Chapter 4 Kinematics
Reference: Bonet, Javier, Antonio J. Gil, and Richard D. Wood. Nonlinear solid mechanics for finite element analysis: statics. Cambridge University Press, 2016.
4.2 The Motion
motion of particle \(\rightarrow\) mapping \(\phi\)
\(\boldsymbol{x} = \phi(\boldsymbol{X},t)\)
4.3 Material and spatial descriptions
Lagrangian (Material, \(u = u(\boldsymbol{X},t)\)) and Eulerian (Spatial, \(u = u(\boldsymbol{x},t)\)) descriptions
The governing equations must be formulated using a spatial description first !!!
Spatial quantities can be expressed in term of initial coordinates. (Description can be transformed)
4.4 Deformation gradient
For two neighboring particles:
relative material position \(\rightarrow\) \(\boldsymbol{F}\) \(\rightarrow\)relative position
deformation gradient tensor: \(\boldsymbol{F} = \frac{\partial \phi}{\partial \boldsymbol{X}} = \nabla_0 \phi\)
\(d\boldsymbol{x} = \boldsymbol{F}d\boldsymbol{X}\) \(\rightarrow\) transforms vectors in reference config into current config
\(\boldsymbol{F} = \frac{\partial\boldsymbol{x}}{\partial \boldsymbol{X}}\) \(\quad\) \(F_{ij} = \frac{\partial x_i}{\partial X_j}\)
Inverse of \(\boldsymbol{F}\) \(\rightarrow\) \(\boldsymbol{F}^{-1} = \frac{\partial\boldsymbol{X}}{\partial \boldsymbol{x}} = \nabla\phi^{-1}\) \(\quad\) \(F_{ji}^{-1} = \frac{\partial X_j}{\partial x_i}\)
4.5 Strain
A general measure of deformation \(\rightarrow\) scalar product of \(d\boldsymbol{X}_1\) and \(d \boldsymbol{X}_2\)
refer to P104 Remark 4.3
\(d\boldsymbol{x}_1\cdot d\boldsymbol{x}_2 = d\boldsymbol{X}_1\cdot \boldsymbol{C}d\boldsymbol{X}_2 = d\boldsymbol{X}_1\cdot \boldsymbol{F}^T\boldsymbol{F}d\boldsymbol{X}_2\)
right Cauchy-Green deformation tensor: \(\boldsymbol{C} = \boldsymbol{F}^T\boldsymbol{F}\) \(\rightarrow\) material tensor quantity
\(d\boldsymbol{X}_1\cdot d\boldsymbol{X}_2 = d\boldsymbol{x}_1\cdot \boldsymbol{b}^{-1}d\boldsymbol{x}_2\)
left Cauchy-Green deformation tensor: \(\boldsymbol{b} = \boldsymbol{F}\boldsymbol{F}^T\) \(\rightarrow\) spatial tensor quantity
Change in scalar product: \(\frac{1}{2}(d\boldsymbol{x}_1\cdot d\boldsymbol{x}_2-d\boldsymbol{X}_1\cdot d\boldsymbol{X}_2) = d\boldsymbol{X}_1\cdot \boldsymbol{E} d\boldsymbol{X}_2 = d\boldsymbol{x}_1\cdot\boldsymbol{e}d\boldsymbol{x}_2\)
Green-Lagrangian strain tensor: \(\boldsymbol{E} = \frac{1}{2}(\boldsymbol{C}-\boldsymbol{I})\)
Almansi-Eluerian strain tensor: \(\boldsymbol{e} = \frac{1}{2}(\boldsymbol{I}-\boldsymbol{b}^{-1})\)
Transformation: \(\boldsymbol{e} = \boldsymbol{F}^{-T}\boldsymbol{E}\boldsymbol{F}^{-1}\)\(\quad\)\(\boldsymbol{E}=\boldsymbol{F}^T\boldsymbol{e}\boldsymbol{F}\)
4.6 Polar decomposition
\(\boldsymbol{F} = \boldsymbol{R}\boldsymbol{U} = \boldsymbol{V}\boldsymbol{R}\)
\(\boldsymbol{R}\) \(\rightarrow\) orthogonal rotation tensor i.e., \(\boldsymbol{R^T}\boldsymbol{R}=\boldsymbol{I}\)
4.7 Volume change
Reference config:
\(d\boldsymbol{X}_i = dX_i\boldsymbol{E}_i\)
\(dV = dX_1 dX_2dX_3\)
\(\boldsymbol{E}_1\cdot (\boldsymbol{E}_2\times\boldsymbol{E}_3)=+1\)
Current config:
\(d\boldsymbol{x}_i = \boldsymbol{F}dX_i = \frac{\partial\phi}{\partial X_i}dX_i\)
\(dv = d\boldsymbol{x}_1\cdot(d\boldsymbol{x}_2\times\boldsymbol{x}_3) = \frac{\partial\phi}{\partial X_1}\cdot(\frac{\partial \phi}{\partial X_2}\times\frac{\partial \phi}{\partial X_3})dX_1dX_2dX_3 = det(\boldsymbol{F})dV = JdV\)
density: \(\rho_0 = \rho J\)
4.8 Distortional component of the deformation gradient
Decompose the deformation gradient into a volumetric part and a distortional part, i.e.,
\(\boldsymbol{F} = \boldsymbol{F}_v\cdot\boldsymbol{F}_d\)
\(J = det(\boldsymbol{F}) = det(\boldsymbol{F}_v)det(\boldsymbol{F}_d)\)
No volume change in distortional(isochoric) part, so
\(det(\boldsymbol{F}_d) = 1\)
So, \(\boldsymbol{F}_d = J^{-\frac{1}{3}}\boldsymbol{F}\) to ensure that \(det(\boldsymbol{F_d})=(J^{-\frac{1}{3}})^3det(\boldsymbol{F})=J^{-1}J=1\)
\(\boldsymbol{F}_v=J^{\frac{1}{3}}\)
The distorrtional part of right Cauchy-Green tensor \(\boldsymbol{C}\):
\(\boldsymbol{C}_d = \boldsymbol{F}^T_d\boldsymbol{F}_d=J^{-\frac{2}{3}}\boldsymbol{C}=det(\boldsymbol{C})^{-\frac{1}{3}}\boldsymbol{C}\)
4.9 Area change
Reference config: \(d\boldsymbol{A}=dA\boldsymbol{N}\) \(\quad\) \(dV = d\boldsymbol{L}\cdot d\boldsymbol{A}\)
Current config: \(d\boldsymbol{a} = da\boldsymbol{n}\) \(\quad\) \(dv = d\boldsymbol{l}\cdot d\boldsymbol{a}\)
\(dv = JdV = Jd\boldsymbol{L}\cdot d\boldsymbol{A} = d\boldsymbol{l}\cdot d\boldsymbol{a} = \boldsymbol{F}d\boldsymbol{L}\cdot d\boldsymbol{a}\)
\(d\boldsymbol{a} = J\boldsymbol{F}^{-T}d\boldsymbol{A}\)
4.10 Linearized kinematics
Linearized deformation gradient
\(D\boldsymbol{F}[\boldsymbol{u}]=\frac{\partial}{\partial \epsilon}|_{\epsilon=0}(\frac{\partial (\phi_t+\epsilon\boldsymbol{u})}{\partial \boldsymbol{X}})=\frac{\partial \boldsymbol{u}}{\partial \boldsymbol{X}} = \nabla_0\boldsymbol{u}\)
Linearized strain
Green-Lagrangian strain:
\(D\boldsymbol{E}[\boldsymbol{u}] = \frac{1}{2}\boldsymbol{F}^T[\nabla \boldsymbol{u}+(\nabla \boldsymbol{u})^T]\boldsymbol{F} = \boldsymbol{F}^T\epsilon\boldsymbol{F}\)
\(\epsilon\) \(\rightarrow\) small strain tensor
Linearized Cauchy-Green deformation tensor
right: \(D\boldsymbol{C}[\boldsymbol{u}] = 2\boldsymbol{F}^T\epsilon\boldsymbol{F}\)
left: \(D\boldsymbol{b}[\boldsymbol{u}] = (\nabla \boldsymbol{u})\boldsymbol{b} + \boldsymbol{b}(\nabla\boldsymbol{u})^T\)
Linearized volume change
\(DJ[\boldsymbol{u}] = Jdiv\boldsymbol{u}=Jtr\boldsymbol{\epsilon}\)
\(D(dv)[\boldsymbol{u}] = (tr\boldsymbol{\epsilon})dv\)
4.11 Velocity and material time derivatives
Rate of deformation
Spin tensor
Rate of change of volume
Superimposed rigid body motions and objectivity
Stress and equilibrium
Cauchy stress tensor
traction vector: \(\boldsymbol{t}(\boldsymbol{n}) = \lim\limits_{\Delta a\rightarrow 0}{\frac{\Delta\boldsymbol{P}}{\Delta a}}\)
\(\boldsymbol{t}(-\boldsymbol{n}) = -\boldsymbol{t}(\boldsymbol{n})\)
\(\boldsymbol{t}(\boldsymbol{n}) = [\displaystyle\sum_{i,j=1}^3\sigma_{ij}(\boldsymbol{e}_i\otimes\boldsymbol{e}_j)]\boldsymbol{n} = \boldsymbol{\sigma}\boldsymbol{n}\)
Cauchy stress tensor: \(\boldsymbol{\sigma}=\displaystyle\sum_{i,j=1}^3\sigma_{ij}(\boldsymbol{e}_i\otimes\boldsymbol{e}_j)\)
Expressed inb terms of principal directions?
refer to P141
Equilibrium
Translational equilibrium
Sum of all forces acting on the body vanishes:
\(\int_{\partial v}\boldsymbol{t}da+\int_{v}\boldsymbol{f}dv=0\)
Further expressed in terms of Cauchy stresses:
\(\int_{\partial v}\boldsymbol{\sigma}\boldsymbol{n}da+\int_{v}\boldsymbol{f}dv=0\)
Using Gauss theorem:
\(\int_{v}(div \boldsymbol{\sigma}+\boldsymbol{f})dv=0\)
\(div \boldsymbol{\sigma}+\boldsymbol{f} = \boldsymbol{0}\) \(\rightarrow\) point-wise spatial equilibrium equation
The pointwise out-of-balance or residual force per volume:
\(\boldsymbol{r} = div \boldsymbol{\sigma}+\boldsymbol{f}\)
Rotational equilibrium
refer to P144
Principle of virtual work
Equilibrium stated by virtual work: \(\delta w = \boldsymbol{r}\cdot \delta \boldsymbol{v} = 0\) \(\rightarrow\) \(\boldsymbol{r} = \boldsymbol{0}\)
per unit volume and time done by the residual force \(\boldsymbol{r}\) during the virtual motion \(\boldsymbol{v}\) (aribitary virtual velocity)
Weak statement of the static equilibrium: \(\delta W(\phi,\delta \boldsymbol{v}) = \int_{v}(div\boldsymbol{\sigma}+\boldsymbol{f})\cdot \delta \boldsymbol{v}dv=0\)
The spatial virtual work equation:
\(\delta W = \int_{v}\boldsymbol{\sigma}:\delta \boldsymbol{d}dv-\int_{v}\boldsymbol{f}\cdot\delta\boldsymbol{v}dv-\int_{\partial v}\boldsymbol{t}\cdot\delta\boldsymbol{v}da=0\)
\(\delta \boldsymbol{d}\) \(\rightarrow\) symmetric virtual rate of deformation
Work conjugacy and alternative stress representations
The kirchhoff Stress Tensor
work conjugate \(\rightarrow\) the product (like \(\boldsymbol{\sigma}\) and \(\boldsymbol{d}\)) gives work per unit current volume
Express the above spatial virtual work equation with respect to the initial volume
\(\int_{V}J\boldsymbol{\sigma}:\delta \boldsymbol{d}dV-\int_{V}\boldsymbol{f}_0\cdot\delta\boldsymbol{v}dV-\int_{\partial V}\boldsymbol{t}_0\cdot\delta\boldsymbol{v}dA=0\)
\(\boldsymbol{f}_0 = J \boldsymbol{f}\) \(\rightarrow\) body force per unit undeformed volume
\(\boldsymbol{t}_0=\boldsymbol{t}(\frac{da}{dA})\)
\(\frac{da}{dA} = \frac{J}{\sqrt{\boldsymbol{n}\cdot\boldsymbol{b}\boldsymbol{n}}} = J\sqrt{\boldsymbol{N}\cdot\boldsymbol{C}^{-1}\boldsymbol{N}}\)
\(\delta W_{int} = \int_{V}\boldsymbol{\tau}:\delta \boldsymbol{d}dV\)
the Kirchhoff stress tensor: \(\boldsymbol{\tau} = J\boldsymbol{\sigma}\)
The work per unit mass is invariant, and \(\rho = \frac{\rho_0}{J}\). so:
\(\frac{1}{\rho}\boldsymbol{\sigma}:\boldsymbol{d}=\frac{1}{\rho_0}\boldsymbol{\tau}:\boldsymbol{d}\)
The First Piola-Kirchhoff Stress Tensor
\(\delta W_{int} = \int_{V}(J\boldsymbol{\sigma}\boldsymbol{F}^{-T}):\delta\dot{\boldsymbol{F}}dV\)
the first Piola-Kirchhoff stress tensor: \(\boldsymbol{P} = J\boldsymbol{\sigma}\boldsymbol{F}^{-T}\)
\(\boldsymbol{P} = \displaystyle\sum_{i,I=1}^{3}P_{i,I}e_i\otimes\boldsymbol{E}_I\) \(\quad\) \(P_{iI}=\displaystyle\sum_{i,I=1}^{3}J\sigma_{ij}(\boldsymbol{F}^{-1})_{Ij}\)
\(\int_{V}\boldsymbol{P}:\delta \dot{\boldsymbol{F}}dv=\int_{V}\boldsymbol{f}\cdot\delta\boldsymbol{v}dv+\int_{\partial V}\boldsymbol{t}\cdot\delta\boldsymbol{v}da\)
Reverse the weak formulation, we can get an equivalent version of differential equilibrium equation:
\(\boldsymbol{r}_0 = J\boldsymbol{r} = DIV\boldsymbol{P}+\boldsymbol{f}_0=\boldsymbol{0} = \boldsymbol{\nabla}_0\boldsymbol{P}:\boldsymbol{I}+\boldsymbol{f}_0 = \frac{\partial \boldsymbol{P}}{\partial \boldsymbol{X}}:\boldsymbol{I}+\boldsymbol{f}_0\)
\(d\boldsymbol{p} = \boldsymbol{\sigma}d\boldsymbol{a} = \boldsymbol{P}d\boldsymbol{A}\) \(\rightarrow\) current force per unit area
\(\boldsymbol{P}\) is unsymmetric two-point tensor
The Second Piola-Kirchhoff Stress Tensor
\(d\boldsymbol{P} = \boldsymbol{F}^{-1}d\boldsymbol{p}\)
Material force vector \(\leftarrow\) Spatial force vector
\(d\boldsymbol{P} = \boldsymbol{S}d\boldsymbol{A}\) \(\quad\) \(\boldsymbol{S} = J\boldsymbol{F}^{-1}\boldsymbol{\sigma}\boldsymbol{F}^{-T}\)
\(\delta W_{int} = \int_{V}\boldsymbol{S}:\delta\dot{\boldsymbol{E}}dV\)
Material virtual work equation:
\(\int_{V}\boldsymbol{S}:\delta\dot{\boldsymbol{E}}dV = \int_{V}\boldsymbol{f}\cdot\delta\boldsymbol{v}dv+\int_{\partial V}\boldsymbol{t}\cdot\delta\boldsymbol{v}da\)
Relations:
\(\boldsymbol{\sigma} = J^{-1}\boldsymbol{P}\boldsymbol{F}^T\) \(\quad\) \(\boldsymbol{\sigma} = J^{-1}\boldsymbol{F}\boldsymbol{S}\boldsymbol{F}^T\)
\(\boldsymbol{S} = \boldsymbol{F}^{-1}\boldsymbol{\tau}\boldsymbol{F}^{-T}\) \(\quad\) \(\boldsymbol{\tau} = J^{-1}\boldsymbol{F}\boldsymbol{S}\boldsymbol{F}^T\)
Piola transformation
refer to P151
Deviatoric and Pressure Components
refer to P153