diff options
Diffstat (limited to 'content/know/concept/cauchy-strain-tensor/index.pdc')
| -rw-r--r-- | content/know/concept/cauchy-strain-tensor/index.pdc | 18 |
1 files changed, 9 insertions, 9 deletions
diff --git a/content/know/concept/cauchy-strain-tensor/index.pdc b/content/know/concept/cauchy-strain-tensor/index.pdc index f150723..cb48377 100644 --- a/content/know/concept/cauchy-strain-tensor/index.pdc +++ b/content/know/concept/cauchy-strain-tensor/index.pdc @@ -82,7 +82,7 @@ we expand the middle term to first order in $\va{a}$: $$\begin{aligned} \va{u}(\va{x} + \va{a}) \approx \va{u}(\va{x}) + a_x \pdv{\va{u}}{x} + a_y \pdv{\va{u}}{y} + a_z \pdv{\va{u}}{z} - = \va{u}(\va{x}) + \va{a} \cdot \nabla \va{u}(\va{x}) + = \va{u}(\va{x}) + (\va{a} \cdot \nabla) \va{u}(\va{x}) \end{aligned}$$ With this, we can now define the "shift" $\delta\va{a}$ @@ -91,7 +91,7 @@ as the difference between $\va{a}$ and $\va{A}$ like so: $$\begin{aligned} \delta{\va{a}} \equiv \va{a} - \va{A} - = \va{a} \cdot \nabla \va{u}(\va{x}) + = (\va{a} \cdot \nabla) \va{u}(\va{x}) \end{aligned}$$ In index notation, we write this expression as follows, @@ -245,8 +245,8 @@ is easy to express using the displacement field $\va{u}$: $$\begin{aligned} \boxed{ \delta(\dd{\va{l}}) - = \dd{\va{l}} \cdot \nabla \va{u} - = (\nabla \vec{u})^\top \cdot \dd{\va{l}} + = (\dd{\va{l}} \cdot \nabla) \va{u} + %= (\nabla \vec{u})^\top \cdot \dd{\va{l}} } \end{aligned}$$ @@ -259,9 +259,9 @@ $$\begin{aligned} = \delta(\va{a} \cross \va{b} \cdot \va{c}) &= \delta\va{a} \cross \va{b} \cdot \va{c} + \va{a} \cross \delta\va{b} \cdot \va{c} + \va{a} \cross \va{b} \cdot \delta\va{c} \\ - &= (\va{a} \cdot \nabla\va{u}) \cross \va{b} \cdot \va{c} - + \va{a} \cross (\va{b} \cdot \nabla\va{u}) \cdot \va{c} - + \va{a} \cross \va{b} \cdot (\va{c} \cdot \nabla\va{u}) + &= (\va{a} \cdot \nabla) \va{u} \cross \va{b} \cdot \va{c} + + \va{a} \cross (\va{b} \cdot \nabla )\va{u} \cdot \va{c} + + \va{a} \cross \va{b} \cdot (\va{c} \cdot \nabla) \va{u} \end{aligned}$$ We can reorder the factors like so @@ -303,7 +303,7 @@ $$\begin{aligned} \delta(\dd{V}) = \delta(\va{c} \cdot \dd{\va{S}}) = \delta\va{c} \cdot \dd{\va{S}} + \va{c} \cdot \delta(\dd{\va{S}}) - = (\va{c} \cdot \nabla\va{u}) \cdot \dd{\va{S}} + \va{c} \cdot \delta(\dd{\va{S}}) + = (\va{c} \cdot \nabla) \va{u} \cdot \dd{\va{S}} + \va{c} \cdot \delta(\dd{\va{S}}) \end{aligned}$$ By comparing this to the previous result for $\delta(\dd{V})$, @@ -311,7 +311,7 @@ we arrive at the following equation: $$\begin{aligned} \nabla \cdot \va{u} (\va{c} \cdot \dd{\va{S}}) - = (\va{c} \cdot \nabla\va{u}) \cdot \dd{\va{S}} + \va{c} \cdot \delta(\dd{\va{S}}) + = (\va{c} \cdot \nabla) \va{u} \cdot \dd{\va{S}} + \va{c} \cdot \delta(\dd{\va{S}}) \end{aligned}$$ Since $\va{c}$ is dot-multiplied at the front of each term, |