PSI - Issue 5

Danuta Miedzińska et al. / Procedia Structural Integrity 5 (2017) 484– 491 Danuta Miedzińska / Structural Integrity Procedia 00 (2017) 000 – 000

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= + ( − ) = + ( − ) = [ ( 1 1 − − , 1) , 0]

(12) (13)

where:

(14)

and is the elastic shear modulus for the fully compacted honeycomb material = 2(1 + ) (15) The relative volume, , is defined as the ratio of the current volume to the initial volume. Typically, = 1 at the beginning of a calculation. The load curves define the magnitude of the average stress as the material changes density (relative volume), see Figure 8. There are two ways to define these curves, a) as a function of relative volume ( ) or b) as a function of volumetric strain defined as: = 1 − (16)

Fig. 8. Stress quantity versus volumetric strain for MAT_026_HONEYCOMB material model by Hallquist (2006).

5. Results and discussion

The achieved results were presented in Table 1 as errors of a homogenization method. The errors were calculated as a percentage difference between results for model after and before homogenization.

Table 1. Calculation results - a percentage difference between results for model after and before homogenization. Model Method error Model Method error LIN_ALL 40% STAT_ALL 44% LIN_4 37% STAT_4 31% LIN_5 18% STAT_5 20% LIN_8 8% STAT_8 7%

The numerical tests of functionally graded materials homogenization were presented in the paper. On the base of achieved results the method errors were calculated for each simulation. It is clearly visible that assuming thinner strips one can get higher accuracy. Unfortunately the characteristic stress distribution around the grains and pores is always lost. The selection of strips thickness should be based on the needed accuracy, accessible computational power and time of analysis.