Notes of FEM Simulation of 3D Deformable Solids

Elasticity in three dimensions

Deformation map and deformation gradient

${\pmb{R}^3 = }$ all vectors with 3 real components.

${\pmb{R}^n = }$ all vectors with n real components.

When the object undergoes deformation, every material point $\vec{X}$ is being displaced to a new deformed location which is, by convention, denoted by a lowercase variable $\vec{x}$ The relation between each material point and its deformation function $\vec{\phi} : R^{3} \rightarrow R^{3} $.
$$
\vec{x} = \vec{\phi}(\vec{X})·1
$$
An important physical quantity derived directly from $\vec{\phi}(\vec{X})$, is the deformation gradient tensor ${ \pmb{F} \in \pmb{R^{3\times3}}}$

If we write ${\vec{X} = (X_1,X_2,X_3)^T} or{\vec{X} = (X,Y,Z)^T} $ and ${\vec{\phi}(\vec{X}) = (\vec{\phi_1}(\vec{X}),\vec{\phi_2}(\vec{X}),\vec{\phi_3}(\vec{X}))^T}$

注意是原每一组点根据三个对应关系被分别转换到三个分量上

for the three components of the vector-valued function ${\vec{\phi}}$, the deformation gradient is written as:
$$
\pmb{F}:= \frac{\partial(\phi_1,\phi_2,\phi_3)}{\partial({X_1,X_2,X_3})} = \left( \begin{matrix}\frac{\partial\phi_1}{\partial X_1} & \frac{\partial\phi_1}{\partial X_2} & \frac{\partial\phi_1}{\partial X_3}\
\frac{\partial\phi_2}{\partial X_1} & \frac{\partial\phi_2}{\partial X_2} & \frac{\partial\phi_2}{\partial X_3}\
\frac{\partial\phi_3}{\partial X_1} & \frac{\partial\phi_3}{\partial X_2} & \frac{\partial\phi_3}{\partial X_3}\end{matrix}\right)
$$
or, in index notation ${ F_{ij} = \phi_{i,j} }$ . In simple terms, the deformation gradient measures the amount of change in shape and size of a material body relative to its original configuration. The magnitude of the deformation gradient can be used to determine the amount of deformation or strain that has occurred, and its orientation can be used to determine the direction of deformation.

Note that, in general, $\pmb{F}$ will be spatially varying across ${\Omega}$, which is the volumetric domain occupied by the object. This domain will be referred to as the reference(or undefined configuration)

Strain energy and hyperelasticity

One of the consequences of elastic deformation is the accumulation of potential energy in the deformed body, which is referred to as strain energy ${E[\phi]}$ in the context of deformable solids. It is suggested that the energy is fully determined by the deformation map of a given configuration.

However intuitive, this statement nevertheless reflects a significant hypothesis that led to this formulation: we have assumed that the potential energy associated with a deformed configuration only depends on the final deformed shape, and not on the deformation path over time that brought the body into its current configuration.

The independence of the strain energy on the prior deformation history is a characteristic property of so-called hyperelastic materials. This property of is closely related with the fact that elastic forces of hyperelastic materials are conservative: the total work done by the internal elastic forces in a deformation path depends solely on the initial and final configurations, not the path itself.

Different parts of a deforming body undergo shape changes of different severity. As a consequence, the relation between deformation and strain energy is better defined on a local scale. We achieve that by introducing an energy density function ${\Psi[\phi;\vec{X}]}$ which measures the strain energy per unit undeformed volume on an infinitesimal domain ${dV}$ around the material point $\vec{X}$. We can then obtain the total energy for the deforming body by integrating the energy density function over the entire domain ${\Omega}$:
$$
E[\phi] = \int_\Omega\Psi[\phi;\vec{X}]d\vec{X}
$$