1.1 Deformation Gradient and Green-Lagrange Strain Tensor
Let the vectors X and x represent the undeformed and deformed configurations while u(X) is the deformation vector between these.
\(X + u = x\)
There’re 2 different paths to reach point A:
Path 1 — > \((X + dX) + u(X + dX)\)
Path 2 — > \(x + dx\)
We know from the definition of derivative:
\(\displaystyle u’ = \frac{du}{dX} = \lim_{dX\rightarrow 0}\frac{u(X + dX) \:-\: u(X)}{dX} = \nabla u\) (*)
Equating these and using (*) :
\(X + dX + u(X + dX) = x + dx\)
\(\begin{aligned}
dx &= dX + u(X + dX) \:-\: (x \:-\: X) \\
&= dX + u(X + dX) \:-\: u(X)\\
&= dX + \nabla u\:dX\\
&=(I + \nabla u)\:dX
\end{aligned}\)
From the definition of deformation gradient (F):
\(\displaystyle I + \nabla u = \frac{dX}{dX} + \frac{du}{dX} = \frac{d(X + u)}{dX} = \frac{dx}{dX} = F\)
Finally, we get the famous equation:
\(\boxed{dx = F\:dX}\)
Let dS and ds represent the undeformed and deformed lengths. Here, we choose to write deformed vector (dx) in terms of undeformed vector (dX). If the opposite were done, we would obtain “Euler-Almansi” strain tensor but we’ll not be using it.
\((dS)^2 = dX \cdot dX\)
\(\begin{aligned}
(ds)^2 &= dx \cdot dx \\
&= (F\:dX) \cdot (F\:dX)\\
&= dX \cdot \underbrace{(F^TF)}_C \cdot dX
\end{aligned}\)
Here, C is defined as “Right Cauchy-Green” tensor. Now, define the difference between the lengths:
\(\begin{aligned}
(ds)^2 \:-\: (dS)^2&=dX \cdot C \cdot dX \:-\: dX \cdot dX \\
&= dX \cdot C \cdot dX \:-\: dX \cdot I \cdot dX\\
&= dX \cdot \underbrace{(C \:-\: I)}_{2E} \cdot dX
\end{aligned}\)
E is called “Green-Lagrange Strain” tensor.
\(\begin{aligned}
E&=\frac{1}{2}(C \:-\: I) \\
&= \frac{1}{2}(F^TF-I)\\
&= \frac{1}{2}\left[(\nabla u + I)(\nabla u + I)^T \:-\: I\right]\\
&= \frac{1}{2}\left[\nabla u + (\nabla u)^T + \nabla u \cdot (\nabla u)^T\right]
\end{aligned}\)
Transform into indicial form:
\(\begin{aligned}
(\nabla u)^T \cdot \nabla u&=\left(\frac{\partial u_t}{\partial x_k}e_k \otimes e_t\right)\left(\frac{\partial u_m}{\partial x_n}e_m \otimes e_n\right) \\
&= \frac{\partial u_t}{\partial x_k}\frac{\partial u_m}{\partial x_n}\:\delta_{tm}\:e_k \otimes e_n\\
&= \frac{\partial u_t}{\partial x_k}\frac{\partial u_t}{\partial x_n}e_k \otimes e_n
\end{aligned}\)
\(\displaystyle (\nabla u)^T \cdot \nabla u=\frac{\partial u_k}{\partial x_i}\frac{\partial u_k}{\partial x_j}e_i \otimes e_j\)
Finally, the Green-Lagrange Strain tensor is found:
\(\boxed{\displaystyle E_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} + \frac{\partial u_k}{\partial x_i}\frac{\partial u_k}{\partial x_j}\right)}\)
1.2 Velocity Gradient
Another significant term is velocity gradient. It’s used for work conjugacy while derivating the virtual work equation. Firstly, begin with the time derivative of deformation gradient:
\(\displaystyle \dot{F} = \frac{dF}{dt} = \frac{d}{dt}\left(\frac{\partial x}{\partial X}\right) = \frac{d}{dX}\underbrace{\left(\frac{\partial x}{\partial t}\right)}_v = \frac{\partial v}{\partial X} = \nabla v\)
Velocity vector is defined according to deformed configuration x. Since \(v = v(x,t)\) and \(x = x(X,t)\) using chain rule:
\(\displaystyle \dot{F} = \frac{\partial v}{\partial X} = \underbrace{\left(\frac{\partial v}{\partial x}\right)}_L\underbrace{\left(\frac{\partial x}{\partial X}\right)}_F\)
\(\displaystyle \dot{F} = L \cdot F\)
\(\boxed{L = \dot{F} \cdot F^{- 1}}\)
L is defined as velocity gradient. Secondly, we need time derivative of Green-Lagrange Strain tensor:
\(\displaystyle \dot{E} = \frac{\partial E}{\partial t} = \frac{1}{2}\frac{d}{dt}\left(F^T \cdot F – I\right) = \frac{1}{2}\left(\dot{F}^T \cdot F + F^T \cdot \dot{F}\right)\)
Since a tensor can be decomposed into symmetric and skew-symmetric parts, the symmetric part of velocity gradient:
\(\begin{aligned}
d&= \frac{1}{2}\left(L^T + L\right)\\
&= \frac{1}{2}\left((\dot{F} \cdot F^{- 1})^T + \dot{F} \cdot F^{- 1}\right)\\
&= \frac{1}{2}\left(F^{- T}\dot{F}^T + \dot{F} \cdot F^{- 1}\right)\\
&= \frac{1}{2}F^{- T}\left(\dot{F}^T + F^T \cdot \dot{F} \cdot F^{- 1}\right)\\
&= \frac{1}{2}F^{- T}\underbrace{\left(\dot{F}^T \cdot F + F^T \cdot \dot{F}\right)}_{2\dot{E}}F^{- 1}
\end{aligned}\)
Finally, the symmetric part of the velocity gradient:
\(\boxed{d = F^{- T} \cdot \dot{E} \cdot F^{- 1}}\)
References:
[1] Holzapfel, G.A. (2000) Nonlinear Solid Mechanics: A Continuum Approach for Engineering.
[2] Reddy, J.N. (2004) An Introduction to Nonlinear Finite Element Analysis.