Bending and wave propagation analysis of axially functionally graded beams based on a reformulated strain gradient elasticity theory

A new size-dependent axially functionally graded(AFG) micro-beam model is established with the application of a reformulated strain gradient elasticity theory(RSGET). The new micro-beam model incorporates the strain gradient, velocity gradient,and couple stress effects, and accounts for the material variation along the axial direction of the two-component functionally graded beam. The governing equations and complete boundary conditions of the AFG beam are derived based on Hamilton's principle. The correctness of the current model is verified by comparing the static behavior results of the current model and the finite element model(FEM) at the micro-scale. The influence of material inhomogeneity and size effect on the static and dynamic responses of the AFG beam is studied. The numerical results show that the static and vibration responses predicted by the newly developed model are different from those based on the classical model at the micro-scale. The new model can be applied not only in the optimization of micro acoustic wave devices but also in the design of AFG micro-sensors and micro-actuators.

Evaluation of Myocardial Strain and Aortic Elasticity in Patients with Bicuspid Aortic Valve

This study evaluated the myocardial strain and aortic elasticity in patients with bicuspid aortic valve（BAV） and then investigated the relation between them. Thirty-nine patients（30 males; mean age 44±19 years; range 6 to 75 years） with BAV were recruited as BAV group, and 29 age- and sex-matched healthy controls（21 males; mean age 42±11 years; range 20 to 71 years） served as control group. Aortic strain, distensibility and stiffness index were derived using M-mode echocardiography. Left ventricular global myocardial strain was acquired with speckle-tracking echocardiography. Correlation between aortic elasticity and myocardial strain was also analyzed. The results showed that aortic stiffness was higher（17.5±14.0 vs. 5.3±2.7, P〈0.001）, and aortic strain（4.9±4.7 vs. 11.0±4.1, P〈0.001） and distensibility（1.8±2.1 vs. 3.7±1.6, P〈0.001） were lower significantly in BAV group than in control group. Global circumferential strain（–19.1±4.2 vs. –22.5±3.7, P〈0.001）, radial stain（29.8±14.9 vs. 38.0±8.8, P〈0.001） and longitudinal stain（–18.4±3.4 vs. –20.8±3.5, P〈0.001） were significantly lower in BAV group than in control group. There was weak association between aortic elasticity and myocardial strain. These findings indicated BAV patients manifest reduced myocardial strain which had weak relationship with aortic elastic lesion.

C^1 natural element method for strain gradient linear elasticity and its application to microstructures

C^1 natural element method （C^1 NEM） is applied to strain gradient linear elasticity, and size effects on mi crostructures are analyzed. The shape functions in C^1 NEM are built upon the natural neighbor interpolation （NNI）, with interpolation realized to nodal function and nodal gradient values, so that the essential boundary conditions （EBCs） can be imposed directly in a Galerkin scheme for partial differential equations （PDEs）. In the present paper, C^1 NEM for strain gradient linear elasticity is constructed, and sev- eral typical examples which have analytical solutions are presented to illustrate the effectiveness of the constructed method. In its application to microstructures, the size effects of bending stiffness and stress concentration factor （SCF） are studied for microspeciem and microgripper, respectively. It is observed that the size effects become rather strong when the width of spring for microgripper, the radius of circular perforation and the long axis of elliptical perforation for microspeciem come close to the material characteristic length scales. For the U-shaped notch, the size effects decline obviously with increasing notch radius, and decline mildly with increasing length of notch.

Exact simulation for direction-dependent large elastic strain responses of soft fibre-reinforced composites

An explicit form of the elastic strain-energy function for direction-dependent large elastic strain behaviors of soft fiber-reinforced composites is first presented based upon a decoupled approach for simulating complex nonlinear coupling effects.From this form,the exact closed-form solutions are then obtained for the uniaxial tension responses in the fiber and cross-fiber directions.With such exact solutions,the issue of simultaneously simulating strongly coupling nonlinear responses in the fiber and cross-fiber directions may be reduced to the issue of separately treating each decoupled uniaxial stress-strain response,thus bypassing usual complexities and uncertainties involved in identifying a large number of strongly coupled adjustable parameters.The numerical examples given are in good agreement with the experimental data for large strain responses.

Study on creep deformation and energy development of underground surrounding rock under four‐dimensional support

There is an urgent need to develop optimal solutions for deformation control of deep high‐stress roadways,one of the critical problems in underground engineering.The previously proposed four‐dimensional support(hereinafter 4D support),as a new support technology,can set the roadway surrounding rock under three‐dimensional pressure in the new balanced structure,and prevent instability of surrounding rock in underground engineering.However,the influence of roadway depth and creep deformation on the surrounding rock supported by 4D support is still unknown.This study investigated the influence of roadway depth and creep deformation time on the instability of surrounding rock by analyzing the energy development.The elastic strain energy was analyzed using the program redeveloped in FLAC3D.The numerical simulation results indicate that the combined support mode of 4D roof supports and conventional side supports is highly applicable to the stability control of surrounding rock with a roadway depth exceeding 520 m.With the increase of roadway depth,4D support can effectively restrain the area and depth of plastic deformation in the surrounding rock.Further,4D support limits the accumulation range and rate of elastic strain energy as the creep deformation time increases.4D support can effectively reduce the plastic deformation of roadway surrounding rock and maintain the stability for a long deformation period of 6 months.As confirmed by in situ monitoring results,4D support is more effective for the long‐term stability control of surrounding rock than conventional support.

Phase field modeling of ferroelastic variant switching in yttria-stabilized t'zirconia with strain gradient elasticity and interface tension

The 6–8 wt%yttria-stabilized zirconia with a tetragonal structure(t’-YSZ)is extensively employed in thermal barrier coatings.The exceptional fracture toughness of t’-YSZ can be attributed to its distinctive ferroelastic toughening mechanism.Microstructure and interface tension play a critical role in ferroelastic variant switching at the micro-and nano-scale.This paper presents an original thermodynamically consistent phase field(PF)theory for analyzing ferroelastic variant switching at the micro-and nano-scale of t’-YSZ.The theory incorporates strain gradient elasticity using higher-order elastic energy and interface tension tensor via geometric nonlinearity to represent biaxial tension resulting from interface energy.Subsequently,a mixed-type formulation is employed to implement the higher-order theory through the finite element method.For an interface in equilibrium,the effects of strain gradient elasticity result in a more uniform distribution of stresses,whereas the presence of interface tension tensor significantly amplifies the stress magnitude at the interface.The introduction of an interface tension tensor increases the maximum value of stress at the interface by a factor of 4 to 10.The nucleation and evolution of variants at a pre-existing crack tip in a mono-phase t’-YSZ have also been studied.The strain gradient elasticity is capable of capturing the size effect of ferroelastic variant switching associated with microstructures in experiments.Specifically,when the grain size approaches that of the specimen,the critical load required for variant switching at the crack tip increases,resulting in greater dissipation of elastic energy during ferroelastic variant switching.Moreover,the interface tension accelerates the evolution of variants.The presented framework exhibits significant potential in modeling ferroelastic variant switching at the micro-and nano-scale.

Diamond semiconductor and elastic strain engineering

Diamond,as an ultra-wide bandgap semiconductor,has become a promising candidate for next-generation microelec-tronics and optoelectronics due to its numerous advantages over conventional semiconductors,including ultrahigh carrier mo-bility and thermal conductivity,low thermal expansion coefficient,and ultra-high breakdown voltage,etc.Despite these ex-traordinary properties,diamond also faces various challenges before being practically used in the semiconductor industry.This review begins with a brief summary of previous efforts to model and construct diamond-based high-voltage switching diodes,high-power/high-frequency field-effect transistors,MEMS/NEMS,and devices operating at high temperatures.Following that,we will discuss recent developments to address scalable diamond device applications,emphasizing the synthesis of large-area,high-quality CVD diamond films and difficulties in diamond doping.Lastly,we show potential solutions to modulate diamond’s electronic properties by the“elastic strain engineering”strategy,which sheds light on the future development of diamond-based electronics,photonics and quantum systems.

Computer simulation of strain-induced morphological transformation of coherent precipitates

The coherent elastic strain-induced morphological transformation of a binarycubic model alloy was simulated with different strain energy parameters. The microscopic diffusionequation was combined with the theory of microscopic elasticity. The results show that when thestrain energy is neglected, the randomly distributed equiaxed particles are obtained with isotropiccharacteristic. It is coarsening that follows the Ostwald ripening mechanism: smaller particlesdwindle and larger particles grow; when the elastic strain is considered, plate precipitates tend toalign along the elastically soft directions <01> with anisotropic characteristic. The particlesgrow in the soft directions and coarsen further; particles dwindle in out of the soft directions.While the coarsening of the particles localized in the same row or column follows the rule: smallerparticles shrink and larger ones grow.

Time-dependent Early-age Behaviors of Concrete under Restrained Condition

To investigate the early-age behaviors of concrete under a restrained condition, a set of apparatus was developed. In this way, the tensile creep and other early-age properties can be investigated in depth. By measuring the modulus of elasticity of concrete, synchronous shrinkage of concrete and steel rings and free shrinkage of concrete, the deformations of concrete ring can be quantified respectively. The experimental results show the tensile stress in concrete is time-dependent, and the stress at cracking is much lower than the tensile strength at that age; the tensile creep plays an important role in relaxing the tensile stress and postponing the cracking of concrete.

Isogeometric nonlocal strain gradient quasi-three-dimensional plate model for thermal postbuckling of porous functionally graded microplates with central cutout with different shapes

This study presents the size-dependent nonlinear thermal postbuckling characteristics of a porous functionally graded material(PFGM) microplate with a central cutout with various shapes using isogeometric numerical technique incorporating nonuniform rational B-splines. To construct the proposed non-classical plate model, the nonlocal strain gradient continuum elasticity is adopted on the basis of a hybrid quasithree-dimensional(3D) plate theory under through-thickness deformation conditions by only four variables. By taking a refined power-law function into account in conjunction with the Touloukian scheme, the temperature-porosity-dependent material properties are extracted. With the aid of the assembled isogeometric-based finite element formulations,nonlocal strain gradient thermal postbuckling curves are acquired for various boundary conditions as well as geometrical and material parameters. It is portrayed that for both size dependency types, by going deeper in the thermal postbuckling domain, gaps among equilibrium curves associated with various small scale parameter values get lower, which indicates that the pronounce of size effects reduces by going deeper in the thermal postbuckling regime. Moreover, we observe that the central cutout effect on the temperature rise associated with the thermal postbuckling behavior in the presence of the effect of strain gradient size and absence of nonlocality is stronger compared with the case including nonlocality in absence of the strain gradient small scale effect.

Dependence of elastic strain field on the self-organized ordering of quantum dot superlattices

A systematic investigation of the strain distribution of self-organized, lens-shaped quantum dot in the case of groffth direction on （001） substrate was presented. The three-dimensional finite element analysis for an array of dots was used for the strain calculation. The dependence of the strain energy density distribution on the thickness of the capping layer was investigated in detail when the elastic characteristics of the matrix material were anisotropic. It is shown that the elastic anisotropic greatly influences the stress, strain, and strain energy density in the quantum dot structures. The anisotropic ratio of the matrix material and the combination with different thicknesses of the capping layer, may lead to different strain energy density minimum locations on the capping layer surface, which can result in various vertical ordering phenomena for the next layer of quantum dots, i.e. partial alignment, random alignment, and complete alignment.

Elastic strain response in the modified phase-field-crystal model

To understand and develop new nanostructure materials with specific mechanical properties, a good knowledge of the elastic strain response is mandatory. Here we investigate the linear elasticity response in the modified phase-field-crystal（MPFC） model. The results show that two different propagation modes control the elastic interaction length and time, which determine whether the density waves can propagate or not. By quantitatively calculating the strain field, we find that the strain distribution is indeed extremely uniform in case of elasticity. Further, we present a detailed theoretical analysis for the orientation dependence and temperature dependence of shear modulus. The simulation results show that the shear modulus reveals strong anisotropy and the one-mode analysis provides a good guideline for determining elastic shear constants until the system temperature falls below a certain value.

Evaluation of rockburst proneness considering specimen shape by storable elastic strain energy

To systematically assess the rockburst proneness considering specimen shape,multiple groups of laboratory tests were performed on 5 rock materials in cylindrical and cuboid shapes.The linear energy storage(LES)law of both cylindrical and cuboid rock specimens under uniaxial compressive load was confirmed,and the energy storage coefficient was found to be unrelated to specimen shape.On the basis of LES law,two rockburst proneness indexes,namely the strain energy storage index(W_(et))and the potential energy of elastic strain(PES),were modified.Subsequently,the W_(et),PES,peak-strength strain energy storage index(W_(et))p,and peak-strength potential energy of elastic strain(PESp)were used to assess the rockburst proneness of the cylindrical and cuboid specimens.In addition,the fragment ejection course of specimens under test was recorded by a high-speed camera.Then,the rockburst proneness judgments obtained from the 4 indexes were compared with the qualitative data during rock destruction.The results show that,under similar stress conditions,specimen shape has an ignorable effect on the rockburst proneness as a whole.The judgment accuracy of the two modified indexes,especially that of the PESp,is favorably improved to evaluate the rockburst proneness of both cylindrical and cuboid specimens.However,misjudgment ofW_(et)^(p)and PESp may still occur in the assessment of rockburst proneness as these two indexes only consider the energy state before rock peak strength and the W_(et)^(p)is formulated in a ratio form.

A new mixed-mode phase-field model for crack propagation of brittle rock

Study on crack propagation process of brittle rock is of most significance for cracking-arrest design and cracking-network optimization in rock engineering.Phase-field model(PFM)has advantages of simplicity and high convergence over the common numerical methods(e.g.finite element method,discrete element method,and particle manifold method)in dealing with three-dimensional and multicrack problems.However,current PFMs are mainly used to simulate mode-I(tensile)crack propagation but difficult to effectively simulate mode-II(shear)crack propagation.In this paper,a new mixed-mode PFM is established to simulate both mode-I and mode-II crack propagation of brittle rock by distinguishing the volumetric elastic strain energy and deviatoric elastic strain energy in the total elastic strain energy and considering the effect of compressive stress on mode-II crack propagation.Numerical solution method of the new mixed-mode PFM is proposed based on the staggered solution method with self-programmed subroutines UMAT and HETVAL of ABAQUS software.Three examples calculated using different PFMs as well as test results are presented for comparison.The results show that compared with the conventional PFM(which only simulates the tensile wing crack but not mode-II crack propagation)and the modified mixed-mode PFM(which has difficulty in simulating the shear anti-wing crack),the new mixed-mode PFM can successfully simulate the whole trajectories of mixed-mode crack propagation(including the tensile wing crack,shear secondary crack,and shear anti-wing crack)and mode-II crack propagation,which are close to the test results.It can be further extended to simulate multicrack propagation of anisotropic rock under multi-field coupling loads.

Experimental Study on the Mechanical Parameters Relating to the Impact Tendency of Coal Sample

Coal burst remains one of the gravest safety risks that will be encountered in mining in the future, because the stress conditions will become more complex as mining depths increase. Various influencing elements exist, and varied geological and mining circumstances might result in diverse coal burst phenomena. The impact propensity of coal has variations as a result of the distinct physical and mechanical qualities of each. To identify the impact propensity of coal and then understand the rules of coal burst occurrence, laboratory tests can be conducted to identify the physical and mechanical parameters affecting coal samples. The mechanical properties, energy absorption, and energy dissipation characteristics of coal samples were examined experimentally in this paper using coal samples that were taken from the mine. On the basis of the evaluation of the impact inclination parameters for four fundamental coal samples, novel impact inclination indicators and the relationship between the fractures in the coal sample and the impact inclination parameters were discussed. The following are the key conclusions: 1) On-site samples of No. 15 coal from the Qi yuan Coal Mine were taken (15 s) and processed in accordance with the guidelines for the coal specimen impact inclination test. The accuracy of the specimen was sufficient for the test. 2) Analysis is done on the mechanical relevance and calculation techniques of the four fundamental coal sample impact tendency characteristics, dynamic failure time (DT), elastic strain energy index (WET), impact energy index (KE), as well as uniaxial compressive strength (RC). 3) Regarding the rock burst danger of rock samples, the potential use of the ratio of pre-peak and post- peak deformation modulus to Kλ and the residual elastic strain energy index CEF as the impact propensity indices of coal samples are discussed. It is possible to utilize two new impact propensity indices to evaluate the impact propensity of coal samples, according to test results that reveal a linear correlation between two new impact inclination indexes and four fundamental impact tendency indexes. 4) The statistical analysis of the crack ratio with the four impact propensity indicators after coal specimen failure, and the correlation among the crack ratio with the indicators, are both done. The findings indicate that the four impact propensity indicators have a linear relationship with the crack ratio of the coal sample surface cracks.

Rockburst mechanism in soft coal seam within deep coal mines

A number of rockburst accidents occurring in soft coal seams have shown that the rockburst mechanism involved in soft coal seams is significantly different from that involved in hard coal seams. Therefore, the method used to evaluate rockburst in hard coal seams is not applicable to soft coal seams. This paper established an energy integral model for the rockburst-inducing area and a friction work calculation model for the plastic area. If the remaining energy after the coal seam is broken in the rockburstinducing area is greater than the friction work required for the coal to burst out, then a rockburst accident will occur. Mechanisms of ‘‘quaking without bursting" and ‘‘quaking and bursting" are clarified for soft coal seams and corresponding control measures are proposed as the optimization of roadway layouts and use of ‘‘three strong systems"(strong de-stressing, strong supporting, and strong monitoring).

Anchoring properties of substrate with a grating surface

The anchoring properties of substrate with a grating surface are investigated analytically. The alignment of nematic liquid crystal （NLC） in a grating surface originates from two mechanisms, thus the anchoring energy consists of two parts. One originates from the interaction potential between NLC molecules and the molecules on the substrate surface, and the other stems from the increased elastic strain energy. Based on the two mechanisms, the expression of anchoring energy per unit area of a projected plane of this grating surface is deduced and called the equivalent anchoring energy formula. Both the strength and the easy direction of equivalent anchoring energy are a function of the geometrical parameters （amplitude and pitch） of a grating surface. By using this formula, the grating surface can be replaced by its projected plane and its anchoring properties can be described by the equivalent anchoring energy formula.

Microscopic phase-field simulation of the coarsening behavior of coherent precipitates in Ni-Al alloys

On the basis of the microscopic phase-field dynamic model and the microelasticity theory, the characteristics of the coarsening behavior of γ＇ phase in Ni-Al alloys have been systematically studied in a certain volume fraction of the precipitates. It was found that the initial irregular shape, randomly distributed γ＇ phase, gradually transformed into cuboidal shape, regularly aligned along the [100] and [010] directions, and a highly preferential selected microstructure was formed during the later stage of precipitation. The volume fraction of the precipitates produced some effects on the precipitate morphology but did not produce an obvious effect on the regularities of precipitate distribution. The coarsening rate constant from the cubic growth law decreased as a function of volume fraction for small volume fractions, remained constant for intermediate volume fractions, and increased as a function of volume fraction for large volume fractions. During the coherent coarsening process, four ＂splitting＂ patterns between γ＇ phases, which belonged to different antiphase domains, were produced via particle aggregation, such as an L-shaped pattern, a doublet, a triplet, and a quartet.

Discrete element modeling of acoustic emission in rock fracture

The acoustic emission （AE） features in rock fracture are simulated numerically with discrete element model （DEM）. The specimen is constructed by using spherical particles bonded via the parallel bond model. As a result of the heterogeneity in rock specimen, the failure criterion of bonded particle is coupled by the shear and tensile strengths, which follow a normal probability distribution. The Kaiser effect is simulated in the fracture process, for a cubic rock specimen under uniaxial compression with a constant rate. The AE number is estimated with breakages of bonded particles using a pair of parameters, in the temporal and spatial scale, respectively. It is found that the AE numbers and the elastic energy release curves coincide. The range for the Kaiser effect from the AE number and the elastic energy release are the same. Furthermore, the frequency-magnitude relation of the AE number shows that the value of B determined with DEM is consistent with the experimental data.

Static and Dynamic Analysis of a Piezoelectric Semiconductor Cantilever Under Consideration of Flexoelectricity and Strain Gradient Elasticity

In this work,the static and dynamic response of a piezoelectric semiconductor cantilever under the transverse end force with consideration of flexoelectricity and strain gradient elasticity is systematically investigated.The one-dimensional governing equations and the corresponding boundary conditions are derived based on Hamilton’s principle.After that,combining with the linearized equations for the conservation of charge,the effects of characteristic length and flexoelectric coefficient on the working performance of a ZnO nanowire are demonstrated as a numerical case,including the static mechanical and electric fields,natural frequencies,and the frequency–response characteristics at resonances.The results indicate that the flexoelectric effect has a great influence on the electric properties of the nanowire,while the strain gradient effect directly contributes to its mechanical properties.To some extent,the increase in characteristic length is equivalent to the stiffness strengthening.The qualitative results and quantitative data are beneficial for revealing the underlying physical mechanism and provide guidance for the design of piezoelectric semiconductor devices.