Редактирование: The mechanics of the Cosserat media
Внимание! Вы не авторизовались на сайте. Ваш IP-адрес будет публично видимым, если вы будете вносить любые правки. Если вы войдёте или создадите учётную запись, правки вместо этого будут связаны с вашим именем пользователя, а также у вас появятся другие преимущества.
Правка может быть отменена. Пожалуйста, просмотрите сравнение версий, чтобы убедиться, что это именно те изменения, которые вас интересуют, и нажмите «Записать страницу», чтобы изменения вступили в силу.
Текущая версия | Ваш текст | ||
Строка 1: | Строка 1: | ||
== Introduction == | == Introduction == | ||
− | [[Файл: | + | [[Файл:Cosserat.png|thumb|200px| Magnetic materials (Kelvin’s medium — special Cosserat medium with particle posessing large spin)]] |
− | |||
Cosserat medium is a continuum whose point bodies (particles) have rotational degrees of freedom. Examples of Cosserat media: heterogeneous materials with granular structure, composites under loading that causes rotation of (sufficiently rigid) grains (superplastic materials, acoustic metamaterials). Cosserat medium is a particular case of complex medium. Its point-body is rigid. There are other more complex media, e.g. where a point-body is deformable (protein chains, porous media, etc.) It is only a first step to the world of enriched continua. Theory is based on the fundamental laws of mechanics (balance of forces, couples, energy) and, for inelastic media, 2nd law of the thermodynamics, symmetry considerations and material frame indifference. Another branch is the microstructural approach. Experimental methods: under development. We need experiments to determine the moduli. Most of them are based on the experiments on waves (mechanics of magnetic and piezoelectric materials, mechanics of granular materials, rotational seismology...). Reduced Cosserat medium: Cosserat medium that does not react to the gradient of rotation. | Cosserat medium is a continuum whose point bodies (particles) have rotational degrees of freedom. Examples of Cosserat media: heterogeneous materials with granular structure, composites under loading that causes rotation of (sufficiently rigid) grains (superplastic materials, acoustic metamaterials). Cosserat medium is a particular case of complex medium. Its point-body is rigid. There are other more complex media, e.g. where a point-body is deformable (protein chains, porous media, etc.) It is only a first step to the world of enriched continua. Theory is based on the fundamental laws of mechanics (balance of forces, couples, energy) and, for inelastic media, 2nd law of the thermodynamics, symmetry considerations and material frame indifference. Another branch is the microstructural approach. Experimental methods: under development. We need experiments to determine the moduli. Most of them are based on the experiments on waves (mechanics of magnetic and piezoelectric materials, mechanics of granular materials, rotational seismology...). Reduced Cosserat medium: Cosserat medium that does not react to the gradient of rotation. | ||
Строка 8: | Строка 7: | ||
== Basic equations == | == Basic equations == | ||
− | + | [[Файл:Cosserat1.png|thumb|200px| Cosserat medium]] | |
<b>Stress tensors</b> | <b>Stress tensors</b> | ||
Строка 14: | Строка 13: | ||
If <math>M_{(n)}</math> is a couple (torque) acting upon a unit surface with normal n, there exist a tensor µ such that <math>M_{(n)} = n \cdot \mu</math>. <math>\mu</math> is called couple stress tensor. <math>\mu</math> works on ∇ω (gradient of the angular velocity in the actual configuration). | If <math>M_{(n)}</math> is a couple (torque) acting upon a unit surface with normal n, there exist a tensor µ such that <math>M_{(n)} = n \cdot \mu</math>. <math>\mu</math> is called couple stress tensor. <math>\mu</math> works on ∇ω (gradient of the angular velocity in the actual configuration). | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
<b>Balance of forces: 1st law of dynamics by Euler</b> | <b>Balance of forces: 1st law of dynamics by Euler</b> | ||
Строка 56: | Строка 39: | ||
<b>Balance of energy. Local form.</b> | <b>Balance of energy. Local form.</b> | ||
− | + | [[Файл:Cosserat2.png|thumb|200px| Reduced Cosserat medium]] | |
<math>\rho \dot U = \tau^T \cdot \cdot \bigtriangledown v - \tau_x \omega + \mu^T \cdot \cdot \bigtriangledown \omega</math> | <math>\rho \dot U = \tau^T \cdot \cdot \bigtriangledown v - \tau_x \omega + \mu^T \cdot \cdot \bigtriangledown \omega</math> | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− |