Damage Evolution Equation Research Articles

Phase field theory for fracture at large strains including surface stresses

Phase field theory for fracture is developed at large strains with an emphasis on a correct introduction of surface stresses at nanoscale. This is achieved by multiplying the cohesion and gradient energies by the local ratio of the crack surface areas in the deformed and undeformed configurations and with the gradient energy in terms of the gradient of the order parameter in the reference configuration. This results in an expression for the surface stresses which is consistent with the sharp surface approach. Namely, the structural part of the Cauchy surface stress represents an isotropic biaxial tension, with the magnitude of a force per unit length equal to the surface energy. The surface stresses are a result of the geometric nonlinearities, even when strains are infinitesimal. They make multiple contributions to the Ginzburg-Landau equation for damage evolution, both in the deformed and undeformed configurations. Important connections between material parameters are obtained using an analytical solution for two separating surfaces, as well as an analysis of the stress-strain curves for homogeneous tension for different degradation and interpolation functions. A complete system of equations is presented in the deformed and undeformed configurations. All the nanoscale phase field parameters are obtained utilizing the existing first principle simulations for the uniaxial tension of Si crystal in the [100] and [111] directions.

Damage characteristics and constitutive model of concrete under uniaxial compression after Freeze-Thaw damage

In order to study the uniaxial compressive damage characteristics and stress–strain behavior of concrete with different degrees of freeze–thaw (F-T) damage, and then effectively evaluate the mechanical properties of concrete under different F-T damage degree. The rapid F–T cycle test was performed on different groups of concrete specimens, and an F–T damage evolution equation was established according to the change law of the relative dynamic elastic modulus. Uniaxial compression tests were performed on specimens with different F–T cycles, and damage evolution during compression was studied using an acoustic emission (AE) system. Finally, the AE characteristic parameters were used to establish a uniaxial compression constitutive model of concrete considering F–T damage. The results showed that the degree of F–T damage increased as the F–T cycles increased. During the concrete compression failure, the AE amplitude showed a more obvious and rapid rising trend when it was close to the critical state point. The proportion of low-amplitude AE events (≤50 dB) gradually decreased, whereas that of medium- and high-amplitude events exhibited a corresponding upward trend. These results indicate that the concrete after F–T damage exhibited more brittle characteristics during the compression failure process. The damage constitutive model established in this study can accurately reflect the damage evolution law of concrete during the entire process of uniaxial compression after F–T cycles. Moreover, the model was sufficiently consistent with test data, providing a reference for the design and safe operation of concrete structures in cold regions.

A New Damage Model of Rock Considering Residual Strength under Seepage Pressure

Seepage pressure changes the rock structure and stress state, inducing rock mass damage and destruction under hydromechanical coupling. To describe the deformation characteristic and the entire stress–strain process, we suppose that rock consists of damaged part, undamaged part, and voids part. The three parts are intermingled to carry external loads and the undamaged part gradually becomes damaged part in compression. A redefinition of the damage variable and evolution equation is proposed incorporating damage mechanics and statistical damage theory. Considering that axial loads of the damage parts equal to residual strength, a new statistical damage softening constitutive model is developed. The model is validated by using the experimental data. The proposed model can reflect the entire stress–strain law very well, especially softening deformation characteristics after the peak. The research results are helpful in the rock mass engineering safety analysis under seepage pressure.

Study of SPRC Impact Resistance Based on the Weibull Distribution and the Response Surface Method.

Silica-fume–polyvinyl-alcohol-fiber-reinforced concrete (SPRC) is a green and environmentally friendly composite material incorporating silica fume and polyvinyl alcohol fiber into concrete. To study the impact resistance of SPRC, compressive-strength and drop hammer impact tests were conducted on SPRC with different silica-fume and polyvinyl-alcohol-fiber contents. The mechanical and impact resistance properties of the SPRC were comprehensively analyzed in terms of the compressive strength, ductility ratio and impact-energy-dissipation variation. Based on the impact resistance of the SPRC, the impact life of SPRC with different failure probabilities was predicted by incorporating the Weibull distribution model, and an impact damage evolution equation for SPRC was established. The impact life of SPRC under the action of silica-fume content, polyvinyl-alcohol-fiber content and failure probability was analyzed in depth by the response surface method (RSM). The research results show that, when the content of silica fume is 10% and the content of polyvinyl alcohol fiber is 1%, the compressive strength and impact resistance of SPRC are the best. The RSM response model can effectively predict and describe the impact life of SPRC specimens under the action of three factors.

Statistical damage constitutive model for the two-component foaming polymer grouting material

Abstract Two-component foaming polymer (TFPU) grouting material is increasingly used in civil engineering. Its compressive strength is key to achieving the desired enhancing effect. The constitutive model of TFPU grouting material is a theoretical basis to evaluate the strength performance, which, however, is not fully understood. Here the uniaxial compression experiment of TFPU samples of different densities (0.11–0.53 g·cm−3) was conducted. Based on the stress–strain curves, the damage evolution equation of each sample was obtained by function fitting, followed by the establishment of statistical damage constitutive model. The model was simplified to a universal function with density as the argument. Results show that the stress–strain curves contain the initial compression stage, linear elastic stage, yield stage, yield plateau stage, and strain hardening stage regardless of the varied density. The variation laws of the damage with strain conform to the form of first-order decay exponential function. The theoretical stress–strain curves are in good agreement with the experimental ones, indicating that the statistical damage constitutive model can well reflect the mechanical behavior of TFPU grouting material. With this constitutive model, the mechanical properties of TFPU grouting material can be obtained according to the density alone, which is more convenient for practical engineering applications.

A new set of physically-based unified viscoplastic constitutive equations for superplastic deformation of Al-Zn-Mg-Cu alloy

A new set of physically-based unified viscoplastic constitutive equations using internal state variable(ISV) method during the superplastic deformation of Al-Zn-Mg-Cu alloy at temperatures ranging from 480 to 530 ℃ and strain rates ranging from 3×10−4s−1 to 3×10−3s−1 has been proposed. The damage evolution equation in this set of constitutive equations takes into account the critical cavity nucleation strain. This set of constitutive equations can predict microstructure evolution and flow stress during superplastic deformation of the investigated Al-Zn-Mg-Cu alloy. The set of constitutive equations have been coupled into crystal plasticity finite element method for determining the evolution of grain size and cavity volume fraction and the lattice rotation field map. The Material parameters are determined by genetic algorithm. The influences of strain, strain rate and temperature on grain size evolution and cavitation behavior were analyzed. The predicted stress-strain behavior, grain size and damage evolution are in good agreements with the experimental results.

A direction-dependent smoothing gradient damage model for anisotropic brittle fracture

We present a novel smeared gradient-enhanced damage model which takes into account direction-dependent damage evolution in two-dimensional brittle materials (e.g., uniaxial fiber-reinforced composites and polycrystalline materials). The recently developed smoothing gradient damage model for isotropic localized failure analysis is revisited by introducing a new damage evolution equation towards anisotropic fracture. To this end, the evolving anisotropic gradient interaction parameter existed in the damage evolution law is redefined by associating with a second-order structural tensor, aiming to possibly drive the directional fracture properties. This structural tensor restrains the crack growth in the direction of predefined fracture plane. In this circumstance, all cleavage or family of cleavage in a grain (for instance in polycrystalline materials) are assumed to be described by a single cleavage plane (or fiber plane) where the resistance of materials is considered to be weakest. Consequently, the directional dependency of cleavage fracture and transgranular fracture are captured precisely. This new definition utilized solely a scalar damage variable is in contrast to most other existing gradient damage models which usually employ multiple damage variables or higher-order gradient terms. Here, we also introduce an effective staggered algorithm aiming to reduce the computational cost, and its desirable features are demonstrated in the numerical experiments for transversely isotropic materials. The predicted crack paths in unidirectional fiber-reinforced composites and polycrystalline materials are in good agreement with reference results derived from other numerical methods.

Dynamic mechanical response and damage evolution of cemented tailings backfill with alkalized rice straw under SHPB cycle impact load

Dynamic disturbances such as blasting excavations cause damage to backfills. To ascertain and improve the dynamic mechanical response of backfill under frequent disturbance, single and cyclic impact load tests were conducted on cemented tailings backfill mixed (CTB)with different contents of alkalized rice straw (ARS) using a split Hopkinson pressure bar. The evolution laws of the dynamic mechanical properties and damage characteristics of CTB with the impact times and the ARS content were analyzed. The results showed that (1) The dynamic compressive strength (DCS) of CTB was improved by the ARS under single impact load with different strain rates. (2) Under the cyclic impact loads, the DCS, dynamic elastic modulus, and peak strain of CTB decreased with the increase in the cyclic impact times. Under the same impact times, the DCS of the backfill first increased and subsequently decreased with the increase in the ARS content, and 0.3% was the best content. (3) The unit volume absorbed energy of CTB was increased by the ARS and the longitudinal wave velocity (LWV) decay rate of CTB was decelerated under the cyclic impact loads. (4) The established damage evolution equation of CTB based on the LWV showed that its cumulative damage degree was increased by the cyclic impacts. Moreover, the incorporation of ARS was beneficial for decelerating and reducing the damage evolution rate and degree. The ARS was wrapped by hydration products in the CTB matrix, and its adhesion with the matrix promoted the ARS to play a bridging role, and inhibited and slowed down the development, propagation degree and propagation rate of cracks in CTB under cyclic impact. When the content of ARS was further increased, the probability of overlapping was significantly improved, which was not conducive to the adhesion between ARS and CTB matrix, resulting in the decline of the mechanical properties of CTB. The research results provide a reference for the understanding of the dynamic mechanical response of CTB with ARS under frequent blasting disturbances.

A Temporal Multiscale Model for Fatigue Damage of Concrete

A temporal multiscale model was developed to characterize the damage that evolves in concrete structures throughout a complete scenario of dynamic fatigue loading. The damage was decomposed into quasistatic and dynamic components, and its evolution was controlled by introducing negative and positive feedback mechanisms. Fatigue damage evolution equations of the power law type were used together with a deduced temporal multiscale scheme to allow a computationally efficient finite-element method (FEM) simulation of the damage that evolves during the whole loading process. The validity and computational efficiency of the FEM model were assessed by comparing its predictions with published experimental data.

Prediction of low-cycle crack initiation life of powder superalloy FGH96 with inclusions based on damage mechanics

The effects of inclusions in powder superalloy FGH96 on low-cycle fatigue life were studied, and a low-cycle crack initiation life prediction model based on the theory of damage mechanics was proposed. The damage characterization parameter was proposed after the construction of damage evolution equations. Fatigue tests of the powder superalloy specimens with and without inclusion were conducted at 530 and 600 °C, and the model verification was carried out for specimens with elliptical, semi-elliptical, polygon and strip-shaped surface/subsurface inclusion. The stress analysis was performed by finite element simulation and the predicted life was calculated. The results showed a satisfying agreement between predicted and experimental life.

Study on Mechanical Behavior and Energy Mechanism of Sandstone under Chemical Corrosion.

Chemical corrosion has a significant impact on the properties of rock materials. To study the mechanical behavior and energy mechanism of rock under chemical corrosion, this paper took the sandstone of Haitangshan tunnel in Fuxin as the research object, used a Na2SO4 solution to simulate different chemical environments, carried out a triaxial loading test on sandstone through the MTS815.02 test system, and analyzed the mechanical parameters and energy damage evolution law of sandstone under different chemical environments. The test results showed that the basic mechanical parameters (peak strength σpk, peak strain εpk, elastic modulus E, cohesion c, and internal friction angle φ) and characteristic stress parameters (closure stress σcc, initiation stress σci, and dilatancy stress σcd) of sandstone first increased and then decreased with the increase of pH in the Na2SO4 solution, Poisson’s ratio µ showed the opposite trend, and the extreme values of all parameters were taken when pH = 7. The influence degree of different pHs on the mechanical parameters of sandstone were as follows: strong acid environment (pH ≤ 4) > strong alkali environment (pH ≥ 10) > weak acid environment (4 ≤ pH < 6) > weak alkali environment (8 ≤ pH < 10) > neutral environment (6 < pH< 8). The total energy and elastic strain energy increased first and then decreased, and the dissipated energy was the opposite. The damage variable decreased first and then increased. With the increasing concentration of the Na2SO4 solution, all the above parameters changed monotonically. Based on the energy theory, the damage evolution equation considering the effect of the Na2SO4 concentration was established. Combined with the test data, the model was verified and the result was good. Under the action of Na2SO4 corrosion, Ca2+ in calcite and Fe2+ in hematite were dissolved and precipitated. With the gradual increase of Ca2+ and Fe2+ concentration, the damage variable increased gradually. The relationship between the two ion concentrations and the damage variable approximately satisfied a linear function.

Evaluation of white sandstone mechanical behaviour and the energy evolution of prepeak unloading damage

Deep high-stress roadway excavation under unloading disturbance inevitably leads to damage deterioration of the surrounding rock, which poses a serious threat to its stability. To explore the energy characteristics of white sandstone damaged by peak front unloading, uniaxial compression tests were conducted on damaged rock samples. The results show that the peak strength and modulus of elasticity of the rock sample gradually decrease with increasing damage degrees. The external work input energy, releasable elastic strain energy and dissipation energy all decreased with increasing damage. Damage evolution curves and equations of the rock samples were obtained based on the damage instantiation model established by the principle of energy dissipation and release. The effects of unloading damage on the fracture characteristics of the rock samples were analysed from both macro and microscopic viewpoints, and the results showed that a micro fracture in the rock is transformed from brittle–ductile damage, while macroscopic damage occurs in the form of a "shear"-"splitting"-"mixed shear-splitting" damage process. This paper has certain research and reference value for understanding the damage evolution characteristics of rocks with peak front unloading damage.

Investigation of ultra-high performance concrete slabs under contact explosions with a calibrated K&C model

Karagozian and Case (K&C) concrete model is extensively adopted in the numerical simulations of ultra-high performance concrete (UHPC) structural members subjected to impulsive loads such as impact and blast. In this study, a calibration of the K&C concrete model was conducted for UHPC in terms of three strength surfaces, equation of state, shear dilatancy, damage evolution and strain rate effect to offer simple and general guidelines on the determination of key model parameters for this new class of concrete. With the calibrated concrete model, a single element method was adopted to verify its accuracy through a comparison to the results from the static tests of the uniaxial compression, direct tension and triaxial compression. Furthermore, the numerical simulations of contact explosion tests on the UHPC slabs with the incorporation of the strain rate effect were performed and the numerical results exhibited good predictions regarding the failure mode, crater and scabbing damage as compared to the test results. More importantly, this proposed numerical model and simulation methodology are reasonable to be generally used for structural members constructed of UHPC materials under contact explosions when lacking sufficient static and dynamic test data. Using the calibrated and validated K&C concrete model, parametric studies were conducted to derive a new empirical equation for predicting the local damage mode of UHPC slabs under contact explosions.

Damage Evolution Equation Reflecting Hydrostatic Pressure Effect and Its Application to Concrete Slabs under Projectile Impact

In this study, triaxial cyclic loading and unloading compression tests were performed on concrete specimens under six confining pressures (0, 5, 10, 20, 30, and 40 MPa). The elastic modulus was reduced to characterize the damage, and the damage-plastic strain relationships under different confining pressures were determined. The results showed that the damage evolution rate declined significantly as the confining pressure increased. The experimental results of the uniaxial-tension cyclic loading and unloading tests were combined to derive a damage evolution equation that reflected the hydrostatic pressure effect, and the material parameters were determined by using experimental data. Finally, the derived damage evolution equation was embedded into the Holmquist-Johnson-Cook model and applied to numerical simulations of a concrete slab under projectile impact. The results indicated that the modified Holmquist-Johnson-Cook model can accurately explain the damage evolution process of a concrete slab under projectile impact. Thus, the proposed equation can describe the damage evolution processes of concrete slabs under both impact compression and tension.

Damage and Fracture Analysis under Multi-axial Stress State Based on Damage Mechanics

In various press forming and deep drawing processes, it is important to anticipate the forming limit in multi-axial stress state. In the uniaxial tension test, it is necessary to take stress triaxiality into account for accurate prediction of failure strain, because after the onset of the necking, the constricted portion, which is actually subjected to loading, is in the multiaxial stress state due to the change in local cross-sectional area. Recently, several damage and fracture models have been proposed that focus on the effect of stress triaxiality on the forming limit, while most of them are phenomenological models. In this study, the constitutive equation and damage evolution equation, which are based on a second-order symmetric damage tensor within the framework of the continuum damage mechanics, are constructed as a function of the stress triaxiality to analyze damage and fracture of materials in multiaxial stress state. Then, FEM analysis of an example problem was carried out to validate this material model.

Phase Field vs Gradient Enhanced Damage Models: A Comparative Study

A comparison of the two different approaches for modelling damage in material in an infinitesimal strain setting is studied. The first approach is the nonlocal gradient enhanced damage model where the damage variable is taken as an independent variable which will be determined based on the local strain measure. Here, the nonlocal integral form is approximated to an implicit or explicit differential form using the Taylor’s series expansion for simpler numerical implementation. The second approach is the phase field damage model where a Helmholtz free energy density function is considered that includes a new energy degradation function along with a phase field non-conserved order parameter. The first variational principle on this energy density functional with respect to the corresponding order parameter variable will reach a stationarity value resulting in the non-conserved Allen-Cahn equation. The relationship of the order parameter with the damage variable gives the Allen-Cahn evolution equation for damage. A 1D bar example is considered for commenting on the similarities and differences of the two approaches to damage based on the mesh convergence studies based on the results obtained for various meshing profiles, changing length scale parameter values and the obtained damage profiles.

Fatigue damage identification method of cement-based nanocomposites based on strain mode

In order to overcome the problems of low sensitivity, low accuracy and long time of traditional methods, a fatigue damage identification method for cement-based nanocomposites based on strain mode was designed in this paper. First, on the basis of calculating the stress concentration coefficient, the mode decomposition method is used to traverse the stress wave and obtain the new stress value. Then, the propagation process of material fatigue damage is separated by the number of load cycles, and the fatigue damage evolution equation is established. Finally, combined with the ultrasonic wave equation, the location and magnitude of fatigue damage are identified according to the position and amplitude of reflected wave. Experimental results show that the proposed method has the highest sensitivity of 0.944, the highest damage identification accuracy of 97.72%, and the minimum identification time is only 11s, indicating that the proposed method has high reliability and timeliness.

Calibration of KCC model for UHPC under impact and blast loadings

Ultra-high performance concrete (UHPC) is a promising material for both civil and military protective structures against the potential impact and blast loadings. However, the existing numerical simulation studies related to the material model calibrations for precisely describing the dynamic behaviors of UHPC structural members under both impact and blast loadings are limited, and the calibration procedures are also not in consensus. This paper aims to propose a comprehensive Karagozian & Case Concrete (KCC) model parameter calibration procedure and reliable parameter generation method to well predict the dynamic behavior of UHPC members subjected to dynamic loadings. First, the failure surface, equation of state (EOS), damage evolution and strain rate effect parameters were comprehensively calibrated based on the existing experimental and analytical data of UHPC, and general guidelines on the determination of key KCC model parameters for UHPC are given. Then, the numerical simulations with both the autogenerated and the calibrated parameters were carried out to predict the dynamic response and damage evolution of UHPC members, i.e., reinforced UHPC beam and column, UHPC-FST beam, and reinforced UHPC panel, under drop hammer impact and field explosion. By comparing the test data, the validation of the proposed parameter generation approach is verified. The present work can provide helpful references for designing and evaluating the UHPC structural members under potential impact and blast loadings.

Experimental analysis and constitutive modeling of anisotropic creep damage in a wrought age-hardenable Al alloy

Many components of aerospace and automotive engineering manufactured from forged age-hardening aluminum alloys operate at high temperature in the range 0.3Tm–0.5Tm, where Tm is the melting temperature. Experimental data for uni-axial creep on specimens sampled from a forged AA2014 alloy component show that inelastic strain rates depend significantly on the orientation of loading with respect to the material flow direction during the manufacturing stage. Based on the recently developed constitutive model of anisotropic creep, hardening/recovery, softening and overageing processes, this paper addresses experimental analysis and modeling tertiary creep regime up to the final stage of creep rupture. The emphasis is placed on the damage evolution equation in order to account for different damage mechanisms depending on the loading mode and loading direction relative to grain axis: cavitation on elongated grain boundaries as well as decohesions and microcracks around and inside coarse grain particles. Experimental data of creep strain rates for longitudinal and transverse loading directions at 130, 150 and 170 °C have been used to calibrate the model. The validation of the model was carried out by comparing experimental and modeled creep curves in the case of a 30°angle between the loading and the material flow direction during forging. The results show that the assumed damage mechanisms and the developed damage evolution equation are able to reflect both the anisotropic accelerated creep and the time to creep rupture for different stress levels and loading directions.

Study on Damage Statistical Constitutive Model of Triaxial Compression of Acid-Etched Rock under Coupling Effect of Temperature and Confining Pressure.

Based on Lemaitre’s strain equivalence hypothesis theory, it is assumed that the strength of acid-etching rock microelements under the coupling effect of temperature and confining pressure follows the Weibull distribution. Under the hypothesis that micro-element damage meets the D-P criterion and based on continuum damage mechanics and statistical theory, chemical damage variables, thermal damage variables and mechanical damage variables were introduced in the construction of damage evolution equations and constitutive models for acid-etching rocks considering the coupled effects of temperature and confining pressure. The required model parameters were obtained by theoretical derivation, and the model was verified based on the triaxial compression test data of granite. Comparing the experimental stress-strain curve with the theoretical stress-strain curve, the results show that they were in good agreement. By selecting reasonable model parameters, the damage statistical constitutive model can accurately reflect the stress-strain curve characteristics of rock in the process of triaxial compression. The comparison between the experimental and theoretical results also verifies the reasonableness and reliability of the model. This model provides a new rock damage statistical constitutive equation for the study of rock mechanics and its application in engineering, and has certain reference significance for rock underground engineering.