Chapter 6. Macro-scale I: ABAQUS implementation of the non-convex yield surface 49 existing material models included in ABAQUS material library can accurately represent the behaviour of the material to be studied, then a UMAT can be used to create one's own material. Through this interface, material models of any complexity can be dened and tested with advanced structural elements, complex loading conditions, contact and friction and so on. It is also extremely convenient if one wants to model materials which present instabilities or localization phenomena.
Now, let us consider a homogeneous 2D square, with eight-node plane stress mesh ele-ments (ABAQUS denomination: CPS8). We set the boundary value problem xing the bottom elements and imposing a displacement on top. From the classical von Mises yield criterion, we modify the UMAT, adding personalized subroutines, increasing gradually the complexity of the model implemented. Therefore, we start implementing the yield surface found with the equation (5.9) and we equip it with the associative ow rule (eq.
3.31) or a ow rule accounting for a stress decomposition into isocoric and volumetric part (eq. 6.4); nally we add isotropic hardening as
Φ =‖T‖ −𝜆 𝑚𝑐𝑟𝑖𝑡(𝜃, 𝜒), (6.1) where𝜆is a multiplying factor, scaling isotropically the yield surface, or with a hardening rule acting locally on the yield surface (eqs. 6.2and6.3). Notice that this last approach is empirical, based on the observation of the eective behaviour of the structure, i.e. an initial softening behaviour, when the rows collapse one after the other, and hardening when compactication happens (refer to chapter1 for the general explanation).
For the complete UMAT le, please look at the AppendixB.
Von Mises yield surface with associative flow rule
As a reference solution, we consider the classical von Mises yield limit (eq. 3.26), together with the associative ow rule (eq. 3.31), shown in Figure 6.1:
Figure 6.1: Final configuration of a squared sheet of polyethylene after a compression test. In the scale of colours the largest eigenvalue of the absolute strain.
Figure 6.2: Final configuration of a homogenized honeycomb structure, where asso-ciative flow rule is applied.
We already see the unstable behaviour of the material, which presents a recess in the middle.
If we add isotropic hardening (see the AppendixB) the result changes slightly:
Chapter 6. Macro-scale I: ABAQUS implementation of the non-convex yield surface 51
Figure 6.3: Final configuration of a homogenized honeycomb structure, where asso-ciative flow rule is applied and isotropic hardening is considered.
We see that the strains are distributed less homogeneously. The recession on the sides is less evident than in the previous case and the general response is stier. In both cases, the nal conguration results are qualitatively wrong: indeed, while expanding laterally, as we can see from both experimental and numerical results presented in the previous chapters, the body shows lateral contraction, up to superimposition of the mesh, when further displacement is imposed.
Finally we implement a distortional hardening that acts locally on the yield surface. As we explained before, with this method, we would like to capture the softening-hardening behaviour that honeycomb structures show when plastic deformations occur, especially during plastic compression. To this purpose we add a term 𝑐(T,𝜀) to the magnitude of the yield stresses so that:
Φ =‖T‖ −(𝑚𝑐𝑟𝑖𝑡(𝜃, 𝜒) +𝑐(T,𝜀)) The function𝑐(T,𝜀)has the following form:
𝑐(T,𝜀) =𝑓(tr(T·𝜀)
‖T‖ )𝑔(𝜒(T)), (6.2)
with
𝑓(𝑥) =
{︃(𝑥−0.4)4−0.44, if 𝑥≥0
0, if 𝑥 <0
𝑔(𝜒) =e(−(𝜒−𝜋)22𝜋/36 )+e(−(𝜒−3/2𝜋)22𝜋/36 ),
(6.3)
where we wrote𝑓(𝑥)instead of 𝑓(tr‖T‖(T·𝜀)) for simplicity.
The function𝑓(𝑥) (Figure6.4) is built so that it triggers localization through softening, until the 80% of the total deformation is reached, and compactication through hard-ening, afterwards. 𝑓(𝑥) < 0 for strains 𝑥 < 0.8 therefore the material will present a softening behaviour, while in the opposite cases we will have hardening.
Figure 6.4: Tuning equation𝑓(𝑥).
Its argument tr‖T‖(T·𝜀) is the plastic strain component which is in the same direction ofT, so that, the described behaviour only happens when you keep loading always in the same direction. On the other hand, this softening-hardening phenomenon, is mostly visible when the structure is subject to compression. That is where the function 𝑔(𝜒)enters.
Figure 6.5: Bell equation 𝑔(𝜒).
𝑔(𝜒)is a bell function which tells when the behaviour captured by𝑓(𝑥)is more eective.
For example, in the cases of pure compression when 𝜒 =𝜋,3/2𝜋, we have a big contri-bution of hardening, and𝑓(𝑥)has a big eect on the yield surface. The overall action is well visible in the following gure:
Chapter 6. Macro-scale I: ABAQUS implementation of the non-convex yield surface 53
Figure 6.6: Combination of tuning and bell equations: 𝑐(T,𝜀)
where the function 𝑐(T,𝜀), in the case of compression, is reported. The material be-haviour will be softened for strains 0.6<‖𝜀‖<0.8, when the rows crash adjacently to each other, and hardened after it, when all the rows have collapsed and compacted.
After implementing such a distortional hardening in ABAQUS, we nd, for the same boundary value problem as in the previous cases, the following nal conguration:
Figure 6.7: Final configuration of a homogenized honeycomb structure, where asso-ciative flow rule is applied and distortional hardening is considered.
On one hand we do not have any more instability problems and singularities in the solution, but, on the other, we see that the localizing behaviour is practically disappeared.
Honeycomb plastic model with stress tensor decomposition flow rule The last set of simulations that we run, are based on a dierent ow rule, which we again equip with isotropic or distortional hardening. For this case, we decompose the stress tensor into its deviatoric and dilatoric part. This will allow us to control how much of
Figure 6.8: Equation tuning the dilatoric and deviatoric contributions ofT: 𝑠1/𝑠2= 10e‖𝜀‖+ 1
In the Appendix B all this modications to the UMAT le of ABAQUS can be found and are further explained step by step.
As rst case, we implement our yield criterion together with the presented non-associative ow rule:
Chapter 6. Macro-scale I: ABAQUS implementation of the non-convex yield surface 55
Figure 6.9: Final configuration of a homogenized honeycomb structure, where non-associative flow rule is applied.
Then we add isotropic hardening:
Figure 6.10: Final configuration of a homogenized honeycomb structure, where non-associative flow rule is applied together with isotropic hardening.
And nally the more complicated model with additional distortional hardening:
Figure 6.11: Final configuration of a homogenized honeycomb structure, where non-associative flow rule is applied and distortional hardening is considered.
From this set of results we can appreciate a more homogeneous distribution of the strains.
We see that the lateral sides of the polygon remain parallel, contrarily to what happened in the previous cases (Figures from6.2to6.7). We notice a small concentration of strains in the upper and lower parts of the body. The last (Figure6.11), more complicated model, nally, does not bring any new informations about the strain distribution, which is, once more, localized close to the upper and lower boundaries.
Conclusions
We implemented the mathematical t of the yield surface of the previous chapter in a UMAT subroutine. We tested the behaviour of the body with associative and non-associative ow rules and isotropic and distortional hardening. We conclude that knowing Φ(T) is not enough to catch the plastic behaviour of the structure, since associative plasticity does not work. Although we can see that softening triggers the localization, all the study is kept on a very empirical level. To determine the ow rule and the evolution of the yield surface, either one performs a huge number of numerical simulations and experiments, or some more ecient approach has to be considered. We will see in chapter 8that the second option leads to better results, is easier to manage and to understand.