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The mechanical behavior of paper materials under compression in the out-of-plane direction is highly nonlinear. If the influence of the surface topography is not taken into account, the stress-strain curve of paper materials in the loading process is a typical example of materials with J-shaped compressive curves. When compression is released, the stress-strain curve in the unloading process is also nonlinear. The main purpose of this paper is to establish a suitable mathematical model and actualize the description of the compression curve for paper and paper stacks. The loading and unloading nonlinearities of paper stress-strain relations can be approximated by using different equations. In this paper, the loading curve of paper is calculated by using the sextic polynomial equation and the unloading curve is described by using the modified exponential function. All the used coefficients for determining the functions are expressed as the functions of the stress at the start point of unloading. The compressive behavior of paper under some given forces are also calculated by using the identified equation and verified by means of the experimental data. For multiple sheets, it is assumed that when the force is the same, the deformation of the paper stack is directly proportional to the number of sheets. Based on this assumption, the force-deformation relation of the paper stack is derived. The comparative analysis of the experimental results demonstrates the effectiveness of the description model.
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 
 
-
 
      -
       



 -
 

      
    
       
       
       
      
  -






 

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

 
 
 



Jian Chen , Dieter Spiehl, Edgar Dörsam, Arash Hakimi Tehrani and Simon Weißenseel
 
 
 
 
 
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-
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
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
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-
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  
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  
    
       
-

      

      -
         
    -
      
      
 
     
     
        
-
       
-
       
      


    

 
   
-

 


     
 
  




   

   
-
-


 

 


 

       
 

  
 
       

 


 

   -
 

   -
  


 
  
  


   

-


  
       






    
 


-



 -

 
 


 
       
       
-
      
-

      
  
  

  
      

-
-
  





       
z  
 -
  





.
     
-
-

 
       
      


 = !+ "∙  + #∙ #+ $∙ $
+%∙ %+ &∙ &+ '∙ '[MPa]
 
-
aii -
 a  
         
a
      -
-
   

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  
 

𝛽
$&'( 𝛽a+b∙
&'( +c∙ &'(
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exp ∙
&'( − exp ∙
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 
  


 
-
 a 

pq-

  
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     
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
 

 

 
F  
 z        -
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 d  d.
-
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
  






  


 
 
 

F F-
z
z-


zn
zn   
n
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

  


  
 
kk

-

 kk  -
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 
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 
-


 
  
 

 
  
 

 
Fn Fn 
     
      
       

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 = 
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 
 

 × 
 
d .
 
       
-
  
 
 
 




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 
-
-

-

-





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
   
-
      
 

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 
 

 
 -
 
d = -

  



   





 
 

        
 -
        
-


ai






    


-


 
 
 -

   
       
-
-

   
        



 
Strain, ε (−)



 β
ε

   
       
  


-
   


spu
 ε β 
   
   
   
  
  
  
      

      

 
-



 6  
   
 
εspu (−)

and εspu


        
 
 a = 
= = -


-

 εrε
      




εspu and εr


 ε εr 
   
   
   
  
  
  

 -
   
  

         

     
 
   

"=
%&'( = 0.49 ∙
&'( − 0.027
 



  
  p = q =    
 = 
 α
ε

 
  


𝛼
$&'( 𝛼&'(
exp 𝛽
&'( − exp 𝛽
/
 
 kF
      -
        
zn
znun        n.

     
         
-

-
 -








-
  

-
 

loading

 F k 
 
  
  
  
  
  
 
k

 F  
k-
 


      k  -
  

Force, Floading (N)


loading

      



",%&'()*+ =2.29×1034 ∙ ",%&'()*+
74∙ 𝑛mm]
 
 kF
  
-

   

        



    -
 


   
    



  



unloading

 F k 
 
  
  
  
  
  
 
   
   
Fk
  
 
      k
F


-

 



unloading under Funloading,

 
       


   

  
 
 

 

-
      

 

        


-
 






-



   






  
-
 
 
      


       
 









 





   
   
   
   
   
   
   
   
   
 
 
 


     





𝜎
$ exp $∙ −1 [MPa]
 
pq


      

"#$ = 0.6357 ∙ exp 13.5392 ∙
"#$ −1 [MPa]
 




spu




  
 
 
pq

   

    
  
 
-
       
        
  

-




    
    -
    
     
       
  -

         
 -

         
  
 

 






 mm MPa

 

  
     

   -
       
-
-
  

      







 










 





    
    +
    +
    
   
   
   
   
   
-

   
      
   


       





     
        
-
 

    

 









 



  



  -
-

-


  

        
  
  






     
   

 -
  
       
 

        
  





      

 

ai i 
a  
d 
d 
F 
F 
F 
F 
F 
Fn  n
Fn  n
k 
k                                                                       
n 
pq 
pq 
pq  
z 
z 
z 
zn n
zn n
 
 
 
 

 
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



 
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, . 

  
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,
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
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 
 
 
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 
 


 

Journal 

 
 .
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Tappi Journal, 
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... However, in [3] such a behavior was observed even for compression of blocks of timber wood. In [4] the influence of the number of stacked paper samples on the stiffness was investigated. In that article even for very small stacks of paper similar progressions could be observed and additionally, it was concluded that the influence of the number of paper sheets is negligible up to 160 layers for reasonable compression which is fulfilled in the present setting. ...
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Windings of power transformers consist of several layers of different materials. The assembled final winding is dried and prepressed. The tasks are to determine the initial heights of the single materials and to predict the stresses in the winding pressring. An automatic simulation tool for modeling the winding compression process is set up. The material model for the winding is a generic model based on the properties of the individual components of the windings. The single component stiffnesses were determined through compression measurements. The Finite Element simulation model is created through Salome and Python and simulated in HOTINT.
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In this paper, the loading force-deformation behavior of paper and paper stacks in the out-of-plane direction is investigated in detail. According to the common literature, the paper is regarded as an elastic material, the paper body is divided into two surface structures and one internal structure. For a given total deformation, the deformations of the surface structure can be calculated by using the Newton-Raphson method. The cubic curve fitting method for calculating the relationship between surface deformation and total deformation is proposed and compared with the linear curve fitting method. Based on the model of one sheet, a new model for calculating the force-deformation relationship of multiple sheets is proposed. Both of the calculated results of the linear and cubic curve fitting methods are compared with experimental results. Then, the deviations of the deformations under different forces will be calculated. The comparative analysis of the experimental results demonstrates the effectiveness of the proposed model and its usability to describe the force-deformation behavior of paper.
Conference Paper
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The surface topography of paper can range from very rough to extremely smooth, which has significant influences on mechanical properties of paper materials, especially the compressive behavior of paper in the out-of-plane direction. Normally, the stress-strain relations of most of the materials are calculated by using the nominal contact area, which is the whole area of the pressure head. The difference between actual and nominal contact area is ignored, but they are very different, and cannot be neglected in all situations. In this paper, a new experimental method for evaluating the relationship between the actual contact area and the normal load is proposed. A carbon paper is introduced in this method, and it is assumed that the measured contact areas between carbon paper and the actually tested paper are the same as the actual contact areas between the pressure head and the tested paper. Based on this assumption, the mechanical behavior of paper in the out-of-plane direction could be discussed by calculating the actual stress-strain relation and deducing the actual modulus. In addition, the force sensitivities of different carbon papers used for showing the actual contact areas were also compared. The calculation results show the crucial differences between the actual and nominal stress-stain behaviors.
Conference Paper
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A number of models have been proposed with the objective of predicting the stress-strain behaviour of paper materials in out-of-plane (ZD) direction, but the simulation research is still quite limited. The mechanical behaviour of paper materials under compression is highly nonlinear, and it is also exhibits quite complicated unloading behaviour when compression is released. The stress-strain curve of paper materials is a typical example of materials with J-shaped compressive curves, which is the same as seen for gasket materials. There is still no model provided in ANSYS or ABAQUS which could be directly used for paper simulation. But the thickness-direction behaviour of gasket materials could be simulated using finite element (FEM) software. The main purpose of this paper is to actualize the compressive simulation of paper materials by using the gasket elements provided in the FEM software. In this paper, the accuracy of this new method was verified by comparing with the experimental results. The compressive simulation of paper under some further selected forces was also calculated and verified based on the material model established. Comparative analysis of the experimental results proves the effectiveness of the simulation.
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The localized deformation patterns developed during in-plane compression and folding of paperboard have been studied in this work. X-ray post-mortem images reveal that cellulose fibres have been reoriented along localized bands in both the compression and folding tests. In folding, the paperboard typically fails on the side where the compressive stresses exists and wrinkles are formed. The in-plane compression test is however difficult to perform because of the slender geometry of the paperboard. A common technique to determine the compression strength is to use the so-called short-span compression test (SCT). In the SCT, a paperboard with a free length of 0.7 mm is compressed. Another technique to measure the compression strength is the long edge test where the motion of the paperboard is constrained on the top and bottom to prevent buckling. A continuum model that previously has been proposed by the authors is further developed and utilized to predict the occurrence of the localized bands. It is shown that the in-plane strength in compression for paperboard can be correlated to the mechanical behaviour in folding. By tuning the in-plane yield parameters to the SCT response, it is shown that the global response in folding can be predicted. The simulations are able to predict the formation of wrinkles, and the deformation field is in agreement with the measured deformation pattern. The model predicts an unstable material response associated with localized deformation into bands in both the SCT and folding.
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