Coordinate Transformation Between Material Points and Spatial Points

In continuum mechanics, two distinct reference frames are used to describe the motion and deformation of a solid body: the material (Lagrangian) frame and the spatial (Eulerian) frame.
The material frame follows each particle of the body, preserving its identity over time, while the spatial frame is fixed in space and observes the evolving field of displacements, velocities, and stresses.
Bridging these two viewpoints requires a well‑defined coordinate transformation, which is essential for formulating balance laws, computing strain measures, and implementing numerical methods such as the finite element method.

Material vs. Spatial Points

Key point – The mapping ( \boldsymbol{\chi} ) and its inverse are the mathematical objects that translate between the two descriptions.

Choosing a Coordinate System

1. Cartesian coordinates

   • Most common; orthonormal basis ( {e_1, e_2, e_3} ).
   • Coordinates ( (X_1, X_2, X_3) ) or ( (x_1, x_2, x_3) ).

2. Curvilinear coordinates (polar, cylindrical, spherical)

   • Useful for problems with symmetry.
   • Require a metric tensor ( g_{ij} ) to handle non‑orthogonal bases.

3. Moving frames

   • For rotating or translating bodies, one may switch between a body‑fixed frame and an inertial frame.
   • The transformation between them is typically a rigid‑body motion.

In most analytical derivations, Cartesian coordinates are assumed, and any change to another system is handled by inserting the appropriate metric factors into the transformation matrix.

Fundamental Transformation Relations

Let a material point have initial coordinates ( \mathbf{X} ) at time ( t_0 ) and current coordinates ( \mathbf{x} ) at time ( t ).
The displacement field is

[
\mathbf{u}(\mathbf{X}, t) = \mathbf{x}(\mathbf{X}, t) - \mathbf{X}.
]

The deformation gradient ( \mathbf{F} ) is the Jacobian of the map ( \boldsymbol{\chi} ):

[
\mathbf{F}(\mathbf{X}, t) = \frac{\partial \mathbf{x}}{\partial \mathbf{X}}
= \mathbf{I} + \frac{\partial \mathbf{u}}{\partial \mathbf{X}},
]

where ( \mathbf{I} ) is the identity tensor.
The inverse gradient, needed when mapping from spatial to material coordinates, is

[
\mathbf{F}^{-1} = \frac{\partial \mathbf{X}}{\partial \mathbf{x}}.
]

The determinant ( J = \det \mathbf{F} ) quantifies local volume change:

[
\mathrm{d}V = J,\mathrm{d}V_0.
]

Linear Transformation Matrix

In many engineering problems the deformation can be approximated by a linear mapping:

[
\mathbf{x} = \mathbf{A},\mathbf{X} + \mathbf{b},
]

• ( \mathbf{A} \in \mathbb{R}^{3\times3} ) – linear transformation matrix (rotation, stretch, shear).
• ( \mathbf{b} \in \mathbb{R}^3 ) – translation vector.

The Jacobian of this map is simply ( \mathbf{A} ), and its determinant ( J = \det \mathbf{A} ) indicates the volumetric change.

Special cases

Illustrative Examples

1. Translation

With a translation vector ( \mathbf{b} = (0.02, 0, 0),\text{m} ):

[
\mathbf{x} = \mathbf{I},\mathbf{X} + \mathbf{b}
\quad\Longrightarrow\quad
\begin{cases}
x_1 = X_1 + 0.02\
x_2 = X_2\
x_3 = X_3
\end{cases}
]

Displacement field: ( \mathbf{u}(\mathbf{X}) = \mathbf{b} ).
Deformation gradient: ( \mathbf{F} = \mathbf{I} ).
Volume change: ( J = 1 ).

2. Rotation About the ( z )-Axis

Rotation by ( \theta = 30^\circ = \pi/6 ):

[
\mathbf{R}_z(\theta) =
\begin{bmatrix}
\cos\theta & -\sin\theta & 0\
\sin\theta & \ \cos\theta & 0\
0 & 0 & 1
\end{bmatrix}.
]

Mapping:

[
\mathbf{x} = \mathbf{R}_z(\pi/6),\mathbf{X}.
]

For ( \mathbf{X} = (1,0,0)^{\top} ):

[
\mathbf{x} = (\cos\frac{\pi}{6}, \sin\frac{\pi}{6}, 0)^{\top}
= (0.866, 0.5, 0)^{\top},\text{m}.
]

Here ( \mathbf{F} = \mathbf{R}_z ) and ( J = \det \mathbf{R}_z = 1 ), confirming that pure rotation preserves volume.

3. General Linear Deformation (Shear + Stretch)

Let

[
\mathbf{A} =
\begin{bmatrix}
1.2 & 0.3 & 0\
0 & 1.0 & 0\
0 & 0 & 0.8
\end{bmatrix},
\qquad \mathbf{b} = \mathbf{0}.
]

Then

[
\mathbf{x} = \mathbf{A},\mathbf{X}, \quad
\mathbf{F} = \mathbf{A}, \quad
J = \det \mathbf{A} = 1.2 \times 1.0 \times 0.8 = 0.96.
]

The body experiences a 4 % volume reduction.
For ( \mathbf{X} = (1,1,1)^{\top} ):

[
\mathbf{x} = (1.5, 1.0, 0.8)^{\top},\text{m}.
]

Applications in Solid Mechanics

Strain Measures

• Green–Lagrange strain (Lagrangian):

  [
  \mathbf{E} = \frac{1}{2}\left(\mathbf{F}^{!\top}\mathbf{F} - \mathbf{I}\right).
  ]

• Eulerian strain:

  [
  \mathbf{e} = \frac{1}{2}\left(\mathbf{I} - \mathbf{F}^{-!\top}\mathbf{F}^{-1}\right).
  ]

Both rely on the deformation gradient ( \mathbf{F} ) obtained from the coordinate transformation.

Momentum Conservation in Lagrangian Form

[
\rho_0 \frac{\partial^2 \mathbf{u}}{\partial t^2}
= \nabla_{! \mathbf{X}}!\cdot \mathbf{P} + \rho_0 \mathbf{b},
]

where ( \mathbf{P} = \mathbf{F},\boldsymbol{\sigma} ) is the first Piola–Kirchhoff stress and ( \boldsymbol{\sigma} ) is the Cauchy stress.
The mapping ( \mathbf{F} ) couples the spatial stress to the material description.

Finite Element Implementation

Within an element, the mapping from natural coordinates ( (\xi, \eta, \zeta) ) to global coordinates ( (x, y, z) ) is

[
\mathbf{x} = \sum_{i=1}^{n} N_i(\xi,\eta,\zeta),\mathbf{x}_i,
]

with shape functions ( N_i ) and nodal positions ( \mathbf{x}_i ).
The Jacobian ( \mathbf{J} = \partial \mathbf{x} / \partial \boldsymbol{\xi} ) is computed to transform integrals from the reference element to the physical element.

Large Deformation Analysis

When deformations are large, the full nonlinear form of ( \mathbf{F} ) must be used; linear approximations ( \mathbf{I} + \nabla \mathbf{u} ) are no longer accurate.
Accurate coordinate transformations become critical for reliable numerical solutions.

Conclusion

• The material point and spatial point provide two complementary lenses on a deforming solid.
• The deformation gradient ( \mathbf{F} ) is the core of the coordinate transformation, linking the two descriptions.
• Linear transformation matrices offer a convenient approximation for many engineering problems, while the Jacobian determinant ( J ) tracks volumetric changes.
• These concepts underpin strain calculations, balance equations, and numerical discretizations such as the finite element method.

Mastering the coordinate transformation between material and spatial points equips engineers and researchers with a unified language to analyze, simulate, and predict the behavior of complex solid bodies under arbitrary loading.