User:Egm6936.s09/EGM 6611 Continuum Mechanics, Fall 2003/Lecture plan


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$$ \displaystyle \int\limits_V {\left[ {2\mu {\rm{\varepsilon :\varepsilon }} + \lambda {{\left( {tr{\rm{\varepsilon }}} \right)}^2}} \right]dV > } 0 \int\limits_ {\left[ {2{\mu _1}{\rm{\varepsilon :\varepsilon }} + {\lambda _1}{{\left( {tr{\rm{\varepsilon }}} \right)}^2}} \right]dV} + \int\limits_ {\left[ {2{\mu _2}{\rm{\varepsilon :\varepsilon }} + {\lambda _2}{{\left( {tr{\rm{\varepsilon }}} \right)}^2}} \right]dV > } 0 $$ (N)
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$$ \displaystyle
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{\mu _1} = \beta {\mu _1} + \left( {1 - \beta } \right){\mu _1} $$ (N)
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$$ \displaystyle
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0 \le \beta \le 1 $$ (N)
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$$ \displaystyle
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\int\limits_ {\left[ {2{\mu _1}\left( {1 - \beta } \right){\rm{\varepsilon :\varepsilon }} + 2{\mu _1}\beta {\rm{\varepsilon :\varepsilon }} + {\lambda _1}{{\left( {tr{\rm{\varepsilon }}} \right)}^2}} \right]dV} + \int\limits_ {t\left[ {2{\mu _2}{\rm{\varepsilon :\varepsilon }} + {\lambda _2}{{\left( {tr{\rm{\varepsilon }}} \right)}^2}} \right]dS > } 0 $$ (N)
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$$ \displaystyle
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{\rm{\varepsilon }} = {{\rm{\varepsilon }}^'} + \frac{1}{k}tr\left( {\rm{\varepsilon }} \right){\rm{I}} $$ (N)
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$$ \displaystyle
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{\rm{\varepsilon :\varepsilon }} = {{\rm{\varepsilon }}^'}{\rm{:}}{{\rm{\varepsilon }}^'} + \frac{1}{k}{\left( {tr{\rm{\varepsilon }}} \right)^2} $$ (N)
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$$ \displaystyle
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{\rm{\varepsilon :\varepsilon }} = {\rm{\omega :\omega }} + {\left( {tr{\rm{\varepsilon }}} \right)^2} + \nabla \bullet \left[ {\left( {{\rm{u}}\nabla } \right) \bullet {\rm{u}} - {\rm{u}}\left( {\nabla  \bullet {\rm{u}}} \right)} \right] $$ (N)
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$$ \displaystyle
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\int\limits_ {\left\{ {2{\mu _1}\left( {1 - \beta } \right)\left[ {{{\rm{\varepsilon }}^'}{\rm{:}}{{\rm{\varepsilon }}^'} + \frac{1}{k}{{\left( {tr{\rm{\varepsilon }}} \right)}^2}} \right] + 2{\mu _1}\beta \left[ {{\rm{\omega :\omega }} + {{\left( {tr{\rm{\varepsilon }}} \right)}^2} + \nabla \bullet \left[ {\left( {{\rm{u}}\nabla } \right) \bullet {\rm{u}} - {\rm{u}}\left( {\nabla  \bullet {\rm{u}}} \right)} \right]} \right] + {\lambda _1}{{\left( {tr{\rm{\varepsilon }}} \right)}^2}} \right\}dV}  + \int\limits_ {t\left[ {2{\mu _2}{\rm{\varepsilon :\varepsilon }} + {\lambda _2}{{\left( {tr{\rm{\varepsilon }}} \right)}^2}} \right]dS > } 0 $$ (N)
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$$ \displaystyle
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\int\limits_ {\left\{ {2{\mu _1}\left[ {\left( {1 - \beta } \right){{\rm{\varepsilon }}^'}{\rm{:}}{{\rm{\varepsilon }}^'} + \beta {\rm{\omega :\omega }}} \right] + \left[ {{\lambda _1} + \frac{k}\left( {1 + \left( {k - 1} \right)\beta } \right)} \right]{{\left( {tr{\rm{\varepsilon }}} \right)}^2}} \right\}dV} + 2{\mu _1}\beta \int\limits_ {\nabla  \bullet \left[ {\left( {{\rm{u}}\nabla } \right) \bullet {\rm{u}} - {\rm{u}}\left( {\nabla  \bullet {\rm{u}}} \right)} \right]dV}  + \int\limits_ {t\left[ {2{\mu _2}{\rm{\varepsilon :\varepsilon }} + {\lambda _2}{{\left( {tr{\rm{\varepsilon }}} \right)}^2}} \right]dS > } 0 $$ (N)
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$$ \displaystyle
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{\rm{n}} \bullet \left[ {\left( {{\rm{u}}\nabla } \right) \bullet {\rm{u}} - {\rm{u}}\left( {\nabla \bullet {\rm{u}}} \right)} \right] = tr\left[ {{\rm{u}} \times \left( {{\rm{u}}\nabla } \right) \times {\rm{n}}} \right] $$ (N)
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$$ \displaystyle
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\int\limits_ {\left\{ {2{\mu _1}\left[ {\left( {1 - \beta } \right){{\rm{\varepsilon }}^'}{\rm{:}}{{\rm{\varepsilon }}^'} + \beta {\rm{\omega :\omega }}} \right] + \left[ {{\lambda _1} + \frac{k}\left( {1 + \left( {k - 1} \right)\beta } \right)} \right]{{\left( {tr{\rm{\varepsilon }}} \right)}^2}} \right\}dV} + $$ (N)
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$$ \displaystyle
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+ \int\limits_S {\left\{ {t\left[ {2{\mu _2}{\rm{\varepsilon :\varepsilon }} + {\lambda _2}{{\left( {tr{\rm{\varepsilon }}} \right)}^2}} \right] + 2{\mu _1}\beta tr\left[ {{\rm{u}} \times \left( {{\rm{u}}\nabla } \right) \times {\rm{n}}} \right]} \right\}dS > } 0 $$ (N)
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$$ \displaystyle
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\int\limits_ {\left[ {2{\mu _1}\varepsilon :\varepsilon + {\lambda _1}{{\left( {tr\varepsilon } \right)}^2}} \right]dV}  + $$
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$$ \displaystyle
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\int\limits_ {\left[ {2{\mu _2}\varepsilon :\varepsilon + {\lambda _2}{{\left( {tr\varepsilon } \right)}^2}} \right]dV > } 0 $$ (N)
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