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:

$${\displaystyle \mathbf {u} (\mathbf {X} ,t)=\mathbf {b} (t)+\mathbf {x} (\mathbf {X} ,t)-\mathbf {X} \qquad {\text{or}}\qquad u_{i}=\alpha _{iJ}b_{J}+x_{i}-\alpha _{iJ}X_{J}}$$

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 ${\displaystyle \nabla _{\mathbf {X} }\mathbf {u} \,\!}$. Thus we have,

$${\displaystyle \nabla _{\mathbf {X} }\mathbf {u} =\nabla _{\mathbf {X} }\mathbf {x} -\mathbf {R} =\mathbf {F} -\mathbf {R} \qquad {\text{or}}\qquad {\frac {\partial u_{i}}{\partial X_{K}}}={\frac {\partial x_{i}}{\partial X_{K}}}-\alpha _{iK}=F_{iK}-\alpha _{iK}}$$

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:

$${\displaystyle \mathbf {U} (\mathbf {x} ,t)=\mathbf {b} (t)+\mathbf {x} -\mathbf {X} (\mathbf {x} ,t)\qquad {\text{or}}\qquad U_{J}=b_{J}+\alpha _{Ji}x_{i}-X_{J}}$$

The spatial derivative, i.e., the partial derivative of the displacement vector with respect to the spatial coordinates, yields the spatial displacement gradient tensor ${\displaystyle \nabla _{\mathbf {x} }\mathbf {U} \,\!}$. Thus we have,

$${\displaystyle \nabla _{\mathbf {x} }\mathbf {U} =\mathbf {R} ^{T}-\nabla _{\mathbf {x} }\mathbf {X} =\mathbf {R} ^{T}-\mathbf {F} ^{-1}\qquad {\text{or}}\qquad {\frac {\partial U_{J}}{\partial x_{k}}}=\alpha _{Jk}-{\frac {\partial X_{J}}{\partial x_{k}}}=\alpha _{Jk}-F_{Jk}^{-1}\,,}$$

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

$${\displaystyle u_{i}=\alpha _{iJ}U_{J}\qquad {\text{or}}\qquad U_{J}=\alpha _{Ji}u_{i}}$$

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:

$${\displaystyle \mathbf {u} (\mathbf {X} ,t)=\mathbf {x} (\mathbf {X} ,t)-\mathbf {X} \qquad {\text{or}}\qquad u_{i}=x_{i}-\delta _{iJ}X_{J}}$$

And in spatial (deformed) coordinates, the displacement may be expressed as:

$${\displaystyle \mathbf {U} (\mathbf {x} ,t)=\mathbf {x} -\mathbf {X} (\mathbf {x} ,t)\qquad {\text{or}}\qquad U_{J}=\delta _{Ji}x_{i}-X_{J}}$$