An Approach to the Mechanics of Fluids

by Frank C. Fung - 1st published in June, 2008.

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Epilog I - A Clarification on { [ 18 ] on [ 18 ] }

Purpose :

• This is an attempt to provide further details on the { [ 18 ] on [ 18 ] } situation .

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The Clarification :

• Let us start-off with a { cubic finite element } , of size [ 1 cm. x 1 cm. x 1 cm. ] , as per the diagram on-the-left below :

  

  Let us now subject the said { cubic finite element } to some stresses , so that :

  • the said { cubic finite element } does come to rest , reaching equilibrium , as per the diagram on-the-right , above .

• Let us now sub-divide the unstrained { cubic finite element } into 4,913 { small cubic finite elements } by doing the following :
  • dividing the [ 1 cm. ] in the { X-direction } into 17 equal parts ,
  • dividing the [ 1 cm. ] in the { Y-direction } into 17 equal parts ,
  • dividing the [ 1 cm. ] in the { Z-direction } into 17 equal parts ;

    so that there are now [ 17 x 17 x 17 ] , or 4,913 { small cubic finite elements } ,

    • as per the diagram on-the-left , below .

  

• For our illustration purposes , let us now look at 3 adjacent { small cubic finite elements } in-the-middle :
  • as identified in { Orange / Green / Cyan colors } , respectively , in the diagram on-the-right , above .

  Let us now mark-off the { nodal points } of the { small cubic finite elements } as { Point Q }'s ,

  • as per the set of 3 diagrams below :

  

  Let us now mark-off the { Centroidal Point } , howsoever it may be defined , for each { small cubic finite elements } as the { Point R }'s ,

  • as per the set of 3 diagrams below :

  

  Let us now mark-off the { Area-Centroids } for each face of the { small cubic finite elements } as { Point P }'s ,

  • as per the set of 3 diagrams below :

  

• For our general { finite element analysis } procedure , we shall now make these statements :
  • { Stress / Forces } are always deemed to be acting on each face of the { six-face finite elements } ,
    • these are therefore assumed to be acting thru the { Point P }'s .

  • The resultant strains are always measured via the movements of the { 8 nodal-points } , relative and/or otherwise :
    • and strains are therefore always attributed to the { Point R }'s .

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The Key Issue :

• There is a key issue here :
  • No matter how finely we sub-divide the original { 1 x 1 x 1 cubic finite element } into { smaller cubic finite elements } ,
    • the { Point P }'s and the { Point R }'s do not meet and never co-incide into { single-points } .

  Extra-care should be taken when formulating the { stress-strain relations } , with this in mind .

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original dated 2008-8-05