Damage Constitutive Relation Research Articles

Mechanical properties and damage constitutive relationship of microwave irradiation of granite under uniaxial compression

Microwaves have great potential to increase the efficiency of hard rock breakage during mechanical excavation in engineering. This study conducted uniaxial compression tests on granite specimens after microwave irradiation at 4.85 kW. The results demonstrated a linear decrease in uniaxial compressive strength and elastic modulus correlating with increasing irradiation durations. Acoustic emission data indicated increased normalized crack closure stress and decreased normalized crack initiation and damage stress, suggesting complex crack propagation and evolution patterns under microwave influence. 3D Digital Image Correlation revealed that as heating time increased, the macrocrack leading to specimen failure shifted from being load-dominated to involving both load and microwave-induced thermal cracks. A simple four-parameter damage constitutive model for granite (a 1, r 1 , a 2, and r 2) effectively described the damage evolution under the coupling effect of microwave and uniaxial load. The model accurately characterized the complete stress–strain curves, including both microcrack compaction and post-peak stages, revealing that initial damage escalates with longer heating, though the progression rate decreases.

Energy dissipation damage constitutive relation of CFRP passively confined coal sample

In view of the stability problem of coal pillars left over during coal resource mining, (Carbon Fiber Reinforced Polymer) CFRP sheet is applied in coal pillar reinforcement. Uniaxial compression tests of CFRP passively confined coal samples are carried out to explore the mechanical response mechanism of passively confined coal samples under different layers, and the energy dissipation damage constitutive relationship of CFRP passively confined coal samples is established based on the energy dissipation principle. The conclusions are: As CFRP layers increased, the local damage of coal samples before the peak evolved from a 'cliff-like jagged' to a 'capillary jagged', with post-peak instability marked by a shift to more 'cliff-like' characteristics. The tests revealed improvements in peak strength and elastic modulus, with a defined functional relationship between these properties and CFRP layers. The energy storage capacity of passively confined coal samples improved with CFRP layers, requiring less axial deformation to achieve equivalent energy levels. The energy dissipation rate showed an initial decrease followed by an increase, with a minimum inflection point, the elastic energy consumption ratio tends to decrease slowly and then rapidly during post-peak instability. A damage constitutive relationship and evolution equation were developed, highlighting that the CFRP sheet significantly inhibits damage, with diminishing effectiveness beyond two layers. The study concludes that three-layer CFRP sheets provide optimal confinement, offering a novel strategy for the reinforcement of coal pillars and the prevention and control of rock burst, without considering the actual coal pillar dimensions and shape. To sum up, the use of CFRP sheet to strengthen coal pillar has considerable potential research value in strengthening coal pillar and improving the recovery rate of coal resources.

Delamination propagation manipulation of composite laminates under low-velocity impact and residual compressive strength evaluation

The poor delamination resistance of laminated composites is always regarded as a potential threat to the service safety of the primary load-bearing structure. However, delamination is inevitable during low-velocity impact (LVI) events, as a result, the residual compressive strength of laminates is significantly reduced. Discrete interleaving is a novel strategy to enhance the damage tolerance of these composites, yet the damage mechanism of laminates modified by this method is more complicated. In this paper, we investigated the delamination failure mechanism of quasi-isotropic laminates through thermal deply experiment, and proposed a discrete interleaving scheme based on its damage characteristics. Accordingly, a series of LVI and compression-after-impact (CAI) tests were conducted to validate the design philosophy. Besides, the damage constitutive relation and evolution model applicable to discrete interleaved laminates were explored in detail, a numerical model embedded strain-rate-dependent progressive damage criterion and modified cohesive zone model was established to further study the damage mechanism. Experimental and numerical results exhibit excellent alignment. The results demonstrate that the CAI strength is enhanced by 16.48 % according to the proposed discrete interleaving method. The main reason is attributed to the employed method can manipulate the delamination propagation during the low-velocity impact loading, and then maneuver the compressive failure mode of laminates to achieve a favorable benign failure (implying higher CAI strength).

Experimental study on the dynamic response of sintered NdFeB and secondary development verification of its Zhu-Wang-Tang damage constitutive model

Currently, there is relatively limited research on the dynamic response of sintered neodymium iron boron (sintered NdFeB). In order to reveal the dynamic response of sintered NdFeB, a Hopkinson pressure bar experimental platform is constructed to study its dynamic response. The results of the study indicate the necessity of establishing a dynamic constitutive model which can reflect its the strain rate sensitivity. Therefore, combining the Weibull distribution with the Zhu-Wang-Tang constitutive model, a constitutive model with damage for sintered NdFeB is created and this model is extended to three-dimensional form, expressing its incremental formulation. Finally, to verify the accuracy of this model in practical applications, the VUMAT subroutine is utilized to compile the damage constitutive relationship of sintered NdFeB. A Hopkinson pressure bar computational model is established, and the results show good consistency between the simulation and the experiment. This study provides a reference for the practical application of the damage constitutive model in actual scenarios.

Fatigue Evaluation of Sulphate-Attacked Industrial Waste-Based Concrete Using Concrete Damaged Plasticity Model

AbstractSulphate attack is a major cause of concrete deterioration in marine environments and its interaction with wave-induced cyclic loading exacerbates the damage. This study has evaluated strengths and fatigue performance (i.e. fatigue life, strain and residual displacement) of sulphate-attacked concrete containing silicomanganese slag, fly ash (FA) and silica fume (SF). Compressive strength, tensile strength and sulphate profile of sulphate-attacked concrete were measured experimentally. Sulphate-induced damage constitutive relations were formulated and used with concrete damaged plasticity (CDP) model to simulate fatigue loading. Experiment showed that incorporating silicomanganese slag lowered sulphate resistance by 4.8–6.6% due to increased sulphate intrusion, but synergy with FA and SF enhanced the resistance by 7.3–13.8% at 365 days. The sulphate penetration depth was 0–20 mm, and the intruded sulphate increased exponentially over time. To evaluate fatigue loading in CDP model, the non-uniform damage was determined as correlation between strength degradation and integral area of sulphate profile. Numerical results were in good agreement with experimental data from literature, with differences of 5.8–26.2% in fatigue life, 9.1–30.1% in fatigue strain and 18.1–41.9% in residual displacement. In long-term deterioration, numerical analysis found that increasing sulphate concentration significantly shortened fatigue life. Despite silicomanganese slag lowered concrete sulphate and fatigue resistance, the inclusion of FA and SF improved the durability and sustainability of concrete for potential marine applications.

Research on the widths of protective coal pillars in main roadway based on the damaged foundation beam model

Long-term stability of the protective coal pillars in coal mines can ensure normal and safe use of roadways; in turn, the integrity of the roadway plays an important role in safe and efficient mining in coal mines. The initial damage and rheological damage caused by coal-seam excavation disturbance have important influences on the selection of reasonable widths of the protective coal pillars. Based on damage mechanics and the theory of elastic foundation beams, a coupled damage constitutive relationship of rock-like materials is proposed in this work by considering the influence of threshold and residual strength to establish the mechanical model of the elastic damage foundation beam. By theoretical analysis and numerical calculation, the differential equation of the deflection curve of the beam is solved, the influence range of the advance abutment pressure under the condition of coupling damage is obtained, and the width of the protective coal pillar is determined. The results show that compared with the elastic foundation beam model, after considering the coupling damage of coal, the influence range of the advance abutment pressure increases with time over a certain duration. At the end of this duration, the influence range of the advance abutment pressure remains almost unchanged. For the case considered in this paper, the long-term stability of the roadway is guaranteed when the width of the coal pillar is 120 m. The potential interference or change related to the specific situation of the site can be studied by changing the relevant parameters in the model to study different working conditions, or the damage of the roof rock beam during the mining process of the working face can be considered, which will be the subject of further research in the future.

Mechanical behavior and damage constitutive relationship of basalt-brucite hybrid fiber reinforced low-heat cement concrete

In this study, the mechanical properties and microstructure of basalt-brucite hybrid fiber reinforced low heat cement concrete was explored. The results indicated that the mechanical properties of low heat cement concrete first increase and then decrease with the increase of hybrid fibers volume fraction, and the cubic compressive strength, splitting tensile strength, and axial compressive strength of the optimal strengthening group, that volume fraction is 0.20% for basalt fiber and 0.15% for brucite fiber, respectively, increased by 40.16%, 70.61%, and 56.16%, respectively. Under a reasonable volume fraction of hybrid fibers, the mechanical properties of concrete could be significantly improved. By comparing the bonding effect between fibers and concrete, fibers exhibited different distribution patterns in concrete matrix, that could improve the compactness of concrete matrix through filling effect, optimize internal structure, and delay the connection and expansion of microcracks. The causes of the bonding effect between fibers and concrete was discussed from the perspective of chemical bonding effects. Based on the above analysis, the reinforcement mechanism of hybrid fibers on concrete was explored. Based on statistical damage theory, a damage constitutive equation for hybrid fiber reinforced low heat cement concrete considering fiber volume fraction was established by modifying the model parameters, and the model curve fitted well with the experimental curve.

Investigating the fatigue performance of conventional reinforced concrete CRTS III ballastless track structures using a fatigue damage constitutive model

Compared to prestressed track structures, the mechanisms underlying fatigue degradation in conventional reinforced concrete track structures under high-cycle train loads remain elusive. This paper introduces a finite element model for CRTS (China Railway Track System) III slab ballastless track. In this model, fatigue damage constitutive relationships for both concrete and reinforcement are incorporated as material subroutines. The study focuses on investigating the fatigue characteristics of composite plate structures, which consist of track slabs and self-compacting concrete (SCC). An accelerated calculation method for high-cycle fatigue issues is utilized. Several key findings emerge from this study. First, the onset of fatigue cracks is observed on the lower surface of the SCC, directly beneath the applied load. These cracks tend to propagate laterally from the most critical location toward the slab edge. Second, a significant interrelationship exists between material fatigue degradation and stress redistribution. Traditional decoupled fatigue analysis methods are likely to overestimate the rate of fatigue evolution at critical points. Third, increasing the stiffness of geotextiles can partially optimize stress distribution within the track structures, thereby extending the time to initial cracking.

Effect of local openings on bearing behavior and failure mechanism of shield tunnel segments

Local failures (loss of concrete or reinforcement) can severely compromise the bearing capacity of shield segments, damaging the tunnel structures. To investigate the effects of local openings on the bearing behavior and failure mechanism, four full-scale bending tests were conducted on specimens with different opening positions and diameters; monitoring of load, displacement, and concrete strain was performed during loading. The test results reveal that both the opening position and diameter significantly influence the bearing characteristics of the segment. The failure process includes four sequential stages distinguished by three critical loads, namely the cracking, failure, and ultimate loads. Subsequently, the numerical model of the local failure segment was established using the elastoplastic damage constitutive relation of the concrete and verified by inversing the full-scale test results. Based on the numerical model, parametric analyses were performed to comprehensively investigate the influences of the opening position, concrete loss, and reinforcement loss on the bending capacity. Furthermore, an analytical model was proposed, indicating that the opening position is the primary factor decreasing the bearing capacity, followed by the opening diameter and reinforcement loss. The results of this study can provide a theoretical basis for the safety assessment and remedial design of subway shield tunnels under extreme breakthrough conditions.

Low-velocity impact and post-impact compression properties of carbon/glass hybrid yacht composite materials

In the field of maritime engineering, low-velocity impact (1-10 ms−1) is typically regarded as a quasi-static event. During navigation, yachts are inevitably subjected to low-velocity impact from floating objects in the water, and may also encounter compression after low-velocity impact. This study employs a combined approach of numerical simulation and experimental testing, aiming to investigate the damage and failure mechanisms of yacht composite laminates under conditions of low-velocity impact and post-impact compression (CAI), and by SEM and ultrasonic C-scanning the damage characteristics of different layup samples was studied in impact and compression tests. In order to conduct a more in-depth study on the damage evolution and failure mechanisms of yacht composite laminates, a three-dimensional damage model is established based on the continuum damage mechanics, and a physico-mechanical failure criterion combining the 3D Hashin and Puck criteria is employed. This criterion is used to capture the initiation points of damage in both fibers and matrix. Additionally, a bi-linear damage constitutive relationship is utilized to describe the damage evolution. Interlayer delamination is addressed through simulating the bonding behavior of interfaces. The numerical results show good correlation with the experimental results, which validates the effectiveness and rationality of the proposed numerical model. The study shows that laying carbon fiber cloth in the middle of the sample exhibits excellent impact resistance; laying carbon fiber cloth at the bottom exhibits better deformation resistance and stronger residual compression performance. Considering the three aspects of deformation, residual strength, and damage area, laying carbon fiber cloth in the first layer exhibits better mechanical properties, which provides useful reference for the design and manufacture of yacht materials and helps to improve the safety and reliability of yachts.

Assessment method for determining rock brittleness based on statistical damage constitutive relations

The brittleness index of rocks is able to provide crucial guidance to the operation of the drilling and hydraulic fracturing. The definition and evaluation method of the rock brittleness has not yet been unified and standardized due to their diversity. We therefore develop an evaluation method of the rock brittleness based on statistical damage relation. A set of uniaxial compression tests on different rock samples are conducted, and corresponding experimental data is collected and analyzed. Then we establish piecewise damage constitutive models based on different combinations of statistical distribution functions, including power function distribution, Weibull distribution, lognormal distribution and logistic distribution. A new evaluation method of rock brittleness based on energy method and the piecewise statistical damage constitutive model is proposed, and the evaluation results show that the increase of damage variable of peak strain will undermine rock brittleness, and the mineral composition contents have influence on the brittleness of rocks. Comparison work between this proposed method and previous brittleness criteria shows that the brittleness index B43 exhibits enhanced stability and consistency in rock brittleness. This study presents a novel method to studying rock brittleness, enhancing current evaluation methods and deepening our understanding of the rock index. Regarding practical application, some field operations involved with the process of drilling or fracturing lead the continuous damage evolution of rocks under loading conditions. The brittleness index from the proposed evaluation method is comparatively reliable and practical for on-site drilling and fracturing.

Experimental and numerical evaluation for tunnel structural stability of fiber concrete lining with different crack features under train load

Cracks are prone to occur in the secondary tunnel lining with the action of train loads, which will cause tunnel failure and adversely affect the long-term safety of tunnels. This paper conducts a series of material tests and numerical simulations to investigate the stability of fiber concrete tunnel linings with different crack features under train load. Through material tests for three different fiber and conventional concrete, the crack extension rule is discovered by combing digital image correlation (DIC) technology. The plastic damage constitutive relation of concrete is used to obtain the damage parameters. Then, the three-dimension numerical model is established to evaluate the stress intensity factor and stable coefficient of safety of tunnel lining. Moreover, four cases are determined to indicate the effect of cracks on the failure mechanisms of the tunnel lining structure. The influence of stability of lining structure with crack at different location, depth, width, inclination angle, train speed and grade of surrounding rock is analyzed.

Investigations into Variations in Meso- and Macro-Physicomechanical Properties of Black Sandstone under High-Temperature Conditions Based on Nuclear Magnetic Resonance

The high-temperature environment has a significant deterioration effect on the meso- and macro-physicomechanical properties of rocks. Black sandstone in Chuxiong Yunnan was chosen as the test sample to perform a high-temperature test, a nuclear magnetic resonance (NMR) test, and a uniaxial compressive test (UCT). After high-temperature heating (25°C–1,200°C), the thermally damaged samples were subjected to NMR tests in both saturated and centrifugal states. The transverse relaxation time T2 spectrum distribution and T2 spectrum area were subsequently obtained. The temperature effect of the mesopore structure was investigated by analyzing the changes in porosity. The T2 spectrum cutoff value was determined, and the temperature effect of the movable fluid migration law for rocks was evaluated. The thermal damage constitutive relationship with porosity as the damage variable was determined, and the evolution law of mesopore structure parameters and macroscopic mechanical parameters was discussed. UCT results showed that the mechanical characteristics of black sandstone were obviously degraded by high temperature, and especially the uniaxial compressive strength (UCS) decreased sharply after the temperature exceeded 900°C. NMR results identified 900°C as an inflection point for the mesostructure change. For temperatures lower than 900°C, the improvement of pore connectivity was not obvious. For temperatures exceeding 900°C, the porosity increased rapidly. The porosity had a great influence on UCS, and there was a clear exponential function relationship between the two. The larger the porosity, the smaller the UCS. It showed that it was an effective angle to study the mesopore characteristics and mechanical properties of high-temperature rocks based on the NMR technique.

Experimental investigation on damage and seepage of red sandstone subjected to cyclic thermal and cold treatment

For geothermal mining projects in cold regions, the problems of surrounding rock and water damage caused by thermal-cold shocks have become the main factors affecting engineering stability and construction safety. In addition, the permeability and damage characteristics of the rock mass after the thermal-cold cyclic shock action are not yet known. Therefore, in this study, seepage tests of red sandstone during complete stress-strain after thermal-cold cyclic shock were carried out to analysis the change in permeability of red sandstone during progressive fracturing. Meanwhile, the permeability model considering damage was further constructed by using the damage constitutive relationship based on statistical damage theory. And found that peak stress and elastic modulus were found to decrease with temperature and cycles, and both initial and maximum permeability increase with temperature and cycles during its gradual fracture. Moreover, the theoretical curve calculated by the permeability model is not much different from the experimental curve, which can effectively characterize the relationship between permeability and damage. The research results can provide reference value for the damage assessment, repair and reinforcement of the wellbore surrounding rock structure after thermal-cold shock cycle in the geothermal engineering of cold regions.

A semiempirical constitutive relationship for rocks under uniaxial compression considering the initial damage recovery

AbstractA better understanding of the constitutive model and damage evolution law is the premise to ensure their feasibility and practicability in rock engineering projects. In this paper, a damage constitutive relationship, which can reflect the initial damage recovery, is established based on the generalized equivalent strain principle and the initial tangent modulus. A Weibull distribution‐based model is also established to represent the deformation after the compaction point. A piecewise expression of the complete stress–strain curve of rocks under uniaxial compression is thereby obtained by the combination of the above two constitutive models. Finally, the effectiveness of the piecewise constitutive relationship is verified using uniaxial compression test data of different types of rocks. The results show that the theoretical curves are in good agreement with the experimental curves. Parameters of the proposed relationship are relatively to be derived, making it convenient for engineering applications.

Phase-field modeling for anisotropic ductile damage of magnesium alloys at finite deformations

The damage anisotropy of an extruded ZK60 Mg alloy is characterized using tensile tests and scanning electronic microscopy. The accumulation of anisotropic deformations leads to the great differences of the dimple evolution and strains at fracture along different loading directions. To introduce the anisotropic deformation information into the damage constitutive relationship, a thermodynamically consistent phase-field model of ductile damage fully coupled with elastoplastic finite deformations is developed in this study. Using the user-defined constitutive relationship and displacement-temperature coupling element, the finite element simulations are conducted. The results show that: (1) ZK60 Mg alloys presents clear R-value difference in 0°, 45°, and 90° tests of intact specimens. The 45° test possesses the greatest R-value (1.50) and the greatest strain at fracture, however, the R-value for 0° is less than 1, indicating the thinning is preferential. (2) The higher ultimate stress leads to a larger average dimension of the dimples, whereas the higher density correlates with a larger elongation ratio at the fracture. The disappearance of the stress-bearing area indicates that the phase-field assumption on stress degradation is completely compatible with the dimple analysis on fractography. (3) The simulation results of the stress-strain relationships and damage paths correlate well with the experimental ductile damage of magnesium alloys at 200 ℃. Slight errors are basically attributed to the modeling parameters and finite element iteration algorithm. The proposed model presents fine applicability and reliability for the predictions of plastic deformations, ductile damage, and fracture of anisotropic Mg alloys.

Study on damage constitutive relationship of mudstone affected by dynamic pile driving

The mudstone around the pile is damaged during pile driving, altering the mechanical properties of the mudstone. This study conducts standard penetration, uniaxial compression, and triaxial shear tests before and after pile driving. It develops a constitutive damage model of mudstone affected by pile driving based on the statistical damage theory and triaxial test data. The findings in this study show that: (1) The standard penetration number (N) of the clay around the pile remains unchanged after driving because of the thixotropic recovery effect. However, the N of mudstone surrounding the pile decreases by 35.1% on average, (2) The average uniaxial compressive strength is 1.56 MPa before pile driving and 1.12 MPa after pile driving, which decreases by 28.2%. After pile driving, c (cohesion) = 217.2 kPa, φ (internal frictional angle) = 21.6°, (3) The calculated value agrees well with the measured one, verifying the rationality of the model, and (4) The mechanical properties of mudstone change considerably after pile driving, and it is unrealistic to use the mechanical parameters of undisturbed mudstone to design the pile. This research provides theoretical support and parameter preparation for calculating pile driven mudstone foundations.

Data-driven enhanced phase field models for highly accurate prediction of Mode I and Mode II fracture

By coupling the elastic and crack surface energy in the total potential function, phase field methods favor a natural track of crack initiation and propagation within the continuum mechanics framework. However, in turn, it raises a critical issue in determining the coupling factor—energy degradation function. This work aims to tackle the problems induced by assigning a degradation function empirically, including the inaccurate reproduction of critical loads and unphysical stiffness reduction prior to fracture. Inspired by the concepts of data-driven computational mechanics, we propose a data-driven phase field scheme to enhance the numerical reproduction of physically consistent fracture responses. This method collaborates a datasets generator, a classifier, a physical-constrained learning algorithm, and the phase field solver in an extensible modular framework, allowing searching for the optimized damage constitutive relation in a carefully designed degradation function space with flexible form. In the case studies, mode I and mode II fracture characteristics of linear elastic and hyperelastic materials are well predicted by the data-driven phase field algorithm, which learns and derives the results from simple scalar datasets, controlling the average error lower than 10%. When introducing the input dataset noise, the model still shows robustness. The results demonstrate the model’s interpretability, thermodynamical consistency, and accuracy.

Tensile low-cycle fatigue performance and life prediction of high-strength bolts

Monotonic tensile tests and tensile low-cycle fatigue tests have been conducted to explore the low-cycle fatigue life of high-strength bolts, which is an important property to guarantee the connections do not occur low-cycle fatigue failure before their connected members under earthquakes. Large-deformation performance and failure modes of bolts were numerically simulated based on the damage constitutive relation which was investigated by the tensile tests of smooth and notched specimens. The results show that complete uniaxial constitutive relation of the bolt material, namely 20MnTiB steel, and the displacement from tensile load response of smooth specimens in the numerical simulation can be better described by the Swift flow stress model. The GTN damage model of the steel was calibrated by tensile tests and finite element models of notched specimens. With the decrease of the notch radius, the tensile capacity of the notched specimen increases while the ductility gets worse. The calibration parameters of GTN damage model can be applied to simulate the load-displacement response and fracture behavior of notched specimens accurately. In order to obtain a more targeted GTN model under different stress states, the quantitative relation between equivalent plastic strain during nucleation and stress triaxiality was proposed, which provided an important reference for establishing fatigue damage models of various bolted joints in the numerical simulation. Both the fatigue failure modes of the bolts under cyclic displacement with constant amplitude and the fatigue life based on the tensile capacity degradation were obtained from the fatigue tests. The fatigue brittle fracture and plastic elongation of the shank are the two fatigue failure modes of the bolt. Typical regional characteristics are presented in the bolt fatigue fracture, with beach-like fatigue bands in the expansion area. Under different failure modes, the capacity degradation can better determine the fatigue life of bolts. The methods for predicting fatigue life were proposed based on cyclic displacement amplitude, equivalent stress amplitude and local plastic strain. A good prediction result was proved by the phenomena that the predicted fatigue life based on the cyclic displacement amplitude and local plastic strain was basically within the scatter band of 1.5. Moreover, the fatigue design parameters of bolts given by the equivalent stress amplitude method provide a reference for establishing a unified fatigue design theory.

Mechanical performance, microstructure, and damage model of concrete containing steel slag aggregate

AbstractReusing industrial solid waste in cement‐based products is an economical and environmentally friendly solution. This study investigated the effect of steel slag on the mechanical properties of concrete mixtures, and development of damage constitutive relationship of steel slag concrete (SSC). Two steel slag aggregates, including fine steel slag aggregate (SSAF) and coarse steel slag aggregate (SSAC) were incorporated into the concrete mixtures to SSC. The portions of natural sand and coarse aggregate each were replaced with SSAF and SSAC at 10%–100% (interval of 20%) by weight. Therefore, effects of SSAF and SSAC on the mechanical properties of SSC mixtures were evaluated separately. Moreover, damage model of SSC mixtures was developed based on Lemaitre's equivalent strain principle and Weibull probability density function. The damage constitutive model was revised using the functional relationship between the toughness index, brittleness index, and distribution parameters. The results indicate that SSC's mechanical properties, crack resistance, and toughness have been significantly improved with increased steel slag content. However, reduction in strength was observed in the SSC specimens containing steel slag above 50% replacement. Nevertheless, these strengths are higher than that of normal concrete. The modified damage constitutive relationship fits better with the experimental results.