An Approach to the Mechanics of Fluids

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

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Section V - General Stress-Strain Relations

Summary for the Section :

• In this Section , we shall look at general Stress-Strain Relations .

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The { Cube } as a { Finite Element } :

• Let us start-off with a cubic { finite element } and mark-off the 6 faces of the { cube } :
  • as [ Face #1 ] thru [ Face #6 ] , respectively ,

    as per this set of 2 diagrams below .

  

• We can then identify , for each of the 6 faces , a set of { 3 orthogonal forces } acting on each of the 6 said-surfaces ,
  • as per this set of 6 diagrams below .

  

  

  • so that there are now a total of 18 { Force Vectors } altogether .

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Shearing Stress-Strain Relations :

• Let us now isolate the situation a bit , to look more closely at the effects of pure { shearing forces } .

  We shall now make the 18 { Vector Forces } mentioned-above all [ zero ] , except for :

  • [ Force S-2Z ] , [ Force S-3X ] , [ Force S-5Z ] and [ Force S-6X ] ,

    so that the 4 { forces } are of-equal-strength and magnitude ,

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

    

    • The diagram on-the-left then shows the strained-shape of original { quadrilateral A-B-D-C } as a result of these pure { shearing forces } .

    Let us now look at these again , the { unstrained element } vs. the { strained element } , in 3-D :

    

• And we simply note here that , under pure { shear } :
  • we started-off with the { cube } as the { finite element } , and

  • we ended-up with a { Symmetric or Quasi-Symmetric Six-Face Object } as the resultant { strained-element } .

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Large Cylindrical Tank with a Drain Pipe :

• Let us now look at a very-large-diameter Cylindrical Tank with circular Drain Pipe in-the-middle ,
  • initially filled with water ,

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

  

  We then open the { drain valve } so that water starts to flow downwards .

• Let us now look a cross-section of { finite elements } , i.e. { Cross-Section A-A } ,
  • as per the diagram on-the-left , below .

  

• Let us now do the following :
  • use a { 17 rows x 17 columns square-mesh } to divide the [ circular area ] into smaller [ sub-areas ] ,

  • adjust the thickness of the { circular-plate cross-section } so that the majority of the { finite elements } here are { cubes } ;
    • as per the diagram on-the-right , above .

• Let us now look at the { finite elements } at a time [ t = Delat-T ] , after the opening of the valve ,

  

• And we ask this question here :
  • which of these { finite elements } would remain a { Symmetric or Quasi-symmetric Six-Face Object } ?

  Our conjecture here is that :

  • The { finite elements } on the { yellow-color-cross } as marked above , might have a chance of remaining { Symmetric or Quasi-Symmetric Six-Face Objects } ,

  • The 9 { finite elements } as marked in { white-color } would most-likely turn into { General-Random-Type Six-Face Objects } .

  And this line-of-thinking would probably remain valid if we were to replace :

  • the { 17 rows x 17 columns square-mesh } above with a { 65,537 rows x 65,537 columns square-mesh } .

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The Key Issue :

• The key issue being raised here is simply this :
  • If we were to force ourselves to replace { GRT-SFO finite elements } with { Symmetric or Quasi-Symmetric Six-Face Objects } ,

    there is an inherent danger that the { faces } would not match exactly as we move from-elment-to-element ,

    • and these discrepancies may need to reconciled with fully in the long-run .

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• In the next 3 Sections , we shall look a preliminary proposal on how to sub-divied a { flow-space } into { finite elements } .

go to the next section : Section VI - { Flow Net } in Soil Mechanics

go to the last section : Section IV - 18 { Degrees-Of-Freedom } associated with a { General-Random-Type Six-Faced Object }

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original dated 2008-6-08