Jiao Haihan Yan Huadong Jin Hui

(1Jiangsu Key Laboratory of Engineering Mechanics, Southeast University, Nanjing 211189, China)(2School of Civil Engineering, Southeast University, Nanjing 211189, China)(3Test and Measuring Academy of Norinco Group, Huayin 714200, China)

Abstract：Based on the Gurson-Tvergaard-Needleman(GTN)damage model considering the defect damage evolution, the influence of void defects caused by the casting process on cast steel nodes’ mechanical properties was studied.Firstly, based on the GTN damage model, the model’s parameter combination of G20Mn5N cast steel was given.Then, the mechanical properties of cast steel nodes were evaluated using the GTN damage model in ABAQUS software, and the influence of model parameters on the failure results was investigated.The results show that the cast steel node considering the GTN damage model fails under 1.93 times of the load.The bearing capacity is lower than that of the bilinear model, and the failure speed is faster.Changes in model parameters will cause a shift in the failure critical point.Meanwhile, the plastic strain index affects the void volume fractions, which shows different variation laws under uniaxial tensile and cyclic loads.Therefore, the GTN damage model establishes the relationship between the micro-defects and macro-mechanical properties of materials, which can better simulate the failure results of structures.

Key words：cast steel node; Gurson-Tvergaard-Needleman(GTN)damage model; bearing capacity; model parameters

Cast steel nodes are a kind of prefabricated integral casting node.They have flexible and various forms and a good mechanical performance and hence are an ideal form for complex structural nodes[1].However, the number of cast steel’s void defects is far more than that of hot-rolled steel because of different production processes[2].The existence of defects destroys the material’s continuity, leading to a decline in the structure’s mechanical properties.

Metal is a ductile material, whose failure is often related to the nucleation, growth, and coalescence of voids caused by a load until macroscopic cracks are formed[3].Therefore, it is necessary to examine the changes of microvoids and their effects on the macro-mechanics of materials to truly reflect the damage evolution process.However, the traditional industrial flaw detection method has low accuracy, making it difficult to determine the distribution and size of micro-defects inside the casting.Moreover, the original bilinear model does not consider internal material damages, causing insufficient assessment of the structural safety and reliability.

In fracture mechanics, current constitutive material models can be divided into two types: Macroscopic models, such as the Lemaitre model, do not consider the micromechanisms of ductile damage[3-4].Gurson[5]coupled the evolution of voids with the equivalent plastic strain of a material based on volume cell models and deduced a relatively complete microscopic damage model.Subsequently, Tvergaard[6]and Needleman et al.[7]modified the model and formed the Gurson-Tvergaard-Needleman(GTN)damage model, which is widely used in the research of metal properties.This model uses the yield function to describe the yield behavior of materials and the void volume fraction to define the failure of materials, which can accurately describe the failure results of metal materials compared with the bilinear model[8].

Since the proposal of the GTN damage model, many scholars have applied it to simulate the damage evolution process of metals.Xu et al.[9]simulated the tensile behavior of corroded reinforcing bars in concrete under a carbonized environment.Liu et al.[10]simulated the effect of MnS inclusions on the initiation and propagation of cracks in as-cast 304 stainless steel at high temperatures.Steglich et al.[11]studied the interaction between the plastic anisotropy and void growth of aluminum alloy 2198.The above instances fully indicate that the GTN damage model promotes the development of micro-damage mechanics and establishes a good connection between experimental verifications and the finite element method(FEM), and its application has become increasingly widespread[12].

In this study, the bearing capacity of cast steel nodes was calculated based on the bilinear model and GTN damage model.The relationship between the model parameters and the failure results of cast steel nodes under different loads was also examined.

1 GTN Damage Model and Its Parameters

1.1 GTN damage model

The yield function of the GTN damage model can be expressed as follows:

(1)

The damage variablef*is a function of the total void volume fractionf, which is expressed as

(2)

The yield surface of the GTN damage model connects the yield with the damage of the material, so the yield surface will gradually shrink with the increase inf*, reflecting the continuous deterioration of materials due to damage evolution, as shown in Fig.1.Whenf*= 0, Eq.(1)degenerates into a von Mises yield function.

Fig.1 Yield surface of the GTN damage model

The total void volume fractionfin the GTN damage model includes the initial void volume fractionf0, growing void volume fractionfgr, and nucleated void volume fractionfnu:

f=f0+fgr+fnu

(3a)

The increment expression is

df=dfgr+dfnu

(3b)

Due to the incompressibility of the matrix material, the growth of the initial voids depends on the plastic strain[13]:

dfgr=(1-f)dεp∶I

(4)

whereεpis the plastic strain andIis the second-order unit tensor.

Generally, nucleation will only occur when the stress exceeds the critical value, and the number of nucleation increases with the strain’s increase[14]:

(5)

Eqs.(1)to(5)constitute the GTN damage model, which can simulate the entire process of void nucleation, growth, coalescence, and fracture.In this model, Eqs.(3)to(5)describe the nucleation and growth of voids, and Eqs.(1)to(2)determine the influence of voids’ coalescence and fracture on the bearing capacity.

1.2 Parameters in the GTN damage model

The GTN damage model has nine parameters, which can be divided into three groups.The first group is the correction parameters, includingq1,q2, andq3; the second is the void volume fractions, includingf0,fc, andff; and the third is the nucleation parameters, includingfN,εN, andSN.To construct the GTN damage model, it is necessary to calibrate the optimal combination of parameters.At present, the common methods to calibrate the parameter values in the GTN damage model are metallographic analysis, cell element method, and finite element reverse method[12].However, even with the proposal of the GTN damage model, it was not easy to obtain all the parameters through experiments because of their numerous material parameters.Therefore, researchers studied on the model parameters’ calibration of different materials and achieved some results[4,10,15-20], which provide a reliable basis for applying the model.

Yan et al.[21-23]calibrated the model parameters of G20Mn5N cast steel using the finite element reverse method combining the three-dimensional X-ray microtomography technique and numerical simulation.The results in Tab.1 show that the numerical simulation results are in good agreement with the experimental results.

Tab.1 Optimal combination of the GTN damage model for G20Mn5N cast steel

2 Evaluation of Cast Steel Node’s Bearing Capacity

2.1 Material properties

The cast steel node selected in this study has four intersecting tubes, located in a large-span steel structure.The overall schematic diagram of the node and the number of each tube are shown in Fig.2.The material of the cast steel node is G20Mn5N cast steel, and its chemical composition and basic mechanical properties are shown in Tabs.2 and 3.

Fig.2 Overall schematic diagram of the cast steel node

Tab.2 Chemical composition of the G20Mn5N cast steel %

Tab.3 Mechanical properties of the G20Mn5N cast steel

2.2 Establishment of the cast steel node model

The FEM software ABAQUS was used to model and calculate the cast steel node with C3D10m solid elements.A fixed constraint is imposed on the main tube port C.A displacement constraint in theXandYdirections is imposed on the main tube port D.According to the requirements of JGJ/T 395—2017[24], the main part of the cast steel node should be in an elastic stage under complex stress.Meanwhile, the local stress concentration area is allowed to transit to the plastic stage.Therefore, a 1 161 284N compressive load is applied to branch tube port A, and a 2 237 085N tensile load is applied to branch tube port B.

There are stress concentration areas at the branch tube ports A and B, so mesh encryption is performed in these areas.Tab.4 and Fig.3 show different mesh densities cases.Fig.4 shows the stress and displacement calculation results of branch tube port B by different cases.The model’s von Mises stress and resultant displacement results stabilize when the branch tube port’s mesh density reaches 9 mm, so this mesh density is selected for calculation.

Tab.4 Different mesh density cases mm2

Fig.3 Mesh density model of the cast steel node

Fig.4 Von Mises stress and resultant displacement results by different mesh densities

2.3 Service limit state

Figs.5 and 6 show the Von Mises stress and the equivalent plastic strain contour of the cast steel node calculated by the bilinear model and GTN damage model under the design load.Branch tube port B yields based on the two constitutive models, but the yielding area is small and most areas are still in the elastic stage.

The maximum Von Mises stress values of cast steel nodes calculated based on the two constitutive models are 364 and 386 MPa.The difference between the two values is minute, and the value based on the GTN damage model is slightly large.Whether the damage is considered or not has little effect on the service limit state of cast steel nodes.The reason is that the main areas of cast steel nodes do not transit to a yielding stage under the design load, and the damage in the nodes has not yet evolved.

(a)

(a)

2.4 Ultimate limit state

The ultimate limit state of the cast steel node mainly refers to the maximum axial force when the node is damaged due to excessive local deformation under a load[25].According to JGJ/T 395—2017[24], the extreme point of the load-displacement curve calculated by the FEM should be taken as the ultimate limit capacity.

Fig.7 shows the load-displacement curves of the branch tube ports A and B calculated by the two constitutive models.Whether the bilinear model or GTN damage model is adopted, branch tube port A yields before branch tube port B, but the relative displacement in the ultimate limit state for branch tube port B is greater than that of branch tube port A.

Fig.7 Load multiple-relative displacement curve of branch tube ports A and B

The load-displacement curves obtained by the two constitutive models have good consistency in the elastic, yielding, and coinciding in the strengthening stage, but there are differences in the degradation stage.The load-displacement curve calculated by the GTN damage model fails earlier, and the strength decreases faster.The ultimate bearing capacity based on the bilinear model and GTN damage model is 1.95 and 1.93 times of the load, respectively, which meets the requirements of JGJ/T 395—2017 that the ultimate limit capacity should be 1.5 times greater than the design load.

Based on this analysis, the damage evolution behavior of materials has a specific impact on the ultimate limit state of the cast steel node.After considering the material performance degradation caused by damage, the failure process of components after reaching the ultimate limit capacity is rapid.Although the stress level of cast steel nodes in the service limit state is in the elastic stage, it is likely to transit the plastic stage under ultimate loads, such as seismic action.The GTN damage model can effectively predict the failure results of cast steel nodes in the ultimate limit state.

3 Model Parameters in the GTN Damage Model

3.1 Effect of model parameters on the load-displacement curves

After evaluating the bearing capacity of the cast steel node, the effects of the model parameters(fc,ff,fN, andεN)on the load-displacement curves of the cast steel node were studied.

Fig.8 shows thatfcdoes not affect the curve’s trend, but for the same void nucleation and growth rate, the largerfcis, the longer the time needed for the void to coalesce.Therefore, the critical failure point will be delayed.

(a)

The influence offfon the curve is similar tofc.When the test conditions and other model parameters are identical, the largerffis, the longer the voids needed nucleate and grow toff, and the critical failure point will move backward.

By increasingfN, the larger the void volume fraction is due to the micro-defects in the material, the faster the critical failure point will be reached, leading to a short fracture displacement.

dfnuis an approximately normal distribution concerning an equivalent plastic strain, so the strain at the maximum point of the void nucleation velocity isεN.The larger theεNis, the smaller the void nucleation velocity is, and the higher the bearing capacity of cast steel nodes is.

3.2 Evolution of the void volume fraction under a load

In this section, we examine the evolution of the void volume fraction(f,fgr, andfnu)and plastic strain index(PEMAG and PEEQ)by the GTN damage model.The uniaxial tensile load and cyclic load applied to the cast steel node are shown in Figs.9 and 10.The evolution of the void volume fraction and plastic strain index at branch tube port B are shown in Figs.11 and 12.

Fig.11 shows the evolution of the variables under auniaxial tensile load.When the materials transit to the yielding stage under the load,fgrandfnurapidly increase after the PEEQ(equivalent plastic strain)accumulates to a certain extent, andfnuincreases faster.fand the PEMAG(plastic strain magnitude)have similar variations, indicating that the GTN damage model effectively couples the damage parameters and the plastic strain of the matrix material.It also explains why the bilinear and GTN damage models’ load-displacement curves differ after the yielding stage.

Fig.12 shows the evolution of these variables under a cyclic load.frises in steps during the load process, and the growth rate becomes increasingly faster.Eq.(4)shows thatfgris related to PEMAG.PEMAG is positive during tension and negative during compression.Therefore,fgrchanges positively and negatively with the cyclic load, and its horizontal stage corresponds to the unloading stage.Eq.(5)shows thatfnuis related to the PEEQ.The PEEQ increases during the loading stage and remains unchanged during the unloading stage, sofnurises in steps during the loading process.

Fig.9 Uniaxial tensile load

Fig.10 Cycle load

Fig.11 Evolution of the void volume fraction and plastic strain index with a uniaxial tensile load

Fig.12 Evolution of the void volume fraction and plastic strain index with a cycle load

4 Conclusions

1)Based on the bilinear model and GTN damage model, the bearing capacity of the cast steel node was evaluated.The results show that there is a specific influence on the ultimate limit state.The cast steel node has an earlier failure time and faster failure speed considering the GTN damage model.

2)The influence of each model parameter on the failure critical point of the cast steel node was compared and analyzed.The results show that asfNincreases, the critical failure point moves forward; asfc,ff, andεNincrease, the critical failure point moves backward.

3)The evolution of the void volume fraction and plastic strain index is similar under the uniaxial tensile load.However, under the cyclic load,fgrandfnuare respectively affected by the PEMAG and PEEQ, which show different evolution laws.