Displacement field (mechanics)

Decomposition

The displacement of a body has two components: a rigid-body displacement and a deformation.

• A rigid-body displacement consists of a simultaneous translation and rotation of the body without changing its shape or size.
• Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration ${\displaystyle {\kappa }_0(\mathcal{B})}$ to a current or deformed configuration ${\displaystyle {\kappa }_t(\mathcal{B})}$ (Figure 1).

A change in the configuration of a continuum body can be described by a displacement field. A displacement field is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. The distance between any two particles changes if and only if deformation has occurred. If displacement occurs without deformation, then it is a rigid-body displacement.

Displacement gradient tensor

Two types of displacement gradient tensor may be defined, following the Lagrangian and Eulerian specifications. The displacement of particles indexed by variable i may be expressed as follows. The vector joining the positions of a particle in the undeformed configuration ${\displaystyle P_i}$ and deformed configuration ${\displaystyle p_i}$ is called the displacement vector, ${\displaystyle p_i-P_i}$, denoted ${\displaystyle u_i}$ or ${\displaystyle U_i}$ below.

Material coordinates (Lagrangian description)

Using ${\displaystyle \mathbf{X}}$ in place of ${\displaystyle P_i}$ and ${\displaystyle \mathbf{x}}$ in place of ${\displaystyle p_i}$, both of which are vectors from the origin of the coordinate system to each respective point, we have the Lagrangian description of the displacement vector: $${\displaystyle \mathbf{u}(\mathbf{X},t)=u_i{\mathbf{e}}_i}$$ where ${\displaystyle {\mathbf{e}}_i}$ are the orthonormal unit vectors that define the basis of the spatial (lab frame) coordinate system.

Expressed in terms of the material coordinates, i.e. ${\displaystyle \mathbf{u}}$ as a function of ${\displaystyle \mathbf{X}}$, the displacement field is: $${\displaystyle \mathbf{u}(\mathbf{X},t)=\mathbf{b}(t)+\mathbf{x}(\mathbf{X},t)-\mathbf{X}\text{or}{u}_i={\alpha }_{iJ}{b}_J+x_i-{\alpha }_{iJ}{X}_J}$$ where ${\displaystyle \mathbf{b}(t)}$ is the displacement vector representing rigid-body translation.

The partial derivative of the displacement vector with respect to the material coordinates yields the material displacement gradient tensor ${\displaystyle {\mathrm{\nabla }}_{\mathbf{X}}\mathbf{u}}$. Thus we have, $${\displaystyle {\mathrm{\nabla }}_{\mathbf{X}}\mathbf{u}={\mathrm{\nabla }}_{\mathbf{X}}\mathbf{x}-\mathbf{R}=\mathbf{F}-\mathbf{R}\text{or}\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{X}_K}=\frac{\mathrm{\partial }{x}_i}{\mathrm{\partial }{X}_K}-{\alpha }_{iK}=F_{iK}-{\alpha }_{iK}}$$ where ${\displaystyle \mathbf{F}}$ is the material deformation gradient tensor and ${\displaystyle \mathbf{R}}$ is a rotation.

Spatial coordinates (Eulerian description)

In the Eulerian description, the vector extending from a particle ${\displaystyle P}$ in the undeformed configuration to its location in the deformed configuration is called the displacement vector: $${\displaystyle \mathbf{U}(\mathbf{x},t)=U_J{\mathbf{E}}_J}$$ where ${\displaystyle {\mathbf{E}}_i}$ are the unit vectors that define the basis of the material (body-frame) coordinate system.

Expressed in terms of spatial coordinates, i.e. ${\displaystyle \mathbf{U}}$ as a function of ${\displaystyle \mathbf{x}}$, the displacement field is: $${\displaystyle \mathbf{U}(\mathbf{x},t)=\mathbf{b}(t)+\mathbf{x}-\mathbf{X}(\mathbf{x},t)\text{or}{U}_J=b_J+{\alpha }_{Ji}{x}_i-X_J}$$

The spatial derivative, i.e., the partial derivative of the displacement vector with respect to the spatial coordinates, yields the spatial displacement gradient tensor ${\displaystyle {\mathrm{\nabla }}_{\mathbf{x}}\mathbf{U}}$. Thus we have, $${\displaystyle {\mathrm{\nabla }}_{\mathbf{x}}\mathbf{U}={\mathbf{R}}^T-{\mathrm{\nabla }}_{\mathbf{x}}\mathbf{X}={\mathbf{R}}^T-{\mathbf{F}}^{-1}\text{or}\frac{\mathrm{\partial }{U}_J}{\mathrm{\partial }{x}_k}={\alpha }_{Jk}-\frac{\mathrm{\partial }{X}_J}{\mathrm{\partial }{x}_k}={\alpha }_{Jk}-F_{Jk}^{-1},}$$ where ${\displaystyle {\mathbf{F}}^{-1}=\mathbf{H}}$ is the spatial deformation gradient tensor.

Relationship between the material and spatial coordinate systems

${\displaystyle {\alpha }_{Ji}}$ are the direction cosines between the material and spatial coordinate systems with unit vectors ${\displaystyle {\mathbf{E}}_J}$ and ${\displaystyle {\mathbf{e}}_i}$, respectively. Thus $${\displaystyle {\mathbf{E}}_J\cdot {\mathbf{e}}_i={\alpha }_{Ji}={\alpha }_{iJ}}$$

The relationship between ${\displaystyle u_i}$ and ${\displaystyle U_J}$ is then given by $${\displaystyle u_i={\alpha }_{iJ}{U}_J\text{or}{U}_J={\alpha }_{Ji}{u}_i}$$

Knowing that $${\displaystyle {\mathbf{e}}_i={\alpha }_{iJ}{\mathbf{E}}_J}$$ then $${\displaystyle \mathbf{u}(\mathbf{X},t)=u_i{\mathbf{e}}_i=u_i({\alpha }_{iJ}{\mathbf{E}}_J)=U_J{\mathbf{E}}_J=\mathbf{U}(\mathbf{x},t)}$$

Combining the coordinate systems of deformed and undeformed configurations

It is common to superimpose the coordinate systems for the deformed and undeformed configurations, which results in ${\displaystyle \mathbf{b}=0}$, and the direction cosines become Kronecker deltas, i.e., $${\displaystyle {\mathbf{E}}_J\cdot {\mathbf{e}}_i={\delta }_{Ji}={\delta }_{iJ}}$$

Thus in material (undeformed) coordinates, the displacement may be expressed as: $${\displaystyle \mathbf{u}(\mathbf{X},t)=\mathbf{x}(\mathbf{X},t)-\mathbf{X}\text{or}{u}_i=x_i-{\delta }_{iJ}{X}_J}$$

And in spatial (deformed) coordinates, the displacement may be expressed as: $${\displaystyle \mathbf{U}(\mathbf{x},t)=\mathbf{x}-\mathbf{X}(\mathbf{x},t)\text{or}{U}_J={\delta }_{Ji}{x}_i-X_J}$$