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Section 1.4 displacement field

Another quantity widely used in the kinematic description of bodies is the difference between the current position and the reference position of the generic point

\begin{equation} \vec{u} = \vec{x} - \vec{X}\label{displ_eq}\tag{1.4.1} \end{equation}

known as displacement field. The relative gradient, known as displacement gradient and indicated with the symbol \(\tens{\nabla u}\text{,}\) is obtained in a similar way to what has already been done for the deformation gradient. Given the field

\begin{align} u_1 \amp = \func{u_1}{X_1, X_2, X_3}\,,\tag{1.4.2}\\ u_2 \amp = \func{u_2}{X_1, X_2, X_3}\,,\tag{1.4.3}\\ u_3 \amp = \func{u_3}{X_1, X_2, X_3}\,.\tag{1.4.4} \end{align}

the following result can be obtained

\begin{equation} \mat{\nabla u} = \left[\begin{array}{ccc} \regulardiff{u_1}{X_1} \amp \regulardiff{u_1}{X_2} \amp \regulardiff{u_1}{X_3}\\ \regulardiff{u_2}{X_1} \amp \regulardiff{u_2}{X_2} \amp \regulardiff{u_2}{X_3}\\ \regulardiff{u_3}{X_1} \amp \regulardiff{u_3}{X_2} \amp \regulardiff{u_3}{X_3} \end{array}\right]\,.\label{Dipl_Grad_X_eq}\tag{1.4.5} \end{equation}

From Equation (1.4.1) the relationship between displacement gradient and deformation can be easily obtained

\begin{equation} \tens{\nabla u} = \tens{F} - \tens{I}\,,\tag{1.4.6} \end{equation}

where \(\tens{I}\text{,}\) the identity tensor, is

\begin{equation*} \mat{I} = \left[\begin{array}{ccc} 1 \amp 0 \amp 0\\ 0 \amp 1 \amp 0\\ 0 \amp 0 \amp 1 \end{array}\right]\,. \end{equation*}
evaluation of the displacement field and its gradient.

With regard to the Transformation 1 already used in the previous examples, the displacement field and its gradient can be evaluated on the basis of the following MATLAB® instructions. Bear in mind that, for the linearity of the transformations involved, the gradients are evaluated as the matrix associated to the transformation. 

T1 = @(X) [-X(2); X(1)];
F = [T1([1; 0]) T1([0; 1])]
Grad_u = F - diag([1 1])

u1 = @(X) T1(X) - X;
Grad_u = [u1([1; 0]) u1([0; 1])]
Listing 1.4.1.

MATLAB® instructions for Transformation 2. 

T2 = @(X) [2*X(1); X(2)];
F = [T2([1; 0]) T2([0; 1])]
Grad_u = F - diag([1 1])

u2 = @(X) T2(X) - X;
Grad_u = [u2([1; 0]) u2([0; 1])]
Listing 1.4.2.

MATLAB® instructions for Transformation 3. 

T3 = @(X) [X(1); 1.5*X(2)];
F = [T3([1; 0]) T3([0; 1])]
Grad_u = F - diag([1 1])

u3 = @(X) T3(X) - X;
Grad_u = [u3([1; 0]) u3([0; 1])]
Listing 1.4.3.

MATLAB® instructions for Transformation 4. 

T4 = @(X) [X(1)+X(2); X(2)];
F = [T4([1; 0]) T4([0; 1])]
Grad_u = F - diag([1 1])

u4 = @(X) T4(X) - X;
Grad_u = [u4([1; 0]) u4([0; 1])]
Listing 1.4.4.