Solid Mechanics - basic kinematic description

Section 4.1 basic kinematic description

As seen in the Section 1.3 the description of the motion of a body in its generic point involves the deformation gradient \(\tens{F} \text{.}\) In turn, the gradient, by the polar decomposition theorem (Section 1.6), admits the decomposition \(\tens{F} = \tens{R} \tens{U} \) . Assuming the hypothesis of rigid body, that is the invariance of the mutual distances of the points that make up the body, the body does not deform and therefore

\begin{equation} \tens{U} = \tens{I}\quad \Rightarrow \quad \tens{F} = \tens{R}\,.\tag{4.1.1} \end{equation}

Furthermore, always for the rigidity hypothesis, the transformation \(\tens{F} \) does not apply only to a neighborhood of the point but is homogeneous and therefore assumes the same value on all points of the body. This further condition allows to express the link between infinitesimal increments \(d\vec{x} = \tens{F} \, d \vec{X} \) in finite terms, that is

\begin{equation} \vec{x} - \vec{x}_o = \tens{F}\left(\vec{X}-\vec{X}_o\right) = \tens{R}\left(\vec{X}-\vec{X}_o\right)\,,\label{rigid_x_eq}\tag{4.1.2} \end{equation}

where \(\vec{X} \) refers to the generic point and \(\vec{X}_o \) assumes the particular meaning of pole of the rigid transformation. The choice of the pole is completely indifferent and any point, inside or outside the body, leads to the same results. If we introduce the \(\vec{x} = \vec{X} + \vec{u} \) in the (4.1.2) we get the description of the rigid motion in terms of the displacement field

\begin{equation} \vec{u} = \vec{u}_o + \left(\tens{R} - \tens{I}\right)\left(\vec{X}-\vec{X}_o\right)\,.\label{rigid_u_eq}\tag{4.1.3} \end{equation}

The latter relationship highlights how the displacement field of a rigid body is definined by two contributions each characterized by 3 degrees of freedom:

a contribution due to pure translation (3 components) shared by all points of the body, contribution given by the displacement of the pole \(\vec{u}_o\text{;}\)
a contribution due to rotation (3 components) which instead varies for each point with respect to the vector \(\left(\vec{X} - \vec{X}_o\right)\text{.}\)

Subsection 4.1.1 linearization

The generic rotation tensor has a representation of the type, see Remark 1.6.3,

\begin{equation*} \mat{R} = \left[\begin{array}{cc} \cos{\varphi} \amp -\sin{\varphi} \\ \sin{\varphi} \amp \cos{\varphi} \end{array}\right]\,. \end{equation*}

whose components depend nonlinearly on the rotation angle for the presence of the trigonometric functions. But it has also been seen that in the case of infinitesimal displacements, see the Insight 1.9.3, the rotation tensor takes the following expression

\begin{equation*} \tens{R} \approx \tens{I} + \text{skew}\tens{\nabla u}\,, \end{equation*}

where, for 2D case,

\begin{equation*} \text{skew}\mat{\nabla u} \approx \left[\begin{array}{cc} 0 \amp -\varphi \\ \varphi \amp 0 \end{array}\right]\,. \end{equation*}

Therefore, in the case of infinitesimal displacements, the relationship (4.1.3) becomes

\begin{align*} \vec{u} \amp = \vec{u}_o + \left(\tens{R} - \tens{I}\right)\left(\vec{X}-\vec{X}_o\right)\\ \amp = \vec{u}_o + \left(\tens{I} + \text{skew}\tens{\nabla u} - \tens{I}\right)\left(\vec{X}-\vec{X}_o\right)\\ \amp = \vec{u}_o + \text{skew}\tens{\nabla u}\,\left(\vec{X}-\vec{X}_o\right)\,, \end{align*}

and then

\begin{equation} \vec{u} = \vec{u}_o + \text{skew}\tens{\nabla u}\,\left(\vec{X}-\vec{X}_o\right)\,.\label{small_rigid_u_eq}\tag{4.1.4} \end{equation}

A component-wise representation provides

\begin{equation} \left[\begin{array}{c} u_1 \\ u_2 \end{array}\right] = \left[\begin{array}{c} {u_1}_o \\ {u_2}_o \end{array}\right] + \left[\begin{array}{cc} 0 \amp -\varphi \\ \varphi \amp 0 \end{array}\right] \left[\begin{array}{c} \left(X_1 - {X_1}_o\right) \\ \left(X_2 - {X_2}_o\right) \end{array}\right]\,.\label{mat_small_rigid_u_eq}\tag{4.1.5} \end{equation}

Remark 4.1.1.

In tensor format the previous formulae are completely generic. Specifying formulae in the matrix format, reference was made to the two-dimensional context that will be used in subsequent applications. Eq. (4.1.5) in particular shows that the two translation components of the pole, \({u_1}_o \) and \({u_2}_o \text{,}\) and the rotation \(\varphi \) constitute the 3 parameters necessary to describe the motion of a rigid body in the plane.

The displacement components vary from point to point on the body, but the rotation of a rigid body does not vary from point to point, however, in subsequent applications, the rotation of the body will also be denoted by \(\varphi_o \) indicating the pole assumed as a reference. Also notation \(\varphi_A \) will used to indicate the roation in point A of a body.

The following MATLAB® instructions define a function for calculating the rigid displacement in the plane of an assigned point, X, with respect to a generic pole, X0. The parameters used are the translation and rotation of the body, u0 and phi0. The considered bodies are one-dimensional and identified by 2 or more points. In the example, a beam defined by 3 points is considered and its displacement is calculated assuming as pole each of these 3 points.

% generic 2D displacement field
rigidDispl = ...
    @(u0, phi0, X0, X)...
    [u0(1)-phi0*(X(2)-X0(2));...
     u0(2)+phi0*(X(1)-X0(1))];

% geometric description of the beam
syms l;
A = [0; 0];
B = [l; 0];
C = [l; l/2];

% displacement field using A as the pole
POLE = A;
syms phiA;
phi0 = phiA;
u0 = sym('uA', [2 1]);
uA = rigidDispl(u0, phi0, POLE, A);
uB = rigidDispl(u0, phi0, POLE, B);
uC = rigidDispl(u0, phi0, POLE, C);

% displacement field using B as the pole
POLE = B;
syms phiB;
phi0 = phiB;
u0 = sym('uB', [2 1]);
uA = rigidDispl(u0, phi0, POLE, A);
uB = rigidDispl(u0, phi0, POLE, B);
uC = rigidDispl(u0, phi0, POLE, C);

% displacement field using C as the pole
POLE = C;
syms phiC;
phi0 = phiC;
u0 = sym('uC', [2 1]);
uA = rigidDispl(u0, phi0, POLE, A);
uB = rigidDispl(u0, phi0, POLE, B);
uC = rigidDispl(u0, phi0, POLE, C);

Listing 4.1.2.

Remark 4.1.3.

An alternative writing of the equation (4.1.4) is obtained by making explicit the structure of the tensor \(\text{skew} \tens{\nabla u} \text{,}\) that is

\begin{equation} \vec{u} = \vec{u}_o + \varphi \,\tens{W}\left(\vec{X}-\vec{X}_o\right)\,,\label{W_small_rigid_u_eq}\tag{4.1.6} \end{equation}

where \(\varphi\) is the angle of rotation and \(\tens{W} \) is a generic skew tensor, see Insight 1.9.5 . Furthermore, using the axial vector associated with the tensor \(\tens{W} \text{,}\) it can also be written

\begin{equation} \vec{u} = \vec{u}_o + \varphi\, \vec{\omega}\times\left(\vec{X}-\vec{X}_o\right)\,.\label{omega_small_rigid_u_eq}\tag{4.1.7} \end{equation}