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The second-order Cauchy stress tensor T {\displaystyle \mathbf {T} } describes the stress experienced by a material at a given point. For any unit vector v {\displaystyle \mathbf {v} } , the product T ⋅ v {\displaystyle \mathbf {T} \cdot \mathbf {v} } is a vector, denoted T ( v ) {\displaystyle \mathbf {T} (\mathbf {v} )} , that quantifies the force per area along the plane perpendicular to v {\displaystyle \mathbf {v} } . This image shows, for cube faces perpendicular to e 1 , e 2 , e 3 {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}} , the corresponding stress vectors T ( e 1 ) , T ( e 2 ) , T ( e 3 ) {\displaystyle \mathbf {T} (\mathbf {e} _{1}),\mathbf {T} (\mathbf {e} _{2}),\mathbf {T} (\mathbf {e} _{3})} along those faces. Because the stress tensor takes one vector as input and gives one vector as output, it is a second-order tensor.

テンソルとは何か数学的定義から物理学への応用までを解説

テンソル多重線形写像ベクトル空間階数
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The second-order Cauchy stress tensor T {\displaystyle \mathbf {T} } describes the stress experienced by a material at a given point. For any unit vector v {\displaystyle \mathbf {v} } , the product T ⋅ v {\displaystyle \mathbf {T} \cdot \mathbf {v} } is a vector, denoted T ( v ) {\displaystyle \mathbf {T} (\mathbf {v} )} , that quantifies the force per area along the plane perpendicular to v {\displaystyle \mathbf {v} } . This image shows, for cube faces perpendicular to e 1 , e 2 , e 3 {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}} , the corresponding stress vectors T ( e 1 ) , T ( e 2 ) , T ( e 3 ) {\displaystyle \mathbf {T} (\mathbf {e} _{1}),\mathbf {T} (\mathbf {e} _{2}),\mathbf {T} (\mathbf {e} _{3})} along those faces. Because the stress tensor takes one vector as input and gives one vector as output, it is a second-order tensor.

テンソルとは何か数学的定義から物理学・機械学習への応用まで

テンソル線形代数多重線形写像物理学
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