Elastic Strain Energy Density Research Articles

A new criterion of coal burst proneness based on the residual elastic energy index

To evaluate the coal burst proneness more precisely, a new energy criterion namely the residual elastic energy index was proposed. This study begins by performing the single-cyclic loading-unloading uniaxial compression tests with five pre-peak unloading stress levels to explore the energy storage characteristics of coal. Five types of coals from different mines were tested, and the instantaneous destruction process of the coal specimens under compression loading was recorded using a high speed camera. The results showed a linear relationship between the elastic strain energy density and input energy density, which confirms the linear energy storage law of coal. Based on this linear energy storage law, the peak elastic strain energy density of each coal specimen was obtained precisely. Subsequently, a new energy criterion of coal burst proneness was established, which was called the residual elastic energy index (defined as the difference between the peak elastic strain energy density and post peak failure energy density). Considering the destruction process and actual failure characteristics of coal specimens, the accuracy of evaluating coal burst proneness based on the residual elastic energy index was examined. The results indicated that the residual elastic energy index enables reliable and precise evaluations of the coal burst proneness.

Effects of pre-existing single crack angle on mechanical behaviors and energy storage characteristics of red sandstone under uniaxial compression

To investigate the effects of the pre-existing single crack angle on the mechanical behaviors and energy storage characteristics of red sandstone, a series of uniaxial compression and single-cyclic loading–unloading uniaxial compression tests were conducted on red sandstone specimens with five crack angles. The relationships among the crack angle and the mechanical properties, failure modes, and energy storage characteristics of red sandstone were analyzed using the acoustic emission (AE), stress–strain curve and curvilinear integral. The results indicate that the peak compressive strength, peak strain, and elastic modulus of specimens increase with the crack angle increasing, and the specimens with five different angles all present “anti-N-type” shear failure. Throughout the loading process, the total input energy density, elastic strain energy density, and dissipated energy density of the red sandstone at five different crack angles exhibit significant quadratic growth in response to the actual unloading stress. Importantly, the elastic strain energy density and dissipated energy density of red sandstone with different crack angles all show highly linear relationships with the total input energy density, and the linear energy storage and dissipation laws of pre-cracked rocks are obtained. Based on the linear energy storage law, the energy densities at peak strength of red sandstone with different crack angles are determined quantitatively. In addition, the linear energy dissipation law is applied to the expression of rock damage, and the damage constitutive equations of uniaxial compression specimens with different crack angles are obtained. Finally, the effects of crack geometric parameters on the energy storage and dissipation characteristics were discussed.

A study on equivalence of nonlinear energy dissipation between first-order computational homogenization (FOCH) and reduced-order homogenization (ROH) methods

Nowadays, studies on the mechanism of macro-scopic nonlinear behavior of materials by accumulation of micro-scopic degradation are attracting more attention from researchers. Among numerous approaches, multiscale methods have been proved as powerful and practical approaches in predicting macro-scopic material status by averaging and homogenizing physical information from associated micro-scopic material behavior. Usually in mechanical problem, the stress, consistent material modulus, and possible material state variables are quantities in interest through the upscaling process. However, the energy-related quantities are not studied much. Some initiative work has been done in the early year including but not limited to the Hill–Mandel condition in multiscale framework, which gives that the macro-scopic elastic strain energy density can be computed by volumetric averaging of that in the micro-scale. However, in the nonlinear analysis, the energy dissipation is an important quantity to measure the degradation status. In this manuscript, two typical multiscale methods, the first-order computational homogenization (FOCH) and reduced-order homogenization (ROH), are adopted to numerically analyze a fiber-reinforced composite material with capability in material nonlinearity. With numerical experiments, it can be shown that energy dissipation is the same for both approaches.

The Shear Stress Determination in Tubular Specimens under Torsion in the Elastic-Plastic Strain Range from the Perspective of Fatigue Analysis.

The comparison of shear stress determination methods in tubular specimens under torsion is presented in this paper. Four methods were analyzed: purely elastic solutions, purely plastic solutions, the midsection approach, and the Chaboche nonlinear kinematic hardening model. Using experimental data from self-designed and conducted fatigue experiments, an interesting insight on this problem was obtained that is not often tackled in the literature. It was shown that there are differences in determined shear stress values, and their level depends on a few factors. The midsection approach and purely plastic solution gave values of surface shear stress very close to the values obtained using the Chaboche nonlinear kinematic hardening model for high strain levels. The purely elastic solution gave proper results for the low strain ranges, close to the cyclic yield limit. Since none of the methods can be trusted in the full range of loading, an important conclusion from these analyses regards the formulated ranges of their applicability. It was also shown that the calculated values of shear stress and plastic and elastic strain energy density determined on this basis have a strong impact on fatigue life predictions. Finally, the influence of predicted values of shear stresses on the interpretation of cyclic hardening phenomena was also presented.

Phase field modelling of fracture and fatigue in Shape Memory Alloys

We present a new phase field framework for modelling fracture and fatigue in Shape Memory Alloys (SMAs). The constitutive model captures the superelastic behaviour of SMAs and damage is driven by the elastic and transformation strain energy densities. We consider both the assumption of a constant fracture energy and the case of a fracture energy dependent on the martensitic volume fraction. The framework is implemented in an implicit time integration scheme, with both monolithic and staggered solution strategies. The potential of this formulation is showcased by modelling a number of paradigmatic problems. First, a boundary layer model is used to examine crack tip fields and compute crack growth resistance curves (R-curves). We show that the model is able to capture the main fracture features associated with SMAs, including the toughening effect associated with stress-induced phase transformation. Insight is gained into the role of temperature, material strength, crack density function and fracture energy homogenisation. Secondly, several 2D and 3D boundary value problems are addressed, demonstrating the capabilities of the model in capturing complex cracking phenomena in SMAs, such as unstable crack growth, mixed-mode fracture or the interaction between several cracks. Finally, the model is extended to fatigue and used to capture crack nucleation and propagation in biomedical stents, a paradigmatic application of nitinol SMAs.

Study on the Regularity of Ore Pressure in Overburden Hard Rock Working Face of Overburden Mining on Reverse Faults

In order to study the mining pressure on the upper coal seam in the fault, the working pressure of the working face with a hard rock roof is based on the production geological conditions of the 3301 working face of Yiqiao Coal Mine. Use flac numerical simulation software to establish a three-dimensional numerical calculation model of mining face, and study the distribution characteristics of mining plastic area, the advance support pressure and elastic strain energy density. And the change curve is treated by origin software fitting function. In the end, it was found that the total volume of the plastic zone and the propulsive distance of the working face showed an exponential function. Before and after the "first square", the total volume of the plastic zone increased more significantly, and the hard roof failure was mainly bending failures; There is a logarithmic function between the advance support pressure and the advancing distance of the working face. The range of 5–14 m in front of the coal wall is a region where the pressure increases sharply. The elastic strain energy density appears relatively high. During the mining process, the advance support of the roadway should be strengthened, and the key monitoring should be carried out. If necessary, corresponding pressure relief measures should be taken to ensure the safe production of the working face.

Upscaling mechanical properties of Al2O3 coated VACNT forest architecture under compression

Effect of oxide coating thickness and strain rate on mechanical properties of vertically aligned carbon nanotubes (VACNT) forest and patterned pillars have been investigated using flat punch nanoindentation. The elastic modulus of 20 nm oxide coated VACNT forest and micro-pillars show a remarkable increase of 135 times and 77 times respectively as compared to uncoated CNT forest samples. Large enhancement in load carrying capacity of oxide-coated VACNT is observed when the strain rate is varied from 50 nm/s to 4000 nm/s in micro compression tests. The elastic strain energy density of the 20 nm coated VACNT forest is estimated to be ~260 times higher than the as-grown VACNT forest. Patterned CNT pillar with coating of 20 nm Al2O3 shows 77 times increment with the strain rate.

Spectroscopic Determination of Ice-Induced Interfacial Strain on Single-Layer Graphene.

Reliably determining the physical properties of ice (e.g., crystal structure, adhesion strength, interfacial state, and molecular orientation) has proven to be both challenging and highly dependent on experiment-specific conditions, including surface roughness, ice formation, water purity, and measurement method. Here, non-destructive measurements of single-layer graphene (SLG) interfaced with bulk ice are used to determine temperature-dependent, ice-induced strain and estimate ice-created strain elastic density in SLG. The use of SLG enables the precise study of interfacial strain by monitoring the 2D Raman mode. Upon ice formation, a clear, ≈2 cm-1 decrease in the 2D mode frequency is observed, which is ascribed to a 0.012% biaxial tensile shear strain at the ice-SLG interface. From this shear strain value, the ice-created SLG elastic strain energy density is estimated to be 2.4 μJ m-2 . In addition to these Raman strain measurements, intentionally ionized water is used to show that water-mediated charging of the SLG surface manifests itself in a distinctly different manner than ice-induced strain. Finally, the localized nature of the Raman probe is used to map SLG regions with and without ice, suggesting that this method cannot only determine ice-induced interfacial strain, but also correlate ice adhesion properties with surface roughness and topology.

Transfer and dissipation of strain energy in surrounding rock of deep roadway considering strain softening and dilatancy

AbstractIn this study, the transfer and dissipation of strain energy in the surrounding rock of a deep roadway were analyzed, considering the objective strain softening and dilatancy behaviors. The strain energy increment was decomposed, and its variation was analyzed based on the incremental plastic flow theory; then, a numerical simulation was conducted for verification and further analysis. The results were verified by the field monitoring data of a coal mine gateway. The results show that in the elastic stage, the volumetric elastic strain energy density Uev decreases, while the shear elastic strain energy density Ues increases. In the plastic stage, both Uev and Ues decrease. The volumetric plastic strain energy density Upv is negative, and its absolute value increases, leading to strain energy accumulation. In contrast, the shear plastic strain energy density Ups is positive and increases, leading to strain energy dissipation. Considering strain softening, the elastic strain energy decreases, the plastic strain energy increases, and the region of strain energy dissipation expands. Considering dilatancy, the plastic strain energy varies more significantly, and the effect of strain softening is amplified. The strain energy is transferred from the deep part to the shallow part of the elastic zone and then to the plastic zone. The preexisting and input strain energies in the plastic zone are transformed into considerable amounts of plastic strain energy and then dissipated. Thereafter, a significant plastic strain appears, leading to the large deformation of the surrounding rock.

Likely striping in stochastic nematic elastomers

For monodomain nematic elastomers, we construct generalised elastic–nematic constitutive models combining purely elastic and neoclassical-type strain energy densities. Inspired by recent developments in stochastic elasticity, we extend these models to stochastic–elastic–nematic forms, where the model parameters are defined by spatially independent probability density functions at a continuum level. To investigate the behaviour of these systems and demonstrate the effects of the probabilistic parameters, we focus on the classical problem of shear striping in a stretched nematic elastomer for which the solution is given explicitly. We find that, unlike the neoclassical case, where the inhomogeneous deformation occurs within a universal interval that is independent of the elastic modulus, for the elastic–nematic models, the critical interval depends on the material parameters. For the stochastic extension, the bounds of this interval are probabilistic, and the homogeneous and inhomogeneous states compete, in the sense that both have a a given probability to occur. We refer to the inhomogeneous pattern within this interval as ‘likely striping’.

Treadmilling stability of a one-dimensional actin growth model

Actin growth is a fundamental biophysical process and it is, at the same time, a prototypical example of diffusion-mediated surface growth. We formulate a coupled chemo-mechanical, one-dimensional growth model encompassing both material accretion and ablation. A solid bar composed of bound actin monomers is fixed at one end and connected to an elastic device at the other. This spring-like device could, for example, be the cantilever tip of an AFM. The compressive force applied by the spring on the bar increases as the solid grows and affects the rate of growth. The mechanical behaviour of the bar, the diffusion of free actin monomers in a surrounding solvent and the kinetic growth laws at the accreting/ablating ends are accounted for. The constitutive response of actin is modeled by a convex but otherwise arbitrary elastic strain energy density function. Treadmilling solutions, characterized by a constant length of the continuously evolving body, are investigated. Existence and stability results are condensed in the form of simple formulas and their physical implications are discussed.

A Modified Bursting Energy Index for Evaluating Coal Burst Proneness and Its Application in Ordos Coalfield, China

Coal burst is a type of dynamic geological hazard in coal mine. In this study, a modified bursting energy index, which is defined as the ratio of elastic strain energy at the peak strength to the released strain energy density at the post-peak stage, was proposed to evaluate the coal burst proneness. The calculation method for this index was also introduced. Two coal mines (PJ and TJH coal mines) located in Ordos coalfield were used to verify the validity of the proposed method. The tests results indicate that modified bursting energy index increases linearly with increasing uniaxial compressive strength. The parameter A, which is used to fit relation between total input and elastic strain energy density, has a significant effect on the modified bursting energy index. A large value of parameter A means more elastic strain energy before the peak strength while a small value indicates most of input energy was dissipated. Finally, the coal burst proneness of these two coal mines was evaluated with the modified index. The results of modified index are consistent with that of laboratory tests, and more reasonable than that from original bursting energy index because it removed the dissipated strain energy from the total input strain energy density.

A New 3D Creep-Fatigue-Elasticity Damage Interaction Diagram Based on the Total Tensile Strain Energy Density Model

Fatigue damage, creep damage, and their interactions are the critical factors in degrading the integrity of most high-temperature engineering structures. A reliable creep-fatigue damage interaction diagram is a crucial issue for the design and assessment of high-temperature components used in power plants. In this paper, a new three-dimensional creep-fatigue-elasticity damage interaction diagram was constructed based on a developed life prediction model for both high-temperature fatigue and creep fatigue. The total tensile strain energy density concept is adopted as a damage parameter for life prediction by using the elastic strain energy density and mean stress concepts. The model was validated by a great deal of data such as P91 steel at 550 °C, Haynes 230 at 850 °C, Alloy 617 at 850 and 950 °C, and Inconel 625 at 815 °C. The estimation values have very high accuracy since nearly all the test data fell into the scatter band of 2.0.

Irradiation-induced internal stresses in polycrystalline [formula omitted]-uranium: a mesoscale mechanical approach

Several advanced nuclear reactor designs incorporate metallic fuel; however, the initial stages of metallic fuel microstructure evolution have not been studied since the 1960s. The development of new fuels and fuel performance models can be greatly accelerated by understanding the physical mechanisms driving the early evolution of fuel microstructure during irradiation. In this work, a mesoscale model based on irradiation-induced growth strains of single crystal α-uranium is developed to understand the initial irradiation-induced deformation of polycrystalline α-uranium. Three-dimensional simulations on single crystal and polycrystalline α-uranium are performed to study irradiation-induced distortion and the development of internal stresses within the material. The elastic strain energy density, von Mises stress and hydrostatic stress are investigated to understand how plasticity and fracture may develop in the material. It is found that at extremely low burnups (e.g., 0.0005 atomic %, or 4.7 MWd/MTU), tensile internal stresses on the order of hundreds of megapascals develop on the grain boundaries and triple points, making either plastic deformation or intergranular cracking likely, depending on the strength of the grain boundaries. In addition, it is found that the application of external stress, even 100 MPa of hydrostatic compression, will not suppress the development of the tensile internal stresses, indicating that plastic flow or fracture will occur during irradiation regardless of other stresses that may arise in service.

The influence of stress ratio on fatigue lifetime of NiTi shape memory alloy

The influence of stress ratio on fatigue lifetime of pseudoelastic NiTi alloy are studied. The stress-, strain- and energy-based criteria were used for analysis under on low-cycle fatigue. Increasing the stress ratio from 0 to 0.5 significantly reduces the fatigue life of the NiTi alloy when used to describe the stress range, strain range and dissipation energy density and increases when using the Odqvist’s parameter. A good agreement between experimental and calculated values of predicted lifetime was obtained. The low–cycle fatigue failure criterion of pseudoelastic NiTi alloy - total elastic strain energy density that takes into account the stress ratio was proposed.

A micro-potential based Peridynamic method for deformation and fracturing in solids: A two-dimensional formulation

A two-dimensional micro-potential based Peridynamic (MPPD) formulation is proposed to investigate deformation and fracturing in solid materials. To this end, square measure of the bond length change is applied and a generalized Peridynamic (PD) strain for plane stress and strain problems is developed to decompose the bond length change into two parts, one resulting from the volumetric PD strain and the other from the deviatoric PD strain. The micro-potential generated in the deformed bond is postulated as a function of both length change components. The nonlocal elastic strain energy density (NESED) at a material point is computed by the integral of the bond potential over the horizon. Through the Frechet derivative of the NESED, a general constitutive relation depending on the micro-potential function is well formulated. The model parameters are calibrated through the equivalence of the NESED with the linear elastic strain energy density. Several specific micro-potential functions and their corresponding constitutive force densities are carefully discussed. The discussions reveal that the proposed MPPD model not only can retrieve the well-known bond based PD model, but also allows specific materials with any Poisson ratio to be represented. Moreover, one failure criterion based on the total micro-bond potential is developed for the present model. The reliability and efficiency of the MPPD model is demonstrated through numerical tests. Simulation results show that the proposed model is capable of investigating deformation and cracking in solids.

“Ʌ”-shaped trend of stress relaxation stability of Inconel718 superalloy with initial stress increasing

Stress relaxation tests were carried out to identify stress relaxation behavior of Inconel718 superalloy under different initial stresses. Experimental results indicate that: increasing initial stress from 260 MPa to 1160 MPa, stress relaxation stability of Inconel718 superalloy rises firstly and then falls. Above all, thermodynamic models of stress relaxation under different initial stresses are constructed, and the release of elastic strain energy density (▵W) and release ratio of elastic strain energy density (η) are identified to quantify thermodynamic characteristics of stress relaxation. The trends of ▵W and η demonstrate that plastic deformation decreases firstly and then increases with initial stress increasing. Besides, constitutive equation and relevant parameters are used to describe kinetic characteristics of stress relaxation behavior. Plastic strain rate increases continuously, but the tendency of stress exponent suggests that work hardening strengthens in the beginning and then weakens remarkably. Due to the coupling effect between the two factors, the accumulation of plastic deformation drops firstly and then rises significantly. Finally, based on microstructure evolution, it can be found that, at low initial stress, stress relaxation consists of two stages, and dislocation movement and grain boundary migration are dominant mechanisms of stress relaxation. While at high initial stress, stress relaxation can be divided into three stages. Twinning is the main mechanism in the first stage, leading to obvious plastic deformation. When it comes to the second stage, dislocation movement and grain boundary migration are dominant reasons. Eventually, stress relaxation achieves the third stage, referred to as the steady state.

A novel Strain-Energy-Density based fatigue criterion accounting for mean stress and plasticity effects on the medium-to-high-cycle uniaxial fatigue strength of plain and notched components

A novel strain-energy–density (SED) based fatigue criterion is here proposed to account for the effect of mean stress and plasticity on the uniaxial fatigue strength of plain and notched components. It is based on the definition of four SED components: ΔW¯el, the elastic SED associated to the stress range, ΔW¯el,max, the maximum elastic SED in the stabilized cycle, ΔW¯pl, the plastic SED dissipated per stabilized cycle, W¯pl,max, the plastic SED dissipated over the cycles until stabilization. The mean stress effect is incorporated in a Walker-like expression, ΔW¯elαW¯el,max1-α, while W¯pl,max is added to the expression of the total SED to include the effect of mean stress relaxation. An energetic approach is proposed to identify the condition of cycle stabilization. The coefficients of the fatigue criterion are calibrated using experimental fatigue data. The criterion is validated by predictions of independent data.

A higher order geometrically nonlinear Cosserat‐shell model with initial curvature effects

AbstractUsing a geometrically motivated 8‐parameter ansatz through the thickness, we reduce a three‐dimensional shell‐like geometrically nonlinear Cosserat material to a fully two‐dimensional shell model. Curvature effects are fully taken into account. For elastic isotropic Cosserat materials, the integration through the thickness can be performed analytically and a generalized plane stress condition allows for a closed‐form expression of the thickness stretch and the nonsymmetric shift of the midsurface in bending. We obtain an explicit form of the elastic strain energy density for Cosserat shells, including terms up to order O(h5) in the shell thickness h. This energy density is expressed as a quadratic function of the nonlinear elastic shell strain tensor and the bending‐curvature tensor, with coefficients depending on the initial curvature of the shell.

Micro-porosity as damage indicator for characterizing cyclic thermal shock-induced anisotropic damage in oxide/oxide ceramic matrix composites

Ceramic matrix composites display complex mechanical behavior under thermo-mechanical loading conditions. The present work focuses on micro-structural evolution and resultant macroscopic property representation of the composites after cyclic thermal shocks. Micro-structural investigation reveals that variations of the hierarchical porosity in the matrix characterize the collective behavior of micro-structural evolution in the material. Cyclic thermal shocks induce thermo-mechanical damage and lead to increasing matrix porosity. Material damage development is driven by elastic strain energy density which is related to the matrix porosity increment. Experiments confirm that the anisotropic damage can be represented by the porosity increment accurately. The porosity provides a meaningful model for the thermal shock damage evolution in the anisotropic composites.