Hello everyone,
First of all, my math background is not of the highest level and I my field of research is continuum mechanics. I have two questions.
1) I am encountered a problem with wolfram mathematica code and the application of laplacian at a tensor and my alternative is to apply laplacian in its other form at the same tensor in order to solve my problem. When I am referring to laplacian's other form I mean the divergence of gradient of a tensor. My problem is that I do not know if the divergence of gradient of a tensor is the same with the gradient of the divergence of the same tensor and if it is, does it hold for every case? All sources I have found until now mention laplacian as the divergence of the gradient of a scalar/vector/tensor but none of them talks about the gradient of the divergence. I found some vector identities in the internet but they do not seem to solve my problem as well.
2) Searching the web, I found that green strain tensor formula (equation 1) holds for every coordinate system and only the deformation gradient changes, depending on the coordinate system. I also found that deformation gradient gives the same results both in cartesian and in cylindrical coordinates, as deformation remains the same no matter what coordinate system someone is using. Therefore, I am trying to check this out in wolfram mathematica, but without success. To be more specific, I am working on cylindrical coordinates and I wrote the displacement fields as well as the deformation gradient in cylindrical coordinates and I calculated the green strain tensor with the equation 1. I did exactly the same thing for cartesian coordinates (changing of course the formula of deformation gradient and the displacement fields to meet cartesian coordinate requirements) and after this I calculated the green strain tensor (again with the equation 1). I turn the results in cylindrical coordinates in order to compare them with each other, but they do not seem to be the same. I have probably made a mistake in my code but in order to be sure about that I need someone's confirmation about the aforementioned knowledge.
This is the Green Strain Tensor formula which holds for every (???) coordinate system (I did not manage to write it with superscript):
E=1/2(Ftranspose*F-I) (eq. 1)
F is the deformation gradient, I is the identity matrix whereas E is the green strain tensor.
Thank you in advance
First of all, my math background is not of the highest level and I my field of research is continuum mechanics. I have two questions.
1) I am encountered a problem with wolfram mathematica code and the application of laplacian at a tensor and my alternative is to apply laplacian in its other form at the same tensor in order to solve my problem. When I am referring to laplacian's other form I mean the divergence of gradient of a tensor. My problem is that I do not know if the divergence of gradient of a tensor is the same with the gradient of the divergence of the same tensor and if it is, does it hold for every case? All sources I have found until now mention laplacian as the divergence of the gradient of a scalar/vector/tensor but none of them talks about the gradient of the divergence. I found some vector identities in the internet but they do not seem to solve my problem as well.
2) Searching the web, I found that green strain tensor formula (equation 1) holds for every coordinate system and only the deformation gradient changes, depending on the coordinate system. I also found that deformation gradient gives the same results both in cartesian and in cylindrical coordinates, as deformation remains the same no matter what coordinate system someone is using. Therefore, I am trying to check this out in wolfram mathematica, but without success. To be more specific, I am working on cylindrical coordinates and I wrote the displacement fields as well as the deformation gradient in cylindrical coordinates and I calculated the green strain tensor with the equation 1. I did exactly the same thing for cartesian coordinates (changing of course the formula of deformation gradient and the displacement fields to meet cartesian coordinate requirements) and after this I calculated the green strain tensor (again with the equation 1). I turn the results in cylindrical coordinates in order to compare them with each other, but they do not seem to be the same. I have probably made a mistake in my code but in order to be sure about that I need someone's confirmation about the aforementioned knowledge.
This is the Green Strain Tensor formula which holds for every (???) coordinate system (I did not manage to write it with superscript):
E=1/2(Ftranspose*F-I) (eq. 1)
F is the deformation gradient, I is the identity matrix whereas E is the green strain tensor.
Thank you in advance