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Continuum Mechanics  Deformation gradient 
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#1
Dec2712, 10:59 AM

P: 14

Hi all,
I am trying to selflearn continuum mechanics, and I have a question regarding the development of the deformation gradient (which ultimately leads to green's deformation tensor). I have attached the specifics of the question in a attached photo. Ultimately, there comes a point when determining the deformation using the change in magnitude of the square of dX and dx: dx^2  dX^2 = dxidxidXadXa However, somehow using a previous equation (dxi = xi,adXa) and the susbtitution property of the kronecker delta, they come up with: dx^2  dX^2 = xi,adXa*xi,bdXb  delta(ab)*dXa*dXb My question is  how was the kronecker delta substituted in? There is no direction associated with magnitudes. Further  where did the subscript "B" come from and what does it represent physically? Any help would be greatly appreciated. 


#2
Dec2712, 05:16 PM

P: 5,462

First the subscript B is introduced as part of the kronecker delta, δ_{ij} or δ_{AB} here.
It is only a dummy and does not really matter since if B ≠ a then δ_{AB} = 0 Which brings us to why the kronecker? Well this is a way of writing the dot product of the vectors dX_{i} and dx_{i} etc. Don't forget that the quantities in this diagram are vectors so additions and multiplicasions are vector additions and multiplications. 


#3
Dec3012, 08:06 PM

P: 14

Thank you for your response. However, I still am trying to wrap my head around the introduction of a new variable. After substiuting out dXa, (dx)^2  (dX)^2 = (xi,a*xi,a  1)*dXa*dXa how does the kronecker delta substitute (dijej = ei) in here to get the next equation? (dx^2)  (dX)^2 = (xi,a*xi,b  dab)*dXa*dXb if a = b, then you get the previous equation. If a =/ b, then you get (dx^2)  (dX)^2 = (xi,a*xi,b)*dXa*dXb which im not sure what that means physically. 


#4
Dec3012, 08:14 PM

P: 14

Continuum Mechanics  Deformation gradient



#5
Jan413, 08:05 PM

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Thanks
PF Gold
P: 5,060

You realize, of course, that the Einstein summation convention is being used here. Thus,
[tex]dX_adX_a=\delta_{a,b}dX_adX_b=(dX_1)^2 +(dX_2)^2 +(dX_3)^2[/tex] In tensorial notation, δ_{i,j} are the components of the identity tensor (aka, unit tensor or metric tensor) I. It is often more revealing to write these equations in dyadic tensor notation. Let dx represent a differential position vector between two material points in the deformed configuration of the body, and let dX represent the differential position vector between these same two material points prior to the deformation. The vectors dx and dX can be related to one another in terms of the deformation gradient tensor F: [tex]\mathbf{dx}=\mathbf{F}\cdot\mathbf{dX}[/tex] The squared length of the vector dx in the deformed configuration of the body can be determined by taking the dot product of dx with itself: [tex]\mathbf{dx}\cdot\mathbf{dx}=\mathbf{dX}\cdot (\mathbf {F^T}\cdot\mathbf{F})\cdot\mathbf{dX}[/tex] The change in the squared length of the differential position vector between the two material points in the deformed and undeformed configurations of the body is given by: [tex]\mathbf{dx}\cdot\mathbf{dx}\mathbf{dX}\cdot\mathbf{dX} =\mathbf{dX}\cdot (\mathbf {F^T}\cdot\mathbf{F}\mathbf{I})\cdot\mathbf{dX}[/tex] The CauchyGreen tensor is defined by: [tex]\mathbf{C}=\mathbf {F^T}\cdot\mathbf{F}[/tex] So [tex]\mathbf{dx}\cdot\mathbf{dx}\mathbf{dX}\cdot\mathbf{dX} =\mathbf{dX}\cdot (\mathbf {CI})\cdot\mathbf{dX}[/tex] It is also possible to define the finite strain tensor E as: [tex]2\mathbf{E}=\mathbf{CI}[/tex] I hope this helps. Chet 


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