Motion, Deformation, and Strain¶

Reference and Current Configurations¶

Definition of Configurations¶

To distinguish which shape in time is used as the basis for describing physical quantities, this manual uses two configurations.

Reference configuration: the undeformed shape at the start of the analysis (time \(0\)), also called the initial configuration. A point in the reference configuration is represented by the material coordinate \(\boldsymbol{X}\), with volume \(V\) and surface \(S\).
Current configuration: the deformed shape at time \(t\). A point in the current configuration is represented by the spatial coordinate \(\boldsymbol{x}\), with volume \(v\) and surface \(s\).

Motion is represented as a mapping between these two configurations, \(\boldsymbol{x} = \boldsymbol{x}(\boldsymbol{X}, t)\), and displacement is defined as \(\boldsymbol{u} = \boldsymbol{x} - \boldsymbol{X}\).

Configuration-Dependent Notation (Uppercase/Lowercase)¶

The case of a symbol distinguishes the configuration to which a physical quantity belongs.

Quantities belonging to the reference configuration are written with uppercase symbols (for example, \(\boldsymbol{X}, \boldsymbol{F}, \boldsymbol{S}, \boldsymbol{E}, V, S\)).
Quantities belonging to the current configuration are written with lowercase symbols (for example, \(\boldsymbol{x}, \boldsymbol{\sigma}, \boldsymbol{e}, v, s\)).
Quantities spanning both configurations, such as the first Piola-Kirchhoff stress \(\boldsymbol{P}\), use conventional notation and are identified explicitly in the text as needed.
In small-deformation theory, the distinction between the two configurations disappears, so only lowercase symbols (\(\boldsymbol{\sigma}, \boldsymbol{\varepsilon}\)) are used.

Reference-Configuration and Current-Configuration Gradients¶

The notation and meaning differ depending on the configuration with respect to which a spatial derivative is taken.

Reference-configuration gradient \(\nabla_X = \partial / \partial \boldsymbol{X}\): gradient with respect to material coordinates. Example: \(\boldsymbol{F} = \partial \boldsymbol{x} / \partial \boldsymbol{X}\).
Current-configuration gradient \(\nabla_x = \partial / \partial \boldsymbol{x}\): gradient with respect to spatial coordinates. Example: \(\boldsymbol{L} = \partial \boldsymbol{v} / \partial \boldsymbol{x}\).

They are related by \(\nabla_X = \boldsymbol{F}^T \nabla_x\). For the convention for time derivatives (material time derivative \(\dot{(\cdot)}\)), see Tensor Notation and Mathematical Foundations.

Description of Motion¶

Motion Mapping and Material/Spatial Descriptions¶

If a continuum is regarded as a collection of material points labeled by their positions \(0\) at time \(\boldsymbol{X}\), motion is represented by a mapping \(\boldsymbol{X}\) that associates each material point \(t\) with its position \(\boldsymbol{x}\) at time \(\phi\):

\[ \boldsymbol{x} = \phi(\boldsymbol{X}, t) \]

Under the convention that the reference configuration is the initial configuration, \(\boldsymbol{X} = \phi(\boldsymbol{X}, 0)\).

Two descriptions are distinguished according to the choice of independent variables.

Material description (Lagrangian description): represents physical quantities as functions of \((\boldsymbol{X}, t)\). In solid mechanics this is a natural choice because the reference configuration is given as a fixed shape.
Spatial description (Eulerian description): represents physical quantities as functions of \((\boldsymbol{x}, t)\). This is the standard description in fluid mechanics.

FrontISTR structural analysis is fundamentally based on the material description, but the Updated Lagrange method also uses quantities in the spatial description because the current configuration is used as the new reference configuration.

Displacement, Velocity, and Acceleration¶

As stated in the previous section, displacement \(\boldsymbol{u}\) is defined by \(\boldsymbol{u} = \boldsymbol{x} - \boldsymbol{X}\). No assumption is made about its magnitude, so finite deformation is included. The velocity and acceleration vectors are defined as material time derivatives of \(\boldsymbol{u}\) while holding the material point fixed:

\[ \boldsymbol{v} = \dot{\boldsymbol{u}}, \qquad \boldsymbol{a} = \dot{\boldsymbol{v}} = \ddot{\boldsymbol{u}} \]

These equations define the velocity and acceleration.

Deformation Gradient and Deformation Tensors¶

Deformation Gradient Tensor¶

The fundamental quantity in finite-deformation continuum mechanics is the deformation gradient tensor \(\boldsymbol{F}\). It is the linear mapping that maps an infinitesimal line element \(d\boldsymbol{X}\) in the reference configuration to an infinitesimal line element \(d\boldsymbol{x}\) in the current configuration, and is defined as the gradient of the motion mapping with respect to material coordinates:

\[ \boldsymbol{F} = \frac{\partial \boldsymbol{x}}{\partial \boldsymbol{X}} = \boldsymbol{I} + \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{X}}, \qquad F_{ij} = \frac{\partial x_i}{\partial X_j} \]

Thus defined (\(d\boldsymbol{x} = \boldsymbol{F}\, d\boldsymbol{X}\)), \(\boldsymbol{F} = \boldsymbol{I}\) in the undeformed state. \(\boldsymbol{F}\) is widely used to define strain and stress and to transform physical quantities between configurations.

Volume Ratio¶

The ratio of an infinitesimal volume \(dV\) in the reference configuration to the corresponding infinitesimal volume \(dv\) in the current configuration is called the volume ratio \(J\):

\[ J = \frac{dv}{dV} = \det \boldsymbol{F} \]

Continuum invertibility requires \(J > 0\). Conservation of mass is expressed as \(\rho_0 = J\rho\).

Right and Left Cauchy-Green Deformation Tensors¶

Because the deformation gradient \(\boldsymbol{F}\) includes rigid-body rotation, measures of deformation itself use quantities from which rigid-body rotation has been removed. The right Cauchy-Green deformation tensor \(\boldsymbol{C}\) (reference configuration) and left Cauchy-Green deformation tensor \(\boldsymbol{b}\) (current configuration) are defined as

\[ \boldsymbol{C} = \boldsymbol{F}^T \boldsymbol{F}, \qquad \boldsymbol{b} = \boldsymbol{F} \boldsymbol{F}^T \]

Both are symmetric tensors and have the same eigenvalues (the squares of the principal stretches). The principal invariants of hyperelastic strain-energy functions are expressed using these tensors.

Strain Tensors¶

Green-Lagrange Strain Tensor¶

As a strain measure referred to the reference configuration, the Green-Lagrange strain tensor \(\boldsymbol{E}\) is defined by

\[ \boldsymbol{E} = \frac{1}{2}(\boldsymbol{C} - \boldsymbol{I}) = \frac{1}{2}(\boldsymbol{F}^T \boldsymbol{F} - \boldsymbol{I}) \]

It is a symmetric tensor representing the change in the inner product of infinitesimal vectors in the reference configuration, is invariant under rigid-body rotation, and satisfies \(\boldsymbol{E} = \boldsymbol{0}\) in the undeformed state. Using \(\boldsymbol{F} = \boldsymbol{I} + \partial \boldsymbol{u} / \partial \boldsymbol{X}\) gives

\[ \boldsymbol{E} = \frac{1}{2}\left\{ \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{X}} + \left(\frac{\partial \boldsymbol{u}}{\partial \boldsymbol{X}}\right)^T + \left(\frac{\partial \boldsymbol{u}}{\partial \boldsymbol{X}}\right)^T \frac{\partial \boldsymbol{u}}{\partial \boldsymbol{X}} \right\} \]

This expresses the strain as the sum of first- and second-order terms in the displacement gradient; the second-order term represents geometric nonlinearity. In the Total Lagrange method it is used as the work-conjugate pair \((\boldsymbol{S}, \boldsymbol{E})\).

Almansi Strain Tensor¶

As a strain measure referred to the current configuration, the Almansi strain tensor \(\boldsymbol{e}\) is defined by

\[ \boldsymbol{e} = \frac{1}{2}(\boldsymbol{I} - \boldsymbol{b}^{-1}) \]

It is related to Green-Lagrange strain by \(\boldsymbol{E} = \boldsymbol{F}^T \boldsymbol{e}\, \boldsymbol{F}\) (pull-back).

Infinitesimal Strain Tensor¶

When the displacement gradient is small (\(|\partial u_i / \partial X_j| \ll 1\)), the second-order term of the Green-Lagrange strain can be neglected. Since the distinction between the two configurations then disappears, it reduces to the infinitesimal strain tensor \(\boldsymbol{\varepsilon}\), with the gradient written in spatial coordinates:

\[ \boldsymbol{\varepsilon} = \frac{1}{2}\left\{ \nabla \boldsymbol{u} + (\nabla \boldsymbol{u})^T \right\}, \qquad \varepsilon_{ij} = \frac{1}{2}\left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \]

FrontISTR uses this strain measure for linear static analysis, modal analysis, frequency-response analysis, and linear dynamic analysis. Finite-deformation analysis uses \(\boldsymbol{E}\) in the Total Lagrange method and the rate-of-deformation tensor \(\boldsymbol{D}\) referred to the current configuration in the Updated Lagrange method.