An improved strain-softening constitutive model of granite considering the effect of crack deformation

This paper presents an improved strain-softening constitutive model considering the effect of crack deformation based on the triaxial cyclic loading and unloading test results.The improved model assumes that total strain is a combination of plastic,elastic,and crack strains.The constitutive relationship between the crack strain and the stress was further derived.The evolutions of mechanical parameters,i.e.strength parameters,dilation angle,unloading elastic modulus,and deformation parameters of crack,with the plastic strain and confining pressure were studied.With the increase in plastic strain,the cohesion,friction angle,dilation angle,and crack Poisson's ratio initially increase and subsequently decrease,and the unloading elastic modulus and the crack elastic modulus nonlinearly decrease.The increasing confining pressure enhances the strength and unloading elastic modulus,and decreases the dilation angle and Poisson's ratio of the crack.The theoretical triaxial compressive stress-strain curves were compared with the experimental results,and they present a good agreement with each other.The improved constitutive model can well reflect the nonlinear mechanical behavior of granite.

Effect of strain rate and deformation temperature on strain hardening and softening behavior of pure copper

The effects of the deformation temperature and the strain rate on the hot deformation behavior of pure copper were investigated based on compression tests. The expressions of strain hardening rate, dynamic recrystallization critical stress, saturated stress, dynamic recovery volume fraction and dynamic recrystallization volume fraction were determined. According to the processing map, the instability regions occur in regions of 400?450 °C, 0.001?0.05 s?1 and 450?750 °C, 0.05?1 s?1. The deformation mechanism in the stability region is dynamic recrystallization. The flow stress was predicted. The results also show that the true stress–true strain curves predicted by the extracted model are in good agreement with the experimental results.

ELASTO-PLASTIC CONSTITUTIVE MODEL OF SOIL-STRUCTURE INTERFACE IN CONSIDERATION OF STRAIN SOFTENING AND DILATION

The behavior of soil-structure interface plays a major role in the definition of soil-structure interaction. In this paper a bi-potential surface elasto-plastic model for soil-structure interface is proposed in order to describe the interface deformation behavior,including strain softening and normal dilatancy. The model is formulated in the framework of generalized potential theory,in which the soil-structure interface problem is regard as a two-dimensional mathematical problem in stress field,and plastic state equations are used to replace the traditional field surface. The relation curves of shear stress and tangential strain are fitted by a piecewise function composed by hyperbolic functions and hyperbolic secant functions,while the relation curves of normal strain and tangential strain are fitted by another piecewise function composed by quadratic functions and hyperbolic secant functions. The approach proposed has the advantage of deriving an elastoplastic constitutive matrix without postulating the plastic potential functions and yield surface. Moreover,the mathematical principle is clear,and the entire model parameters can be identified by experimental tests. Finally,the predictions of the model have been compared with experimental results obtained from simple shear tests under normal stresses,and results show the model is reasonable and practical.

Strain Softening and Hardening Behavior in AZ61 Magnesium Alloy

The deformation behavior of AZ61 Mg alloy during hot deformation has been investigated in wide temperature and strain rate range by a Gleeble simulator. Specimens are deformed in compression in the temperature range of 523-673 K and at strain rates of 0.001-1 s-1. It is found that the flow curves exhibit a peak and then decrease towards steady-state of classical DRX, which decrease with rising temperature and decreasing strain rate. The deformation behavior of the specimens can be attributed to the occurrence of strain hardening and softening. As stress decreases, the strain hardening rate declines at a fast rate when temperature rises or strain rate decreases. The shapes of θ-σ curves indicate some important features such as subgrain formation, the criticai stress, the peak stress and steady stress. The onset of DRX can be determined by the point of inflection on θ-σ or Inθ-σ curves.

Estimation of compaction grouting pressure in strain softening soils

A new method was proposed to predict the limited compaction grouting pressure for the soft soils. Theoretical basis of the method considered the conical shear failure above the grout bulb. Using the Mohr-Coulomb yield criterion as the initial yield function, the limited compaction grouting pressure was determined, according to the softening elastic-plastic model based on the conventional triaxial compression tests to simulate the strain softening soils. The small strain in the elastic zone and large stain in the plastic zone and the rational yield function for the strain softening phase stage, the analytical solutions to the compaction grouting pressure were presented. The results indicate reasonable agreement and show a good potential of the proposed method for rationally optimizing the design of compaction grouting operations.

Damage constitutive model for strain-softening rock based on normal distribution and its parameter determination

Firstly, using the damage model for rock based on Lemaitre hypothesis about strain equivalence, a new technique for measuring strength of rock micro-cells by adopting the Mohr-Coulomb criterion was developed, and a statistical damage evolution equation was established based on the property that strength of micro-cells is consistent with normal distribution function, through discussing the characteristics of random distributions for strength of micro-cells, then a statistical damage constitutive model that can simulate the full process of rock strain softening under specific confining pressure was set up. Secondly, a new method to determine the model parameters which can be applied to the situations under different confining pressures was proposed, by deeply studying the relations between the model parameters and characteristic parameters of the full stress-strain curve under different confining pressures. Therefore, a unified statistical damage constitutive model for rock softening which can reflect the effect of different confining pressures was set up. This model makes the physical property of model parameters explicit, contains only conventional mechanical parameters, and leads its application more convenient. Finally, the rationality of this model and its parameters-determining method were identified via comparative analyses between theoretical and experimental curves.

Influences of strain softening and seepage on elastic and plastic solutions of circular openings in nonlinear rock masses

Considering the influence of strain softening, the solutions of stress, displacement, plastic softening region radius and plastic residual region radius were derived for circular openings in nonlinear rock masses subjected to seepage. The radial stress distribution curve, ground reaction curve, and relation curve between plastic softening region radius and supporting force in three different conditions were drawn respectively. From the comparisons among these results for different conditions, it is found that when the supporting force is the same, the displacement of tunnel wall considering both seepage and strain softening is 85.71% greater than that only considering seepage. The increase values of radial displacement at 0.95 m and plastic softening region radius at 6.6 m show that the seepage and strain softening have the most unfavorable effects on circular opening stability in strain softening rock masses.

Analytical solutions of stress and displacement in strain softening rock mass around a newly formed cavity

The closed form solutions of the stress and displacement in strain softening rock mass around a newly formed cavity are derived with a three step-wise elasto-plastic model. Hoek-Brown criterion is adopted as the yielding criterion of rock mass. Damage factors are proposed to account for degradation of the material parameters to reflect the degree of strain softening. The surrounding rock mass around the cavity is divided into three regions: elastic region, strain softening region and residual state region. The analytical solutions of stress, strain, displacement and radius of each region are obtained. The effects of the strain softening and shear dilatancy behavior on the results are investigated with parametric studies. The results show that the radii of the residual state region and strain softening region in the surrounding rock mass with higher damage degree are larger. The radii of the residual state region and strain softening region are 1-2 times and 1.5-3 times of the cavity radius, respectively. The radial and tangential stresses decrease with the increase of the damage factor. The displacement of the cavity wall for the case with maximum plastic bulk strain is nearly twice than that with no dilation. Rock mass moves more toward the center for the case with larger damage factor and shear dilation. The area of the plastic region is larger when the damage factors are considered. The displacements in the surrounding rock mass increase with the increase of the damage factors and shear dilation factors. The solutions can be applied to the stability analysis and support design of the underground excavation.

Effect of strain hardening and strain softening on welding distortion and residual stress of A7N01-T4 aluminum alloy by simulation analysis

The effect of strain hardening and strain softening behavior of flow stress changing with temperature on welding residual stress, plastic strain and welding distortion of ATN0 1-T4 aluminum alloy was studied by finite simulation method. The simulation results show that the weld seam undergoes strain hardening in the temperature range of 180-250 ℃, however, it exhibits strain softening at temperature above 250 ℃ during welding heating and cooling process. As a result, the strain hardening and strain softening effects counteract each other, introducing slightly influence on the welding residual stress, residual plastic strain and distortion. The welding longitudinal residual stress was determined by ultrasonic stress measurement method for the flat plates of A7N01-T4 aluminum alloy. The simulation results are well accordant with test ones.

A vector sum analysis method for stability evolution of expansive soil slope considering shear zone damage softening

Slope stability analysis is a classical mechanical problem in geotechnical engineering and engineering geology.It is of great significance to study the stability evolution of expansive soil slopes for engineering construction in expansive soil areas.Most of the existing studies evaluate the slope stability by analyzing the limit equilibrium state of the slope,and the analysis method for the stability evolution considering the damage softening of the shear zone is lacking.In this study,the large deformation shear mechanical behavior of expansive soil was investigated by ring shear test.The damage softening characteristic of expansive soil in the shear zone was analyzed,and a shear damage model reflecting the damage softening behavior of expansive soil was derived based on the damage theory.Finally,by skillfully combining the vector sum method and the shear damage model,an analysis method for the stability evolution of the expansive soil slope considering the shear zone damage softening was proposed.The results show that the shear zone subjected to large displacement shear deformation exhibits an obvious damage softening phenomenon.The damage variable equation based on the logistic function can be well used to describe the shear damage characteristics of expansive soil,and the proposed shear damage model is in good agreement with the ring shear test results.The vector sum method considering the damage softening behavior of the shear zone can be well applied to analyze the stability evolution characteristics of the expansive soil slope.The stability factor of the expansive soil slope decreases with the increase of shear displacement,showing an obvious progressive failure behavior.

MICROMECHANICAL MODELLING OF STRAIN SOFTENING IN MICROCRACK-WEAKENED QUASI-BRITTLE MATERIALS

By using the concept of domain of microcrack growth(DMG),the micromechanisms of damage in quasi-brittle materials subjected to triaxial either tensile or compressive loading are investigated and the complete strew-strain relation including four stages is obtained from micromechanical analysis.The regime of pre-peak nonlinear hardening corresponds to the distributed damage,i.e.the stable propagation of microcracks.After the attainment of the ultimate strength of load-bearing capacity, some microcracks experience the second unstable growth and the distributed damage is transmitted to the localization of damage.These analyses improve our understanding of the hardening and softening behaviors of quasi-brittle materials.

Instability criterion for the system composed of elastic beam and strain-softening pillar based on gradient-dependent plasticity

A mechanical model is proposed for the system of elastic beam and strain-softening pillar where strain localization is initiated at peak shear stress. To obtain the plastic deformation of the pillar due to the shear slips of multiple shear bands, the pillar is divided into several narrow slices where compressive deformation is treated as uniformity. In the light of the compatibility condition of deformation, the total compressive displacement of the pillar is equal to the displacement of the beam in the middle span. An instability criterion is derived analytically based on the energy principle using a known size of localization band according to gradient dependent plasticity. The main advantage of the present model is that the effects of the constitutive parameters of rock and the geometrical size of structure are reflected in the criterion. The condition that the derivative of distributed load with respect to the deflection of the beam in the middle span is less than zero is not only equivalent to, but also even more concise in form than the instability criterion. To study the influences of constitutive parameters and geometrical size on stability, some examples are presented.

Geotechnical stability analysis considering strain softening using micro-polar continuum finite element method

Geotechnical stability analyses based on classical continuum may lead to remarkable underestimations on geotechnical safety.To attain better estimations on geotechnical stability,the micro-polar continuum is employed so that its internal characteristic length(lc)can be utilized to model the shear band width.Based on two soil slope examples,the role of internal characteristic length in modeling the shear band width of geomaterial is investigated by the second-order cone programming optimized micro-polar continuum finite element method.It is recognized that the underestimation on factor of safety(FOS)calculated from the classical continuum tends to be more pronounced with the increase of lc.When the micro-polar continuum is applied,the shear band dominated by lc is almost kept unaffected as long as the adopted meshes are fine enough,but it does not generally present a slip surface like in the cases from the classical continuum,indicating that the micro-polar continuum is capable of capturing the non-local geotechnical failure characteristic.Due to the coupling effects of lc and strain softening,softening behavior of geomaterial tends to be postponed.Additionally,the bearing capacity of a geotechnical system may be significantly underestimated,if the effects of lc are not modeled or considered in numerical analyses.

Alternative splicing of the PECTINESTERASE gene encoding a cell wall-degrading enzyme affects postharvest softening in grape

The firmness of table grape berries is a crucial quality parameter. Despite extensive research on postharvest fruit softening, its precise molecular mechanisms remain elusive. To enhance our comprehension of the underlying molecular factors, we initially identified differentially expressed genes(DEGs) by comparing the transcriptomes of folic acid(FA)-treated and water-treated(CK) berries at different time points. We then analyzed the sequences to detect alternatively spliced(AS) genes associated with postharvest softening. A total of 2,559 DEGs were identified and categorized into four subclusters based on their expression patterns, with subcluster-4 genes exhibiting higher expression in the CK group compared with the FA treatment group. There were 1,045 AS-associated genes specific to FA-treated berries and 1,042 in the CK-treated berries, respectively. Gene Ontology(GO) annotation indicated that the AS-associated genes in CK-treated berries were predominantly enriched in cell wall metabolic processes,particularly cell wall degradation processes. Through a comparison between treatment-associated AS genes and subcluster-4 DEGs, we identified eight genes, including Pectinesterase 2(VvPE2, Vitvi15g00704), which encodes a cell wall-degrading enzyme and was predicted to undergo an A3SS event. The reverse transcription polymerase chain reaction further confirmed the presence of a truncated transcript variant of VvPE2 in the FA-treated berries.Our study provides a comprehensive analysis of AS events in postharvest grape berries using transcriptome sequencing and underscores the pivotal role of VvPE2 during the postharvest storage of grape berries.

Influence of thermomechanical treatment on recrystallization and softening resistance of Cu-6.5Fe-0.3Mg alloy

The recrystallization and softening resistance of a Cu-6.5Fe-0.3Mg(mass fraction,%)alloy prepared by Process 1(cold rolling heat treatment)and Process 2(hot/cold rolling heat treatment)were studied using Vickers hardness tests,tensile tests,scanning electron microscopy and transmission electron microscopy.The softening temperature,hardness and tensile strength of the alloy prepared by Process 2 were 110°C,HV 15 and 114 MPa higher,respectively,than those of the alloy prepared by Process 1 after aging at 300°C.The recrystallization activation energy of the alloys prepared by Process 1 and Process 2 were 72.83 and 98.11 kJ/mol,respectively.The pinning effects of the precipitates of the two alloys on grain boundaries and dislocations were basically the same.The softening mechanism was mainly attributed to the loss of dislocation strengthening.The higher Fe fiber density inhibited the average free migration path of dislocations and grain boundary migration in the alloy,which was the main reason for higher softening temperature of the alloy prepared by Process 2.

Ground reaction curves for circular excavations in non-homogeneous,axisymmetric strain-softening rock masses

Fast methods to solve the unloading problem of a cylindrical cavity or tunnel excavated in elasto-perfectly plastic, elasto-brittle or strain-softening materials under a hydrostatic stress feld can be derived based on the self-similarity of the solution. As a consequence, they only apply when the rock mass is homogeneous and so exclude many cases of practical interest. We describe a robust and fast numerical technique that solves the tunnel unloading problem and estimates the ground reaction curve for a cylindrical cavity excavated in a rock mass with properties depending on the radial coordinate, where the solution is no longer self-similar. The solution is based on a continuation-like approach(associated with the unloading and with the incremental formulation of the elasto-plastic behavior), fnite element spatial discretization and a combination of explicit sub-stepping schemes and implicit techniques to integrate the constitutive law, so as to tackle the diffculties associated with both strong strain-softening and elasto-brittle behaviors. The developed algorithm is used for two practical ground reaction curve computation applications. The frst application refers to a tunnel surrounded by an aureole of material damaged by blasting and the second to a tunnel surrounded by a ring-like zone of reinforced(rock-bolted) material.

Expansion of spherical cavity of strain-softening materials with different elastic moduli of tension and compression

An expansion theory of spherical cavities in strain-softening materials with different moduli of tension and com-pression was presented. For geomaterials,two controlling parameters were introduced to take into account the different moduli and strain-softening properties. By means of elastic theory with different moduli and stress-softening models,general solutions cal-culating Tresca and Mohr-Coulomb materials' stress and displacement fields of expansion of spherical cavity were derived. The effects caused by different elastic moduli in tensile and compression and strain-softening rates on stress and displacement fields and development of plastic zone of expansion of cavity were analyzed. The results show that the ultimate expansion pressure,stress and displacement fields and development of plastic zone vary with the different elastic moduli and strain-softening prop-erties. If classical elastic theory is adopted and strain-softening properties are neglected,rather large errors may be the result.

THE ASYMPTOTIC SOLUTION TO THE ANTIPLANE SHEAR DYNAMICCRACK-TIP FIELD IN AN ELASTIC STRAIN-SOFTENING VISCOPLASTIC MATERIAL

The elastic strain softening-viscoplastic model is given in this paper. Using this model, the asymptotic stress and strain equations surrounding the tip of a propagating crock are given and numerical results ale obtained under antiplane shear. The analysis and calculation show that at the crack tip the strain possesses logarithmic singularity (ln(R/r))(1/(n+1)) while the stress is like (ln(R/r))(-n/(n+1)), therefore the asymptotic behaviour of the elastic strain-softening viscoplastic field is revealed under the antiplane shear.

Elevated temperature strain softening characteristics of 12Cr1MoV steel friction welding pipe joints

The strain softening characteristics analysis of 12Cr1MoV streel friction welding pipe joints.which were removed from heating apparatus pipes ager over 9 and is yearly of service, was undertaken. It is concluded that the evident softening did not occur in the weld metal and HAZ during serice. whereas it did in the base metal 2mm apart from the center of the weld metal. The difference is due to the restriction of the pipe hoop, which is formed on the joint during friction welding. The bainite. microstructure in the weld metal and HA Z, which is harder than the pearlite, pecrostpucture in the base metal, makes contpibution too. ThIS phenomenon is peculiar to the friction weldings pipes.

THE DYNAMIC RESPONSE OF A STRAIN-SOFTENING BEAM

The dynamic response of a strain-softening beam subjected to a transverse impulsive on its tip is investigated.A softening moment-curvature relation is assumed for the beam and a closed form solution is obtained for a special kind of load,which shows that there exists a softening region in the beam and this re- gion propagates along the beam.This result indicates that,except for the possible discrete softening points with rotation discontinuity caused by the deformation localization,the existence of the softening region and its travelling along the beam are the essential features of the dynamic response of a strain-softening beam.The results also show that the failure of the beam should take place under a special load and the critical condition on which the dynamic failure occurs is given.