Strain and interface energy of ellipsoidal inclusions subjected to volumetric eigenstrains: shape factors

Thermodynamic modeling of the development of non-spherical inclusions as precipitates in alloys is an important topic in computational materials science. The precipitates may have markedly different properties compared to the matrix. Both the elastic contrast and the misfit eigenstrain may yield a remarkable generation of elastic strain energy which immediately influences the kinetics of the developing precipitates. The relevant thermodynamic framework has been mostly based on spherical precipitates. However, the shapes of actual particles are often not spherical. The energetics of such precipitates can be met by adapting the spherical energy terms with shape factors. The well-established Eshelby framework is used to evaluate the elastic strain energy of inclusions with ellipsoidal shapes (described by the axes a, b, and c) that are subjected to a volumetric transformation strain. The outcome of the study is two shape factors, one for the elastic strain energy and the other for the interface energy. Both quantities are provided in the form of easy-to-use diagrams. Furthermore, threshold elastic contrasts yielding strain energy shape factors with the value 1.0 for any ellipsoidal shape are studied.

First-Principles Calculations to Investigate the Third-Order Elastic Constants and Mechanical Properties of Mg, Be, Ti, Zn, Zr, and Cd

We present theoretical studies for the third-order elastic constants of Mg, Be, Ti, Zn, Zr, and Cd with a hexagonal-close-packed (HCP) structure. The method of homogeneous deformation combined with first-principles total-energy calculations is employed. The deformation gradient F i j is applied to the crystal lattice vectors r i , and the elastic strain energy can be obtained from the first-principles calculation. The second- and third-order elastic constants are extracted by a polynomial fit to the calculated energy-strain results. In order to assure the accuracy of our method, we calculated the complete set of the equilibrium lattice parameters and second-order elastic constants for Mg, Be, Ti, Zn, Zr, and Cd, and our results provide better agreement with the previous calculated and experimental values. Besides, we have calculated the pressure derivatives of SOECs related to third-order elastic constants, and high-pressure effects on elastic anisotropy, ductile-to-brittle criterion, and Vickers hardness are also investigated. The results show that the hardness model H v = 1.877 k 2 G 0.585 is more appropriate than H v = 2 k 2 G 0.585 − 3 for HCP metals under high pressure.

Experimental and Numerical Investigation of Failure Characteristics and Mechanism of Granites with Different Joint Angles

The uniaxial compression tests were conducted on granite samples with different joint dip angles to more favorably explore the influences of the nonconsecutive joint on mechanical properties and deformation characteristics of the rock mass. The stress-strain curves, deformation and strength characteristics, and energy evolution process of the samples were analyzed. Numerical simulation using particle flow code (PFC) is employed to study the crack propagation process. The mode of jointed and fractured rock was investigated. The research results showed a significant reduction in both the peak strength and elastic modulus of jointed samples compared with intact ones: the peak strength and elastic modulus drop to the minimum at the joint dip angle of about 45°, especially for the peak strength, which takes up about 55% of the intact samples. The fractured samples’ total energy, elastic strain energy, and dissipated energy during the uniaxial compression drop significantly relative to intact samples. The proportion of the fracture modes varies with different joint dip angles, in which the ratio of shear cracks grows at first and then declines, with the highest balance at the dip angle of 45°. The damage stress’s sensitivity to the dip angle change is greater than that of the peak stress, with reduction amplitude more extensive than the latter.

Roof deformation and collapse of stamps with isolated grooves: a contact mechanics approach

This paper has revisited the roof deformation and collapse of stamps with isolated grooves based on a contact mechanics approach, with emphasis on establishing the non-adhesive and adhesive contact solutions for surfaces containing a shallow rectangular groove with the effects of applied load and interfacial adhesion taken into account. By solving singular integral equations and using the energy release rate approach, closed-form solutions are derived analytically for the deformed groove shapes, interfacial stress distributions and equilibrium relations between load and contact size, which reduce to the previously proposed solutions without adhesion or without applied load. Finite element analysis is performed to validate the non-adhesion solutions, while experiment results of stamp collapse reported in the literature are adopted to examine the adhesion solutions. By introducing the Johnson parameter a to represent a competition between surface energy and elastic strain energy of the groove, four kinds of contact behaviors of the groove roof can be characterized appropriately: non-adhesion, weak adhesion, intermediate adhesion and strong adhesion. Hysteresis loop and energy loss due to distinct load/unloading paths are revealed in the cases of intermediate and strong adhesion. We also provided the critical applied pressure to achieve roof collapse and the corresponding equilibrium contact size for full range of a.

Evolution of bone tissue based on angiogenesis as a crucial factor: New mathematical attempt

In this paper, we present a novel theoretical model of bone evolution. The model takes into account growth of the blood vessels network as a real issue during the process. The stimulation of this growth is controlled by the signals from overloaded dying cells. A system of integro-differential equations describes changes in the density of blood vessels, bone cells and Young modulus, all of which define how the bone tissue evolves over time. The model considers several biomechanical signals such as changes in the density of elastic strain energy and nutrients. Two examples of changing bone tissue were examined to test the model: bone healing around a tooth implant and formation of osteophytes during osteoarthritis. In both cases, the effects of mechanical loading in the bone tissue evolution which are in agreement with the medical observations can be observed.

Mechanism of Hydraulic Fracturing Cutting Hard Basic Roof to Prevent Rockburst

Rockburst is globally regarded as one of the most severe and complicated mining dynamic disasters to predict or control. Generally, the occurrence mechanism of rockbursts can be considered as a process of the elastic strain energy accumulation, emancipation, transmission, and occurrence. Tracing to the source, the reasons for large accumulation of elastic strain energy in coal and rock mass are the high stress of the roof layer that loads on the coal and rock masses around the mining space coupling effect with the natural horizontal tectonic stress. In this study, using the minimum energy theory and elasticity theory, the analytical formula for calculating elastic strain energy of the roof cantilever beam structure acting on the coal body load in front of the working face is deduced. Accordingly, we achieved a method of using hydraulic fracturing to improve the roof structure. In detail, we use a high-pressure jet to cut the cantilever roof structure, which can make a prelocated fracture surface, and then utilize the packers to make sure that the injected high-pressure fracturing fluid is propagating along the prelocated fracture surface and can cut off the cantilever roof structure eventually to prevent rockbursts in advance. Due to the rockburst occurrence mechanism and the quantitatively elastic strain energy analytical formula, a preconditioning water jet cutting induced fracture surface to create orientation-controllable hydraulic fracture strategy is proposed to guard against the high hazard caused by the massive elastic strain energy, which accumulated in the coal body in front of the working face and coal pillar.

Study on Energy Evolution and Damage Constitutive Model of Siltstone

In order to study the energy evolution characteristics and damage constitutive relationship of siltstone, the conventional triaxial compression tests of siltstone under different confining pressures are performed, and the evolution laws of input energy, elastic strain energy and dissipative energy of siltstone with axial strain and confining pressure are analyzed. According to the test results, the judgment criterion of the rock damage threshold is improved, and an improved three-shear energy yield criterion is proposed., The damage constitutive equation of siltstone is established based on the damage mechanics theory through the principle of minimum energy consumption and by considering the residual strength of rock, and lastly, the rationality of the model is verified by experimental data. The results reveal that (1) both the input energy and dissipative energy gradually increase with the increase of axial strain, and the elastic strain energy first increases and then decreases with the increase of axial strain, and reaches its maximum at the peak. (2) The input energy and dissipation energy increase exponentially with the increase of the confining pressure, and the elastic strain energy increases linearly with the increase of confining pressure. (3) According to the linear relationship between the sum of shear strain energy and hydrostatic pressure, an improved three-shear energy yield criterion is established. (4) The model curve can better describe the strain softening stage and the residual strength characteristics of siltstone. The relative standard deviation between the model results and the test results is only 4.35%, which verifies the rationality and feasibility of the statistical damage constitutive model that is established in this paper.

Investigation on Pore-Fracture of Coal and Its Influence Mechanism on Tensile Failure Behavior of Coals with Bursting Proneness

Both computed tomography (CT) and notched semicircular bend (NSCB) tests are performed for coals with high and medium bursting proneness to extract the scientific expression of pore-fracture and its influence mechanism on the tensile failure behavior. The acoustic emission (AE) parameters in the sample during loading and failure are monitored, and the influence mechanism of pore-fracture on tensile failure behavior of coal is analyzed. The result illustrates that the spatial distribution feature of the pore-fracture in coals with high and medium bursting proneness is extremely different. The deformation and failure mode of the coals are affected by many factors, loading mode, notch depth and width, mechanical properties of matrix and minal, spatial distribution feature of pore-fracture, etc. The influence of primary pore-fracture in the coal on the extension and penetration of the secondary fracture could be divided into two types: bifurcation and promotion, which would cause different local damage in the sample and affect the final failure mode. The feature of acoustic emission parameters indicates that the deformation and failure process of a sample under loading could be divided into four stages: compaction stage, elastic deformation stage, displacement plastic growth stage, and post peak failure stage, which is the result of comprehensive action of many factors. The evolution process of secondary fracture is accompanied by the dissipation of elastic strain energy and the intensification of internal damage of coal, which reflects the failure process of coal.

Study on the Meso-Energy Damage Evolution Mechanism of Single-Joint Sandstone under Uniaxial and Biaxial Compression

Energy conversion and release occur through the entire deformation and failure process in jointed rock masses, and the accumulation and dissipation of rock mass energy in engineering can reveal the entire process of deformation and instability. This study uses PFC2D to carry out numerical simulation tests on single-joint sandstone under uniaxial compression and biaxial compression, respectively, and analyse the influence of joint inclination, length, and confining pressure on the meso-energy conversion process and phase evolution of jointed sandstone. Through analysis, it is found that the input meso total strain energy is transformed into meso dissipated energy and meso-elastic strain energy. Macroscopic and microscopic joint sandstone law is consistent with the overall energy evolution; and the difference is reflected in two aspects: (1) the microlevel energy evolution has no initial compaction energy consumption section and (2) the linear energy storage section before the macroenergy evolution peak can be subdivided into two sections in the meso-level energy evolution. Under uniaxial compression, the energy values at the characteristic points of the meso-level energy evolution phases first asymmetrically decrease and then increase with the increase of the joint inclination. The initiation point of jointed sandstone is significantly affected by the length of the joint, and the degradation effect of the meso-energy at the damage point and peak point weakens with the increase of the joint length. Comparing the data obtained from the PFC numerical simulation with the experimental data, it is found that the error is small, which shows the feasibility of the numerical model in this paper. Under biaxial compression, the accumulation rate of meso-elastic strain at the peak point of the jointed sandstone first decreases and then increases with the joint inclination angle. After the peak of jointed sandstone, the rate of sudden change of meso-energy change decreases with the increase of joint length. The conditions of high confining pressure will promote the meso-accumulated damage degree of the jointed sandstone before the peak, while inhibiting the meso-energy and the mutation degree of the damage after the peak. The higher the confining pressure, the more obvious the joint length and inclination effect characteristics of the elastic strain energy at the peak point of the jointed sandstone.