Damage Evolution Equation Research Articles

A thermodynamically consistent strain difference-based nonlocal damage mechanics approach for failure analysis of quasi-brittle materials

The nonlocal integral approaches are established methods known for their effectiveness in dealing with the challenges of ill-posed behaviour and mesh sensitivity during the strain softening process, especially within classical continuum theories. The present work extends the strain difference-based nonlocal approach to continuum damage mechanics (CDM) theories for predicting the failure behaviour of quasi-brittle materials. The strain difference approach is adopted for two reasons: (i) it maintains physical realism and indirectly incorporates nonlocal effects while computing both the damage variable and the damage energy release rate, and (ii) the constitutive model maintains a symmetry nature in constructing the nonlocal part of the stiffness matrix, reducing the computational cost in strain-softening problems. The state and damage evolution equations for the present nonlocal approach are formulated within the thermodynamic framework. Numerical examples involving 2D and 3D cases are solved: (i) to study the influence of nonlocal parameter ‘α’ on the damage distribution while capturing the post-peak behaviours, (ii) to demonstrate the effectiveness of the present methodology by obtaining mesh-independent results and accurately predicting damage propagation paths, and (iii) to compare the experimental results and numerical structural response for coarse mesh, highlighting both computational efficiency and reduction of mesh-bias dependency. The results obtained by the strain difference-based nonlocal damage model agree well with the experimental results in the literature.

FEM and field tests to study the dynamic response of composite pavement surrounding embedded tram tracks to moving loading: implications to fatigue cracking

Under long-term tram loads, cracking occurs on the composite pavement surrounding an embedded tram track. However, in the current pavement design of the shared right-of-way section of a tram track, the trackside pavement fatigue characteristics are not considered. The investigation of the real dynamic strain response of the trackside composite pavement under tram loads is crucial for pavement system design. In this paper, a three-dimensional finite element model of tram-track- composite pavement was established. The rationality of this model was verified by a field pavement dynamic test. The influence of the track irregularity and tram speed on the dynamic characteristics of the pavement surrounding the track was explored. The trackside pavement strain distribution along the longitudinal and vertical directions of the track had been studied. The high-frequency strain components caused by the random shortwave track irregularity excitation were found in the strain frequency domain curves in different directions, and the track irregularity amplifies the influence of the tram speed on the surrounding pavement dynamic response. The loading frequency is an important input for determining the dynamic model and fatigue characteristics of asphalt materials. At the same time, the lab test loading frequency and fatigue life corresponding to the mixture were calculated by the mechanistic coupled model and empirical damage evolution equation. It provides an analytical base for the design of pavement surrounding embedded tram tracks.

Experimental study of the damage characteristics of rocks containing non-penetrating cracks under cyclic loading

The damage evolution process of non-penetrating cracks often causes some unexpected engineering disasters. Gypsum specimens containing non-penetrating crack(s) are used to study the damage evolution and characteristics under cyclic loading. The results show that under cyclic loading, the relationship between the number of non-penetrating crack(s) and the characteristic parameters (cyclic number, peak stress, peak strain, failure stress, and failure strain) of the pre-cracked specimens can be represented by a decreasing linear function. The damage evolution equation is fitted by calibrating the accumulative plastic strain for each cycle, and the damage constitutive equation is proposed by the concept of effective stress. Additionally, non-penetrating cracks are more likely to cause uneven stress distribution, damage accumulation, and local failure of specimen. The local failure can change the stress distribution and relieve the inhibition of non-penetrating crack extension and eventually cause a dramatic destruction of the specimen. Therefore, the evolution process caused by non-penetrating cracks can be regarded as one of the important reasons for inducing rockburst. These results are expected to improve the understanding of the process of spalling formation and rockburst and can be used to analyze the stability of rocks or rock structures.

Fatigue performance analysis of stainless steel cruciform joint with angular misalignment

AbstractWelding distortion causes the change of stress distribution of load‐carrying cruciform stainless steel welded joints, and the weld with angle less than 90° between steel plates became the fatigue performance control weld because of stress magnification. The effect of angular misalignment on fatigue performance of stainless steel cruciform welded joint was experimentally and numerically studied. The fatigue damage evolution equations, under different stress amplitudes, were derived using relative deformation increment as damage variable based on continuum damage mechanics to explain the process of damage transformation from micro to macro crack and finally rapid fracture. The hot spot stress magnification factor was positively correlated with the welding distortion angle and exhibited two discernible stages of rise and fall with increasing nominal stress taking the proportional limit as boundary. A proposed formula considering both misalignment angle and applied stress aimed to enhance accuracy in fatigue performance evaluation compared to current methods.

A microdamage model for FCC single crystals considering a mixed failure mechanism of slip and cleavage

In this study, a microdamage model is newly proposed to predict the failure process of FCC single crystals. Two micro damage mechanisms, slip and cleavage, are both involved in the model. A microscopic damage variable is first defined at the slip plane scale, and a damage coupled crystal plasticity constitutive equation in a co-rotational tensor framework is then established. A damage driving force consisting of generalized energies related to the resolved shear stress and the positive resolved normal stress is proposed to reflect the mixed failure mechanism of FCC single crystals induced by slip and cleavage. The damage evolution equation is then derived in the framework of thermodynamics. An explicit algorithm for the damage coupled crystal plasticity finite element method is given. The proposed microdamage model is utilized to predict the slip-dominant fracture behavior of a single crystal copper and the cleavage-dominant fracture behavior of a nickel-based single crystal alloy. The damage evolution process of slip planes is given and the influence mechanism is revealed. The calculated results agree well with the experimental data, which verifies the good capability of the proposed model in describing and predicting the multi failure processes of FCC metals. Finally, a 2D fracture simulation of a single crystal notched specimen under shear loading is conducted, and the effect of damage parameters on the fracture modes is also discussed.

Validation of a damage constitutive model based on logistic model dedicated to the mechanical behavior of the rock

The establishment of mechanical models of rocks can provide crucial guidance to the surrounding rocks stability analysis in the rock engineering. According to statistical damage theory and Mohr–Coulomb strength criterion, the Logistic model is used to yield the damage evolution equation and further establish statistical damage constitutive model for describing the whole failure process of rocks. Then, the rationality of the proposed damage model is proved by comparing with the different test curves, and damage evolution laws are summarized into four stages, including: slow-growth stage, rapid-growth stage, constant-speed-similar growth stage and speed-reducing growth stage. Physical meanings of model parameters r′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${r}{\\prime}$$\\end{document} and m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m$$\\end{document} are also discussed. Finally, to verify the application of this proposed model, this damage constitutive model is developed and implemented to simulate the tunnel excavation and analyze the stability of surrounding rocks by using Fortran, python and Abaqus. The displacement data on site is used to prove the superiority of proposed damage constitutive model compared with the existing ones in Abaqus. The research outcomes presented in this paper can also provide useful references for the theory and application of rock mechanics.

Research on printing parameters and salt frost resistance of 3D printing concrete with ferrochrome slag and aeolian sand

Aeolian sand (AS) and ferrochrome slag (FS), an industrial byproduct, are abundant in Inner Mongolia. Rational use of FS and AS resources can lead to better economic benefits. In this study, FS and AS were used as fine aggregates to prepare 3D printed concrete (3DPFAC). The effect of different printing parameters on the mechanical properties of 3DPFAC was investigated and the optimal printing parameters were determined. The durability of 3DPFAC in salt freeze-thaw cycle (FTC) environments was analyzed by FTC tests. The mass loss rate, relative dynamic elastic modulus, and compressive strength loss rate are used to characterize the performance degradation law of 3DPFAC and the mechanism of performance degradation is revealed by SEM, XRD, and bubble spacing analyzer. We conclude that the optimal printing parameters for 3DPFAC are 20 mm print nozzle diameter, 8 mm print layer height, 50 mm/s print travel speed, and consecutive print interval time. Under different production processes, the degree of damage and deterioration of concrete is 3D-Y (loading along the Y direction) > 3D-X (loading along the X direction) > JZ (pouring specimens). After 400 FTCs, the mass loss rates for JZ, 3D-X and 3D-Y were 3.02 %, 3.12 % and 3.56 %, respectively, the relative dynamic elastic modulus decreased to 71.78 %, 66.18 % and 58.64 %, respectively, with compressive strength loss rates of 21.0 %, 22.4 % and 24.0 %. Finally, the damage evolution equation for 3DPFAC is derived using the Weibull distribution model, and the lifetime prediction is performed based on the Weibull distribution damage model. The results can provide a theoretical basis for the application of FS and AS in the 3DPC domain.

Dynamic damage constitutive model for UHPC with nanofillers at high strain rates based on viscoelastic dynamic constitutive model and damage evolution equation

This study established a dynamic damage constitutive model for ultra-high performance concrete (UHPC) with nanofillers, based on a viscoelastic dynamic constitutive model and a damage evolution equation. Ten types of nanofillers, including particle, tube and flake nanofillers, were incorporated to modify UHPC. The split Hopkinson pressure bar was used to obtain the relationship between stress and strain of UHPC specimens at a strain rate of 200/s-800/s. The experimental results indicated that the dynamic compressive strength of UHPC with nanofillers at strain rates of approximately 200/s, 500/s, and 800/s can reach 172.8 MPa, 219.6 MPa, and 275.9 MPa, respectively, reflecting an increase of 85.2 %, 76.5 %, and 53.9 % compared with the blank UHPC. The established dynamic damage constitutive model considered the damage accumulation with strains under dynamic loading. The fitting coefficients of the dynamic damage constitutive model, when compared against experimental results, range from 0.8796 to 0.9963, showing a higher accuracy compared with traditional Zhu-Wang-Tang (ZWT) viscoelastic model, especially at a strain rate of approximately 200/s.

Experimental Study on the Mechanical Properties of Hybrid Basalt-Polypropylene Fibre-Reinforced Gangue Concrete

Incomplete data indicate that coal gangue is accumulated in China, with over 2000 gangue hills covering an area exceeding 200,000 mu and an annual growth rate surpassing 800 million tons. This accumulation not only signifies a substantial waste of resources but also poses a significant danger to the environment. Utilizing coal gangue as an aggregate in the production of coal-gangue concrete offers an effective avenue for coal-gangue recycling. However, compared with ordinary concrete, the strength and ductility of coal-gangue concrete require enhancement. Due to coal-gangue concrete having higher brittleness and lower deformation resistance than ordinary concrete, basalt fibre (BF) is a green, high-performance fibre that exhibits excellent bonding properties with cement-based materials, and polypropylene fibre (PF) is a flexible fibre with high deformability; thus, we determine if adding BF and PF to coal-gangue concrete can enhance its ductility and strength. In this paper, the stress–strain curve trends of different hybrid basalt–polypropylene fibre-reinforced coal-gangue concrete (HBPRGC) specimens under uniaxial compression are studied when the matrix strengths are C20 and C30. The effects of BF and PF on the mechanical and energy conversion behaviours of coal-gangue concrete are analysed. The results show that the ductile deformation of coal-gangue concrete can be markedly enhanced at a 0.1% hybrid-fibre volume content; HBPRGC-20-0.1 and HBPRGC-30-0.1 have elevations of 53.66% and 51.45% in total strain energy and 54.11% and 50% in dissipative energy, respectively. And HBPRGC-20-0.2 and HBPRGC-30-0.2 have elevations of 31.95% and 30.32% in total strain energy and −3.46% and 28.71% in dissipative energy, respectively. With hybrid-fibre volume content increased, the elastic modulus, the total strain energy, and the dissipative energy all show a downward trend. Therefore, 0.1% seems to be the optimum hybrid-fibre volume content for well-enhancing the ductility and strength of coal-gangue concrete. Finally, the damage evolution and deformation trends of coal-gangue concrete doped with fibre under uniaxial action are studied theoretically, and the constitutive model and damage evolution equation of HBPRGC are established based on Weibull theory The model and the equation are in good agreement with the experimental results.

Application of concrete damage evolution equation considering confining pressure and strain rate effects in projectile penetration

Repeated loading and unloading tests were carried out on concrete under different confining pressures and strain rates to characterize the damage using the decrease in elastic modulus, and a damage evolution equation considering the effects of confining pressure and strain rate was established. At the same time, the yield criterion and equation of state of the Holmquist–Johnson–Cook (HJC) model were modified by combining the experimental data of high confining pressure. The modified model was validated by performing single-element numerical simulations for different stress states. Then, experiments were carried out on the penetration of tungsten alloy projectiles into concrete slabs, with penetration velocities ranging from 172 m/s to 302 m/s. The depth of penetration and crater area were obtained. Numerical simulations were subsequently carried out using the proposed model. The results showed that this model can describe the behaviour of concrete under dynamic loading. Finally, the damage evolution mechanism of concrete during the penetration process is analysed in conjunction with the proposed damage evolution equation.

Simulation of hydraulic fracture propagation based on implicit gradient non-local model

Fracture extension simulation is an important tool for studying hydraulic fracture characteristics. To solve the problem of mesh sensitivity of strain and fluid pressure during hydraulic crack propagation, the equivalent strain in the damage evolution equation was non-localized, and implicit gradient was performed. A hydraulic-mechanical coupling model based on implicit gradient non-local damage was established. Numerical simulation program was further developed. Comparison of simulation results with experimental data on ‘L’-brittle material verified the validity of the program. The fracture propagation process of hydraulic fracturing in a perforated well was simulated. On this basis, the influence of internal length scale parameters and mesh size of the non-local model on the calculation results was analyzed. The influence of mesh size on hydraulic fracture morphology and damage variables is reduced by introducing an implicit gradient nonlocal model, which effectively alleviates the localization effect in hydraulic-mechanical coupling damage simulation. This new model achieves the nonlocalization of the fluid field by introducing the relationship among porosity-permeability and deformation and damage, which avoids the mesh sensitivity of equivalent strain, damage, fluid pressure, and improves the numerical stability. It provides a new method for simulating hydraulic fracture propagation and studying the failure behavior of porous media materials.

Achieving irreversibility in damage evolution: Extended gradient damage model with decoupled damage profile and cohesive law

The irreversibility of damage evolution is a cornerstone of classical damage mechanics. Unfortunately, the existing gradient damage models in the literature cannot always fulfill this critical condition. Therefore, the Extended Gradient Damage (EGD) model is proposed in the present work in response to this challenge. In the EGD model, the governing equation of damage evolution is developed in the form that the damage evolution and cohesive law are fully decoupled. Thus, the irreversible damage evolution could be easily established by defining a well-behaved damage profile function without any intervention from cohesive law. Other than that, complex cohesive laws for different materials could be well captured in the EGD model with the help of the integral transformation technique. Moreover, the variational form of the proposed EGD model is developed by taking advantage of the dissipative force term, and the principle of thermodynamics is proven to be fulfilled. In the end, the proposed EGD model is validated by four numerical examples.

Coupling model for the gradient loading creep behavior of nickel-based single-crystal superalloys

The emergency situation caused by the abnormal service of an aeroengine, such as gradient loading, directly leads to blade rupture. Considering the effect of gradient loading on creep, a coupling creep model of nickel-based single crystals under gradient loading conditions is proposed based on crystal plastic theory and thermodynamic damage theory. Internal state variables—elastic strain, plastic strain and damage—are introduced to describe the thermodynamic irreversible process of a system. An anisotropic creep damage evolution equation represented by the damage strain energy release rate Y is deduced by the energy conversion method. The gradient loading creep behavior predicted by the finite element calculation is in good agreement with the experimental results.

Damage Evolution of Polypropylene-Basalt Hybrid Fiber Ceramsite Concrete under Chloride Erosion and Dry-Wet Cycle.

To investigate the influence of polypropylene-basalt hybrid fibers (PBHFCC) on the durability of ceramsite concrete, this study determined the appearance change, mass loss rate, relative dynamic elastic modulus, compressive strength and splitting tensile strength of ceramsite concrete with four kinds of hybrid fibers volume admixture under chloride erosion and dry-wet cycles. The results reveal that under this effect, the apparent damage of each group of specimens increased with the growth of the erosion time. The quality, compressive strength and splitting tensile strength of the specimens all increased gradually during the erosion age period of the first 72 d and gradually decreased after 72 d. The relative dynamic elastic modulus was similarly mutated in 48 d. When the hybrid fiber content of the specimens is 0.15 vol %, the enhancement effect of ceramsite concrete is better than that of the other three amounts. The relative dynamic elastic modulus value is used as a damage variable to establish the damage equation, and the damage evolution equation of PBHFCC considering the volume of hybrid fiber under chloride erosion and dry-wet cycle is derived. The conclusions can be used as a reference for the durability design and construction of PBHFCC.

Acoustic emission characteristics of recycled brick-concrete aggregate concrete under freeze-thaw-load coupling action and damage constitutive model

Replacing natural aggregate with construction waste to prepare concrete is an effective way of its resource utilization, which is in line with China 's " carbon peaking and carbon neutrality " strategy and the green development needs of the construction industry. Due to the existing technology is difficult to completely separate waste bricks and waste concrete, this paper mixes recycled brick aggregate and recycled concrete aggregate in a ratio of 1:1 into a mixture, and replaces the natural aggregate with a replacement rate of 30% to prepare Recycled Brick-Concrete Aggregate Concrete (RB-CAC). Concrete in northern China is often subjected to freeze-thaw and load coupling, which accelerates the damage degradation of RB-CAC. Therefore, it is necessary to study its damage constitutive model. In this paper, the effects of different stress levels (0%, 20%, 40%, 60% of flexural strength) and freeze-thaw cycles (0 times, 40 times, 80 times, 120 times, 160 times) on the apparent morphology, mass loss, relative dynamic elastic modulus and acoustic emission characteristics of RB-CAC were analyzed by freeze-thaw-load coupling test, constitutive relation test under uniaxial compression and acoustic emission characteristic test. The dynamic evolution of its frost resistance and damage process was analyzed and evaluated. Finally, combined with the knowledge of damage mechanics, the elastic modulus is defined as the damage variable generated by the freeze-thaw-load coupling effect. Based on the mesoscopic damage theory, the ratio of the number of failure interfaces to the total number of interfaces is defined as the damage variable generated by uniaxial pressure. The damage evolution equation of RB-CAC and the damage constitutive model of freeze-thaw times and stress level are established.

Creep mechanical properties and creep damage model of sandstone considering temperature effect based on acoustic emission

AbstractThis work studies the creep mechanical properties and creep damage model of high‐temperature heat treated sandstone. Acoustic emission (AE) technology is used to monitor the crack evolution of sandstone in uniaxial compression and creep experiments in real time. The influence of temperature on the mechanical properties of sandstone is analyzed through AE count, cumulative AE count, and AE absolute energy. The relationship between cumulative AE count and axial strain is discussed, and the cumulative AE count increases exponentially with increasing axial strain. Moreover, an appropriate damage evolution equation is established based on the cumulative AE count, and a nonlinear viscoelastic‐plastic model capable of describing the decay creep, steady‐stage creep, and accelerated creep stages of high‐temperature heat treated sandstone is constructed. Through the model validation and parameter inversion, the new creep damage model can be employed to study the creep instability of geotechnical engineering under high temperature environment.

Modelling of damage and failure within polymeric adhesives

AbstractWithin this contribution, we propose a fully thermo‐mechanical coupled isotropic damage model for polymeric adhesives under finite strains. This model is based on the multiplicative decomposition of the deformation gradient into mechanical and thermal parts. We consider rate‐dependent damage behaviour (e.g. creep damage) of the material by applying a Perzyna‐type ansatz for the damage evolution equations. To overcome the pronounced mesh dependencies which would result from the usage of a local damage model, we make use of a gradient‐extended damage formulation. Besides the main aspects of model, we show the thermodynamically consistent derivation of constitutive quantities as well as the numerical treatment of the governing equations. Finally, we show selected numerical examples to demonstrate the capabilities of the model.

Study on Mechanical Properties of Deep Expansive Soil and Coupling Damage Model of Freeze–Thaw Action and Loading

This study is primarily intended to present a damage constitutive equation under the combined action of confining pressures and freeze–thaw cycles by subjecting deep expansive clay to the consolidated undrained triaxial tests. We study the influence of the numbers of freeze–thaw cycles on various mechanical indexes of soil by using the TSZ-2 instrument (fully automatic triaxial instrument). As the number of freeze–thaw cycles increases, the ultimate peak stress of the soil decreases, and then, the effect of the freeze–thaw effect on the shear strength gradually weakened. By combining the expression method of the damage variable under the action of loading alone with the expression method under the action of freeze–thaw cycles alone, we brought in the damage evolution equation to obtain the damage constitutive equation under the combined action of confining pressures and freeze–thaw cycles. The stress values under three confining pressures (100 kPa, 200 kPa, and 300 kPa) can be determined by using the final damage constitutive model. The measured data with a water content of 17% and six freeze–thaw cycles were compared with the theoretical data. The actual strength values were 118.4 kPa, 152.3 kPa, and 184.1 kPa, and the theoretical strength values were 120 kPa, 150 kPa, and 186 kPa. The fitting degree of the strength value was as high as 99%, which verifies the feasibility of this model. This study can serve as an available reference for well wall construction and disaster prediction in deep coal mining.

A nonlinear fatigue damage accumulation model under variable amplitude loading considering the loading sequence effect

Damage accumulation is nonlinear in the actual fatigue process, and the loading sequence is an important factor under variable amplitude loading. In this paper, a nonlinear fatigue damage accumulation model under variable amplitude loading considering the loading sequence effect is proposed. First, the cumulative rule of fatigue damage under variable amplitude loading is analyzed. It is suggested that the exponential function based on the load cycle ratio can be used to characterize the load sequence effect. Then, a nonlinear fatigue damage evolution equation of materials under constant amplitude load is proposed by taking the static toughness of the material as the damage parameter. Furthermore, the exponent of the load sequence influence parameter is determined as a function of the cycle life and elastic modulus. Then, a fatigue damage accumulation model under variable amplitude loading that considers the effect of the loading sequence is obtained based on the damage equivalence principle. Finally, the model is validated and compared through fatigue test data of five metal materials. It is found that the prediction accuracy of the proposed model is significantly higher than the other two models. In contrast, the difficulty of using the proposed model is lower than that of similar models.

Research on compressive damage mechanism of concrete based on material heterogeneity

To reveal the influence of material heterogeneity on the mechanical properties of concrete, computed tomography (CT) tests of concrete under uniaxial compression were carried out. A CT scan test of concrete under uniaxial compressive stress was carried out, and CT scan images of a loading stage were obtained. Based on the theory of damage evolution, a mapping relationship between elastic modulus and CT number was constructed, andthe “numerical concrete model of material heterogeneity” was established. The equation for damage evolution and the constitutive equation for subsection damage based on porosity were constructed. Then, the influence of heterogeneity on the damage mechanical properties of concrete under uniaxial compressive stress was studied. The results show that material heterogeneity has a more significant effect on the mechanical properties of concrete due to damage. In addition, the elastic modulus of the material heterogeneity samples is smaller than that of the structural heterogeneity samples, and the peak strength of the material heterogeneity samples is 10.7% lower than that of the structural heterogeneity samples. The numerical concrete model of material heterogeneity can effectively reflect the initial defects of the concrete. Furthermore, the crack propagation of the sample is orderly. The crack is evident around the aggregate and has good directivity. The shear crack characteristics are also noticeable. Meanwhile, the main crack is roughly at a 45-degree angle with the sample, which is closer to the actual physical CT test.