Linear Elastic Solution Research Articles

Mechanical integrity of carbon nanotubes for bending and torsion

The mechanical integrity of single-walled carbon nanotubes for bending and torsional deformations is evaluated using molecular dynamics simulations. A cantilever type nanotube loaded by a follower lateral force at the free end of the tube exhibits an unstable local buckling near the fixed end. A hysteresis loop is observed, accompanied by an irreversible thermodynamically stable topological transition from four hexagons to two pairs of pentagons and heptagons. The finite deformation can be described by a linear elastic solution almost up to the point of buckling. The initial torsional deformation gives a linear relationship between the torsional moment and the angle of twist per unit length. After buckling, an anti-plane distorted structure is generated, whose rigidity is weaker that of the original structure. The critical bending stress and the shear stress in torsion depend not only on the geometry of the cross section but also on the tube length.

Direct Measurements of Fracture Toughness and Crack Growth in Polysilicon MEMS

ABSTRACTDirect measurements of Mode-I critical stress intensity factor and crack tip displacements were conducted in the vicinity of atomically sharp edge cracks in polycrystalline silicon MEMS using our in situ Atomic Force Microscopy (AFM)/Digital Image Correlation (DIC) method. The average Mode-I critical stress intensity factor for various fabrication runs was 1.00 ± 0.1 MPa√m. The experimental crack tip displacement fields were in very good agreement with linear elastic fracture mechanics solutions. By means of an AFM, direct experimental evidence of incremental crack growth in polycrystalline silicon was obtained for the first time via spatially resolved crack growth measurements. The incremental crack growth in brittle polysilicon is attributed to its locally anisotropic polycrystalline structure which also results in different local and macroscopic (apparent) stress intensity factors.

Irreversible constitutive law for modeling the delamination process using interfacial surface discontinuities

An irreversible cohesive–decohesive constitutive law is postulated for interfacial surface discontinuities to predict initiation and progression of delamination. An exponential function is used for the constitutive law that naturally satisfies a multi-axial stress criterion for the onset of delamination and a mixed mode fracture criterion for the progression of delamination. The constitutive equations are made thermomechanically consistent by including a damage parameter to prevent the restoration of the previous cohesive state between the interfacial surfaces. To demonstrate the capability to predict delamination and the irreversibility capability of the constitutive law, steady-state delamination growth is simulated for quasi-static loading–unloading cycle of various fracture test specimens. The finite element results are in good agreement with either experimental data available in the literature or with linear elastic fracture mechanics analytical solutions which are valid for sufficiently long cracks.

Predictive modeling of nanoindentation-induced homogeneous dislocation nucleation in copper

Nanoscale contact of material surfaces provides an opportunity to explore and better understand the elastic limit and incipient plasticity in crystals. Homogeneous nucleation of a dislocation beneath a nanoindenter is a strain localization event triggered by elastic instability of the perfect crystal at finite strain. The finite element calculation, with a hyperelastic constitutive relation based on an interatomic potential, is employed as an efficient method to characterize such instability. This implementation facilitates the study of dislocation nucleation at length scales that are large compared to atomic dimensions, while remaining faithful to the nonlinear interatomic interactions. An instability criterion based on bifurcation analysis is incorporated into the finite element calculation to predict homogeneous dislocation nucleation. This criterion is superior to that based on the critical resolved shear stress in terms of its accuracy of prediction for both the nucleation site and the slip character of the defect. Finite element calculations of nanoindentation of single crystal copper by a cylindrical indenter and predictions of dislocation nucleation are validated by comparing with direct molecular dynamics simulations governed by the same interatomic potential. Analytic 2D and 3D linear elasticity solutions based on the Stroh formalism are used to benchmark the finite element results. The critical configuration of homogeneous dislocation nucleation under a spherical indenter is quantified with full 3D finite element calculations. The prediction of the nucleation site and slip character is verified by direct molecular dynamics simulations. The critical stress state at the nucleation site obtained from the interatomic potential is in quantitative agreement with ab initio density functional theory calculation.

Shear crack propagation along weak planes in solids: a finite deformation analysis incorporating the linear harmonic potential

Recent molecular dynamics simulations of dynamic crack propagation have shown that a shear crack propagating at a sub-Rayleigh speed may have a finite crack opening, but the crack opening vanishes once the crack tip velocity exceeds the shear wave speed. This observation is at odds with classical linear elastic solutions which indicate that a pure shear crack should have zero opening. To understand this discrepancy, we develop in this paper a finite deformation continuum theory incorporating the linear harmonic potential to describe the deformation of a crack in a solid with triangular lattice structure. Using the asymptotic method of Knowles [Eng. Fract. Mech. 15 (1981) 469], we show that there is indeed a finite crack opening for a dynamic, sub-Rayleigh shear crack. This opening is on the order of the lattice constant, and is attributable to the geometric nonlinearity of finite deformation near the crack tip. We also show in another paper that, once the crack tip velocity exceeds the shear wave speed, Knowles’ asymptotic method gives a vanishing crack opening. These conclusions based on the continuum analysis for sub-Rayleigh and super-shear cracks agree well with the molecular dynamics simulations.

Orientation dependence of the plastic slip near notches in ductile FCC single crystals

Results from experiments conducted on copper FCC single crystals are reported. Two symmetric crystallographic orientations and four nonsymmetric crystallographic orientations were tested. The slip line fields that form near a pre-existing notch in these specimens were observed. The changes in these patterns as the orientation of the notch in the crystal is rotated in an {101} plane are discussed. Sectors of similar slip line patterns are identified and the type of boundaries between these sectors are discussed. A type of sector boundary called mixed kink is identified. Specimen orientations that differ by 90° are found to have different slip line patterns, contrary to the predictions of perfectly plastic slip line theory. The locations of the first slip lines to form are compared to the predictions obtained using anisotropic linear elastic stress field solutions and the initial plane-strain yield surfaces. It is found that comparison of these surface slip line fields to plane strain crack tip solutions in the annular region between 350 and 750 μm is justified. The differences in anisotropic elastic solutions for orientations that are 90° apart explain the lack of agreement with perfectly plastic slip line theory.

A Summary of Asymptotic Finite Deformation Results for a Point Load on a Hyperelastic Half-Space

The nonlinearly elastic Boussinesq problem uses the exact equations of finite elasticity to mathematically model the deformation produced in a homogeneous, isotropic, elastic half-space by a point force normal to the undeformed boundary. For this core problem of elasticity and engineering, the 1885 linear elasticity solution of Boussinesq is still used in a variety of applications despite the fact that the linear solution predicts physically unrealistic behavior in the primary region of interest beneath the load. In this paper, we aim to summarize two recent SIAM Journal of Applied Mathematics papers (D. A. Polignone Warne and P. G. Warne SIAM J. Appl. Math. 62 107-128 (2001) and SIAM J. Appl. Math. at press (2002)). These two papers develop asymptotic analyses for the nonlinearly elastic tensile point load problem for a half-space composed of either a general incompressible or compressible material, respectively. We also wish to present several new closed-form asymptotic solutions in the case of a tensile point load acting on an incompressible half-space. Finally, we comment on our approach to treating the complementary problem involving a compressive point load. Here we summarize the governing equations, conservation laws, hypotheses, and asymptotic tests needed to determine whether an isotropic hyperelastic material can support a finite deflection under a tensile point load. A variety of particular constitutive models in nonlinear elasticity are tested, and it is found that a material must be sufficiently stiff in order to support the load. Thus, it follows that many of the well-known strain-energy models for compressible hyperelastic materials proposed in the literature are unable to do so. For models which may sustain a tensile point load, we determine either the full asymptotic solution or the remaining equations and conditions for this solution. For classes of material models motivated by studies of Beatty and Jiang, we solve the resulting boundary value problem numerically in a compressible hyperelasticity setting, and we derive some new results including several closed-form explicit asymptotic solutions for a family of incompressible material models.

A novel technique for characterizing elastic properties of thin biological membrane

We present a new technique to characterize the elastic properties of a thin biological membrane. An effective instrument that embodies video enhanced microscope was developed primarily to provide the capability of simultaneously measuring both the applied force and the resultant displacement of the biological membrane under a central point force. A theoretical linear elastic solution was applied to quantitatively interpret the measured central deflection of the membrane under a central point load. The Young’s modulus of raw and boiled Leghorn egg inner tissue can be easily determined once the applied point load and the central deflection, together with the essential dimensions are known. The viscoelasticity property of natural tissue was manifest in the experiment. The experimental results have verified that the novel technique is applicable to determine the Young’s modulus and other viscoelatic properties of thin biological tissue.

On the range of applicability of the Reissner-Mindlin and Kirchhoff-Love plate bending models

We show that the Reissner-Mindlin plate bending model has a wider range of applicability than the Kirchho-Love model for the approximation of clamped linearly elastic plates. Under the assumption that the body force density is constant in the transverse direction, the Reissner-Mindlin model solution converges to the three-dimensional linear elasticity solution in the relative energy norm for the full range of surface loads. However, for loads with a significant transverse shear eect, the Kirchho-Love model fails.

An Asymptotic Finite Deformation Analysis for an Isotropic Compressible Hyperelastic Half-Space Subjected to a Tensile Point Load

The nonlinearly elastic Boussinesq problem is to find the deformation produced in a homogeneous, isotropic, elastic half-space by a point force normal to the undeformed boundary, using the exact equations of elasticity. For this core problem of elasticity and engineering, the 1885 linear elasticity solution of Boussinesq is still used in a variety of applications. In [SIAM J. Appl. Math., 62 (2001), pp. 107--128], we addressed the case of a tensile point load under the constraint of incompressible finite elasticity. Here we consider an analogous asymptotic analysis of this problem within the context of compressible finite elasticity. Asymptotic tests are developed to determine whether an isotropic hyperelastic material can support a finite deflection under a tensile point load. The results are then applied to a variety of particular constitutive models for compressible nonlinearly elastic materials. It is found that, for many of the well-known strain energy models for compressible hyperelastic materials proposed in the literature, a tensile point load cannot be supported. For models which may sustain a tensile point load, we determine the remaining equations and conditions for the asymptotic solution, and numerically compute this solution for a particular case.

Theoretical investigation of elastoplastic notch fields under triaxial stress constraint

In this paper, an exact elastic-plastic solution has been obtained based on the J2-deformation theory of plasticity for a plate having a circular hole under biaxial tension and triaxial stress constraint in linear elastic strain-hardening materials. The theoretical solution shows that a linear elastic solution of the equivalent strain can be used to linear elastic-power hardening plastic situation just by a simple variable replacement. Then a strain equivalent rule (SER) is proposed to predict the elastoplastic notch fields by use of the elastic solution. Validations against theoretical analyses and finite element calculation for various combinations of material properties, triaxial stress constraints, load levels show that the SER can be used to predict stress-strain distributions in the whole plastic zone effectively and conveniently.

Singularities near three-dimensional corners in composite laminates

The high interlaminar stresses, which appear in laminated composites due to the boundary layer effect near the free edge, play an important role in the analysis and design of advanced structures. Moreover, they are also the dominant effect causing delamination. Even if the singular behavior of such structures is investigated in many works, most of them deal either with 2D, or with pseudo-3D problems, i.e. problems of two variables in a three-dimensional space. However, some numerical and experimental findings indicate that laminated plates exhibit a tendency to delaminate at corners, an effect impossible to be determined by a two-dimensional analysis. The aim of the present paper is to investigate stress singularities in a laminated composite wedge under consideration of real three-dimensional corner effects. A weak formulation, as well as a finite element approximation technique introduced in the past for isotropic problems is extended here to cover anisotropic material properties. This formulation leads to a quadratic eigenvalue problem, which is solved iteratively using the Arnoldi method. The first singular terms in the asymptotical expansion of the linear-elastic solution near the vertex of the wedge are obtained as eigenpairs of this eigenvalue problem. The order and mode of singularity are reported for all wedge angles and different fiber orientations for angle and cross-ply laminates. All calculations are based on a typical for some high modulus graphite-epoxy systems orthotropic material model.

Direction-sensitive strain-mapping with carbon nanotube sensors

Single-wall carbon nanotubes (SWNTs) embedded in a polymer can be used as mechanical sensors because the position of the D* Raman band is strongly dependent on the strain transferred from the matrix to the nanotubes. The unpolarized Raman spectrum of the nanotubes has high strain sensitivity if the nanotubes are oriented along the principal strain axis in the polymer, whereas with polarized Raman, even unoriented nanotubes exhibit a strong wavenumber shift in the Raman spectrum with strain. These methods are demonstrated here by measuring the stress distribution around a circular hole in SWNT/polymer composites under uniaxial tension. In both cases the results fit the classical linear elasticity solution.

Adherence of a Rectangular Flat Punch Onto a Clamped Plate: Transition From a Rigid Plate to a Flexible Membrane

A linear elastic solution is proposed for the adhesion/delamination of a constrained thin film adhered to a rectangular flat punch. As the punch is pulled away by an external load, the film deforms and gradually delaminates until a line contact is left prior to complete separation. This is in sharp contrast with the finite pull-off contact radius as predicted by the classical Johnson-Kendall-Roberts theory for adhesion between solid bodies. In order to portray the transition from a platelike to a membranelike behavior, the film thickness and stiffness are allowed to span a wide range of values. Simple experiments demonstrated the validity of the theory.

Stress fields around defects and fibers in a polymer using carbon nanotubes as sensors

Raman spectroscopy was used to map the stress distribution in the vicinity of discontinuities in a polymer using single-wall nanotubes seeded in the specimen. In the case of a hole in a polymer matrix subjected to unidirectional stress, the experimental stress field compared well with the classical linear elasticity solution. For a single glass fiber embedded in a polymer, the tangential thermal residual stress in the vicinity of the fiber was picked up by Raman spectroscopy and is in satisfactory agreement with a standard two-phase concentric cylinder model.

Overview of the International Comparative Assessment Study of Pressurized Thermal-Shock in Reactor Pressure Vessels (RPV PTS ICAS)

This paper summarizes the recently completed International Comparative Assessment Study of Pressurized Thermal-Shock in Reactor Pressure Vessels (RPV PTS ICAS). The ICAS project brought together international experts to perform comparative evaluations of methodologies employed in the assessment of RPV integrity under prototypical PTS conditions. ICAS grew out of a strong interest expressed by the participants to proceed with further evaluations after completion of the earlier FALSIRE II project. FALSIRE focused on evaluation of structural analysis and fracture assessment methods on the basis of experimental and analytical results. The ICAS problem statement defined transient thermal–mechanical conditions postulated to result from loss-of-coolant accidents in a Western type four-loop RPV with cladding on the inner surface. The primary focus was on the behavior of relatively shallow cracks located under and through the cladding. The assessment activities were divided into three tasks: Deterministic Fracture Mechanics (DFM), Probabilistic Fracture Mechanics (PFM) and Thermal-Hydraulic Mixing (THM). The results showed that a best-estimate methodology for RPV integrity assessment could benefit from a reduction of the uncertainties in each phase of the process. Within the DFM task, where account was taken of material properties and boundary conditions, reasonable agreement was obtained in linear-elastic and elastic–plastic analyses. Results from linear-elastic and J-estimation analyses were shown to provide conservative estimates of peak crack driving force when compared with those from complex 3D finite element analyses. For the PFM task, linear-elastic solutions were again shown to be conservative with respect to elastic–plastic solutions (by a factor of 2 to 4). Scatter in solutions obtained using the same computer code was generally attributable to differences in input parameters, e.g. standard deviations for the initial value of RT NDT, as well as for nickel and copper content. In the THM task, while there was a high degree of scatter during the early part of the transient, reasonable agreement was obtained during the latter part of the transient. Generally, the scatter was due to differences in analytical approaches used by the participants, which included correlation-based engineering methods, system codes and three-dimensional computational fluids dynamics codes. Based on concluding discussions from ICAS participants, a project is being organized to develop a computer software tool named QUAMET (acronym for QUAlification METhodology) for future use in qualifying codes and analysts engaged in the structural integrity assessment of RPVs.

Thin-Film-Edge-Induced Stresses in Substrates

ABSTRACTWe have obtained an analytic solution of linear elasticity for the stress distribution under a thin film edge in isotropic substrates of finite thickness and of infinite extent in the other two directions. The following boundary conditions are considered. At the film/substrate interface and the free surfaces at the top and bottom of the substrate, the normal stress component vanishes. Far from the film edge on the side without the film, all stress components are zero. Far from the film edge under the film, the stress distribution is in accordance with that given by the bimetallic strip theory. Two examples of comparison between theory and experiments were given to demonstrate the validity of this solution. The infrared photoelastic stress fringe pattern obtained by a dark-field plane polariscope in a Si substrate under an oxide film edge was successfully reproduced. The calculated and experimental results of Raman-shift of Si under patterned CoSi2 line structures also showed good agreement with each other.

Behaviour of deep shafts in rock considering nonlinear elastic models

Some models that consider the non-linearity of the stress-strain behaviour of rocks are reviewed and discussed in this work. The models are an extension of the classic elastic theory where the constant elastic modulus is replaced by a function, which relates the modulus of elasticity and the confining pressure. The analytical solutions for determining the stresses and strains induced by vertical excavations are compared with the linear elastic solution, considering the excavation at three different depths (250, 1000 and 2000 m). It is verified that the stresses and strains given by the non linear analysis can be very different from those given by the linear elastic analysis, depending on the relation between the elastic modulus and the confining pressure.

Cytoindentation for Obtaining Cell Biomechanical Properties

A novel biomechanical testing methodology was developed to obtain the intrinsic material properties of an individual cell attached to a rigid substrate. With use of a newly designed cell-indentation apparatus (cytoindenter), displacement-controlled indentation tests were conducted on the surface of individual MG63 cells and the corresponding surface reaction force of each cell was measured. The cells were modeled with a linear elasticity solution of half-space indentation and the linear biphasic theory on the assumption that the viscoelastic behavior of each cell was due to the interaction between the solid cytoskeletal matrix and the cytoplasmic fluid. To obtain the intrinsic material properties (aggregate modulus, Poisson's ratio, and permeability), the data for experimental surface reaction force and deformation were curve-fitted with use of solutions predicted with a linear biphasic finite element code in conjunction with optimization routines. The MG63 osteoblast-like cells had a compressive aggregate modulus of 2.05 ± 0.89 kPa, which is two to three orders of magnitude smaller than that of articular cartilage, six to seven orders smaller than that of compact bone, and quite similar to that of leukocytes. The peremeability was 1.18 ± 0.65 (×10-10) m4/N-s, which is four to six orders of magnitude larger than that of cartilage. The Poisson's ratio was 0.37 ± 0.03. The intrinsic material properties of the individual cell in this study can be useful in precisely quantifying mechanical stimuli acting on cells. This information is also needed for theories attempting to establish mechanotransductional relationships.

The flow stress contribution of the dislocation 'forest' in bcc lattices

The athermal flow stress contribution by junction reactions of a glide dislocation with a 'forest' of density rho is usually written as tau = alpha mu b rho 1/2. The strength coefficient alpha has been computed for the bcc lattice by virtual displacement of triple nodes following the method of Puschl et al. (1982, Physica status solidi (a), 74, 211). In the isotropic approximation, the values for glide dislocations with 0, 30, 60 and 90o character are as alpha = 0.18, 0.21, 0.19 and 0.22 respectively. A comparison with values calculated previously by Frydman is made, and good agreement is found when differences in the cut-off radii of the linear elastic solution are accounted for.