Size-dependent fracture and higher order J-integral for solids characterized by strain gradient elasticity

Elastic fracture is governed by the material's strain energy released rate and depends on the applied loads and the stiffness of the structure. The effect of stiffness on fracture as a function of structural size is typically modeled using strain-based elastic fracture mechanics, but recent experimental evidence indicated that when the size of the structure is on the order of the higher order material length scale parameters, elastic strain gradients would stiffen the structure. In this paper, the higher order J-integral and energy-released rate for the analysis of fracture of strain gradient stiffened structures are developed. The effects of beam size on the fracture behaviors of strain gradient stiffened cantilevers on a substrate were analyzed using the higher order J-integral and the energy-released rate. Analyses revealed that the fracture load is elevated to more than 1.4 times of the un-stiffened case when the bending stiffness is doubled by strain gradient stiffening. Elastic fracture is shown to have an added dependence on the size of the structure when strain gradient stiffening is non-negligible.     

Isolated concentration waves of point defects under pulsed laser irradiation

A model is proposed for the first time for the propagation of an isolated (soliton-like) concentration wave of point defects in a crystal exposed to laser pulses. It is shown that this isolated concentration wave is formed as a result of a nonlinear concentration dependence of the defect source function which is caused by a reduction in the activation energy for defect formation near clusters when the elastic stress field is taken into account. Conditions for the excitation of an isolated concentration wave, its profile, and propagation velocity are determined. Pis’ma Zh. Tekh. Fiz. 25, 90–94 (August 26, 1999)     

Third-order elastic constants and pressure derivatives of the second-order elastic constants of hexagonal boron nitride

The second and third order elastic constants and pressure derivatives of second order elastic constants of hexagonal boron nitride have been obtained using the deformation theory. The strain energy derived using the deformation theory is compared with the strain dependent lattice energy obtained from elastic continuum model approximation to get the expressions for second and third order elastic constants. Higher order elastic constants are a measure of anharmonicity of crystal lattice. The six second-order elastic constants and the ten non-vanishing third order elastic constants and six pressure derivatives of hexagonal boron nitride are obtained in the present work and are compared with available experimental values. The second order elastic constant C ₁₁ which corresponds to the elastic stiffness along the basal plane of the crystal is greater than C ₃₃. Since C ₃₃ being the stiffness tensor component along the c-axis of the crystal, this result is expected from a layer-like material like boron nitride (BN). The third order elastic constants of hexagonal BN are generally one order of magnitude greater than the second-order of elastic constants as expected of a crystalline solid. The pressure derivative dC ₃₃/dp obtained in the present study is greater than dC ₁₁/dp which indicates that the compressibility along c-axis is higher than that along ab-plane of hexagonal BN.     

A full tangent stiffness field-boundary-element formulation for geometric and material non-linear problems of solid mechanics

The field-boundary-element method naturally admits the solution algorithm in the incompressible regimes of fully developed plastic flow. This is not the case with the generally popular finite-element method, without further modifications to the method such as reduced integration or a mixed method for treating the dilatational deformation. The analyses by the field-boundary-element method for geometric and material non-linear problems are generally carried out by an incremental algorithm, where the velocities (or displacement increments) on the boundary are treated as the primary variables and an initial strain iteration method is commonly used to obtain the state of equilibrium. For problems such as buckling and diffused tensile necking, involving very large strains, such a solution scheme may not be able to capture the bifurcation phenomena, or the convergence will be unacceptably slow when the post-bifurcation behaviour needs to be analysed. To avoid this predicament, a full tangent stiffness field-boundary-element formulation which takes the initial stress–velocity gradient (displacement gradient) coupling terms accurately into account is presented in this paper. Here, the velocity field both inside and on the boundary are treated as primary variables. The large strain plasticity constitutive equation employed is based on an endochronic model of combined isotropic/kinematic hardening plasticity using the concepts of material director triad and the associated plastic spin. A generalized mid-point radial return algorithm is presented for determining the objective increments of stress from the computed velocity gradients. Numerical results are presented for problems of diffuse necking, involving very large strains and plastic instability, in initially perfect elastic–plastic plates under tension. These results demonstrate the clear superiority of the full tangent stiffness algorithm over the initial strain algorithm, in the context of the integral equation formulations for large strain plasticity.     

Chemically Tuning Mechanics of Graphene by BN

With finite bandgaps, g‐BNC, a boron nitride monolayer (g‐BN) phase within a graphene layer, is a promising semiconductor for next generation electronics. We report its mechanics dependence of the g‐BN concentration, including the high order elastic constants and mechanical failure, through a first‐principles study based on density functional theory. The in‐plane stiffness as well as third order elastic constants of graphene can be linearly tuned with g‐BN concentration. The longitudinal mode elastic constants are sensitive to the BN modification, in contrast to the shear mode elastic constants. This study may provide guidance in optimizing the mechanics of graphene‐based nanodevices.     

Longitudinal and transverse waves in anisotropic elastic materials

Summary It is shown that a necessary and sufficient condition for a longitudinal wave to propagate in the direction n in an anisotropic elastic material is that the elastic stiffness C ₁₁ (n) is a stationary value (maximum, minimum or saddle point) at n. Explicit expressions of all n and the corresponding elastic stiffness C ₁₁ (n) for which a longitudinal wave can propagate are presented for orthotropic, tetragonal, trigonal, hexagonal and cubic materials. As to longitudinal waves in triclinic and monoclinic materials, only few explicit expressions are possible. We also present necessary and sufficient conditions for a transverse wave to propagate in the direction n. As an illustration, explicit expressions of all n, the polarization vector a and the wave speed c for which a transverse wave can propagate in cubic and hexagonal materials are given. The search for n in hexagonal materials confirms the known fact that a transverse wave can propagate in any direction. A longitudinal wave is necessarily accompanied by two transverse waves. However, a transverse wave can propagate without being accompanied by a longitudinal wave.     

Effect of pressure on the structural and elastic properties of ZnS and MgS alloys in the B3 and B1 phases

From the first-principles calculations, we have investigated the elastic stiffness coefficients C₁₁, C₁₂, C₄₄ and the bulk modulus B of the II-VI semiconductors ZnS and MgS under hydrostatic pressure. The calculations are based on the density functional theory within the generalized gradient approximation (GGA) for exchange-correlation interaction. For the structural properties we have shown that ZnS adopt the rocksalt (NaCl or B1) structure over 11.87 GPa pressure, the same character is adopted by MgS over 0.8 GPa. The elastic coefficients have the same behavior for the different structures of alloys; they increase with increasing pressure values. Our results for the structural parameters and equilibrium phase elastic constants are in good agreement with the available theoretical and experimental data.     

Effect of rotationally restrained and Pasternak foundation on buckling of an orthotropic rectangular Mindlin plate

This article, based on first-order shear deformation theory, presents the buckling analysis of a rotationally restrained orthotropic rectangular Mindlin plate resting on a Pasternak elastic foundation. Thus, the Mindlin–Reissner plate theory is employed for which the governing equations are solved by the Rayleigh–Ritz method. Uniformly distributed in-plane loads are applied to two simply supported opposite edges of the plate and the other two edges have rotationally restrained conditions without loading. Finally, the effects of plate parameters, such as foundation stiffness coefficients, aspect ratios, and ratio of elastic modulus in the x to y direction on the buckling loads are presented. The results show that the buckling load would increase when the ratio of the elastic modulus in the x to y direction increases and the plate is close to isotropic. The variation of buckling load versus changing ratio of elastic modulus in the x to y direction in the state of without elastic foundation and with clamp support is more than the rest of the state.     

Free vibration analysis of double-bonded isotropic piezoelectric Timoshenko microbeam based on strain gradient and surface stress elasticity theories under initial stress using differential quadrature method

The size-dependent effect on free vibration of double-bonded isotropic piezoelectric Timoshenko microbeams using strain gradient and surface stress elasticity theories under initial stress is presented. This article is developed for isotropic piezoelectric material. Due to the high surface-to-volume ratio, surface stress has an important role with micro- and nanoscale materials. Thus, the Gurtin–Murdoch continuum mechanic approach is used. Governing equations of motion are derived by Hamilton's principle and solved by the differential quadrature method. The effects of pre-stress load, surface residual stress, surface mass density, surface piezoelectrics, Young's modulus of surface layers, three material length scale parameters, thickness to material length scale parameter ratios, various boundary conditions, and two elastic foundation coefficients are investigated. It is concluded that the effect of pre-stress load in greater modes is negligible for higher aspect ratios and this effect is similar to lower aspect ratios. Also, the size-dependent effect on the dimensionless natural frequency for strain gradient theory is higher than that for modified couple stress theory and classical theory, which is due to increasing stiffness of the Timoshenko microbeam model. Moreover, the results show that dimensionless natural frequency affects more by considering the material length scale parameters with respect to surface effect. The results are compared with the obtained results from the literature and show good agreement between them. It is concluded that the amplitude of the transverse displacements (w₀) for a microbeam (MB) is more than the transverse displacements (w₁) for a piezoelectric microbeam (PMB). On the other hand, using a piezoelectric layer for PMB, the amplitude of the transverse displacements (w₁) reduces considerably with respect to MB, in which this effect leads to increase the stiffness of the microbeam and stability of microstructures. With considering the piezoelectric layer, the obtained results can be used to control the amplitude and vibration of microstructures, prevent the resonance phenomenon, design smart structures, and can be employed for micro-electro-mechanical systems and nano-electro-mechanical systems.     

Characterization and modeling of stiffness reduction in SCS-6-Ti composites under low cycle fatigue loading

The stiffness reduction and evolution of microstructural damage of a unidirectional silicon carbide fiber reinforced titanium matrix composite under tension-tension fatigue were investigated. Tests were conducted under load control with maximum applied stresses ranging from 750 to 945 MPa. The crack density of the interfacial reaction layer and matrix, matrix crack length, and interfacial debonding length as a function of fatigue cycles and applied stress levels were measured. The results showed that the composites exhibited an initial regime with slow stiffness reduction, followed by a rapid stiffness drop regime and a plateau regime with minimal change in stiffness for the applied stress levels used in this study. The residual stiffness at N = 10⁶ cycles is independent of the applied stress levels, while the microstructural damage accumulation varied with the applied stresses. A partial crack shear-lag model was also developed to predict the residual stiffness as a function of fatigue damage accumulation. Analytical simulation indicated that the profile of the stiffness reduction curves was dominated by the matrix crack density, while the extent of stiffness reduction was dominated by the matrix crack length.     

Study of infrared spectroscopy and elastic properties of fine and coarse grained nickel–cadmium ferrites

The elastic properties of Cd _x Ni_1−x Fe₂O₄ (x = 0.2, 0.4, 0.6 and 0.8) spinel ferrite system synthesized by wet-chemical technique, have been studied by infra-red spectroscopy and X-ray diffraction pattern analysis before (W) and after high temperature annealing (AW). The average particle size for wet-samples was within the range 4–5 nm, which is much lower than the average particle size found for AW samples (≈85 nm). The force constants for tetrahedral and octahedral sites determined by infrared spectral analysis, lattice constant and X-ray density values by X-ray diffraction pattern analysis; have been used to calculate elastic constants. The elastic moduli for W-samples are found to be larger as compared to AW samples, which are explained on the basis of grain size reduction effect. The average crystallite size calculated from elastic data is in agreement to that determined from X-ray diffraction data analysis.     

Density functional study of structural and electronic properties of GaPn (2?≤?n?≤?12) clusters

Low-lying equilibrium geometric structures of GaP _n (n = 2–12) clusters obtained by an all-electron linear combination of atomic orbital approach, within spin-polarized density functional theory, are reported. The binding energy, dissociation energy, and stability of these clusters are studied within the local spin density approximation (LSDA) and the three-parameter hybrid generalized gradient approximation (GGA) due to Becke–Lee–Yang–Parr (B3LYP). Ionization potentials, electron affinities, hardness, and static dipole polarizabilities are calculated for the ground-state structures within the GGA. It is observed that the gallium atoms of the symmetric ground-state structures prefer to occupy the peripheral positions. It is found that the relative ordering of the isomers is influenced by the nonlocal exchange-correlation effects for small clusters. Generalized gradient approximation extends bond lengths and widens the energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), as compared to the LSDA gap. The odd–even oscillations in the dissociation energy, the second differences in energy, the HOMO–LUMO gaps, the ionization potential, the electron affinity, and the hardness are more pronounced within the GGA. The stability analysis based on the energies clearly shows the GaP₅ and GaP₇ clusters to be endowed with special stabilities.     

Microstructure–Stiffness Relationships of Common Yew and Norway Spruce

Abstract: Yew (Taxus baccata L.) exhibits among conifers a unique macroscopic elastic behaviour. For example, it shows a comparatively low longitudinal elastic modulus related to its comparatively high density. We herein explore the microstructural origin of these peculiarities, aiming at the derivation of microstructure–stiffness relationships. We measure stiffness properties of yew at different hierarchical levels and compare them to corresponding stiffnesses of Norway spruce (Picea abies [L.] Karsten). Cell wall stiffness is investigated experimentally by means of nanoindentation in combination with microscopy and thermogravimetric analysis. On the macroscopic level, we perform uniaxial tension and ultrasonic tests. Having at hand, together with previously reported stiffnesses, a consistent data set of mechanical, chemical and physical properties across hierarchical levels of wood, we discuss influences of microstructural characteristics at different scales of observation. Moreover, a micromechanical model is applied to predict trends of effects of the microstructure on the investigated stiffness properties. On the cell wall level, particularly, the amount of cellulose and its orientation – which was earlier reported to be distinctly different for yew and spruce – result in differences between the two considered species. On the macroscopic scale, model predicted effects of the annual ring structure on transverse stiffness and shear stiffness are found to be smaller than effects of the microfibril angle and mass density.     

Design optimization of machinery mounting systems with an elastic support structure

An elastic support-based optimization model is developed for a machinery mounting system comprising a vibrating machine supported on an elastic structure by multiple resilient mounts. This model is used to investigate the design optimization of an X?Y motion stage mounting system used in microelectronics wire-bonding equipment. By varying the stiffness coefficients of the resilient mounts while constraining the dynamic displacement amplitudes of the X?Y motion stage, the total force transmitted from the X?Y motion stage (the vibrating machine) to the equipment table (the elastic support structure) is minimized at each frequency interval in the frequency range of interest for different stiffnesses of the equipment table. The results show that, when the equipment table is relatively flexible, the total transmitted force minimized by the model developed is significantly lower than that minimized using a conventional rigid support-based optimization model at some critical frequency. When the equipment table is sufficiently rigid, both models provide almost the same predictions of the total transmitted force.     

Analytic and numerical studies on mode I and mode II fracture in elastic-plastic materials with strain gradient effects

This paper presents a study on fracture of materials at microscale (∼1 μm) by the strain gradient theory (Fleck and Hutchinson, 1993; Fleck et al., 1994). For remotely imposed classical K fields, the full-field solutions are obtained analytically or numerically for elastic and elastic-plastic materials with strain gradient effects. The analytical elastic full-field solution shows that stresses ahead of a crack tip are significantly higher than their counterparts in the classical K fields. The sizes of dominance zones for mode I and mode II near-tip asymptotic fields are 0.3l and 0.5l,while strain gradient effects are observed within land 2l to the crack tip, respectively, where l is the intrinsic material length in strain gradient theory and is on the order of microns in strain gradient plasticity (Fleck et al., 1994; Nix and Gao, 1998; Stolken and Evans, 1997). The Dugdale–Barenblatt type plasticity model is obtained to provide an estimation of plastic zone size for mode II fracture in materials with strain grain effects. The finite element method is used to investigate the small-scale-yielding solution for an elastic-power law hardening solid. It is found that the size of the dominance zone for the near-tip asymptotic field is the intrinsic material lengthl. For mode II fracture under the small-scale-yielding condition, transition from the remote classical K _IIfield to the near-tip asymptotic field in strain gradient plasticity goes through the HRR field only when K _IIis relatively large such that the plastic zone size is much larger than the intrinsic material length l. For mode I fracture under small-scale-yielding condition, however, transition from the remote classical K _I field to the near-tip asymptotic field in strain gradient plasticity does not go through the HRR field, but via a plastic zone. This revised version was published online in July 2006 with corrections to the Cover Date.     

Scattering of 3He and 4He Atoms from 4He Clusters

We develop a microscopic theory of elastic and inelastic scattering and sticking of ³ He and ⁴ He atoms impinging on superfluid ⁴ He clusters. A number of essential aspects must be observed in a physically meaningful and reliable theory of such scattering processes. These are all connected with multiparticle processes, particularly the possibility of energy loss. These processes are (a) the coupling to low-lying excitations which is manifested in a finite imaginary part of the self energy, (b) the interaction with roton–like excitations and the broadening of the roton minimum due to finite–size effects, and (c) the discreteness of low–lying excitations in helium clusters.     

Elasticity of Fractal Inspired Interconnects

The use of fractal‐inspired geometric designs in electrical interconnects represents an important approach to simultaneously achieve large stretchability and high aerial coverage of active devices for stretchable electronics. The elastic stiffness of fractal interconnects is determined analytically in this paper. Specifically, the elastic energy and the tensile stiffness for an order n fractal interconnect of arbitrary shape are obtained, and are verified by the finite element analysis and experiments.     

Evaluation of damping and elastic properties of composites and composite structures by the resonance technique

This paper describes the resonance technique for determining the stiffness and damping properties of a composite or composite structure. Pultruded GRP composites and optical fibre cables (multi-component structures) were investigated. The resonance frequencies (natural frequencies) of a material, or a system, are a function of its elastic properties, dimensions and mass. This concept is used to determine the stiffness of a vibrated material by the resonance technique, which applies only very low stresses through the application of acoustic energy. This makes it applicable to measure the stiffness of multi-element cables. The damping properties, in terms of Q ^–1 (internal friction) were determined by both a free exponential decay curve and half-peak bandwidth methods. The influence of specimen length and measurement set-up was investigated. The applicability and accuracy of the resonance technique for a composite structure were discussed. The measured elasticity of optical cables was found to be in good agreement with the derived theoretical value.     

A Hybrid T Matrix/Boundary Element Method for Elastic Wave Scattering from a Defect Near a Non-planar Surface

The in-plane P-SV scattering of elastic waves by a defect and a close non-planar surface is considered. A hybrid T matrix/boundary element approach is used, where a boundary integral equation is used for the non-planar surface and the Green’s tensor in this integral equation is chosen as the one for the defect and thus incorporates the transition (T) matrix of the defect. The integral equation is discretized by the boundary element method in a standard way. Also models of ultrasonic probes in transmission and reception are included. In the numerical examples the defect is for simplicity chosen as a circular cavity. This cavity is located close to a non-planar surface, which is planar except for a smooth transition between two planar parts. It is illustrated that the scattering by the cavity and the non-planar surface becomes quite complicated, and that shielding and masking may appear.     

Determination of Elastic Properties of Resected Human Breast Tissue Samples Using Optical Coherence Tomographic Elastography

Abstract: The present study is concerned with the application of optical coherence tomographic (OCT) elastography technique for quantitative assessment of the elastic properties of resected human breast tissue samples subjected to axial compressive loading in vitro. Three classes of breast tissue samples, namely normal, benign (fibroadenoma) and malignant (invasive ductal carcinoma), were considered. A speckle tracking technique based on two‐dimensional cross correlation was employed to track the speckle motion between original (pre‐compressed) and the displaced (post‐compressed) OCT images of the tissue samples for the measurement of displacement and strain maps. The overall data reduction approach for quantitative assessment of elastic properties was validated against the results of gelatin phantoms containing activated charcoal particles as scattering centres. Results are presented in the form of OCT images and displacement and axial strain maps for normal, benign and malignant breast tissue samples. Based on the stress–strain relationship obtained for these three classes, the values of stiffness coefficients were reported in terms of modulus of elasticity. Results of the study reveal significant differences between the two‐dimensional displacement vector maps of normal and cancerous breast tissue samples. The stiffness of benign tissue samples is found to be about two times higher than that of normal tissue samples, whereas for malignant samples, it is about four times higher, thereby signifying appreciable differences in the stiffness of cancerous and normal tissue samples.