Three-dimensional Strain Fields Research Articles

Damage analysis by physically nonlinear composite beam theory

Damaging constitutive models are incorporated into the variational asymptotic beam sectional analysis method (VABS). Under the governing of the virtual work principle, Newton’s method is applied to solve for the finite cross-sectional warping field and the damage variables according to continuum damage mechanics models from the literature. These solutions result in mesh-insensitive predictions of the damaged beam stiffness and three-dimensional stress and strain fields in the damaged state. The present theory is implemented using the finite element method to capture the high-fidelity elastic-to-damaged beam constitutive behaviors by the cross-sectional analyses. The predicted beam constitutive relations and local fields are compared with results from finite element analyses (FEA) using solid elements and plate elements, and experiments.

Microscopic full-field three-dimensional strain measurement during the mechanical testing of additively manufactured porous biomaterials

A custom-designed micro-digital image correlation system was used to track the evolution of the full-surface three-dimensional strain field of Ti6Al4V additively manufactured lattice samples under mechanical loading. The high-magnification capabilities of the method allowed to resolve the strain distribution down to the strut level and disclosed a highly heterogeneous mechanical response of the lattice structure with local strain concentrations well above the nominal global strain level. In particular, we quantified that strain heterogeneity appears at a very early stage of the deformation process and increases with load, showing a strain accumulation pattern with a clear correlation to the later onset of the fracture.The obtained results suggest that the unique opportunities offered by the proposed experimental method, in conjunction with analytical and computational models, could serve to provide novel important information for the rational design of additively manufactured porous biomaterials.

High-resolution three-dimensional structural microscopy by single-angle Bragg ptychography.

Coherent X-ray microscopy by phase retrieval of Bragg diffraction intensities enables lattice distortions within a crystal to be imaged at nanometre-scale spatial resolutions in three dimensions. While this capability can be used to resolve structure-property relationships at the nanoscale under working conditions, strict data measurement requirements can limit the application of current approaches. Here, we introduce an efficient method of imaging three-dimensional (3D) nanoscale lattice behaviour and strain fields in crystalline materials with a methodology that we call 3D Bragg projection ptychography (3DBPP). This method enables 3D image reconstruction of a crystal volume from a series of two-dimensional X-ray Bragg coherent intensity diffraction patterns measured at a single incident beam angle. Structural information about the sample is encoded along two reciprocal-space directions normal to the Bragg diffracted exit beam, and along the third dimension in real space by the scanning beam. We present our approach with an analytical derivation, a numerical demonstration, and an experimental reconstruction of lattice distortions in a component of a nanoelectronic prototype device.

An efficient modelling approach based on a rigorous cross-sectional analysis for analysing box girder bridge superstructures

A novel analysis technique is introduced for efficient modelling of box girder bridge decks. The general three-dimensional equations used to accurately define the deformation of these complex beam-like slender structures are decoupled into a two-dimensional cross-sectional problem and a one-dimensional beam problem through decomposition of the three-dimensional strain field. The two-dimensional cross-sectional problem is solved by a two-dimensional finite element analysis considering in-plane as well as out-of-plane warping displacements of the beam section. This gives an accurate constitutive relationship of the one-dimensional beam problem without making any major assumptions as is often done in usual beam theories. The one-dimensional beam problem is solved by a one-dimensional beam finite analysis and the results obtained are used to recover three-dimensional stress, strain and displacement fields accurately. Numerical examples of box girder bridge deck systems having thin-walled sections are solved by the proposed approach to show its performance.

On Saint-Venant's Problem for Helicoidal Beams

This paper proposes a novel solution strategy for Saint-Venant's problem based on Hamilton's formalism. Saint-Venant's problem focuses on helicoidal beams and its solution hinges upon the determination of the subspace of the system's Hamiltonian matrix associated with its null and pure imaginary eigenvalues. A projection approach is proposed that reduces the system Hamiltonian matrix to a matrix of size 12 × 12, whose eigenvalues are identical to the null and purely imaginary eigenvalues of the original system, with the same Jordan structure. A fundamental theoretical result is established: Saint-Venant's solutions exist because rigid-body motions create no strains. Indeed, the solvability conditions for the governing equations of the problem are satisfied because a matrix identity holds, which expresses the fact that rigid-body motions create no strains. Because it avoids the identification of the Jordan structure of the original system, the implementation of the proposed projection for large, realistic problems is straightforward. Closed-form solutions of the reduced problem are found and three-dimensional stress and strain fields can be recovered from the closed-form solution. Numerical examples are presented to demonstrate the capabilities of the analysis. Predictions are compared to exact solutions of three-dimensional elasticity and three-dimensional FEM analysis.

Modeling of three-dimensional elastic strain fields by point-source method

The work objective is to investigate the possibility and efficiency of three-dimensional numerical models of the elastic stress fields in the deformed solids. The field point-source method (PSM) designated as the method of fundamental solutions (MFS) in the foreign literature is used to develop these models. The PSM system generation for modeling fields of different physical nature is described. The concept of a point source of the elastic displacement field in the deformed solid is introduced. The research result is a developed PSM equations system that can be applied for solving three-dimensional problems in the elasticity theory, including the classical first and second boundary value problems in the elasticity theory (when either the voltage or bias is set on the boundary), as well as a mixed boundary problem (when on one part of the boundary, the displacement is set, and on the other - the voltage). The PSM properties are studied when solving standard problems, and the Dirichlet problem for a circular domain. The dependences of the numerical solution error on the problem parameters, in particular, on such as the charge number simulating the desired field, the remoteness of the charges from the solution domain boundaries, are obtained. The test problem of calculating the deformation field in the ball domain is solved. Upon the results obtained, the following conclusion is made. In the numerical solution of three-dimensional elasticity problems, a decreasing exponential dependence of the PSM error on the square root of the simulating charge number is observed. This property provides a numerical solution with a very low relative error that implies the PSM application perspectiveness in the numerical solution of the elasticity problems, including three-dimensional problems.

On the solution of Almansi–Michell’s problem

This paper develops a Hamiltonian formalism for the solution of Almansi–Michell’s problem that generalizes the corresponding solution of Saint-Venant’s problem. Saint-Venant’s and Almansi–Michell’s problems can be represented as homogenous and non-homogenous Hamiltonian systems, respectively. The solution of Almansi–Michell’s problem is determined by the coefficients of the Hamiltonian matrix but also by the distribution pattern of the applied loading. The solution proceeds in two steps: first, for the homogenous problem, a projective transformation is constructed based on a symplectic matrix and second, the effects of the external loading are taken into account by augmenting this projection. With the help of this projection, the three-dimensional governing equations of Almansi–Michell’s problem are reduced to a set of one-dimensional beam-like equations, leading to a recursive solution process. Furthermore, the three-dimensional displacement, strain, and stress fields can be recovered from the one-dimensional solution. Numerical examples show that the predictions of the proposed approach are in excellent agreement with exact solutions of two-dimensional elasticity and three-dimensional FEM analysis.

In Vivo Measurement of Cervical Spinal Cord Deformation During Traumatic Spinal Cord Injury in a Rodent Model.

The spinal cord undergoes physical deformation during traumatic spinal cord injury (TSCI), which results in biological damage. This study demonstrates a novel approach, using magnetic resonance imaging and image registration techniques, to quantify the three-dimensional deformation of the cervical spinal cord in an in vivo rat model. Twenty-four male rats were subjected to one of two clinically relevant mechanisms of TSCI (i.e. contusion and dislocation) inside of a MR scanner using a novel apparatus, enabling imaging of the deformed spinal cords. The displacement fields demonstrated qualitative differences between injury mechanisms. Three-dimensional Lagrangian strain fields were calculated, and the results from the contusion injury mechanism were deemed most reliable. Strain field error was assessed using a Monte Carlo approach, which showed that simulated normal strain error experienced a bias, whereas shear strain error did not. In contusion injury, a large region of dorso-ventral compressive strain was observed under the impactor which extended into the ventral region of the spinal cord. High tensile lateral strains under the impactor and compressive lateral strains in the lateral white matter were also observed in contusion. The ability to directly observe and quantify in vivo spinal cord deformation informs our knowledge of the mechanics of TSCI.

An investigation into axial impacts of the cervical spine using digital image correlation

Background contextHigh-energy impacts are commonly encountered during sports such as rugby union. Although catastrophic injuries resulting from such impacts are rare, the consequences can be devastating for all those involved. A greater level of understanding of cervical spine injury mechanisms is required, with the ultimate aim of minimizing such injuries. PurposeThe present study aimed to provide a greater understanding of cervical spine injury mechanisms, by subjecting porcine spinal specimens to impact conditions based on those measured in vivo. The impacts were investigated using high-speed digital image correlation (DIC), a method not previously adopted for spinal impact research. Study designThis was an in vitro biomechanical study. MethodsEight porcine specimens were impacted using a custom-made rig. The cranial and caudal axial loads were measured at 1 MHz. Video data were captured with two cameras at 4 kHz, providing measurements of the three-dimensional deformation and surface strain field of the specimens using DIC. ResultsThe injuries induced on the specimens were similar to those observed clinically. The mean±standard deviation peak caudal load was 6.0±2.1 kN, which occurred 5.6±1.1 ms after impact. Damage observable with the video data occurred in six specimens, 5.4±1.1 ms after impact, and the peak surface strain at fracture initiation was 4.6±0.5%. ConclusionsThis study has provided an unprecedented insight into the injury mechanisms of the cervical spine during impact loading. The posture represents a key factor in injury initiation, with lordosis of the spine increasing the likelihood of injury.

237 グラフェンナノリボン電気伝導特性の三次元ひずみ場依存性

237 グラフェンナノリボン電気伝導特性の三次元ひずみ場依存性

Nonlinear three-dimensional beam theory for flexible multibody dynamics

In flexible multibody systems, it is convenient to approximate many structural components as beams or shells. Classical beam theories, such as Euler–Bernoulli beam theory, often form the basis of the analytical development for beam dynamics. The advantage of this approach is that it leads to a very simple kinematic representation of the problem: the beam’s section is assumed to remain plane and its displacement field is fully defined by three displacement and three rotation components. While such an approach is capable of capturing the kinetic energy of the system accurately, it cannot represent the strain energy adequately. This paper presents a different approach to the problem. Based on a finite element discretization of the cross-section, an exact solution of the theory of three-dimensional elasticity is developed. The proposed approach is based on the Hamiltonian formalism and leads to an expansion of the solution in terms of extremity and central solutions. Kinematically, the problem is decomposed into an arbitrarily large rigid-section motion and a warping field. The sectional strains associated with the rigid-section motion and the warping field are assumed to remain small. As a consequence of this kinematic decomposition, the governing equations of the problem fall into two distinct categories: the equations describing geometrically exact beams and those describing local deformations. The governing equations for geometrically exact beams are nonlinear, one-dimensional equations, whereas a linear, two-dimensional analysis provides the detailed distribution of three-dimensional stress and strain fields. Within the stated assumptions, the solutions presented here are the exact solution of three-dimensional elasticity for beams undergoing arbitrarily large motions.

Dynamic finite-strain modelling of the human left ventricle in health and disease using an immersed boundary-finite element method.

Detailed models of the biomechanics of the heart are important both for developing improved interventions for patients with heart disease and also for patient risk stratification and treatment planning. For instance, stress distributions in the heart affect cardiac remodelling, but such distributions are not presently accessible in patients. Biomechanical models of the heart offer detailed three-dimensional deformation, stress and strain fields that can supplement conventional clinical data. In this work, we introduce dynamic computational models of the human left ventricle (LV) that are derived from clinical imaging data obtained from a healthy subject and from a patient with a myocardial infarction (MI). Both models incorporate a detailed invariant-based orthotropic description of the passive elasticity of the ventricular myocardium along with a detailed biophysical model of active tension generation in the ventricular muscle. These constitutive models are employed within a dynamic simulation framework that accounts for the inertia of the ventricular muscle and the blood that is based on an immersed boundary (IB) method with a finite element description of the structural mechanics. The geometry of the models is based on data obtained non-invasively by cardiac magnetic resonance (CMR). CMR imaging data are also used to estimate the parameters of the passive and active constitutive models, which are determined so that the simulated end-diastolic and end-systolic volumes agree with the corresponding volumes determined from the CMR imaging studies. Using these models, we simulate LV dynamics from enddiastole to end-systole. The results of our simulations are shown to be in good agreement with subject-specific CMR-derived strain measurements and also with earlier clinical studies on human LV strain distributions.

Dynamical effects in strain measurements by dark-field electron holography.

Here, we study the effect of dynamic scattering on the projected geometric phase and strain maps reconstructed using dark-field electron holography (DFEH) for non-uniformly strained crystals. The investigated structure consists of a {SiGe/Si} superlattice grown on a (001)-Si substrate. The three-dimensional strain field within the thin TEM lamella is modelled by the finite element method. The observed projected strain is simulated in two ways by multiplying the strain at each depth in the crystal by a weighting function determined from a recently developed analytical two-beam dynamical theory, and by simply taking the average value. We demonstrate that the experimental results need to be understood in terms of the dynamical theory and good agreement is found between the experimental and simulated results. Discrepancies do remain for certain cases and are likely to be from an imprecision in the actual two-beam diffraction conditions, notably the deviation parameter, and points to limitations in the 2-beam approximation. Finally, a route towards a 3D reconstruction of strain fields is proposed.

High-performance hybrid CPU and GPU parallel algorithm for digital volume correlation

We present a hybrid Message Passing Interface (MPI) and graphics processing unit (GPU)-based parallel digital volume correlation (DVC) algorithm for measuring three-dimensional (3D) displacement and strain fields inside a material undergoing motion or deformation. Our algorithm achieves resolution comparable to that achieved in two-dimensional (2D) digital image correlation (DIC), in time that is commensurate with the image acquisition time, in this case, using microcomputed tomography ([Formula: see text] CT) for scanning images. For DVC, the volume of data and number of correlation points both grow cubically with the linear dimensions of the image. We turn to parallel computing to gain sufficient processing power to scale to high resolution, and are able to achieve more than an order-of-magnitude increase in resolution compared with previous efforts that are not based on a parallel framework.

Evaluation of Residual Stress Development at the Interface of Plasma Electrolytically Oxidized and Cold-Worked Aluminum

Fatigue failure in hard oxide-coated aluminum is usually driven by rapid short crack propagation from the interface through the substrate; mitigation of this is possible by introducing interfacial compressive stresses. Combining cold work with hard oxide coating can improve their performance under conditions of simultaneous wear, corrosion, and fatigue. Three-dimensional strain fields in an aluminum alloy with combined cold work and PEO coating have been measured and mechanisms for stress redistribution presented. These comprise material consumption, expansive growth of oxide layers, and local annealing.

Anomalous coherent diffraction of core-shell nano-objects: A methodology for determination of composition and strain fields

A procedure to retrieve the three-dimensional strain and composition fields of single-crystalline nanoparticles using a combination of anomalous and coherent x-ray diffraction was developed. The applicability of the method was demonstrated using numerically generated data for Ag/Au core-shell nanowires for which the epitaxy-induced strain field was simulated by the use of the finite element method for infinitely long nanowires with either abrupt (``as-prepared'') or diffuse (``as-annealed'') interfaces between core and shell. First, the influence of the elemental distribution and the associated inhomogeneous strains on the x-ray diffraction patterns were described. Next, the diffraction patterns as determined for different x-ray energies were inverted by applying phase retrieval algorithms. Only if diffraction patterns at different energies are available and inverted can the concentrations and displacements be obtained subject to specific assumptions.

108 グラフェンナノリボン電気伝導特性に及ぼす三次元ひずみ場の影響

108 グラフェンナノリボン電気伝導特性に及ぼす三次元ひずみ場の影響

In situ coherent X-ray diffraction of isolated core–shell nanowires

Anomalous coherent X-ray diffraction in Bragg geometry is a promising technique to study three-dimensional composition and strain fields in nanostructures. This paper presents results of coherent X-ray diffraction experiments on single Ag/Au core–shell nanowires with a typical diameter of 100 to 300nm in the as-prepared state and during in-situ heating. Experimental coherent diffraction patterns are compared with finite-element simulations. Moreover, experimental constraints owing to radiation damage are discussed.

Mapping Three-Dimensional Stress and Strain Fields within a Soft Hydrogel Using a Fluorescence Microscope

Three-dimensional cell culture is becoming mainstream as it is recognized that many animal cell types require the biophysical and biochemical cues within the extracellular matrices to perform truly physiologically realistic functions. However, tools for characterizing cellular mechanical environment are largely limited to cell culture plated on a two-dimensional substrate. We present a three-dimensional traction microscopy that is capable of mapping three-dimensional stress and strain within a soft and transparent extracellular matrix using a fluorescence microscope and a simple forward data analysis algorithm. We validated this technique by mapping the strain and stress field within the bulk of a thin polyacrylamide gel layer indented by a millimeter-size glass ball, together with a finite-element analysis. The experimentally measured stress and strain fields are in excellent agreements with results of the finite-element simulation. The unique contributions of the presented three-dimensional traction microscopy technique are: 1), the use of a fluorescence microscope in contrast with the confocal microscope that is required for the current three-dimensional traction microscopes in the literature; 2), the determination of the pressure field of an incompressible gel from strains; and 3), the simple forward-data-analysis algorithm. Future application of this technique for mapping animal cell traction in three-dimensional nonlinear biological gels is discussed.

Membrane Curvatures and Stress-strain Full Fields of Axisymmetric Bulge Tests from 3D-DIC Measurements. Theory and Validation on Virtual and Experimental results

The bulge test is mostly used to analyze equibiaxial tensile stress state at the pole of inflated isotropic membranes. Three-dimensional digital image correlation (3D-DIC) technique allows the determination of three-dimensional surface displacements and strain fields. In this paper, a method is proposed to determine also the membrane stress tensor fields for in-plane isotropic materials, independently of any constitutive equation. Stress-strain state is then known at any surface point which enriches greatly experimental data deduced from the axisymmetric bulge tests. Our method consists, first in calculating from the 3D-DIC experimental data the membrane curvature tensor at each surface point of the bulge specimen. Then, curvature tensor fields are used to investigate axisymmetry of the test. Finally in the axisymmetric case, membrane stress tensor fields are determined from meridional and circumferential curvatures combined with the measurement of the inflating pressure. Our method is first validated for virtual 3D-DIC data, obtained by numerical simulation of a bulge test using a hyperelastic material model. Afterward, the method is applied to an experimental bulge test performed using as material a silicone elastomer. The stress-strain fields which are obtained using the proposed method are compared with results of the finite element simulation of this overall bulge test using a neo-Hookean model fitted on uniaxial and equibiaxial tensile tests.