User:Egm6936.s09/Solid shell in cartesian coordinates

 Could you look at the wiki article Word macros to try to install macros into MS Word that would help you write wiki articles; look at the last section of this article. This method is different from the use of WikEd. You need to learn how to install macro codes into MS Word. Try this method out and let me know whether it is better for you. Egm6321.f09 12:53, 30 September 2009 (UTC)

Let $$  (   \xi^1  ,   \xi^2   ,   \xi^3   ) $$ be cartesian coordinates, and consider $$\displaystyle \mathcal B_0 \equiv \mathcal B$$, i.e., the initial configuration $$\displaystyle \mathcal B_0$$ is identical to the material configuration $$\displaystyle \mathcal B$$, then $$\displaystyle \mathbf{E} _i \equiv \mathbf{G} _i$$, for $$\displaystyle i=1,2,3$$, where $$\displaystyle \equiv$$ means "equivalent to" or "identical to".

HW: Redraw Fig.2 and Fig.3, and write all Eqs.(2.1)-(2.11) in component form.

For example, take the vector $$\displaystyle \mathbf{X}$$, which is the position vector of the material point $$\displaystyle X \in \mathcal B_0 \equiv \mathcal B$$. In cartesian components, the position vector $$\displaystyle \mathbf{X}$$, can be expressed as follows:


 * $$\displaystyle \mathbf{X} = X^i \mathbf{E} _i \equiv \xi^i \mathbf{G} _i$$,

 This component form is what I wanted you to write, i.e., you need to have the vector and the basis vectors in the expression, not just the components alone. Egm6321.f09 12:53, 30 September 2009 (UTC)

In the above equation, we have

$$\displaystyle X^i \equiv \xi^i$$, since $$\displaystyle \mathbf{E} _i \equiv \mathbf{G} _i$$.

The material three-dimensional continuum of the shell geometry is described by
 * {| style="width:100%" border="0"

{\textbf{X}}\left( {X_1 ,X_2 ,X_3 } \right) = \frac{1}{2}\left[ {\left( {1 + X^3 } \right){\textbf{X}}_u \left( {X_1 ,X_2 } \right) + \left( {1 - X^3 } \right){\textbf{X}}_l \left( {X_1 ,X_2 } \right)} \right]\,\,\,\,\,\,\,\,\,\,\left( {X_1 ,X_2 ,X_3 } \right) \in \,[ - 1,1] \times [ - 1,1] \times [ - 1,1] $$
 * style= |
 * }

With the component form of the vectors $$\displaystyle \mathbf{X}$$, $$\displaystyle \mathbf{X}_u$$, and $$\displaystyle \mathbf{X}_l$$, given by


 * $$\displaystyle

\mathbf{X} = X^i \mathbf{E} _i \equiv \xi^i \mathbf{G} _i , \quad \mathbf{X}_u = X^i_u \mathbf{E} _i , \quad \mathbf{X}_l = X^i_l \mathbf{E} _i $$, then the cartesian components ( $$\displaystyle X^i \, X^i_u \ , X^i_l $$ ) of ( $$\displaystyle \mathbf{X} \, \mathbf{X}_u \ , \mathbf{X}_l $$ ), respectively, are related to each other as follows:


 * {| style="width:100%" border="0"

\displaystyle X^i \left( {X_1 ,X_2 ,X_3 } \right) = \frac{1}{2}\left[ {\left( {1 + X^3 } \right)X_u^i \left( {X_1 ,X_2 } \right) + \left( {1 - X^3 } \right)X_l^i \left( {X_1 ,X_2 } \right)} \right] $$
 * style="width:95%" |$$
 * style="width:95%" |$$
 * style= | (2.1)
 * }

We can rewrite (2.1) as follows:
 * {| style="width:100%" border="0"

\displaystyle X^i \left( {X_1 ,X_2 ,X_3 } \right) = \underbrace{ \frac{1}{2} \left[ {X_u^i \left( {X_1 ,X_2 } \right) + X_l^i \left( {X_1 ,X_2 } \right)} \right] }_{X^i_m (X_1 ,X_2)} \, + X^3 \underbrace{ \frac{1}{2} \left[ {X_u^i \left( {X_1 ,X_2 } \right) - X_l^i \left( {X_1 ,X_2 } \right)} \right] }_{\delta_{i3}} = X_m^i \left( {X_1 ,X_2 } \right) + X^3 \delta _{i3} $$
 * style="width:95%" |$$
 * style="width:95%" |$$
 * style= | (2.2)
 * }

Similarly, in the current (deformed) configuration, the geometry of the solid shell is described by:
 * {| style="width:100%" border="0"

\displaystyle {\textbf{x}}\left( {X_1 ,X_2 ,X_3 } \right) = \frac{1}{2}\left[ {\left( {1 + X^3 } \right){\textbf{x}}_u \left( {X_1 ,X_2 } \right) + \left( {1 - X^3 } \right){\textbf{x}}_l \left( {X_1 ,X_2 } \right)} \right] $$
 * style="width:95%" |$$
 * style="width:95%" |$$
 * style= |
 * }

or in Cartesian coordinates:
 * {| style="width:100%" border="0"

\displaystyle x^i \left( {X_1 ,X_2 ,X_3 } \right) = \frac{1}{2}\left[ {\left( {1 + X^3 } \right)x_u^i \left( {X_1 ,X_2 } \right) + \left( {1 - X^3 } \right)x_l^i \left( {X_1 ,X_2 } \right)} \right] = \frac{1}{2}\left[ {x_u^i \left( {X_1 ,X_2 } \right) + x_l^i \left( {X_1 ,X_2 } \right)} \right]\, + \frac{1}{2}X^3 \left[ {x_u^i \left( {X_1 ,X_2 } \right) - x_l^i \left( {X_1 ,X_2 } \right)} \right] $$
 * style="width:95%" |$$
 * style="width:95%" |$$
 * style= |
 * }


 * {| style="width:100%" border="0"

\displaystyle x^i \left( {X_1 ,X_2 ,X_3 } \right) = \,x_m^i \left( {X_1 ,X_2 } \right) + \frac{1}{2}X^3 h\left( {X_1 ,X_2 } \right) $$ where $$h\left( {X_1 ,X_2 } \right)$$ is the shell thickness.
 * style="width:95%" |$$
 * style="width:95%" |$$
 * style= | (2.3)
 * }

The material configuration is related to the deformed configuration by the displacement field $$\textbf{u}$$ as follows:
 * {| style="width:100%" border="0"

\displaystyle \textbf{x} \left( {X_1 ,X_2 ,X_3 } \right) = \textbf{X} \left( {X_1 ,X_2 ,X_3 } \right) + \textbf{u} \left( {X_1 ,X_2 ,X_3 } \right) $$ or in Cartesian coordinates:
 * style="width:95%" |$$
 * style="width:95%" |$$
 * style= |
 * }
 * {| style="width:100%" border="0"

\displaystyle x^i \left( {X_1 ,X_2 ,X_3 } \right) = X^i \left( {X_1 ,X_2 ,X_3 } \right) + u^i \left( {X_1 ,X_2 ,X_3 } \right) $$
 * style="width:95%" |$$
 * style="width:95%" |$$
 * style= |  (2.4)
 * }

The convected basis vectors $$\textbf{G}_i$$ in the material configuration $$B0$$ are related to the position vector $$\textbf{X}$$ and the convected coordinates $$X_i$$ by
 * {| style="width:100%" border="0"

{\textbf{G}}_i \left( \xi \right) = \frac = \frac = {\textbf{E}}_i \,\,\,\,\,\,  \Leftrightarrow G_i^k  =  \delta _i^k $$
 * style="width:95%" |$$
 * style="width:95%" |$$
 * style= |  (2.5)
 * }

and satisfy the following relations:
 * {| style="width:100%" border="0"

{\textbf{G}}_i \cdot {\textbf{G}}_j = G_i^k G_j^k  = \delta _i^k \delta _j^k  = \delta _{ij} $$
 * style="width:95%" |$$
 * style="width:95%" |$$
 * style= |  (2.6)
 * }

Similarly, the convected basis vectors $$\textbf{g}_i$$ in the current configuration $$Bt$$ are obtained using (2.4) and (2.5) as followings:
 * {| style="width:100%" border="0"

{\textbf{g}}_i = {\textbf{G}}_i  + \frac\,\,\,\,\, \Leftrightarrow \,\,\,g_i^k  = G_i^k  + \frac = \delta _i^k  + \frac $$ and satisfy the following relations:
 * style="width:95%" |$$
 * style="width:95%" |$$
 * style= |  (2.7)
 * }
 * {| style="width:100%" border="0"

g_{ij} {\rm{ = g}}_i {\rm{g}}_j = g_i^k g_j^k  = \left( {\delta _i^k  + \frac} \right)\left( {\delta _j^k  + \frac} \right) = \delta _i^k \delta _j^k  + \delta _i^k \frac + \delta _j^k \frac + \frac\frac $$ or equivalent to
 * style="width:95%" |$$
 * style="width:95%" |$$
 * style= |
 * }
 * {| style="width:100%" border="0"

g_{ij} = \delta _{ij}  + \frac + \frac + \frac\frac $$
 * style="width:95%" |$$
 * style="width:95%" |$$
 * style= | (2.8)
 * }

The deformation gradient expressed in convected basis vectors $$\textbf{g}_i$$ and $$\textbf{G}^i$$ takes the form (why we have this formular??)
 * {| style="width:100%" border="0"

{\textbf{F}} = {\textbf{g}}_k \otimes {\textbf{G}}^k $$ or in Cartesian coordinates:
 * style="width:95%" |$$
 * style="width:95%" |$$
 * style= |
 * }


 * {| style="width:100%" border="0"

F_{ij} = \left[ {{\textbf{g}}_k \otimes {\textbf{G}}^k } \right]_{ij}  = g_k^i G_j^k  = \left( {\delta _k^i  + \frac} \right)\delta _j^k  = \delta _k^i \delta _j^k  + \delta _j^k \frac = \delta _j^i  + \frac $$
 * style="width:95%" |$$
 * style="width:95%" |$$
 * style= | (2.9)
 * }