CHAPTER 7 FE ANALYSES OF GROUTED TUBULAR JOINTS USING
7.6 D ETERMINATIONS OF THE MATERIAL CONSTANTS
In this section, the procedures for the determinations of the key parameters for the CDM approach are introduced. The effect of element size is discussed.
ff f f f
7.6.1 Determinations of the material constants
In addition to the usual parameters for plasticity, three material constants are required to define the initiation and the evolution of the damage, e.g., one dimensional damage initiation stain, εd; one dimensional plastic displacement at fracture, uf0; and the maximum damage, Dc. The values of εd and Dc can be obtained explicitly from uniaxial tests by monitoring the deterioration of the elasticity modulus as the specimen is damaged to failure. However, this type of data is often not available. Moreover, the onset of necking in the coupon test destroys the uni-axial state of the stress (Yun, 1996) and hence it is impossible to get the real fracture strain from the nominal stress-strain curve.
The values for the three constants are then adjusted to reproduce the experimental, nominal strain-stress diagram.
Damage initiation stain εd
Lemaitre (1985) indicates that ductile plastic damage generally begins when necking starts while other study reveals that the coupon specimen begins to neck when the tensile load reaches the maximum value during a test (Yun, 1996). Since when necking just starts, the stress localization is still insignificant and the uniaxial state of stress is not destroyed yet, it is logical to assume that the strain is still uniformly distributed across the coupon cross section at this moment and the corresponding measured strain could represent the real strain at the onset of necking. Hence, εd is taken as the measured true strain at the point when the tensile load reaches the maximum value in each coupon test.
In the present experimental investigation, three coupons have been tested for each type of pipe and the average value has been adopted for the corresponding pipe, as summarized in Table 7-2.
True strain-stress curve
To start the iteration to reproduce the experimental nominal strain-stress diagram, we must get the whole flow curve, including the part after necking. However, because the onset of necking destroys the uniaxial state of the stress, it is impossible to determine a uniaxial true stress-strain curve relation by a standard tensile test once necking has started.
If the nominal strain-stress curve is still used to derive the true strain-stress curve after necking, a wrong “softening” behavior of the true stain-stress will be observed and should be discarded, as the “softening” part indicated in Figure 7-9. In the present study, a power law has been used to represent the true stress-strain relation after necking:
kεm
σ= (7.15)
Where k and m are the hardening constants determined from the known true stress-strain curve before the necking of each material and the determined values of k and m for each pipe are summarized in Table 7-2. The true strain-stress used in the present analyses is the solid line demonstrated in Figure 7-9.
0 200 400 600 800
0.00 0.10 0.20 0.30 0.40
Strain
Stress
Nominal strain-stress True stain-stress
Figure 7-9 Illustration of true stress-strain curve adopted for analyses
Dc and uf0
Coupon tests have been carried out for all the specimens and the corresponding complete stress-strain diagrams have been obtained. The FE models are created for these coupon
εd
Necking point From nominal
Strain-stress
From Power law
Softening
specimens, and the true-stress curves based on the test data are applied to the models.
The values of Dc and uf0 are assumed and varied to match the experimental nominal stress-strain diagram as demonstrated in Figure 7-10.
0 150 300 450 600
0 0.1 0.2 0.3 0.4
Nominal Strain
Nominal Stress
Test
FE(without damage) FE(with damage)
Figure 7-10 Comparison of experimental and analytical nominal stress-strain diagram
Since the value of Dc only decides when the fracture occurs while the value of uf0 only influences the softening path, during the iterations, the value of Dc is assumed to be 1 first and the value of uf0 is varied to find the softening path fitting the test results the best.
Once the value of uf0 is determined, the value of Dc is varied so that the FE model would fracture at the same nominal strain as that for the corresponding coupon.
7.6.2 Effect of the element size
As indicated early, the application of the plastic displacement instead of the plastic strain to the damage material model minimizes the mesh dependency of numerical results. To investigate the effect of the mesh density on the determinations of the damage parameters, additional analyses have been carried out for the FE models with different mesh sizes.
Four mesh sizes have been adopted for the analyses, e.g., 0.75mm, 1.5mm, 3mm and 6mm. For each FE model, the corresponding Dc and uf0 have been determined
Rupture
respectively through the procedure described in 7.6.1. The determined parameters for the FE models with the specified element sizes are summarized in Table 7-1.
Table 7-1 Effect of element size characteristic length of element
(mm) 0.75mm 1.5mm 3mm 6mm
Dc 0.2 0.2 0.2 0.2
uf0 0.8 1.4 2 2.7
As the table shows, the critical damage Dc is independent of the mesh size and also in accordance with the value reported by Lemaitre (1985) (0.2 to 0.8). On the other hand, the plastic displacement at fracture still depends on the element size. Hence, in the later joint analyses, the same mesh size has been adopted for the modeling of joint as that for the modeling of the corresponding coupon to maintain the validity of the determined parameters.
The determined parameters for each circular hollow section (CHS) used are summarized in Table 7-2, together with the corresponding characteristic length of the element (L) used. These parameters are adopted for the FE analyses of the fully grouted joints incorporating the CDM approach.
Table 7-2 Identified damage parameters Specification
(mm) Specimen L
(mm) εd Dc uf0
(mm) k m
324*8.0 X5-G-T/X6-G-T 3 0.170 0.2 2.2 919 0.24
324*12.5 X3-G-T/X4-G-T 3 0.152 0.2 2.0 950 0.22
406*21.4 X2-G 3 0.150 0.2 1.9 892 0.22
457*8.0 X7-G-T 3 0.160 0.2 2.2 949 0.23
508*15.09 X1-G 3 0.144 0.2 2.4 893 0.26