User:Tmhoang81/Energy Principles


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$$ \displaystyle \boldsymbol\nabla \cdot {\boldsymbol{\sigma }} + {\mathbf{b}} = 0\,\,in\,\,V,\,\,\,\,{\boldsymbol{\sigma }} \cdot {\mathbf{n}} = {{\mathbf{t}}^*}\,\,on\,\,{S_1},\,\,\,\,{\mathbf{u}} = {{\mathbf{u}}^*}\,\,on\,\,{S_2}
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$$ (N)
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$$ \displaystyle
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\int\limits_V {\left( {\boldsymbol\nabla \cdot {\boldsymbol{\sigma }} + {\mathbf{b}}} \right) \cdot \delta {\mathbf{u}}dV = 0\,\, \leftrightarrow \,\,} \int\limits_V {\left[ {\boldsymbol\nabla  \cdot \left( {{\boldsymbol{\sigma }} \cdot \delta {\mathbf{u}}} \right) - {\boldsymbol{\sigma :}}\left( {\delta {\mathbf{u}}} \right)\overleftarrow \boldsymbol\nabla  } \right]dV + \int\limits_V {\mathbf{b}}  \cdot \delta {\mathbf{u}}dV = 0} $$ (N)
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$$ \displaystyle
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\leftrightarrow \,\,\int\limits_V {\boldsymbol\nabla \cdot \left( {{\boldsymbol{\sigma }} \cdot \delta {\mathbf{u}}} \right)dV}  - \int\limits_V {{\boldsymbol{\sigma :}}\delta \left( {{\mathbf{u}}\overleftarrow \boldsymbol\nabla  } \right)dV + \int\limits_V {\mathbf{b}}  \cdot \delta {\mathbf{u}}dV = 0\,\,}  \leftrightarrow \,\,\int\limits_S {{\mathbf{n}} \cdot \left( {{\boldsymbol{\sigma }} \cdot \delta {\mathbf{u}}} \right)dS}  - \int\limits_V {{\boldsymbol{\sigma :}}\delta {\mathbf{e}}dV + \int\limits_V {\mathbf{b}}  \cdot \delta {\mathbf{u}}dV = 0\,\,} $$ (N)
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$$ \displaystyle
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\leftrightarrow \,\,\delta \pi = \int\limits_V {{\boldsymbol{\sigma :}}\delta {\mathbf{e}}dV}  - \int\limits_V {\mathbf{b}}  \cdot \delta {\mathbf{u}}dV - \int\limits_ {{{\mathbf{t}}^*} \cdot \delta {\mathbf{u}}dS}  = \int\limits_V {\delta WdV - \int\limits_V {\mathbf{b}}  \cdot \delta {\mathbf{u}}dV - } \int\limits_ {{{\mathbf{t}}^*} \cdot \delta {\mathbf{u}}dS}  = 0 $$ (N)
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$$ \displaystyle
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\delta W = {\boldsymbol{\sigma :}}\delta {\mathbf{e}}\,\, \leftrightarrow \,\,{\boldsymbol{\sigma }} = \frac\,\, \leftrightarrow \,\,{\mathbf{C:e}} = \frac\,\, \leftrightarrow W = \frac{1}{2}{\mathbf{e:C:e}} = \frac{1}{2}{\boldsymbol{\sigma :e}}\,\,\left( {if\,{\boldsymbol{\sigma }} = {{\boldsymbol{\sigma }}_0} = const\, \to \,\,W = {{\boldsymbol{\sigma }}_0}{\mathbf{:e}}\,\,} \right) $$ (N)
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$$ \displaystyle
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- \delta \phi = {\mathbf{b}} \cdot \delta {\mathbf{u}}\,\& \, - \delta \psi  = {{\mathbf{t}}^*} \cdot \delta {\mathbf{u}} $$ (N)
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$$ \displaystyle
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\delta \pi = \delta \left( {\int\limits_V {\left( {W + \phi } \right)dV}  + \int\limits_ {\psi dS} } \right) = 0 $$ (N)
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$$ \displaystyle
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\pi = \int\limits_V {\left( {W + \phi } \right)dV}  + \int\limits_ {\psi dS} $$ (N)
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$$ \displaystyle
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\leftrightarrow \,\,\delta \pi = \delta \left( {\int\limits_V {WdV - \int\limits_V {\mathbf{b}}  \cdot {\mathbf{u}}dV - } \int\limits_ {{{\mathbf{t}}^*} \cdot {\mathbf{u}}dS} } \right) = 0 $$ (N)
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$$ \displaystyle
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\pi = \int\limits_V {WdV - \int\limits_V {\mathbf{b}}  \cdot {\mathbf{u}}dV - } \int\limits_ {{{\mathbf{t}}^*} \cdot {\mathbf{u}}dS}  \to \,\min $$ (N)
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$$ \displaystyle
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\int\limits_V {\left[ {{\mathbf{\varepsilon }} - sym\left( {{\mathbf{u}}\overleftarrow \boldsymbol\nabla } \right)} \right]{\mathbf{:}}\delta {\boldsymbol{\sigma }}dV = 0\,\, \leftrightarrow \,\,} \int\limits_V {{\mathbf{\varepsilon :}}\delta {\boldsymbol{\sigma }}dV}  - \int\limits_V {\delta {\boldsymbol{\sigma :}}\left( {{\mathbf{u}}\overleftarrow \boldsymbol\nabla  } \right)dV}  = \int\limits_V {{\mathbf{\varepsilon :}}\delta {\boldsymbol{\sigma }}dV}  - \int\limits_V {\left[ {\boldsymbol\nabla  \cdot \left( {\delta {\boldsymbol{\sigma }} \cdot {\mathbf{u}}} \right) - \left( {\boldsymbol\nabla  \cdot \delta {\boldsymbol{\sigma }}} \right) \cdot {\mathbf{u}}} \right]dV = 0} $$ (N)
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$$ \displaystyle
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\leftrightarrow \,\,\int\limits_V {{\mathbf{\varepsilon :}}\delta {\boldsymbol{\sigma }}dV} - \int\limits_S {{\mathbf{n}} \cdot \left( {\delta {\boldsymbol{\sigma }} \cdot {\mathbf{u}}} \right)dS}  + \int\limits_V {\left( {\boldsymbol\nabla  \cdot \delta {\boldsymbol{\sigma }}} \right) \cdot {\mathbf{u}}dV}  = \,\,\int\limits_V {{\mathbf{\varepsilon :}}\delta {\boldsymbol{\sigma }}dV}  - \int\limits_ {\delta {\mathbf{t}} \cdot {{\mathbf{u}}^*}dS}  = 0 $$ (N)
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$$ \displaystyle
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\boldsymbol\nabla \cdot \delta {\boldsymbol{\sigma }} = 0\,\,in\,\,V,\,\,\,\,\,\delta {\boldsymbol{\sigma }} \cdot {\mathbf{n}} = 0\,\,on\,\,{S_1},\,\,\,\,\,\delta {\boldsymbol{\sigma }} \cdot {\mathbf{n}} = \delta {\mathbf{t}}\,\,on\,\,{S_2} $$ (N)
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$$ \displaystyle
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\delta {\pi ^*} = \int\limits_V {{\mathbf{\varepsilon :}}\delta {\boldsymbol{\sigma }}dV} - \int\limits_ {\delta {\mathbf{t}} \cdot {{\mathbf{u}}^*}dS}  = \int\limits_V {\delta {W^*}dV}  - \int\limits_ {\delta {\mathbf{t}} \cdot {{\mathbf{u}}^*}dS}  = 0 $$ (N)
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$$ \displaystyle
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\delta {W^*} = {\mathbf{e:}}\delta {\boldsymbol{\sigma }}\,\, \leftrightarrow \,\,{\mathbf{e}} = \frac\,\, \leftrightarrow \,\,{{\mathbf{C}}^{ - 1}}{\mathbf{:\sigma }} = \frac\,\, \leftrightarrow {W^*} = \frac{1}{2}{\boldsymbol{\sigma :}}{{\mathbf{C}}^{ - 1}}{\mathbf{:\sigma }} = \frac{1}{2}{\boldsymbol{\sigma :e}} = W $$ (N)
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$$ \displaystyle
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W \ne {W^*}\,\,but\,\,\,W + {W^*} = {\boldsymbol{\sigma :e}} $$ (N)
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$$ \displaystyle
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\delta {\pi ^*} = \delta \left( {\int\limits_V {{W^*}dV} - \int\limits_ {{\mathbf{t}} \cdot {{\mathbf{u}}^*}dS} } \right) = 0\,\, \leftrightarrow \,\,{\pi ^*} = \int\limits_V {{W^*}dV}  - \int\limits_ {{\mathbf{t}} \cdot {{\mathbf{u}}^*}dS}  \to \min $$ (N)
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$$ \displaystyle
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\delta \pi = \int\limits_V {{\boldsymbol{\sigma :}}\delta {\mathbf{e}}dV}  - \int\limits_V {\mathbf{b}}  \cdot \delta {\mathbf{u}}dV - \int\limits_ {{{\mathbf{t}}^*} \cdot \delta {\mathbf{u}}dS}  - \int\limits_ {{\mathbf{t}} \cdot \delta {{\mathbf{u}}^*}dS}  = \int\limits_V {\delta WdV - \int\limits_V {\mathbf{b}}  \cdot \delta {\mathbf{u}}dV - } \int\limits_ {{{\mathbf{t}}^*} \cdot \delta {\mathbf{u}}dS}  - \int\limits_ {{\mathbf{t}} \cdot \delta {{\mathbf{u}}^*}dS}  = 0 $$ (N)
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$$ \displaystyle
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\delta {\pi ^*} = \int\limits_V {{\mathbf{\varepsilon :}}\delta {\boldsymbol{\sigma }}dV} - \int\limits_ {\delta {{\mathbf{t}}^*} \cdot {\mathbf{u}}dS}  - \int\limits_ {\delta {\mathbf{t}} \cdot {{\mathbf{u}}^*}dS}  + \int\limits_V {\left( {\boldsymbol\nabla  \cdot \delta {\boldsymbol{\sigma }}} \right) \cdot {\mathbf{u}}dV}  = \int\limits_V {\delta {W^*}dV}  - \int\limits_ {\delta {{\mathbf{t}}^*} \cdot {\mathbf{u}}dS}  - \int\limits_ {\delta {\mathbf{t}} \cdot {{\mathbf{u}}^*}dS}  + \int\limits_V {\left( {\boldsymbol\nabla  \cdot \delta {\boldsymbol{\sigma }}} \right) \cdot {\mathbf{u}}dV} $$ (N)
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$$ \displaystyle
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\int\limits_V {dWdV = \int\limits_V {\mathbf{b}} \cdot d{\mathbf{u}}dV + \int\limits_ {{{\mathbf{t}}^*} \cdot d{\mathbf{u}}dS}  + \int\limits_ {{\mathbf{t}} \cdot d{{\mathbf{u}}^*}dS}  = 0,\,\,\,\,\,dW = {\boldsymbol{\sigma :}}d{\mathbf{e}}\,\,\,\,} $$ (N)
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$$ \displaystyle
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\int\limits_V {d{W^*}dV} = \int\limits_ {d{{\mathbf{t}}^*} \cdot {\mathbf{u}}dS}  + \int\limits_ {d{\mathbf{t}} \cdot {{\mathbf{u}}^*}dS}  - \int\limits_V {\left( {\boldsymbol\nabla  \cdot d{\boldsymbol{\sigma }}} \right) \cdot {\mathbf{u}}dV}  = \int\limits_ {d{{\mathbf{t}}^*} \cdot {\mathbf{u}}dS}  + \int\limits_ {d{\mathbf{t}} \cdot {{\mathbf{u}}^*}dS}  + \int\limits_V {d{\mathbf{b}} \cdot {\mathbf{u}}dV} ,\,\,\,\,d{W^*} = {\mathbf{e:}}d{\boldsymbol{\sigma }} $$ (N)
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$$ \displaystyle
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\boldsymbol\nabla \cdot \left( {{\boldsymbol{\sigma }} + d{\boldsymbol{\sigma }}} \right) + \left( {{\mathbf{b}} + d{\mathbf{b}}} \right) = 0\,\, \to \,\,\boldsymbol\nabla  \cdot \left( {d{\boldsymbol{\sigma }}} \right) + d{\mathbf{b}} = 0 $$ (N)
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$$ \displaystyle
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\int\limits_V {\mathbf{b}} \cdot {{\mathbf{u}}^'}dV + \int\limits_ {{{\mathbf{t}}^*} \cdot {{\mathbf{u}}^'}dS}  = \int\limits_V {{\boldsymbol{\sigma :}}{{\mathbf{\varepsilon }}^'}dV}  = \int\limits_V {{{\mathbf{\varepsilon }}^'}{\mathbf{:C:\varepsilon }}dV}  = \int\limits_V {{\mathbf{\varepsilon :C:}}{{\mathbf{\varepsilon }}^'}dV}  = \int\limits_V {{{\boldsymbol{\sigma }}^'}{\mathbf{:\varepsilon }}dV}  = {\int\limits_V {\mathbf{b}} ^'} \cdot {\mathbf{u}}dV + \int\limits_ {{{\mathbf{t}}^{'*}} \cdot {\mathbf{u}}dS} $$ (N)
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$$ \displaystyle
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{A_{I,I}} + {A_{II,II}} + {A_{I,II}} = {A_{II,II}} + {A_{I,I}} + {A_{II,I}}\,\, \to \,\,{A_{I,II}} = {A_{II,I}} $$ (N)
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$$ \displaystyle
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{P_i}{\Delta _{ji}} = {P_j}{\Delta _{ij}}\,\, \leftrightarrow \,\,{\alpha _{ji}} = \frac = \frac = {\alpha _{ij}} $$ (N)
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$$ \displaystyle
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\left( {{{\mathbf{b}}^'} = {\mathbf{b}},{{\mathbf{t}}^{'*}},{{\mathbf{u}}^'}} \right) $$ (N)
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$$ \displaystyle
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{{\mathbf{t}}^{'*}} = {{\mathbf{t}}^*}\,\,on\,\,{S_1} - S_1^'\,\,\,and\,\,\,\,{{\mathbf{t}}^{'*}} = {{\mathbf{t}}^*} + \Delta {{\mathbf{t}}^*}\,\,on\,\,S_1^' $$ (N)
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$$ \displaystyle
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\int\limits_V {\mathbf{b}} \cdot {{\mathbf{u}}^'}dV + \int\limits_ {{{\mathbf{t}}^*} \cdot {{\mathbf{u}}^'}dS}  = {\int\limits_V {\mathbf{b}} ^'} \cdot {\mathbf{u}}dV + \int\limits_ {{{\mathbf{t}}^{{*^'}}} \cdot {\mathbf{u}}dS}  = \int\limits_V {\mathbf{b}}  \cdot {\mathbf{u}}dV + \int\limits_{{S_1} - S_1^'} {{{\mathbf{t}}^*} \cdot {\mathbf{u}}dS}  + \int\limits_{S_1^'} {\left( {{{\mathbf{t}}^*} + \Delta {{\mathbf{t}}^*}} \right) \cdot {\mathbf{u}}dS} $$ (N)
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$$ \displaystyle
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\int\limits_V {\mathbf{b}} \cdot {{\mathbf{u}}^'}dV + \int\limits_ {{{\mathbf{t}}^*} \cdot {{\mathbf{u}}^'}dS}  = \int\limits_V {\mathbf{b}}  \cdot {\mathbf{u}}dV + \int\limits_ {{{\mathbf{t}}^*} \cdot {\mathbf{u}}dS}  + \int\limits_{S_1^'} {\Delta {{\mathbf{t}}^*} \cdot {\mathbf{u}}dS}  = 2U + \int\limits_{S_1^'} {\Delta {{\mathbf{t}}^*} \cdot {\mathbf{u}}dS} $$ (N)
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$$ \displaystyle
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2U = \int\limits_V {\mathbf{b}} \cdot {\mathbf{u}}dV + \int\limits_ {{{\mathbf{t}}^*} \cdot {\mathbf{u}}dS} $$ (N)
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$$ \displaystyle
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2{U^'} = {\int\limits_V {\mathbf{b}} ^'} \cdot {{\mathbf{u}}^'}dV + \int\limits_ {{{\mathbf{t}}^{{*^'}}} \cdot {{\mathbf{u}}^'}dS = } \int\limits_V {\mathbf{b}} \cdot {{\mathbf{u}}^'}dV + \int\limits_ {{{\mathbf{t}}^*} \cdot {{\mathbf{u}}^'}dS}  + \int\limits_{S_1^'} {\Delta {{\mathbf{t}}^*} \cdot {{\mathbf{u}}^'}dS = } 2U + \int\limits_{S_1^'} {\Delta {{\mathbf{t}}^*} \cdot {\mathbf{u}}dS}  + \int\limits_{S_1^'} {\Delta {{\mathbf{t}}^*} \cdot {{\mathbf{u}}^'}dS} $$ (N)
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$$ \displaystyle
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\to \,\,2{U^'} - 2U = 2\Delta U = \int\limits_{S_1^'} {\Delta {{\mathbf{t}}^*} \cdot \left( {{\mathbf{u}} + {{\mathbf{u}}^'}} \right)dS} \approx \,\,\Delta {{\mathbf{t}}^*} \cdot \left( {{\mathbf{u}} + {{\mathbf{u}}^'}} \right)S_1^' = \Delta {{\mathbf{f}}^*} \cdot \left( {{\mathbf{u}} + {{\mathbf{u}}^'}} \right) $$ (N)
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$$ \displaystyle
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\Delta {{\mathbf{f}}^*} = S_1^'\Delta {{\mathbf{t}}^*} $$ (N)
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$$ \displaystyle
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\frac = \frac{1}{2}\left( {{\mathbf{u}} + {{\mathbf{u}}^'}} \right)\,\, $$ (N)
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$$ \displaystyle
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\frac = {\mathbf{u}} $$ (N)
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