Three-dimensional Strain Fields Research Articles

3D observations provide striking findings in rubber elasticity

The mechanical response of rubbers has been ubiquitously assumed to be only a function of the imposed strain. Using innovative X-ray measurements capturing the three-dimensional spatial volumetric strain fields, we demonstrate that rubbers and indeed many common engineering polymers undergo significant local volume changes. But remarkably, the overall specimen volume remains constant regardless of the imposed loading. This strange behavior which also leads to apparent negative local bulk moduli is due to the presence of a mobile phase within these materials. Combining X-ray tomographic observations with high-speed radiography to track the motion of the mobile phase, we have revised classical thermodynamic frameworks of rubber elasticity. The work opens broad avenues to understand not only the mechanical behavior of rubbers but a large class of widely used engineering polymers.

Energy evolution and strain localization in fractured sandstone under freeze-thaw cycling and uniaxial loading-unloading

In cold regions, combined cyclic freeze-thaw (F-T) and loading-unloading can cause rock slope instability and sliding. The macro-failure of rock masses originates from the expansion of strain localization areas driven by energy. Therefore, a deep understanding of the energy evolution and strain distribution can reveal the instability process. Along these lines, in this work, intact and intermittently jointed sandstones were subjected to a cyclic F-T environment and subsequently carried out uniaxial graded loading-unloading to obtain their mechanical parameters and X-ray computed tomography images. By investigating the energy density, it was found that the input, elastic, dissipated, and damage energies all increased with loading according to a quadratic polynomial function, with the energies obeying a linear allocation law. Meanwhile, F-T changed the energy storage capacity and energy dissipation coefficient of the rock by causing meso-damage. To measure the meso-scale strain localization characteristics, the advanced digital volume correlation technique was also used to obtain three-dimensional strain fields. In terms of local strain, the failure behavior of intact specimens under the application of low and high numbers of F-T cycles was controlled by potential fracture bands and frost-heave cracks, respectively, while fractured specimens were mainly dependent on the macro-fracture arrangement. However, both frost heaving and thawing transformed sandstone from brittle to ductile. In terms of statistical strain, the equivalent strain (εeq) showed a log-normal distribution and continued to deviate to the right due to loading-unloading. Meanwhile, the normal strain (εxx, εyy, εzz) exhibited a normal distribution, where fractured sandstone mainly developed along with the slip direction. Finally, the intrinsic correlations among the energy parameters, meso-damage, and macro-failure were discussed. From our analysis, three stages of energy allocation and transformation were revealed, namely the uniform allocation of damage energy, banded accumulation of damage energy, and fracture caused by the elastic energy release.

Reconstruction of three-dimensional strain field in an asymmetrical curved core–shell hetero-nanowire

Crystal orientation and strain mapping of an individual curved and asymmetrical core–shell hetero-nanowire (NW) is performed based on transmission electron microscopy. It relies on a comprehensive analysis of scanning nanobeam electron diffraction data obtained for 1.3 nm electron probe size. The proposed approach also handles the problem of appearing twinning defects on diffraction patterns and allows for the investigation of materials with high defect densities. Based on the experimental maps and their comparison with finite element simulations, the entire core–shell geometry including full three-dimensional strain distribution within the curved core–shell NW are obtained. Our approach represents, therefore, a low-dose quasi-tomography of the strain field within a nanoobject using only a single zone axis diffraction experiment. Our approach is applicable also for electron beam-sensitive materials for which performing conventional tomography is a difficult task.

Weak form quadrature element analysis of spatial geometrically exact composite beams with torsional warping

Spatial thin-walled laminated composite beams under large deformations with torsional warping effects are studied. A novel two-dimensional cross-section model for composite beams with shear and torsion-warping deformations is proposed wherein the three-dimensional warping and coupling effects caused by inhomogeneous anisotropic materials are considered. A weak form quadrature element formulation is established for nonlinear analysis of geometrically exact composite beams with torsional warping and shear deformations. By combining the two-dimensional cross-section model and the nonlinear one-dimensional beam model, the three-dimensional stress and strain fields can be obtained. Numerical examples are provided to investigate the torsional warping effects and verify the effectiveness of the present formulation. Results of the present beam formulation which are achieved at a significantly lower computational cost are in excellent agreement with those of shell and solid models of commercial codes.

3D deformation and strain fields in drying kaolinite obtained from tracking internal bubbles using X-ray CT and ANN

AbstractDrying fine-grained sediments experience shrinkage and desiccation cracking that may dramatically alter their mechanical and hydraulic properties. This study adopts X-ray computed tomography (CT) to monitor the three-dimensional (3D) internal deformation and strain fields, and their relationships with desiccation crack formation, for drying kaolinite samples contained in plastic containers. Two kaolinite samples, one dried at room temperature and the other oven-dried at 60 °C, were CT scanned at several intervals during the drying process. From sequential CT scans for the same sample, entrained gas bubbles were extracted and used as tracking markers for deformation and strain field measurements. Since the bubble morphology continuously changed during the drying process, an artificial neural network (ANN) model was developed to link bubbles in sequential scans for the same sample. The tracking algorithm was trained with manually linked bubbles and optimised by comparing different combinations of bubble information, e.g. bubble location, size and shape. The drying samples experienced primarily vertical displacement before the air-entry value, while horizontal displacement occurred during vertical crack formation. Internal vertical and horizontal strains were generally uniform, indicating a limited impact of non-uniform sample drying and substrate constraint.

Experimental study on the spatial fracture behaviour of red sandstone containing weak inclusions with non-unitary geometric features under uniaxial compression

The failure of weakly filled rock masses is an important factor in the safety and stability of rock engineering. To investigate the spatial fracture behaviour of red sandstone with non-unitary geometric weakly filled joints, uniaxial compression experiments were carried out. The 3D DIC and acoustic emission (AE) technology were used to obtain the three-dimensional spatial deformation information and AE parameters of specimens containing both planar and circular weak inclusions, respectively. The main results are as follows: (1) The results show that the ductility of planar weak inclusions has a certain deterioration effect on the bearing capacity of the specimens, and the extent of this deterioration effect is not constant. (2) When the ductile length of the planar weak inclusions increases, the three-dimensional spatial displacement and strain fields of the specimens are divided by the planar weak inclusions, showing different degrees of deformation behaviour and different distribution patterns in the upper and lower parts. (3) The three-dimensional spatial fracture characteristics of specimens with different ductile lengths of planar weak inclusions will be dominated by the ductility, and the main failure mode of the specimen will gradually evolve from shear failure to splitting failure. The “guiding effect” of circular hole weak inclusions on the main crack is no longer evident when the ductility of planar weak inclusions is at its maximum. (4) The growth trend of the cumulative AE counts can be divided into four stages, and the stage characteristics are significantly influenced by the ductile length of the planar weak inclusions; different failure modes show different AE signal distribution characteristics. In addition, the key points of stability control of rock masses containing weak inclusions in practical engineering are discussed.

Investigation of carbon-aramid hybrid braided composites using digital volume correlation

Hybrid braided composites combine two or more reinforcing fibres in a matrix. Limited research on hybrid braided composites has been conducted. The Digital Image Correlation (DIC) method has been used to understand hybrid braided composites' external deformation and strain. However, DIC measurements do not provide sufficient information on the internal behaviour of hybrid braids. Therefore, this study focuses on understanding the behaviour of hybrid braids Digital Volume Correlation (DVC). DVC is a micro-computed tomography (Micro-CT) based volumetric strain measurement technique that allows for the internal strain fields of an object. Tensile loading of the Carbon-Aramid hybrid braided samples is achieved using an integrated material testing stage. DVC is used to generate three-dimensional displacement and strain fields and understand the internal strain behaviour of hybrid braided composites. The peak and strain observed for the sample was around −0.065 (µm/µm) and 0.017 (µm/µm), respectively. The effect of contrast-enhancing tracking particles on the DVC resulting DVC strain fields were also examined. Samples with and without contrast particles were subjected to in-situ loading and then analyzed using the DVC process. From this comparison, it was found that the addition of contrast-enhancing particles improves the volumetric strain results from the DVC measurements.

Microstructure evolution and precipitation behavior of Al-Mg-Si alloy during initial aging

The microstructure evolution and precipitation behavior of Al-Mg-Si alloy during initial aging were studied using hardness testing, conductivity testing, differential scanning calorimetry (DSC), and high resolution transmission electron microscopy (HRTEM). The results show that the precipitation sequence of the Al-Mg-Si alloy during initial aging can be represented as: supersaturated solid solution → spherical Mg/Si clusters → needle-like Guinier Preston (GP) zone → β″. Clusters are completely coherent with the Al matrix. The GP zone with relatively complete independent lattice parameters that differ slightly from the Al matrix parameters, is oriented along the direction of <111>Al and lying on {111}Al plane. The strength of the Al-Mg-Si alloy is greatly enhanced by the three-dimensional strain field that exists between the β″ phase and the two {200}Al planes. After aging at 170 °C for 6 h, the hardness reaches the peak of 127 HV and remains for a long time. At this stage, the electrical conductivity keeps relatively stable due to the formation of coherent precipitates (Mg/Si clusters/GP zones) and the reduction in solute atom concentration in the Al matrix. The severe coarsening and decreased number density of the β″ phase during the over-aging stage result in a significant decrease in the hardness.

In situ micromechanical analysis of discontinuous fiber-reinforced composite material based on DVC strain and fiber orientation fields

The influence of fiber orientation and its distribution on the global and local mechanical behavior of a short fiber-reinforced thermoplastic (SFT) composite is studied based on an in situ tensile test using synchrotron X-ray tomography. The in situ test data are utilized to compute three-dimensional (3D) strain fields inside the material at various loading steps through an in-house digital volume correlation (DVC) code. The DVC analysis finds that, although the global strain fields seem relatively uniform, the internal strain fields are locally non-uniform because of the complex reinforcement architecture. The local non-uniformity is further investigated at a single fiber level to find a correlation between the orientation of fibers, fiber volume fraction, and strain field. The investigation is performed on micro-scale subvolumes, in which all the fibers are individually segmented from the 3D tomographic data. It is found that the fibers aligned in the direction transverse to loading impede transversal shrinkage, resulting in a locally small compressive strain in that direction. The in situ test has revealed that the average fiber orientation does not change noticeably even after the global strain reaches 18%. This small change implies that initially-determined fiber orientation is crucial to the locally anisotropic mechanical behavior of a SFT-like material even undergoing large deformation. On the other hand, the loading-directional strain is more affected by the local volume fraction than the fiber orientation. The local volume fraction also mainly determines the stiffness of the subvolume that can be estimated from an analytical homogenization model. The effect of the local volume fraction on the subvolume stiffness is identified from multivariate analysis.

Experimental study on failure characteristics of rock-fiber concrete composite under compression load

To study the failure characteristics of the rock-fiber concrete composite structure under compressive load, the experimental research was conducted with the aid of the 3D-DIC full-field strain test experimental system. The three-dimensional space deformation field information of the composite layer was obtained. The deformation characteristics, damage characteristics and failure modes of the composite layer were analyzed. (1) The local strain field and the overall strain field of the composite layer will increase to different degrees when subjected to compressive load. When the failure is approaching, the local strain field in the crack initiation area will increase rapidly. (2) The changes in the three-dimensional displacement field and strain field of the composite layer will retain the characteristics of brittle materials. The horizontal strain field and the vertical strain field show different distribution patterns, respectively. (3) The damage change of the specimen undergoes four stages: micro damage, damage accumulation, damage manifestation and complete damage. (4) The test results show that the binding site of the composite can play a certain effect of crack arrest.

A computational framework to obtain nonlinearly elastic constitutive relations of special Cosserat rods with surface energy

We present a computational framework to obtain nonlinearly elastic constitutive relations of special Cosserat rods with surface energy. The framework allows the rod’s cross-section to have arbitrary shape and the rod material to obey arbitrary three-dimensional material model for its bulk volume and arbitrary surface energy model for its lateral surface. The kinematics of the Helical Cauchy–Born rule is used to first construct a family of six-parameter (corresponding to the six strain measures of rod theory) helical rod configurations in which the generated three-dimensional strain field is uniform in the rod’s arc-length coordinate. This uniformity allows the three-dimensional nonlinear equations of elasticity to reduce to just the rod’s cross-sectional plane wherein the cross-section’s boundary also experiences traction due to the inherent surface energy in the rod’s lateral surface. The solution of this cross-sectional problem yields the induced in-plane displacement and out-of-plane warping in the cross-section for arbitrarily prescribed strain measures of the rod. We then derive expressions for the induced internal contact force, moment and stiffnesses of the rod in terms of the solution of the deformed cross-section and the prescribed strains of the rod. A finite element formulation is presented to solve the nonlinear cross-sectional deformation problem and further obtain the induced internal contact force, moment and stiffnesses. The presented framework will be useful for modeling of nanorods where surface energy plays a dominant role. Several numerical examples are presented using the presented framework to illustrate the effect of surface energy parameters on the deformation of the nanorod’s cross-section and the nanorod’s bending, torsional, extensional and shearing stiffnesses. We also derive improved analytical formulas for the extensional and bending stiffnesses of isotropic rectangular nanorods in their reference state. These formulas reduce to the existing widely used formulas for a special choice of material parameters, i.e., when the surface Poisson’s ratio and the bulk Poisson’s ratio match thus highlighting the limitation of the existing formulas.

Accurate internal deformation measurement of an indentation test using micro-CT and self-adaptive digital volume correlation.

Mechanical indentation testing is a widely used technique for determining local mechanical properties of materials. Accurate measurement of internal deformation in the indentation test is necessary for further study of material properties. Therefore, an in situ experimental measurement strategy combining micro-CT imaging and self-adaptive digital volume correlation (SA-DVC) is proposed. Unlike conventional DVC, SA-DVC can automatically identify the optimal subvolume size for each calculation point, which can effectively minimize measurement errors. The efficacy of the proposed method is first verified by the simulated indentation experiment. Then, it is used to analyze the deformation of epoxy resin composite in a real indentation experiment. Measurement results indicate that the proposed method can estimate three-dimensional displacement and strain fields with enhanced accuracy, and further application of the obtained measurement results on material parameter identification and stress field reconstruction is expected.

Method for the biomechanical analysis of aqueous veins and perilimbal sclera by three-dimensional photoacoustic imaging and strain field calculation

A method motivated by the eye’s aqueous veins is described for the imaging and strain calculation within soft biological tissues. A challenge to the investigation of the biomechanics of the aqueous vein—perilimbal sclera tissue complex is resolution of tissue deformations as a function of intraocular pressure and the subsequent calculation of strain (a normalized measure of deformation). The method involves perfusion of the eye with a contrast agent during conduction of non-invasive, optical resolution photoacoustic microscopy. This imaging technique permits three-dimensional displacement measurements of tracked points on the inner walls of the veins which are used in a finite element model to determine the corresponding strains. The methods are validated against two standard strain measurement methods. Representative porcine globe perfusion experiments are presented that demonstrate the power of the method to determine complex strain fields in the veins dependent on intraocular pressure as well as vein anatomy. In these cases, veins are observed to move radially outward during increases in intraocular pressure and to possess significant spatial strain variation, possibly influenced by their branching patterns. To the authors’ knowledge, these are the only such quantitative, data driven, calculations of the aqueous vein strains available in the open literature.

High resolution three-dimensional strain measurements in human articular cartilage

An unresolved challenge in osteoarthritis research is characterising the localised intra-tissue mechanical response of articular cartilage. The aim of this study was to explore whether laboratory micro-computed tomography (micro-CT) and digital volume correlation (DVC) permit non-destructive quantification of three-dimensional (3D) strain fields in human articular cartilage. Human articular cartilage specimens were harvested from the knee, mounted into a loading device and imaged in the unloaded and loaded states using a micro-CT scanner. Strain was measured throughout the cartilage volume using the micro-CT image data and DVC analysis. The volumetric DVC-measured strain was within 5% of the known applied strain. Variation in strain distribution between the superficial, middle and deep zones was observed, consistent with the different architecture of the material in these locations. These results indicate DVC method may be suitable for calculating strain in human articular cartilage.

Experimental demonstration of coupled multi-peak Bragg coherent diffraction imaging with genetic algorithms

Bragg coherent diffraction imaging has the potential to provide significant insight into the structure-properties relationship for crystalline materials by imaging, with nanoscale resolution, three-dimensional strain fields within individual grains and nanoparticles. The capability of present-day synchrotrons to locate and measure a multiplicity of Bragg reflections from a single grain makes it possible to recover the full strain tensor with nanometer resolution. Recent methods for coupling reconstructions from several peaks to determine the strain tensor have been developed and applied to synthetic data, but have not been applied to experimental data. Here, using a coupled genetic reconstruction algorithm, we reconstruct an experimental data set and demonstrate improvements in the ability to resolve vector-valued displacement fields internal to the particle as compared to what is achieved with a noncoupled approach. The coupled approach developed in this work was also validated on simulated data sets. In both simulated and experimental data, reconstructions from our coupled Bragg peak algorithm show improvements over the noncoupled independent reconstruction method of $5%$ in terms of accuracy and $53%$ in terms of consistency.

A thermo-mechanical terrestrial model of Arctic coastal erosion

Although the Arctic comprises one-third of the global coastline and has some of the fastest eroding coasts, current tools for quantifying permafrost erosion are unable to explain the episodic, storm-driven erosion events that occur in this region. In this paper, we present a novel multi-physics finite element model for the numerical simulation of Arctic coastal permafrost degradation: the terrestrial component of the Arctic Coastal Erosion (ACE) model. This model is comprised of two main ingredients: (1) a solid mechanics model that calculates the three-dimensional (3D) stress, strain and displacement fields of the underlying permafrost developing in response to a frozen water content dependent plasticity model, and (2) a novel thermal model governing the 3D heat conduction and solid–liquid phase change occurring within the permafrost. These two physics sets are coupled via a sequential thermo-mechanical coupling scheme developed within the AlbanyLCM open-source finite element code. Unlike prior approaches, our modeling methodology enables failure from any allowable deformation (block failure, thermo-denudation, thermo-abrasion); moreover, failure modes develop from constitutive (rather than empirical) relationships inherent in the underlying finite element model. Elements are dynamically removed from the underlying finite element mesh so as to simulate transient permafrost erosion events. Our thermo-mechanical terrestrial model is evaluated on a pseudo-realistic problem in which a slice of permafrost is exposed to realistic oceanic and atmospheric forcing boundary condition data occurring at Drew Point, Alaska in July 2018.

Component-wise approach to reinforced concrete structures

With the rapid development of engineering constructions, especially transportation facilities, the structural models for the simulation of large-scale structures shall be eventually enhanced for predicting the complete three-dimensional stress and strain fields in reinforced concrete-made components. This paper proposes a component-wise approach for the modeling of reinforced concrete structures in which steel rebars and the concrete part are considered as two independent one-dimensional entities. Lagrange polynomials are used to express the cross-section deformations and different component-wise subdomains are joined by simply imposing displacement continuity at the chosen Lagrange points along the component boundary. The Finite Element (FE) method is applied to provide numerical solutions whereas Carrera Unified Formulation (CUF) is used to generate the related stiffness matrices in a compact and straightforward way. The classical case of homogenized beam solutions, as well as the one in which a virtual layer is associated with the steel zones, are implemented too. The three solutions are compared for a number of reinforced concrete beam problems, from single to double reinforced beam, including a T-shape cross-section. A final study considering transverse stiffeners (steel stirrups) is investigated. These stiffeners are modeled component-wise as well. Results clearly show the advantages and superiority of the component-wise FE-CUF based model to completely capture the three-dimensional strain and stress states, including shear ones, of reinforced concrete structures.

Influences of structural anisotropy and heterogeneity on three-dimensional strain fields and cracking patterns of a clay-rich rock

In this study, three-dimensional non-uniform strain fields and cracking patterns of a clay-rich rock are investigated. Uniaxial compression creep tests are performed on samples drilled in five different orientations with respect to bedding planes. A digital volume correlation method is applied to X-ray micro-tomographic images taken during the creep tests to calculate both full local strain fields and average global strains. Both instantaneous and time-dependent strains, respectively, induced by applied stress and creep mechanism are analysed. Strain heterogeneity inside samples is quantified by calculating strain concentration zones, which are correlated with the presence of stiff inclusions. Cracking patterns of samples are characterized through reconstructed X-ray micro-tomographic images. Correlations between major cracking patterns and strain concentration zones are investigated.

Dynamic analyses of viscoelastic three-dimensional structures with advanced one-dimensional finite elements

This paper presents mechanical responses of viscoelastic structures with complicated geometries. The structures are discretized with finite beam elements, whose kinematic theories are derived according to the Carrera Unified Formulation. The governing equations are solved in the Laplace domain to avoid the computation of the convolution integral, which mathematically simulates the viscoelastic constitutive law. The three-dimensional displacement, strain, and stress fields are transformed in the time domain through a numerical inversion algorithm. The numerical simulations are performed on a beam with a hole at the center of its cross-section and on a thick-walled cylindrical shell subjected to various loading and boundary conditions. The comparisons between the present results, analytical predictions, and three-dimensional finite element solutions demonstrate the accuracy of the formulation.

Assessment of the strain depth sensitivity of Moiré sampling Scanning Transmission Electron Microscopy Geometrical Phase Analysis through a comparison with Dark-Field Electron Holography

In this study, the Moiré sampling Scanning Transmission Electron Microscopy Geometrical Phase Analysis (or STEM Moiré GPA) strain characterization method is compared to the well-established Dark-Field Electron Holography technique on a thin film stack grown by Molecular Beam Epitaxy. While experimental data obtained with the two techniques are, overall, in good qualitative agreement, small statistically relevant differences are locally observed between the two methods. The results obtained from both techniques are further confronted with Finite Element Method (FEM) mechanical simulations modeling the strain relaxation phenomena from a thin lamella. The FEM simulation highlights a non-uniform deformation field along the beam propagation direction with a higher deformation level near the surface of the lamella compared to the center of the same lamella. The center-surface strain differences obtained from modeling are consistent with the experimentally derived differences accounting for the fact that the SMG method is sensitive to the strain state of the surface of the lamella with a very narrow depth-of-field, and the DFEH technique is measuring the strain state of the center of the same lamella averaging over a large section of the thickness. The depth-of-field difference between both methods can be reasonably related to their respective contrast mechanisms (STEM vs Conventional Transmission Electron Microscopy). As the SMG method is using a convergent probe, the narrow depth-of-field might be used to sense the deformation field over different sections of the lamella using the defocus and potentially retrieve the three-dimensional strain field.