise to substantial changes in the yield and flow stresses and ductility. Atomic-scale computer simulation is able to provide detail of how these effects are influenced by obstacle structure, applied stress, strain rate and temperature. Some recent results from modelling dislocations interacting with obstacles are described. Emphasis is placed on dislocation interaction with voids, copper precipitates and dislocation loops in the BCC metal iron and stacking fault tetrahedra in FCC copper. In the latter case, the importance of surfaces in reactions in TEM foils is highlighted. It is shown that while some atomic processes can be represented adequately by the continuum theory of crystal defects, others cannot.

The approach developed in preceding papers is extended to derive the equilibrium positions of n edge dislocations in a linear pile-up stressed by a constant applied loading against an interface in a bimetallic solid. As n →∞, the dislocations in the inner region are located with sufficient accuracy that the stress distribution at the interface can be evaluated by a simple computational procedure. Such a prediction is impossible using a conventional continuum dislocation theory.