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Could you finally solve your problem? I have the same issue. Could you please help me?
I have developed in the past in the university of sherbrooke a program which visualizes tetrahedra elements on a workstation under unix operating system with the GL graphics library: the 3D meshes are colored, lighted, shrinked and rotated (movement) using reendering and double buffering for hidden planes in order to check the FE mesh. If you want I can exchange the listing of the program with one of your books.
With the recent available hardwares you can use the opengl (or else) library to do the equivalent work with the corresponding functions. The representation of the displacements, stresses and/or strains can be made with: calculating the stress inside the element and use a 2D or 3D representation of these values with a valid graphics library. This will visualize the ranges of areas values in 2D or the volumes values in 3D.
In 2013, I haven't found any valid calculation of the constant strain tetrahedron shape functions. After that I have developed an efficient method to compute them. This is the modification that I have made to my background knowledges. Regards
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Besides the excellent Notes recommended by Prof. Suo, I found resources provided by Prof. A.F. Bower which explains concisely the question I asked.
There are three rules of mixtures: voigt, reuss, and hybrid. voight assumes constant strain, reuss assumes constant stress and hybrid for UD composites assumes uniform normal strain along fiber direction and all the other stress components are uniform. The most applicable formulas for UD composites are hybrid rules of mixtures.
The only avaiable experiments are applied to a one- directinal composite but the formula can be extended approximately to a 3D one. Your case can be processed with two main formulae: the first one you have cited and (1/Ey)=(Vm/Em)+(Vf/Ef) when taking Ec=Ec. I am convinced that Mr Wenbin has forgotted the following formula Nu=Nuxy=Num*Vm+Nuf *Vf which is cited by Tsai in micromechanics. It is evident that when you have a tensile case the compression case is consequence in the remain direction. The only remark is that they have taken Nu16=0 when using 1, 2 and 6 directions corresponding to x, y and xy. There are other formulae using the angles for each ply laminate calculating the stresses and deformations at each ply then you have Nuxy= - epsilonx/epsilony and else. For a 90 degree it is easier to interchange Ex by Ey. You can also use (1/Gxy)=(Vm/Gm)+(Vf/Gf) using x and y axes and Nux*Ey=Nuy*Ex.
In the volumic case I have used: Nuyx*Ey=Nuxy*Ex, Nuyz*Ey=Nuzy*Ex and Nuzx*Ez=Nuxz*Ex.
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Thank You very much for the notes!
In reply to Although I am not clear what
Dear Wenbin Yu,
Thanks for your reply.
What I mean by stating bidirectional lamina, we put fibers in two directions orthogonal to each other (for example 0 and 90 degree with respect to applied principle stress.
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Although I am not clear what you mean by bi-directional composites, the rules of mixtures you cited are only correct if the Poisson's ratio is zero. My tutorial on micromechanics at https://cdmhub.org/resources/1039/download/Micromechanics.pdf will give you some idea how to compute the material properties for bidirectional composites. There are four types of rigorous rules of mixtures you can use for your problem. If you want to use more sophisticated methods, you can always use the SwiftComp code at https://cdmhub.org/resources/scstandard.
In reply to 3rd generation
This is Marcelo’s response: “The Zeitgeist is now different and so is the style and individual scope. There are many extremely talented people exploring new areas, both in the foundations and the applications. Perhaps what has been lost is a certain sense of grandeur, a kind of pioneering spirit. Continuum Mechanics is no longer regarded by the applied scientists as a flight of fancy. Its place is assured and its terminology widely accepted, thanks to the works of the first two generations. In ten or twenty years from now you will be able to look back and see what the third generation has achieved and how it defined itself in relation to the previous ones.
I collected a few items on constitutive models from my teaching notes.
In reply to "Why" to tensor transformation rules
Thank you very much for your kind words. I have just updated the notes on vector.
In these notes, wish to show rules of calculation. I also wish to show where vectors come from. I show several methods to create new vectors:
- Cartesian product of scalar sets
- Subspaces and their intersections and sums
- Maps from an arbitrary set to a vector space
There are so many giants amongst his generation, who have shaped modern continuum/solid mechanics. But does there exist a third generation of the continuum mechanics given that an era might be closed? --- "His passing may be considered as the closing of an era, namely, the second generation of the continuum mechanics"
In reply to Vector
The best note I've seen on Vectors, Thank you very much!
I particurly like the concrete examples showing how change of basis changes a vector's components. The note provide a clear "why" to the vector transformation rules. I assume tensor is similar, because of the change of basis, a tensor's components also change, and the change of the components obeys the rules of tensor transformation rules. I have found Your note on "tensor" as well, looking forward to reading that, hope I will be able find (appreciate) a clear "why" for tensor as well (which is not usually explained in the text books or classes, and we are often taught to memorize these rules as a student). Thanks!