Linear Elastic Deformation Research Articles

A note on the linear deformations close to the boundaries of a cellular material

Previously reported measurements of uniaxial compression of cellular acoustic materials have shown an intriguing loss of stiffness in the regions close to the boundaries. The present contribution attempts to further investigate if these effects may be modelled by assuming a local alteration of the microstructure in these regions close to the boundaries resulting from cutting a block of material into smaller samples. Assuming that fewer struts contribute locally to the mechanical behaviour, implies a reduced equivalent porosity, or relative density, in the boundary region. The approach explored here consists in randomly removing a portion of the struts constituting the microstructure of the cells because of damage incurred in the cutting process. The analysis is performed using a linear elastic deformation behaviour at the boundary of cellular acoustic materials.For the modelling of the microstructure, the Kelvin cell geometry is chosen. Applying the assumption that struts are damaged in the process of cutting a block of material into smaller samples, as an explanation for the observed high-strain boundary regions, it is shown that the experimental observations may be qualitatively reproduced. It is also shown that the equivalent density in the boundary regions may be estimated from the ratio of the boundary compressive modulus to that of the interior region of the sample. Through the presented results, the approximation errors in compression stiffness measurements performed for a real material may be estimated. Although providing a viable explanation of the experimentally observed behaviour of the boundary regions, it should be understood that the proposed approach primarily offers a simple, accurate equivalent description of this phenomenon.

Experimental Study on the Relationship between Compression Stability and Deformation Localization of Viscous/Brittle Rock‐Like Materials

Rock‐like materials often exhibit irregular failure deformation under long‐term service conditions, and the deformation and failure of asphalt and concrete materials is a serious problem that leads to subgrade failure. In this study, two different viscous/brittle rock‐like materials were prepared by the in situ loading and optical speckle synchronous monitoring test method, and the evolution characteristics of the deformation field were studied during compression. The formation process of the compression deformation localization of rock‐like materials and their relationship with stability were analyzed. A quantitative description of the compression deformation stage and localization characteristics of the viscous/brittle rock‐like materials is presented. The results can be summarized as follows. At the initial stage of compression, the deformation localization zone of viscous/brittle rock‐like materials begins to expand from the middle area to the surrounding area. Preliminary results of the deformation localization of the linear elastic deformation stage were obtained. The failure cloud image is completely formed at the peak, which is consistent with the failure physical map. The deformation process of compression can be quantitatively described using the deformation localization characteristics of rock‐like materials.

Tensile Behavior of Tetragonal Zirconia Micro/Nano-Fibers and Beams In-Situ Tested in Push-to-Pull Devices

The tensile mechanical behaviour of tetragonal zirconia micro/nano-fibers and beams were studied with a push-to-pull (PTP) device equipped in an in-situ nanoindenter. Polycrystalline and oligocrystalline micro/nano-fibers exhibited some degree of plastic strain before fracture with the tensile strength ranging from ~0.9 GPa to 1.4 GPa. Single-crystal beams generally experienced linear elastic deformation with tensile strength of ~2.1-3.4 GPa. No martensitic transformation induced shape memory strain was detected in the zirconia fibers and beams. Further variation of dopant concentration and crystal orientation was explored for single-crystalline beams and their significance in controlling the tensile strength was assessed and discussed.

Щодо розрахунку деформацій зсуву в призматичних стержнях із полімерних матеріалів за умов розтягу з крученням

The process of creep of prismatic rods made of linear-viscoelastic polymeric materials under combined loading is considered. Defining equations that determine the relationship between strains, stresses and time are given in the form of a superposition of shear and bulk strain. The object of study is prismatic bars made of fiberglass ST-1. The area of linearity of the model is substantiated on the basis of the hypothesis of the existence of the creep function, which is built on the yield curves, a single diagram of long-term deformation and the statistical value of the quantile of statistics. The region of linear-elastic deformation is recognized based on the fulfillment of the condition of existence of a single creep function. The defining equations of the model contain a set of functions and coefficients determined from the basic experiments. On the basis of the relations between the kernels of the one-dimensional stress state, the parameters of the kernels under the condition of a complex stress state are determined. The linearity of viscoelastic properties is given by the Boltzmann-Voltaire equations. The fractional-exponential kernels of heredity are chosen as the kernels of heredity. The obtained values of the core parameters are used to calculate the creep deformations of prismatic bars made of ST-1 fiberglass under conditions of simultaneous tensile tension.

Considerations on the Thermoelastic Effect in proximity of crack tips on Titanium and Aluminium: a new formulation

Thermoelastic Stress Analysis (TSA) is an experimental technique used for describing the stress state and the mechanical behaviour of materials subjected to linear-elastic deformations. The classic TSA theory consists of a simple relation between temperature and stress variations and was successfully applied in fracture mechanics for SIF and crack tip evaluation. This theory for some materials, such as titanium and aluminium, is no longer valid since the mechanical properties of the material cannot be neglected. The research framework lies in the field of the validity hypothesis of thermoelastic stress analysis, in this regard, the aim of this work is to present a new thermoelastic formulation that includes the second-order effects to investigate the thermoelastic effect in the proximity of crack tips, on Titanium and Aluminium. Moreover, error analysis has been carried out for investigating the differences between the proposed approach and the classical one.

Experimental investigation on static and dynamic characteristics of granulated rubber-sand mixtures as a new railway subgrade filler

Because of their low unit weight, strong linear elastic deformation ability, good durability and high energy dissipation, mixtures of waste tire rubber and sand have been increasingly used as a new type of subgrade filler in railway engineering. To investigate static and dynamic characteristics of granulated rubber-sand mixtures as a new filler, consolidated undrained monotonic and cyclic triaxial tests were carried out. The monotonic triaxial tests reveal that the shear strength of mixtures first increases and then decreases with increasing granulated rubber content and is maximized when the rubber content is approximately 10%. Based on the theory of an equivalent intergranular void ratio, a new equation for estimating the peak deviatoric stress of mixtures is proposed. The cyclic triaxial tests reveal that the addition of granulated rubber can constrain the generation and accumulation of dynamic pore water pressure and improve the liquefaction resistance of mixtures. The change in shear modulus is inversely proportional to the change in granulated rubber content and frequency, and directly proportional to the change in confining pressure. A new equation is also established for calculating the maximum shear modulus considering the effect of the equivalent intergranular void ratio and confining pressure. By comparing the maximum shear modulus of the new railway subgrade filler with that of other subgrade filler, it is concluded that a granulated rubber content of less than 20% can meet engineering requirements related to shear stiffness. The damping ratio is correlated with granulated rubber content, negatively correlated with confining pressure and insensitive to loading frequency. The relationship between the damping ratio and shear strain conforms to the equation proposed by Hardin and Drnevich. Based on the analysis of static and dynamic characteristics of granulated rubber-sand mixtures with different rubber content, the optimum granulated rubber content should be approximately 10%.

InSitu Scanning Transmission Electron Microscopy Observations of Fracture at the Atomic Scale.

The formation, propagation, and structure of nanoscale cracks determine the failure mechanics of engineered materials. Herein, we have captured, with atomic resolution and in real time, unit cell-by-unit cell lattice-trapped cracking in two-dimensional (2D) rhenium disulfide (ReS_{2}) using insitu aberration corrected scanning transmission electron microscopy (STEM). Our real time observations of atomic configurations and corresponding strain fields in propagating cracks directly reveal the atomistic fracture mechanisms. The entirely brittle fracture with non-blunted crack tips as well as perfect healing of cracks have been observed. The mode I fracture toughness of 2D ReS_{2} is measured. Our experiments have bridged the linear elastic deformation zone and the ultimate nm-sized nonlinear deformation zone inside the crack tip. The dynamics of fracture has been explained by the atomic lattice trapping model. The direct visualization on the strain field in the ongoing crack tips and the gained insights of discrete bond breaking or healing in cracks will facilitate deeper insights into how atoms are able to withstand exceptionally large strains at the crack tips.

Mechanical behavior of tetragonal zirconia nanopillars subjected to uniaxial loading: A molecular dynamics study

A novel stress-strain relation with two stages of linear elastic deformation is observed in [001]-oriented tetragonal zirconia nanopillars subjected to tensile loading via molecular dynamics simulation. This phenomenon results from a phase transformation from the tetragonal structure to the monoclinic one. Detailed explorations including the crystallographic structural analysis and the atomic strain calculation have been made to further elucidate the stress-strain curve. The lattice orientation strongly affects the plastic deformation mechanism, i.e., the [001]- and [011]-oriented nanopillars experience the phase transformation under tensile loading, while the brittle fracture is induced for the orientation lying along the [111] direction. Complementary uniaxial compressive tests are performed to study the loading direction dependence. The deformation mechanism differs for the nanopillars with the same lattice orientation but different loading directions; for instance, under compressive loading, the plastic deformation behavior for [001]-oriented nanopillar is governed by intense dislocation activity rather than phase transformation. Additionally, a significant temperature effect is observed, with Young's modulus decreasing linearly from 323.97 to 283.55 GPa as the temperature increases from 300 to 1400 K. This work will help deepen the understanding of the tetragonal-to-monoclinic transformation and nanoscale mechanical behavior of zirconia.

Lifting of Dust Particles under the Action of Laser Radiation on a Chondritic Target and the Possibility of Modeling Plasma–Dust Processes on the Lunar Surface

The first investigation devoted to experimental modeling of the lift of dust particles by shock wave over a target surface has been performed on the Saturn setup for studying the interaction of laser radiation with a porous chondritic target surface bearing finely dispersed talc particles. Results of experimental modeling can be used to describe the lift of dust particles from the zones of nonlinear and linear elastic deformation of a regolith substance characterizing a meteoroid impact on the Moon’s surface.

Mathematical Modeling of the Dynamics and Destruction of Magmatic Bombs Forming During the Eruption of Submarine Volcanos

Theoretical investigations into the dynamics and destruction of magmatic bombs forming during the eruption of submarine volcanos have been conducted. Continuum mechanics methods have been used for mathematical modeling of the dynamics and destruction of objects under investigation. In this case, it has been assumed that the mechanical behavior of the magmatic bomb material obeys the law of linear elastic deformation of a solid body.

Evaluation of internal microcrack evolution in red sandstone based on time–frequency domain characteristics of acoustic emission signals

Rocks are commonly used as building stone and construction materials in many engineering applications. To study the internal microcrack evolution characteristics of red sandstone specimens during deformation and failure, a uniaxial compression experiment was conducted based on RMT-150C rock mechanics, PCI-II acoustic emission (AE), and the strain acquisition testing systems. For eight red sandstone specimens, the relationships among axial stress, axial strain, acoustic emission events rate, and average frequency centroid were established. The variability of the AE events rate and average frequency centroid of the specimens was analyzed for five cracking levels. Additionally, the relationship between the loading stress corresponding to the minimum average frequency centroid of the AE signals and the peak stress during deformation and failure of the red sandstone specimens was discussed, and critical failure precursor characteristics revealed. Moreover, statistical analysis of the AE signal distribution characteristics of different frequency bands and amplitudes during rock deformation and failure was performed. Based on the difference between the AE signal duration ratio and the specimen rise time, a method for extracting AE characteristic signals during deformation and failure was proposed. Time-frequency analysis was performed on the extracted AE characteristic signals and results showed that the signals were closely related to the axial stress–strain change in the specimens, and were affected by the degree and development of internal microcracks. The wave peaks of the crack closure and linear elastic deformation stage AE characteristic signal frequency domain waveforms are relatively scattered. The wave peaks of the frequency-domain waveforms of the AE characteristic signals during stable and unstable crack growth stages are relatively concentrated.

A strain gradient linear viscoelasticity theory

In this paper, a strain gradient viscoelastic theory is proposed strictly, which can be used to describe the cross-scale mechanical behavior of the quasi-brittle advanced materials. We also expect the theory to be applied to the description for the cross-scale mechanical behavior of advanced alloy metals in linear elastic deformation cases. In the micro-/nano-scale, the mechanical properties of advanced materials often show the competitive characteristics of strengthening and softening, such as: the strength and hardness of the thermal barrier coatings with nanoparticles and the nanostructured biological materials (shells), as well as the strength of nanocrystalline alloy metals which show the characteristics of positive-inverse Hall-Petch relationship, etc. In order to characterize these properties, a strain gradient viscoelastic theory is established by strictly deriving the correspondence principle. Through theoretical derivation, the equilibrium equations and complete boundary conditions based on stress and displacement are determined, and the correspondence principle of strain gradient viscoelasticity theory in Laplace phase space is obtained. With the help of the high-order viscoelastic model, the specific form of viscoelastic parameters is presented, and the time curve of material characteristic scale parameters in viscoelastic deformation is obtained. When viscoelasticity is neglected, the strain gradient viscoelasticity theory can be simplified to the classical strain gradient elasticity theory. When the strain gradient effect is neglected, it can be simplified to the classical viscoelastic theory. As an application example of strain gradient viscoelastic theory, the solution to the problem of cross-scale viscoelastic bending of the Bernoulli-Euler beam, is analyzed and presented.

Droplets on Soft Surfaces Exhibit a Reluctance to Coalesce due to an Intervening Wetting Ridge

AbstractMicroscale interactions with deformable substrates are of fundamental interest for studying self‐assembly processes and the mobility of cells on soft surfaces, with applications in traction force microscopy. The behavior of microscale water droplets on a soft polymer substrate is investigated. Droplets formed by condensation on the soft substrate are reluctant to coalesce, which leads to coverage of the surface with clusters of droplets assembled in a honeycomb‐like pattern. Cryogenically fixed in this state, scanning electron microscopy of these droplets reveals the presence of an intervening wetting ridge of the polymer that acts as a barrier between neighboring droplets and prevents coalescence. A linear elastic deformation model is developed to predict this surface profile and corroborate the observed behavior.

Finite element modeling and simulation of the fiber–matrix interface in fiber reinforced metal matrix composites

The properties of fiber–matrix interface in the fiber reinforced metal matrix composites (MMCs) have been investigated in the past with the use of push-out test. The finite element (FE) analysis which is used to model the push-out test helps the composite’s designer in optimizing the interfacial properties. A two dimensional planar FE model is presented for investigating the properties of interface. Cohesive layer concept is used to model the interface with a specified thickness. The initial response of the cohesive element is assumed to be linear elastic. The maximum nominal stress criterion is used to define the damage initiation and the damage evolution is defined in terms of fracture energy. The simulation results show that the cohesive layer behaves almost like the matrix for the linear elastic deformation prior to damage. It is concluded that the values of Young’s modulus and shear modulus for the cohesive layer can be taken equal to that of the matrix material for the finite element simulation of the debonding behavior of these MMCs. Furthermore, it is observed that the shear stress at which damage initiates can be determined by using the cohesive layer concept for the interface. Once a damage initiation criterion is met, material damage occurs in accordance with the defined damage evolution law. The complete failure of the interface takes place when the stress becomes zero. However, the interface behavior needs to be defined in a more sophisticated way so as to model the stepwise debonding and failure evolution once the complete debonding has occurred.

Elastic wave propagation and scattering in prestressed porous rocks

Poro-acoustoelastic theory has made a great progress in both theoretical and experimental aspects, but with no publications on the joint research from theoretical analyses, experimental measurements, and numerical validations. Several key issues challenge the joint research with comparisons of experimental and numerical results, such as digital imaging of heterogeneous poroelastic properties, estimation of acoustoelastic constants, numerical dispersion at high frequencies and strong heterogeneities, elastic nonlinearity due to compliant pores, and contamination by boundary reflections. Conventional poro-acoustoelastic theory, valid for the linear elastic deformation of rock grains and stiff pores, is modified by incorporating a dual-porosity model to account for elastic nonlinearity due to compliant pores subject to high-magnitude loading stresses. A modified finite-element method is employed to simulate the subtle effect of microstructures on wave propagation in prestressed digital cores. We measure the heterogeneity of samples by extracting the autocorrelation length of digital cores for a rough estimation of scattering intensity. We conductexperimental measurements with a fluid-saturated sandstone sample under a constant confining pressure of 65 MPa and increasing pore pressures from 5 to 60 MPa. Numerical simulations for ultrasound propagation in the prestressed fluid-saturated digital core of the sample are followed based on the proposed poro-acoustoelastic model with compliant pores. The results demonstrate a general agreement between experimental and numerical waveforms for different stresses, validating the performance of the presented modeling scheme. The excellent agreement between experimental and numerical coda quality factors demonstrates the applicability for the numerical investigation of the stress-associated scattering attenuation in prestressed porous rocks.

Numerical Analysis of Fracture Characteristics of Elliptic Porous Sandstone and its Filling Gypsum Based on Statistical Damage Theory

In view of the fact that less attention has been paid to the failure characteristics of rock mass structures with different elliptical shapes, based on the statistical damage theory, the finite element calculation model of sandstone with elliptical holes and its filled gypsum is established, the meso-parameters are calibrated by laboratory test results, and the fracture damage process, peak intensity and acoustic emission law are numerical simulated. Based on the acoustic emission results, the concept of damage degree is defined, and the damage process of the specimen is quantitatively described. The results show that with the increase of k value, the "main crack" produced by tensile stress in the hole gradually evolves into a tension-shear compound "wing crack". After the grain "," main crack "and" wing crack "developed to a certain extent, shear failure appeared on the left and right sides of the hole, and then developed into horizontal inclined tension-shear composite failure. The peak strength of elliptical pores and sandstone filled with gypsum sandstone increased at first and then decreased. The peak strength of filled gypsum sandstone increased by 6.92%, 4.61%, 11.21%, 13.64%, 15.31%, 10.6% and 11.50%, respectively, compared with that of unfilled gypsum sandstone, and the peak strength of sandstone filled with gypsum sandstone increased by 6.92%, 4.61%, 11.21%, 13.64%, 15.31%, 10.6% and 11.50%, respectively. Damage and variation of concrete during loading. There are four stages: linear elastic deformation stage, crack initiation stage, crack acceleration stage and damage stable development stage. With the increase of k value, the maximum damage degree increases at first and then decreases, and the damage degree of sandstone filled with gypsum is larger than that of unfilled sandstone sample as a whole.

Insights into the Effect of Spontaneous Fluid Imbibition on the Formation Mechanism of Fracture Networks in Brittle Shale: An Experimental Investigation.

In this paper, the high-temperature/high-pressure triaxial testing system of rocks is used to study the effect of spontaneous fluid imbibition on the formation mechanism of fracture networks, by means of acoustic emission (AE) monitoring and ultrasound measurement. After the water–shale interaction, the rock mechanical parameters such as rock strength, elastic modulus, cohesion, and internal friction angle of shales significantly decrease as the imbibition time increases, indicating that the fluid has a strong influence on the mechanical properties of brittle shales. The stress–strain curves of the wet and dry shales and their AE characteristics are quite different: (i) the stress–strain curve of wet shale samples shows multiple fluctuations before macroscopic failure, and its cumulative AE number curve presents a step-like jump many times that corresponds to the local microcracking; (ii) the stress–strain curve of dry shale samples mainly shows the characteristic of linear elastic deformation during early loading, which has less AE event number, and the step-like jump is not observed in all the AE curves. The dry shale only has a large number of AE events until it is close to macroscopic failure. Nuclear magnetic resonance, mineral composition, and microstructure analysis show that Chengkou shale generally develops micro–nanoscale pores with a small pore throat, and thus strong capillary spontaneous absorption occurs. The shale–water interaction includes both chemical and physical effects, which affect the key parameters such as acoustic velocity, frictional force on the surfaces of artificial fracture, fracability, and other mechanical properties. This paper provides new insights to the investigation on the formation mechanism of artificial fracture networks in brittle shales.

Deformation Wave Theory and Application to Optical Interferometry.

A method to diagnose the deformation status of solid objects under loading is discussed. The present method is based on a recent field theory of deformation and fracture and optical interferometry known as the Electronic Speckle-Pattern Interferometry (ESPI). Using one of the most fundamental principles of physics referred to as symmetry in physics, this field theory formulates all stages of deformation and fracture on the same theoretical basis. In accordance with the formalism, the theory has defined the criteria for different stages of deformation (linear elastic, plastic and fracturing stages) expressed by certain spatiotemporal features of the differential displacement (the displacement occurring during a small time interval). The ESPI is used to visualize the differential displacement field of a specimen as two-dimensional, full-field interferometric fringe patterns. This paper reports experimental evidence that demonstrates the usefulness of the present method. A tensile load is applied to an aluminum-alloy plate specimen at a constant pulling rate and the resultant in-plane displacement field is visualized with a two-dimensional ESPI setup. The differential displacement field is obtained at each time step and the interferometric fringe patterns are interpreted based on the criterion for each stage of deformation. It has been found that the criteria of linear elastic deformation, plastic deformation and fracturing stage are clearly observed in the corresponding fringe patterns and that the observations are consistent with the loading characteristics.

Experimental Study and Numerical Simulation on Rock-Like Materials Containing Weak Layers and Holes

In order to study the fracture mechanism of rock materials containing weak layers and holes, the rock-like specimens with holes and cracks were made by using cement mortar, and the numerical simulation was carried out based on the statistical damage theory. Then the damage constitutive equation is deduced based on the acoustic emission results of numerical simulation. Finally, the failure mechanism of “through layer” and “along layer” crack propagation modes were discussed. Results show that: The failure mode of rock-like specimens includes tensile cracks (T), shear cracks (S) and remote cracks (R). The pre-existing crack angle changes the failure mode of the rock-like specimens. The stress–strain curves of rock-like specimens are divided into three stages: linear elastic deformation, nonlinear deformation and residual deformation. The numerical model is established based on the statistical damage theory, and the numerical results are similar to the test results. Based on the acoustic emission results of numerical simulation, the damage constitutive equation is derived, which can characterize the damage process of rock-like specimens under different conditions. The damage evolution is divided into four stages: linear elastic deformation stage, crack initiation stage, crack acceleration stage and steady development stage, and the damage factor ranges from 0.72 to 0.87. The mechanism of “through layer” and “along layer” crack propagation modes are discussed and we consider that the tensile strength and inclination angle of weak layers are the key influencing factors.

Thermodynamic Consistency of Plate and Shell Mathematical Models in the Context of Classical and Non-Classical Continuum Mechanics and a Thermodynamically Consistent New Thermoelastic Formulation

Inclusion of dissipation and memory mechanisms, non-classical elasticity and thermal effects in the currently used plate/shell mathematical models require that we establish if these mathematical models can be derived using the conservation and balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the mathematical models. Thermodynamic consistency ensures thermodynamic equilibrium during the evolution of the deformation. When the mathematical models are thermodynamically consistent, the second law of thermodynamics facilitates consistent derivations of constitutive theories in the presence of dissipation and memory mechanisms. This is the main motivation for the work presented in this paper. In the currently used mathematical models for plates/shells based on the assumed kinematic relations, energy functional is constructed over the volume consisting of kinetic energy, strain energy and the potential energy of the loads. The Euler’s equations derived from the first variation of the energy functional for arbitrary length when set to zero yield the mathematical model(s) for the deforming plates/shells. Alternatively, principle of virtual work can also be used to derive the same mathematical model(s). For linear elastic reversible deformation physics with small deformation and small strain, these two approaches, based on energy functional and the principle of virtual work, yield the same mathematical models. These mathematical models hold for reversible mechanical deformation. In this paper, we examine whether the currently used plate/shell mathematical models with the corresponding kinematic assumptions can be derived using the conservation and balance laws of classical or non-classical continuum mechanics. The mathematical models based on Kirchhoff hypothesis (classical plate theory, CPT) and first order shear deformation theory (FSDT) that are representative of most mathematical models for plates/shells are investigated in this paper for their thermodynamic consistency. This is followed by the details of a general and higher order thermodynamically consistent plate/shell thermoelastic mathematical model that is free of a priori consideration of kinematic assumptions and remains valid for very thin as well as thick plates/shells with comprehensive nonlinear constitutive theories based on integrity. Model problem studies are presented for small deformation behavior of linear elastic plates in the absence of thermal effects and the results are compared with CPT and FSDT mathematical models.