For a triangular element, the displacements and velocities of the three nodes are known, the shape function is known too. How can I obtain the deformation gradient and the velocity gradient at the integration point?
If this is a linear element, these gradients are approximated as constant throughout the element.
This is pretty classic finite element B-matrix stuff. You need to express the gradients of the master element variables (typically xi and eta) with respect to the current configuration. Take the gradient of the displacement field (or velocity field) with respect to the master element variables and use the chain rule to get the gradients with respect to the real spatial variables (typically x and y). I've worked through this for fun, but don't have the patience to type it up here.
Keep in mind that you will have to add the 2-D identity tensor to the displacement gradient you find to construct the deformation gradient.
It sounds like you will need a higher order element, then, in order to have a basis function set that is second order and produces nonconstant gradients. The process I outlined (look in any FEM book for details based on your basis fns) is the same and can be evaluated at the integration points.
Linear element?
If this is a linear element, these gradients are approximated as constant throughout the element.
This is pretty classic finite element B-matrix stuff. You need to express the gradients of the master element variables (typically xi and eta) with respect to the current configuration. Take the gradient of the displacement field (or velocity field) with respect to the master element variables and use the chain rule to get the gradients with respect to the real spatial variables (typically x and y). I've worked through this for fun, but don't have the patience to type it up here.
Keep in mind that you will have to add the 2-D identity tensor to the displacement gradient you find to construct the deformation gradient.
Matt
In reply to Linear element? by Matt Lewis
Thanks.
Matt Lewis, Thanks for your help.
I will use an element which have non-constant strain throughout the element.
Maybe I need some time to figure out that.
Best Regards.
Higher order
Yang,
It sounds like you will need a higher order element, then, in order to have a basis function set that is second order and produces nonconstant gradients. The process I outlined (look in any FEM book for details based on your basis fns) is the same and can be evaluated at the integration points.
Matt Lewis
Los Alamos, New Mexico
To find strain gradient you
To find strain gradient you will need a higher order element as mentioned by Matt Lewis.
You can check out this paper: http://onlinelibrary.wiley.com/doi/10.1002/nme.2918/abstract
where strain gradients were used