Based on the above results, it finds that the creep failure locates at the fillet at compressive and tensile loads. But due to the different stress distribution, the creep and stress triaxiality development are different. Figure 9 shows the maximum CEEQ and the stress triaxiality change with time for the same node at the fillet at compressive and tensile loads. It shows that the CEEQ and stress triaxiality at the tensile load are larger than those at compressive load, which proves that the brazing joint has a better creep resistance at compressive load. Therefore, the damage in the fillet at compressive state is far smaller than that at tensile state, as shown in Figure 10. At tensile load, the damage reaches the failure value 0.99 within 557 h. But at compressive load, it takes 27,300 h to reach the failure value 0.99. The damage rate at the fillet at tensile load is 50 times as much as that at compressive load. See Figure 9(a), the stress triaxiality almost keeps stable with time changing, while the CEEQ increases with the time changing (see Figure 9(b)), proving that the growth of creep strain is the dominant for damage. At compressive load, the stress triaxiality is negative value (~-0.8), therefore the damage grows very slowly. At tensile load, the stress triaxiality is a positive and accelerates the damage.

Figure 9: The maximum CEEQ (a) and stress triaxiality (b) change with time at the fillet for tensile and compressive loads.

Figure 10: Damage change with time at the fillet at compressive and tensile loads.

Figure 11 shows the effect of the applied load on the damage at the fillet. It shows that the damage increases with the applied load increasing. In total, the damage value at compressive load is smaller than that at tensile state. At tensile state, as the load increases to 0.3 MPa, the damage value has reached the failure value 0.99, while it is still very small at compressive state. At the compressive load, the damage reaches the failure 0.99 until the load is 0.57 MPa. Figure 12 shows the effect of applied load on CEEQ at the fillet. It shows that the local maximum CEEQ increases with the applied load increasing. The CEEQ changes little as the applied load varies from 0.1 MPa to 0.4 MPa, but then it increases suddenly as the applied load increases from 0.4 MPa to 0.57 MPa. At 0.57 MPa, the maximum local CEEQ is near 5 %. It also shows that the local maximum CEEQ at tensile load are larger than those at compressive load. Figure 13 shows the load effect on the equivalent creep strain of the whole panel structure. It shows that the equivalent creep strain increases slightly as the load increases from 0.1 to 0.4 MPa, but it has a sudden increase from 0.5 to 0.6 MPa. As the load is 0.57 MPa, the creep strain has been 2.84 %. Based on results of Figures 11–13, it concludes that for this model the limit load at tensile load and compressive load is 0.3 and 0.57 MPa, respectively.

Figure 11: Effect of the applied load on the damage at compressive and tensile loads.

Figure 12: Effect of the applied load on the local maximum CEEQ at compressive and tensile loads.

Figure 13: Effect of the applied load on the equivalent creep strain for the whole panel structure.

Lattice truss sandwich structure is also a type of cellular material. In recent years, great attention has been paid on its creep strength. Two analytical models have been developed to describe the creep behavior of the cellular material.

One is developed by Gibson and Ashby (GA model) [33]:
$\stackrel{.}{\dot{\mathrm{\epsilon}}=A\frac{0.6}{n+2}\left(\frac{1.7(2n+1)}{n}\right){\mathrm{\sigma}}^{n}{\mathrm{\rho}}^{-(3n+1)/2}exp\left(-\frac{Q}{RT}\right)}$(5)

where *σ* is the uniaxial stress applied to the cellular material, $\mathrm{\rho}$ is the relative density. And the other parameters are related to the power law creep equation for the solid material of which the cellular material consists:
$\stackrel{.}{{\dot{\mathrm{\epsilon}}}_{b}=A{\mathrm{\sigma}}_{b}^{n}}exp\left(-\frac{Q}{RT}\right)$(6)

where ${\dot{\mathrm{\epsilon}}}_{b}$ is the uniaxial strain rate, *A* is the creep constant, ${\mathrm{\sigma}}_{b}$ is the uniaxial stress, *n* is the stress component, *Q* is the creep activation energy, *R* is the gas constant and *T* is the temperature.

The other model is developed by Hodge and Dunand (HD model) [34]:
$\dot{\mathrm{\epsilon}}=A{\left(\frac{\mathrm{\rho}}{3}\right)}^{-{n}^{.}}{\mathrm{\sigma}}^{n}exp\left(-\frac{Q}{RT}\right)$(7)

Here we compare the creep strain rate between the two models and our FEM result, as shown in Figure 14. It shows that the GA model is relatively closed to the present FEM result, while it has a big discrepancy compared to HD model. Because the truss bears the bending load mainly during the tensile or compressive loads, which is more suitable for GA model while HD model is used for the struts bear the compressive stress. But still there is a discrepancy between GA model and the present FEM. Because GA model is developed for the foam material, and the creep constitute equation is related to the relative density and independent on the dimension of truss, which leads to a large error.

Figure 14: Load dependent creep rate by GA, HD, modified GA models and FEM.

In fact, the size effect on mechanical properties for the lattice truss sandwich structure is very notable. For this, Boonyongmaneerat and Dunand [35] developed a modified GA model (eq. 8) which considers the effect of strut dimension by a single parameter, the strut aspect ratio *a*= 2*d*/*k* (2*d* and *k* are strut length and width).
${\stackrel{.}{\dot{\mathrm{\epsilon}}=\frac{3.1}{n+2}\left(\frac{9.3\cdot (2n+1)}{n}\right)}}^{n}\cdot \left[{\left(\frac{a-1}{2}\right)}^{2+n}{\left(\frac{a+1}{2}\right)}^{2n-1}\right]\cdot K{\mathrm{\epsilon}}^{n}$(8)

Figure 14 shows a comparison between our FEM result and the modified GA model. It shows that the modified model is much closed to FEM than GA model. But still there is a discrepancy because the modified GA model only considers the effect of the strut length and width. The other parameters such as truss thickness, the inclination angle and the thickness of face sheet have been ignored. In addition, in the above three models, the node dimension *b* shown in Figure 2 has also been ignored, and the node is assumed as rigid and the four trusses deform by creep bending, which also brings some error. As proved by Jiang et al. [28, 29], the size effect, including truss dimension, face sheet thickness and inclination angle, has a great effect on mechanical strength. We found that the as-brazed residual stresses increase as the face sheet thickness increasing. With truss thickness and truss length increase, the residual stresses decrease first and then increase. Therefore in the future, the size effect on creep should be investigated fully, and the GA model still needs to be improved to show the effect of all dimensions. In fact, it is very difficult to get an analytical model considering all the parameters. Therefore, a sensitivity study of parameters affecting creep deformation by finite element calculation to the unit cell should be performed in the future.

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