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stiffness matrix of incompatible elements

Zoeb Kaizar Lakdawala's picture

 am a student of M. Tech. 2nd year. My project topic for dissertation is "Numerical & Experimental Estimation of Fracture Toughness of Some Materials using Compact Tension Specimen". In this dissertation I also wanted to develop a code which directly gives me the values of stress intensity factor just like ansys.

Actually I have develop an matlab programm (using linear quadrilateral element) which gives me the value of displacement at the cracktip using that i would be finding the displacement extrapolation method i would be calculating stress intensity factors

But the major hindrance in my matlab code is that when i am calculating the stresses at the crack tip there is large deviation in cracktip stresses shown by matlab code and ansys. Looking at the ansys theory reference it was found that apart from four shape function i.e, N1, N2, N3 and N4 they are using two more shape function N5 and N6 whose values are (1-s^2) and (1-t^2) s for x-direction and t for y-direction respectively.

Also such type of shape function is described in R. D. Cook, Bathe and Zienkiewics books which they are calling as non-conformal elements which are used to remove the shear lock effect of linear quadrilateral elements and having co-efficient as a1 a2 for x-direction and a3 and a4 for y-direction. But nothing has been stated about the values of ai (i varies from 1 to 4).

Hence Sir, I would be most thankful to you if you could give me guidelines regarding how to solve such type of FEM problem having non-conformal elements or u could give me reference where Problems are solved using Non-conformal elements I would be most thankful to you


Zoeb Kaizar Lakdawala
M. Tech (Industrial Process Equipment Design)
S. V. National Institute of Technology, Surat
Mobile : 09904288552


Using a "non-conformal" elements or a typical displacement
based element is really the same. If you take your example, your linear
quadrilateral element (Q4) has a stiffness matrix which is 8x8. With
the corresponding incompatible element (you add 2 incompatible modes),
the stiffness matrix becomes 12x12 where you have 4 additional degrees
of freedom (a1,a2,a3 and a4).

However, since those additional
degrees of freedom(dof) do not correspond to any nodes, the applied
loads corresponding to those dof should be 0 and you solve your problem
like you would with a typical linear quadrilateral element.

linear analyses, you can also perform a static condensation of the
stiffness matrix to save some computation time. This is well explained in any of the book you cited.

In a nonlinear analysis, you need
the values of all the dof to carry to the next iterations and
increments. So I'm not sure static condensation should be used. Does
anyone have any comment on that? 

Also, related to another post
(, for this type of "non-conformal"
elements, there is no compatibility of displacements along the element

Hope this help you understand how to use non-conformal elements and hopefully you can solve your problem.


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