Deformation index

<h2 id="definition">Definition</h2>
<p>Futamura's deformation index 
  
    
      
        n
      
    
    {\displaystyle n}
  
 can be defined as follows.  
  
    
      
        p
      
    
    {\displaystyle p}
  
 is the parameter whose value is controlled (ie held constant).  
  
    
      
        E
      
    
    {\displaystyle E}
  
 is Young's modulus of linear elasticity.  
  
    
      
        ϵ
      
    
    {\displaystyle \epsilon }
  
 is the strain.  
  
    
      
        σ
      
    
    {\displaystyle \sigma }
  
 is the stress.  
</p>

p
        =
        ϵ
        
          E
          
            
              n
              2
            
          
        
        =
        σ
        
          E
          
            
              
                n
                2
              
            
            −
            1
          
        
      
    
    {\displaystyle p=\epsilon E^{\frac {n}{2}}=\sigma E^{{\frac {n}{2}}-1}}

<p>Particular choices of 
  
    
      
        n
      
    
    {\displaystyle n}
  
 yield particular modes of control and determine the units of 
  
    
      
        p
      
    
    {\displaystyle p}
  
.  For 
  
    
      
        n
        =
        0
      
    
    {\displaystyle n=0}
  
, we get strain control: 
  
    
      
        p
        =
        ϵ
        =
        σ
        
          E
          
            −
            1
          
        
      
    
    {\displaystyle p=\epsilon =\sigma E^{-1}}
  
.  For 
  
    
      
        n
        =
        1
      
    
    {\displaystyle n=1}
  
, we get energy control: 
  
    
      
        w
        =
        
          
            1
            2
          
        
        
          p
          
            2
          
        
        =
        
          
            1
            2
          
        
        
          ϵ
          
            2
          
        
        E
        =
        
          
            1
            2
          
        
        
          σ
          
            2
          
        
        
          E
          
            −
            1
          
        
      
    
    {\displaystyle w={\frac {1}{2}}p^{2}={\frac {1}{2}}\epsilon ^{2}E={\frac {1}{2}}\sigma ^{2}E^{-1}}
  
.  For 
  
    
      
        n
        =
        2
      
    
    {\displaystyle n=2}
  
, we get stress control: 
  
    
      
        p
        =
        ϵ
        E
        =
        σ
      
    
    {\displaystyle p=\epsilon E=\sigma }
  
.
</p>
<h2 id="history">History</h2>
<p>The parameter was originally proposed by <a href="/facts/Shingo_Futamura/BqntRej9">Shingo Futamura</a>, who won the <a href="/facts/Melvin_Mooney_Distinguished_Technology_Award/YffjRP3g">Melvin Mooney Distinguished Technology Award</a> in recognition of this development. Futamura was concerned with predicting how changes in viscoelastic dissipation were affected by changes to compound stiffness. Later, he extended applicability of the approach to simplify finite element calculations of the coupling of thermal and mechanical behavior in a tire.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> William Mars adapted Futamura's concept for application in fatigue analysis.
</p>
<h2 id="analogy-to-polytropic-process">Analogy to polytropic process</h2>
<p>Given that the deformation index may be written in a similar algebraic form, it may be said that the deformation index is in a certain sense analogous to the polytropic index for a <a href="/facts/Polytropic_process/eIBfR9fI">polytropic process</a>.
</p>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Futamura, Shingo (1 March 1991). "Deformation Index—Concept for Hysteretic Energy-Loss Process". Rubber Chemistry and Technology. 64 (1): 57–64. doi:10.5254/1.3538540. Retrieved 4 August 2022. <a href="https://meridian.allenpress.com/rct/article/64/1/57/91469/Deformation-Index-Concept-for-Hysteretic-Energy" target="_blank">https://meridian.allenpress.com/rct/article/64/1/57/91469/Deformation-Index-Concept-for-Hysteretic-Energy</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Mars, William V. (1 June 2011). "Analysis of Stiffness Variations in Context of Strain-, Stress-, and Energy-Controlled Processes". Rubber Chemistry and Technology. 84 (2): 178–186. doi:10.5254/1.3570530. Retrieved 19 August 2022. <a href="https://doi.org/10.5254/1.3570530" target="_blank">https://doi.org/10.5254/1.3570530</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Futamura, Shingo; Goldstein, Art (2004). "A Simple Method of Handling Thermomechanical Coupling for Temperature Computation in a Rolling Tire". Tire Science and Technology. 32 (2): 56–68. doi:10.2346/1.2186774. Retrieved 7 October 2022. <a href="https://meridian.allenpress.com/tst/article/32/2/56/129391/A-Simple-Method-of-Handling-Thermomechanical" target="_blank">https://meridian.allenpress.com/tst/article/32/2/56/129391/A-Simple-Method-of-Handling-Thermomechanical</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Deformation index open-in-new

Deformation index