Кіріспе
Математикалық тұжырымдама
Mathematical Concept
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off diagonal. Thus it can be expressed as the vector
As another example:
The stress tensor (in matrix notation) is given as
In Voigt notation it is simplified to a 6 dimensional vector:
The strain tensor, similar in nature to the stress tensor—both are symmetric second order tensors , is given in matrix form as
Its representation in Voigt notation is
where , , and are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance
is preserved. Likewise, a three dimensional symmetric fourth order tensor can be reduced to a 6×6 matrix.
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off diagonal. Thus it can be expressed as the vector
As another example:
The stress tensor (in matrix notation) is given as
In Voigt notation it is simplified to a 6 dimensional vector:
The strain tensor, similar in nature to the stress tensor—both are symmetric second order tensors , is given in matrix form as
Its representation in Voigt notation is
where , , and are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance
is preserved. Likewise, a three dimensional symmetric fourth order tensor can be reduced to a 6×6 matrix.
Математикада Войгт белгісі немесе көп сызықты алгебрадағы Войгт формасы – симметриялық тензорды оның ретін төмендету арқылы көрсету тәсілі. Бұл идеяның бірнеше түрі мен байланысты атаулары бар: Мандель белгісі, Мандель–Войгт белгісі және Най белгісі. Келвин белгісі – Хельбигтің лорд Келвиннің бұрынғы идеяларын қайта жаңартуы. Бұл жердегі айырмашылық тензордың таңдалған элементтеріне қосылатын нақты салмақтарда. Аталым қолданылатын саланың дәстүріне қарай өзгеруі мүмкін. Мысалы, 2×2 симметриялық тензор X-тің тек үш ерекше элементі бар: екеуі диагональде, ал үшіншісі диагональден тыс. Сондықтан оны вектор түрінде көрсетуге болады.
Mathematical Concept
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off diagonal. Thus it can be expressed as the vector
As another example:
The stress tensor (in matrix notation) is given as
In Voigt notation it is simplified to a 6 dimensional vector:
The strain tensor, similar in nature to the stress tensor—both are symmetric second order tensors , is given in matrix form as
Its representation in Voigt notation is
where , , and are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance
is preserved. Likewise, a three dimensional symmetric fourth order tensor can be reduced to a 6×6 matrix.
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off diagonal. Thus it can be expressed as the vector
As another example:
The stress tensor (in matrix notation) is given as
In Voigt notation it is simplified to a 6 dimensional vector:
The strain tensor, similar in nature to the stress tensor—both are symmetric second order tensors , is given in matrix form as
Its representation in Voigt notation is
where , , and are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance
is preserved. Likewise, a three dimensional symmetric fourth order tensor can be reduced to a 6×6 matrix.
Тағы бір мысал:
Mathematical Concept
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off diagonal. Thus it can be expressed as the vector
As another example:
The stress tensor (in matrix notation) is given as
In Voigt notation it is simplified to a 6 dimensional vector:
The strain tensor, similar in nature to the stress tensor—both are symmetric second order tensors , is given in matrix form as
Its representation in Voigt notation is
where , , and are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance
is preserved. Likewise, a three dimensional symmetric fourth order tensor can be reduced to a 6×6 matrix.
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off diagonal. Thus it can be expressed as the vector
As another example:
The stress tensor (in matrix notation) is given as
In Voigt notation it is simplified to a 6 dimensional vector:
The strain tensor, similar in nature to the stress tensor—both are symmetric second order tensors , is given in matrix form as
Its representation in Voigt notation is
where , , and are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance
is preserved. Likewise, a three dimensional symmetric fourth order tensor can be reduced to a 6×6 matrix.
Серпімділік тензоры (матрицалық жазбада) былай берілген:
Mathematical Concept
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off diagonal. Thus it can be expressed as the vector
As another example:
The stress tensor (in matrix notation) is given as
In Voigt notation it is simplified to a 6 dimensional vector:
The strain tensor, similar in nature to the stress tensor—both are symmetric second order tensors , is given in matrix form as
Its representation in Voigt notation is
where , , and are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance
is preserved. Likewise, a three dimensional symmetric fourth order tensor can be reduced to a 6×6 matrix.
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off diagonal. Thus it can be expressed as the vector
As another example:
The stress tensor (in matrix notation) is given as
In Voigt notation it is simplified to a 6 dimensional vector:
The strain tensor, similar in nature to the stress tensor—both are symmetric second order tensors , is given in matrix form as
Its representation in Voigt notation is
where , , and are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance
is preserved. Likewise, a three dimensional symmetric fourth order tensor can be reduced to a 6×6 matrix.
Войгт белгісінде ол 6 өлшемді векторға дейін жеңілдетіледі:
Mathematical Concept
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off diagonal. Thus it can be expressed as the vector
As another example:
The stress tensor (in matrix notation) is given as
In Voigt notation it is simplified to a 6 dimensional vector:
The strain tensor, similar in nature to the stress tensor—both are symmetric second order tensors , is given in matrix form as
Its representation in Voigt notation is
where , , and are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance
is preserved. Likewise, a three dimensional symmetric fourth order tensor can be reduced to a 6×6 matrix.
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off diagonal. Thus it can be expressed as the vector
As another example:
The stress tensor (in matrix notation) is given as
In Voigt notation it is simplified to a 6 dimensional vector:
The strain tensor, similar in nature to the stress tensor—both are symmetric second order tensors , is given in matrix form as
Its representation in Voigt notation is
where , , and are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance
is preserved. Likewise, a three dimensional symmetric fourth order tensor can be reduced to a 6×6 matrix.
Серпімділік тензоры, табиғаты жағынан серпімділік тензорына ұқсас – екеуі де симметриялық екінші реттік тензорлар, матрицалық түрде былай берілген:
Mathematical Concept
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off diagonal. Thus it can be expressed as the vector
As another example:
The stress tensor (in matrix notation) is given as
In Voigt notation it is simplified to a 6 dimensional vector:
The strain tensor, similar in nature to the stress tensor—both are symmetric second order tensors , is given in matrix form as
Its representation in Voigt notation is
where , , and are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance
is preserved. Likewise, a three dimensional symmetric fourth order tensor can be reduced to a 6×6 matrix.
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off diagonal. Thus it can be expressed as the vector
As another example:
The stress tensor (in matrix notation) is given as
In Voigt notation it is simplified to a 6 dimensional vector:
The strain tensor, similar in nature to the stress tensor—both are symmetric second order tensors , is given in matrix form as
Its representation in Voigt notation is
where , , and are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance
is preserved. Likewise, a three dimensional symmetric fourth order tensor can be reduced to a 6×6 matrix.
Оның Войгт белгісіндегі көрсетілуі:
Mathematical Concept
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off diagonal. Thus it can be expressed as the vector
As another example:
The stress tensor (in matrix notation) is given as
In Voigt notation it is simplified to a 6 dimensional vector:
The strain tensor, similar in nature to the stress tensor—both are symmetric second order tensors , is given in matrix form as
Its representation in Voigt notation is
where , , and are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance
is preserved. Likewise, a three dimensional symmetric fourth order tensor can be reduced to a 6×6 matrix.
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off diagonal. Thus it can be expressed as the vector
As another example:
The stress tensor (in matrix notation) is given as
In Voigt notation it is simplified to a 6 dimensional vector:
The strain tensor, similar in nature to the stress tensor—both are symmetric second order tensors , is given in matrix form as
Its representation in Voigt notation is
where , , and are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance
is preserved. Likewise, a three dimensional symmetric fourth order tensor can be reduced to a 6×6 matrix.
мұнда , , және инженерлік иілу деформациялары. Серпімділік пен деформацияны көрсетудің әртүрлі тәсілдерін пайдаланудың артықшылығы – скалярлық инвариантты сақтау болып табылады. Сол сияқты, үш өлшемді симметриялық төртінші реттік тензорды 6×6 матрицаға дейін азайтуға болады.
Mathematical Concept
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off diagonal. Thus it can be expressed as the vector
As another example:
The stress tensor (in matrix notation) is given as
In Voigt notation it is simplified to a 6 dimensional vector:
The strain tensor, similar in nature to the stress tensor—both are symmetric second order tensors , is given in matrix form as
Its representation in Voigt notation is
where , , and are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance
is preserved. Likewise, a three dimensional symmetric fourth order tensor can be reduced to a 6×6 matrix.
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off diagonal. Thus it can be expressed as the vector
As another example:
The stress tensor (in matrix notation) is given as
In Voigt notation it is simplified to a 6 dimensional vector:
The strain tensor, similar in nature to the stress tensor—both are symmetric second order tensors , is given in matrix form as
Its representation in Voigt notation is
where , , and are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance
is preserved. Likewise, a three dimensional symmetric fourth order tensor can be reduced to a 6×6 matrix.
Қолданбалар
Бұл жазу физик Волдемар Войт және ғалым Джон Най есімдерімен аталған. Мысалы, материалдарды модельдеуге арналған құраушы модельдерді қолданатын есептеулерде, сондай-ақ жалпыланған Гук заңы, шекті элементтер талдауы және диффузиялық МРТ сияқты жағдайларда бұл пайдалы. Гук заңы 81 компоненті бар (3×3×3×3) симметриялық төртінші реттік қатаңдық тензорына ие, бірақ төртінші реттік тензорды симметриялық екінші реттік тензорға қолданғанда, нәтиже басқа симметриялық екінші реттік тензор болуы керек, сондықтан 81 элементтің барлығы тәуелсіз емес. Войт нотациясы мұндай төртінші реттік тензорды 6×6 матрица арқылы көрсетуге мүмкіндік береді. Дегенмен, Войт формасы квадраттардың қосындысын сақтамайды, бұл Гук заңы үшін геометриялық маңызға ие. Осы себепті салмақтар енгізіледі (карталауды изометрияға айналдыру үшін). Войт нотациясы мен Мандель нотациясының инварианттылығы туралы талқылауды Helnwein (2001) еңбегінен табуға болады.
The notation is named after physicist Woldemar Voigt & John Nye (scientist). It is useful, for example, in calculations involving constitutive models to simulate materials, such as the generalized Hooke's law, as well as finite element analysis, and Diffusion MRI. Hooke's law has a symmetric fourth order stiffness tensor with 81 components (3×3×3×3), but because the application of such a rank 4 tensor to a symmetric rank 2 tensor must yield another symmetric rank 2 tensor, not all of the 81 elements are independent. Voigt notation enables such a rank 4 tensor to be represented by a 6×6 matrix. However, Voigt's form does not preserve the sum of the squares, which in the case of Hooke's law has geometric significance. This explains why weights are introduced (to make the mapping an isometry). A discussion of invariance of Voigt's notation and Mandel's notation can be found in Helnwein (2001).