Ecuațiile Beltrami

Ecuațiile Beltrami , cunoscute și sub numele de ecuațiile Beltrami-Michell , sunt ecuații utilizate pentru soluționarea unei probleme elastice tridimensionale generice tratate în teoria elasticității . Pentru soluționarea problemei elastice în solide, una dintre abordările care pot fi urmate este așa-numita metodă de solicitare . Ziceri $\sigma$ ${\ displaystyle \ sigma}$ $\ sigma$ componentele vectorului de stres și d $\varepsilon$ ${\ displaystyle \ varepsilon}$ $\ varepsilon$ componentele deformării dețin următoarele ecuații numite respectiv ecuații de echilibru și congruență :

${\frac {\partial \sigma _{xx}}{\partial x}}+{\frac {\partial \sigma _{xy}}{\partial y}}+{\frac {\partial \sigma _{xz}}{\partial z}}+F_{x}=0$ ${\ displaystyle {\ frac {\ partial \ sigma _ {xx}} {\ partial x}} + {\ frac {\ partial \ sigma _ {xy}} {\ partial y}} + {\ frac {\ partial \ sigma _ {xz}} {\ partial z}} + F_ {x} = 0}$ ${\ displaystyle {\ frac {\ partial \ sigma _ {xx}} {\ partial x}} + {\ frac {\ partial \ sigma _ {xy}} {\ partial y}} + {\ frac {\ partial \ sigma _ {xz}} {\ partial z}} + F_ {x} = 0}$

${\frac {\partial \sigma _{yx}}{\partial x}}+{\frac {\partial \sigma _{yy}}{\partial y}}+{\frac {\partial \sigma _{yz}}{\partial z}}+F_{y}=0$ ${\ displaystyle {\ frac {\ partial \ sigma _ {yx}} {\ partial x}} + {\ frac {\ partial \ sigma _ {yy}} {\ partial y}} + {\ frac {\ partial \ sigma _ {yz}} {\ partial z}} + F_ {y} = 0}$ ${\ displaystyle {\ frac {\ partial \ sigma _ {yx}} {\ partial x}} + {\ frac {\ partial \ sigma _ {yy}} {\ partial y}} + {\ frac {\ partial \ sigma _ {yz}} {\ partial z}} + F_ {y} = 0}$

${\frac {\partial \sigma _{zx}}{\partial x}}+{\frac {\partial \sigma _{zy}}{\partial y}}+{\frac {\partial \sigma _{zz}}{\partial z}}+F_{z}=0$ ${\ displaystyle {\ frac {\ partial \ sigma _ {zx}} {\ partial x}} + {\ frac {\ partial \ sigma _ {zy}} {\ partial y}} + {\ frac {\ partial \ sigma _ {zz}} {\ partial z}} + F_ {z} = 0}$ ${\ displaystyle {\ frac {\ partial \ sigma _ {zx}} {\ partial x}} + {\ frac {\ partial \ sigma _ {zy}} {\ partial y}} + {\ frac {\ partial \ sigma _ {zz}} {\ partial z}} + F_ {z} = 0}$

${\frac {\partial ^{2}\varepsilon _{xy}}{\partial x\partial y}}={\frac {\partial ^{2}\varepsilon _{xx}}{\partial y^{2}}}+{\frac {\partial ^{2}\varepsilon _{yy}}{\partial x^{2}}}$ ${\ displaystyle {\ frac {\ partial ^ {2} \ varepsilon _ {xy}} {\ partial x \ partial y}} = {\ frac {\ partial ^ {2} \ varepsilon _ {xx}} {\ partial y ^ {2}}} + {\ frac {\ partial ^ {2} \ varepsilon _ {yy}} {\ partial x ^ {2}}}}$ ${\ displaystyle {\ frac {\ partial ^ {2} \ varepsilon _ {xy}} {\ partial x \ partial y}} = {\ frac {\ partial ^ {2} \ varepsilon _ {xx}} {\ partial y ^ {2}}} + {\ frac {\ partial ^ {2} \ varepsilon _ {yy}} {\ partial x ^ {2}}}}$

unde x, y, z sunt coordonatele unui sistem ortez de referință cartezian, F sunt componentele forței externe aplicate de-a lungul celor trei axe. Folosind legea lui Hooke ajungem la ecuațiile Beltrami-Michell pentru problema elastică tridimensională

${\begin{cases}\nabla ^{2}\sigma _{xx}+\left({\frac {1}{1+\nu }}\right){\frac {\partial ^{2}\Theta }{\partial x^{2}}}=-\left({\frac {\nu }{1-\nu }}\right)\nabla \cdot F-2{\frac {\partial F_{x}}{\partial x}}\\\nabla ^{2}\sigma _{yy}+\left({\frac {1}{1+\nu }}\right){\frac {\partial ^{2}\Theta }{\partial y^{2}}}=-\left({\frac {\nu }{1-\nu }}\right)\nabla \cdot F-2{\frac {\partial F_{y}}{\partial y}}\\\nabla ^{2}\sigma _{zz}+\left({\frac {1}{1+\nu }}\right){\frac {\partial ^{2}\Theta }{\partial z^{2}}}=-\left({\frac {\nu }{1-\nu }}\right)\nabla \cdot F-2{\frac {\partial F_{z}}{\partial z}}\\\nabla ^{2}\sigma _{xy}+\left({\frac {1}{1+\nu }}\right){\frac {\partial ^{2}\Theta }{\partial x\partial y}}=-\left({\frac {1}{1+\nu }}\right)\left({\frac {\partial F_{x}}{\partial y}}+{\frac {\partial F_{y}}{\partial x}}\right)\\\nabla ^{2}\sigma _{yz}+\left({\frac {1}{1+\nu }}\right){\frac {\partial ^{2}\Theta }{\partial y\partial z}}=-\left({\frac {1}{1+\nu }}\right)\left({\frac {\partial F_{y}}{\partial z}}+{\frac {\partial F_{z}}{\partial y}}\right)\\\nabla ^{2}\sigma _{xz}+\left({\frac {1}{1+\nu }}\right){\frac {\partial ^{2}\Theta }{\partial x\partial z}}=-\left({\frac {1}{1+\nu }}\right)\left({\frac {\partial F_{x}}{\partial z}}+{\frac {\partial F_{z}}{\partial x}}\right)\end{cases}}$ ${\ displaystyle {\ begin {cases} \ nabla ^ {2} \ sigma _ {xx} + \ left ({\ frac {1} {1+ \ nu}} \ right) {\ frac {\ partial ^ {2 } \ Theta} {\ partial x ^ {2}}} = - \ left ({\ frac {\ nu} {1- \ nu}} \ right) \ nabla \ cdot F-2 {\ frac {\ partial F_ {x}} {\ partial x}} \\\ nabla ^ {2} \ sigma _ {yy} + \ left ({\ frac {1} {1+ \ nu}} \ right) {\ frac {\ partial ^ {2} \ Theta} {\ partial y ^ {2}}} = - \ left ({\ frac {\ nu} {1- \ nu}} \ right) \ nabla \ cdot F-2 {\ frac { \ partial F_ {y}} {\ partial y}} \\\ nabla ^ {2} \ sigma _ {zz} + \ left ({\ frac {1} {1+ \ nu}} \ right) {\ frac {\ partial ^ {2} \ Theta} {\ partial z ^ {2}}} = - \ left ({\ frac {\ nu} {1- \ nu}} \ right) \ nabla \ cdot F-2 { \ frac {\ partial F_ {z}} {\ partial z}} \\\ nabla ^ {2} \ sigma _ {xy} + \ left ({\ frac {1} {1+ \ nu}} \ right) {\ frac {\ partial ^ {2} \ Theta} {\ partial x \ partial y}} = - \ left ({\ frac {1} {1+ \ nu}} \ right) \ left ({\ frac { \ partial F_ {x}} {\ partial y}} + {\ frac {\ partial F_ {y}} {\ partial x}} \ right) \\\ nabla ^ {2} \ sigma _ {yz} + \ left ({\ frac {1} {1+ \ nu}} \ right) {\ frac {\ partial ^ {2} \ Theta} {\ partial y \ partial z}} = - \ left ({\ frac {1 } {1+ \ nu}} \ right) \ left ({\ frac {\ parts al F_ {y}} {\ partial z}} + {\ frac {\ partial F_ {z}} {\ partial y}} \ right) \\\ nabla ^ {2} \ sigma _ {xz} + \ left ({\ frac {1} {1+ \ nu}} \ right) {\ frac {\ partial ^ {2} \ Theta} {\ partial x \ partial z}} = - \ left ({\ frac {1} {1+ \ nu}} \ right) \ left ({\ frac {\ partial F_ {x}} {\ partial z}} + {\ frac {\ partial F_ {z}} {\ partial x}} \ right ) \ end {cases}}}$ ${\ displaystyle {\ begin {cases} \ nabla ^ {2} \ sigma _ {xx} + \ left ({\ frac {1} {1+ \ nu}} \ right) {\ frac {\ partial ^ {2 } \ Theta} {\ partial x ^ {2}}} = - \ left ({\ frac {\ nu} {1- \ nu}} \ right) \ nabla \ cdot F-2 {\ frac {\ partial F_ {x}} {\ partial x}} \\\ nabla ^ {2} \ sigma _ {yy} + \ left ({\ frac {1} {1+ \ nu}} \ right) {\ frac {\ partial ^ {2} \ Theta} {\ partial y ^ {2}}} = - \ left ({\ frac {\ nu} {1- \ nu}} \ right) \ nabla \ cdot F-2 {\ frac { \ partial F_ {y}} {\ partial y}} \\\ nabla ^ {2} \ sigma _ {zz} + \ left ({\ frac {1} {1+ \ nu}} \ right) {\ frac {\ partial ^ {2} \ Theta} {\ partial z ^ {2}}} = - \ left ({\ frac {\ nu} {1- \ nu}} \ right) \ nabla \ cdot F-2 { \ frac {\ partial F_ {z}} {\ partial z}} \\\ nabla ^ {2} \ sigma _ {xy} + \ left ({\ frac {1} {1+ \ nu}} \ right) {\ frac {\ partial ^ {2} \ Theta} {\ partial x \ partial y}} = - \ left ({\ frac {1} {1+ \ nu}} \ right) \ left ({\ frac { \ partial F_ {x}} {\ partial y}} + {\ frac {\ partial F_ {y}} {\ partial x}} \ right) \\\ nabla ^ {2} \ sigma _ {yz} + \ left ({\ frac {1} {1+ \ nu}} \ right) {\ frac {\ partial ^ {2} \ Theta} {\ partial y \ partial z}} = - \ left ({\ frac {1 } {1+ \ nu}} \ right) \ left ({\ frac {\ parts al F_ {y}} {\ partial z}} + {\ frac {\ partial F_ {z}} {\ partial y}} \ right) \\\ nabla ^ {2} \ sigma _ {xz} + \ left ({\ frac {1} {1+ \ nu}} \ right) {\ frac {\ partial ^ {2} \ Theta} {\ partial x \ partial z}} = - \ left ({\ frac {1} {1+ \ nu}} \ right) \ left ({\ frac {\ partial F_ {x}} {\ partial z}} + {\ frac {\ partial F_ {z}} {\ partial x}} \ right ) \ end {cases}}}$

unde este $\nu$ ${\ displaystyle \ nu}$ $\ nu$ este raportul lui Poisson și $\Theta =\sigma _{xx}+\sigma _{yy}+\sigma _{zz}$ ${\ displaystyle \ Theta = \ sigma _ {xx} + \ sigma _ {yy} + \ sigma _ {zz}}$ ${\ displaystyle \ Theta = \ sigma _ {xx} + \ sigma _ {yy} + \ sigma _ {zz}}$ . Aceste ecuații, cu condiții relative la graniță, definesc problema elastică tridimensională

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