Hello!
I have tested performance of C3D8I and CSS8 elements with composite layups in the scope of homogenization accuracy and thus their suitability for complete 3D modeling of composite structures with arbitrary layups. In order to do this I compared effective properties of a "brick" of such elements with specific layups calculated through Micromechanics Plugin for Abaqus/CAE (QA00000046185) with 3D properties obtained in the same way on the true layered model of the laminate and also with 3D properties obtained with analytical homogenization technics. For all of the tested stacking sequences all these three methods resulted in the same values of plain stiffness properties, i.e. elastic moduli E1 and E2, Poisson's ratio Nu12 and shear modulus G12. The remaining transverse stiffness properties (E3, G13, G23, Nu13, Nu23) as well as coefficients of thermal expansion (a11, a22, a33) notably deviated from the true layered model and analytical homogenization in some cases. The most probable mechanism of this deviation will be discussed below. It is necessary now to give small considerations of the computational homogenization theory by an example of shear.
The result of computational homogenization (i.e. obtaining effective properties of a composite material) depends on what the system of boundary conditions is chosen for RVE. There are options (see the attached picture).
In the case of free periodic boundary conditions RVE deforms in such way that its opposite faces are unconstrained and can warp periodically. This often helps to create homogeneous strains in the constituents and eliminate edge effects. In the case of constrained periodic boundary conditions opposite faces of RVE are constrained to be flat surfaces. Great care must be taken in latter case due to possible presence of edge effects or deformation patterns which do not agree with actual material behavior. Free periodic BCs can also be considered as "true" BCs in the sense that stiffness properties of a 3D array of RVEs in the case of constrained periodic BCs tend to stiffness properties of a single RVE with free periodic BCs as the array size gets bigger. This property is extremely useful for periodic materials because it is sufficient to take a minimal periodic cell of the material and apply free periodic BCs to get true macroscale properties. For symmetrical RVEs differences in deformation mentioned above occur only in shear. As a summary, free periodic BCs are appropriate for most cases, and when you apply constrained periodic BCs you must clearly understand a reason of their usage.
In the case of transverse shear in composite laminates there are analytical ways to compute effective shear moduli which correspond to both of boundary conditions mentioned above. Volume homogenization of stiffness tensor Cijkl and subsequent extraction of shear moduli G13 and G23 yields to result which has been obtained from laminate RVE with constrained periodic BCs. On the other hand, volume homogenization of compliance tensor Jijkl and subsequent extraction of shear moduli G13 and G23 yields to result corresponding to free periodic boundary conditions. In other words, the way of obtaining effective transverse shear moduli from the compliance tensor implies deformation pattern in which straight lines normal to the laminate midsurface can become broken or curved lines after interlaminar shear, representing the assumption that softer layers undergo more shear and stiffer layers undergo less shear. On the other hand, the way of obtaining effective transverse shear moduli from the stiffness tensor implies deformation pattern in which straight lines normal to the laminate midsurface remain straight after interlaminar shear, representing the assumption that the value of shear strain for all plies is equal and doesn’t depend on layer stiffness. First deformation pattern seems to be very real for composite laminates, while second is not.
Now let’s return to performance of C3D8I and CSS8 elements with composite layups. I’ve discovered that all transverse stiffness properties in this case are such that they are obtained from direct homogenization of stiffness tensor. This means implicit usage of the constrained deformation pattern for shear moduli G13 and G23, which is incorrect for laminates in my opinion. Effective properties E3, Nu13, Nu23 must be homogenized in conjunction with in-plane behavior of laminates due to Poisson’s effect, which is not accurately predicted by direct homogenization of stiffness tensor. The effective coefficients of thermal expansion are also imprecise for some reason, may be due to imprecise stiffness properties. Latter is quite strange because of existing of precise analytical methodic for computation thermal expansion properties of laminates, at least for in-plane directions.
One can rise a question if the difference in the values of true transverse properties and properties obtained using composite layup approach is quite small? As long as we consider composite laminates in normal conditions (with resin is in glassy state), the difference usually is less than 5-6%. But as soon as we start dealing with composites with resin in rubbery state, the difference may become significant. Correct modeling of transverse behavior of composites with rubbery state of resin is extremely important in the field of modeling technological processes and calculating spring-in distortions of composite structures.
To illustrate the case where composite layup method fails, I attach two python scripts (please rename file extensions from .txt to .py):
- composite_layup_model.py creates a model of a laminate through composite layup assigned to continuum solid elements.
- true_layered_model.py creates a model of a laminate in which every ply is modeled by separate layer of solid elements with homogeneous solid sections.
Both models are to be analyzed with Micromechanics Plugin for Abaqus/CAE (QA00000046185) to calculate effective laminate properties (of course, ply material properties and stacking sequences are the same in both models). Material properties correspond to plies of carbon fiber and resin in rubbery state. Comparing results one can see:
- 73.4% difference in modulus G13,
- 10.7% difference in modulus G23,
- 8.3% difference in modulus E3,
- 7.3% difference in coefficient of thermal expansion a22,
- 6.2% difference in Poisson’s ratio Nu13,
- -5.6% difference in Poisson’s ratio Nu23.
I would be very interested to know what the Abaqus developers think about this issue. Is it possible to improve the method of calculating transverse stiffness properties and CTE’s of composite layups assigned to continuum solid elements?
Dear @RM , @RM , @CJ , @JS , please read this post.
Best regards,
Mikhail Kozlov
