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#body #configuration #continuum-mechanics #particle #position #region

We shall use the term “body” to be a mathematical abstraction of an “object that occurs in nature”. A *body* \(\mathcal B\) is composed of a set of *particles* \(p\) (or *material points*). In a given *configuration* of the body, each particle is located at some definite point \(\mathbf{y}\) in three-dimensional space. The set of all the points in space, corresponding to the locations of all the particles, is the *region* \(\mathcal R\) occupied by the body in that configuration. A particular body, composed of a particular set of particles, can adopt different configurations under the action of different stimuli (forces, heating etc.) and therefore occupy different regions of space under different conditions. Note the distinction between the *body*, a *configuration* of the body, and the *region* the body occupies in that configuration; we make these distinctions rigorous in what follows. Similarly note the distinction between a *particle* and the *position* in space it occupies in some configuration.

In order to appreciate the difference between a configuration and the region occupied in that configuration, consider the following example: suppose that a body, in a certain configuration, occupies a circular cylindrical region of space. If the object is “twisted” about its axis (as in torsion), it continues to occupy this same (circular cylindrical) region of space. Thus the region occupied by the body has not changed even though we would say that the body is in a different “configuration”.

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Tags

#limitless

Question

Which are the technology villains?

Answer

Through our educational platform Kwik Learning, we have students in 195 countries and have generated tens of millions of podcast downloads. Our community has expressed a growing concern about their overreliance on technology and they come to us to upgrade their brains to find relief from these “four horsemen” of our age: digital deluge, digital distraction, digital dementia, and digital deduction. It’s important to note that overload, distraction, forgetfulness, and default thinking have been around for ages. While technology doesn’t cause these conditions, it has great potential to amplify them. The benefits of the digital age are plentiful, but let’s take a look at how the advances in technology that help you, can possibly also hinder you.

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Through our educational platform Kwik Learning, we have students in 195 countries and have generated tens of millions of podcast downloads. Our community has expressed a growing concern about their overreliance on technology and they come to us to upgrade their brains to find relief from these “four horsemen” of our age: digital deluge, digital distraction, digital dementia, and digital deduction. It’s important to note that overload, distraction, forgetfulness, and default thinking have been around for ages. While technology doesn’t cause these conditions, it has great potential to amplify them. The benefits of the digital age are plentiful, but let’s take a look at how the advances in technology that help you, can possibly also hinder you.

Tags

#limitless

Question

Describe the rat experiment and the relation between downtime and and memory. Which digital villain can be related to this experiment?

Answer

In a University of California, San Francisco, study on the effect of downtime, researchers gave rats a new experience and measured their brain waves during and after the activity. Under most circumstances, a new experience will express new neural activity and new neurons in the brain—that is, if the rat is allowed to have downtime. With downtime, the neurons made their way from the gateway of memory to the rest of the brain, where long-term memory is stored. The rats were able to record memories of their experiences, which is the basis for learning.

Doesn’t that make you wonder what happens if you don’t have downtime? There is a growing body of evidence that suggests that if we never let our mind wander or be bored for a moment, we pay a price—poor memory, mental fog, and fatigue.

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田村虎蔵 - Wikipedia 田村虎蔵 出典: フリー百科事典『ウィキペディア（Wikipedia）』 ナビゲーションに移動 検索に移動 [imagelink] [emptylink] 田村虎蔵(1910年頃) 田村 虎蔵（たむら とらぞう、1873年 5月24日 - 1943年 11月7日 ）は、日本の音楽教育家、作曲家。鳥取県 岩美郡 馬場村（現・岩美町 馬場）生まれ、蒲生小学校 卒業、 鳥取高等小学校 、鳥取県尋常師範学校 。東京音楽学校 卒。 東京音楽学校、東京高等師範学校 助教授。言文一致唱歌 を提唱し、納所弁次郎 らと「幼年唱歌 」（1900年）「尋常小学唱歌 」などを編集。1922年西洋に渡り音楽教育事情を研究、帰国後は、東京市 音楽担当視学 となる。弟子に堀内敬三 がいる。

田村虎蔵先生をたたえる碑 東京都新宿区筑土八幡町2-1 筑土八幡神社入口。金太郎の一節と由緒の書かれた碑が建てられている。

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田村虎蔵先生記念刊行会編 目黒書店 1933年 『名作唱歌選集』故田村虎蔵先生記念会編 音楽之友社 1950年 伝記[編集 ] 丸山忠璋 『言文一致唱歌の創始者田村虎蔵の生涯』音楽之友社 1998年 史蹟[編集 ] 田村虎蔵先生之生地 鳥取県 岩美郡 岩美町 馬場 碑と像が建てられている。 田村虎蔵旧居跡 東京都 新宿区 筑土八幡町 4-24 区指定史蹟。由緒の書かれた案内板がある。 <span>田村虎蔵先生をたたえる碑 東京都新宿区筑土八幡町2-1 筑土八幡神社 入口。金太郎の一節と由緒の書かれた碑が建てられている。 関連項目[編集 ] 鉄道唱歌 電車唱歌 帝国音楽学校 島崎赤太郎 わらべ館 八波則吉 朝鮮藝術賞 みんなの童謡 唱歌 (教科) 花咲か爺 日本音楽文化協会 平忠度 広瀬武夫 外部リンク[編集 ] コトバンク わらべ館 典拠管理 WorldCat Identities ISNI : 0000 0000 8235 7732 LCCN : nr2003005715 Musi

Question

Art. 96. A expressão "legislação tributária" compreende

Answer

[default - edit me]

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Question

[default - edit me]

Answer

as leis, os tratados e as convenções internacionais, os decretos e as normas complementares que versem, no todo ou em parte, sobre tributos e relações jurídicas a eles pertinentes.

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Tags

#continuum-mechanics

Question

When introducing the concepts of the continuum theory of materials, Rohan Abeyaratne encourages the reader to pay special attention to the distinctions between the different concepts introduced. List some of them.

Answer

The reader is encouraged to pay special attention to the distinctions between the different concepts introduced here. These concepts include the notions of

- a body,
- a configuration of the body,
- a reference configuration of the body,
- the region occupied by the body in some configuration,
- a particle (or material point),
- the location of a particle in some configuration,
- a deformation,
- a motion,
- Eulerian and Lagrangian descriptions of a physical quantity,
- Eulerian and Lagrangian spatial derivatives, and
- Eulerian and Lagrangian time derivatives (including the material time derivative).

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The reader is encouraged to pay special attention to the distinctions between the different concepts introduced here. These concepts include the notions of a body, a configuration of the body, a reference configuration of the body, the region occupied by the body in some configuration, a particle (or material point), the location of a particle in some configuration, a deformation, a motion, Eulerian and Lagrangian descriptions of a physical quantity, Eulerian and Lagrangian spatial derivatives, and Eulerian and Lagrangian time derivatives (including the material time derivative).

Tags

#body #configuration #continuum-mechanics #particle #position #region

Question

Introduce the concepts of body, particles, configuration and region regarding continuum mechanics.

Answer

We shall use the term “body” to be a mathematical abstraction of an “object that occurs in nature”. A *body* \(\mathcal B\) is composed of a set of *particles* \(p\) (or *material points*). In a given *configuration* of the body, each particle is located at some definite point \(\mathbf{y}\) in three-dimensional space. The set of all the points in space, corresponding to the locations of all the particles, is the *region* \(\mathcal R\) occupied by the body in that configuration. A particular body, composed of a particular set of particles, can adopt different configurations under the action of different stimuli (forces, heating etc.) and therefore occupy different regions of space under different conditions. Note the distinction between the *body*, a *configuration* of the body, and the *region* the body occupies in that configuration; we make these distinctions rigorous in what follows. Similarly note the distinction between a *particle* and the *position* in space it occupies in some configuration.

In order to appreciate the difference between a configuration and the region occupied in that configuration, consider the following example: suppose that a body, in a certain configuration, occupies a circular cylindrical region of space. If the object is “twisted” about its axis (as in torsion), it continues to occupy this same (circular cylindrical) region of space. Thus the region occupied by the body has not changed even though we would say that the body is in a different “configuration”.

status | not learned | measured difficulty | 37% [default] | last interval [days] | |||
---|---|---|---|---|---|---|---|

repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

We shall use the term “body” to be a mathematical abstraction of an “object that occurs in nature”. A body \(\mathcal B\) is composed of a set of particles \(p\) (or material points). In a given configuration of the body, each particle is located at some definite point \(\mathbf{y}\) in three-dimensional space. The set of all the points in space, corresponding to the locations of all the particles, is the region \(\mathcal R\) occupied by the body in that configuration. A particular body, composed of a particular set of particles, can adopt different configurations under the action of different stimuli (forces, heating etc.) and therefore occupy different regions of space under different conditions. Note the distinction between the body, a configuration of the body, and the region the body occupies in that configuration; we make these distinctions rigorous in what follows. Similarly note the distinction between a particle and the position in space it occupies in some configuration. In order to appreciate the difference between a configuration and the region occupied in that configuration, consider the following example: suppose that a body, in a certain configuration, occupies a circular cylindrical region of space. If the object is “twisted” about its axis (as in torsion), it continues to occupy this same (circular cylindrical) region of space. Thus the region occupied by the body has not changed even though we would say that the body is in a different “configuration”.

[unknown IMAGE 4730646564108]

Tags

#continuum-mechanics #has-images

Question

Define body, region, particle and configuration the body "more formally".

Answer

More formally, in continuum mechanics a *body* \(\mathcal B\) is a collection of elements which can be put into one-to-one correspondence with some *region* \(\mathcal R\) of Euclidean point space. An element \(p \in \mathcal B\) is called a *particle* (or *material point*). Thus, given a body \(\mathcal B\), there is necessarily a mapping \(\chi\) that takes particles \(p \in \mathcal B\) into their geometric locations \(y \in \mathcal R\) in three-dimensional Euclidean space:

\(y = \chi(p) \quad \textrm{where} \quad p \in \mathcal B, \mathbf{y} \in \mathcal R.\)

The mapping \(\chi\) is called a *configuration* of the body \(\mathcal B\); \(\mathbf y\) is the *position* occupied by the particle \(p\) in the configuration \(\chi\); and \(\mathcal R\) is the region occupied by the body in the configuration \(\chi\). Often, we write \(\mathcal R = \chi \left( \mathcal B \right)\).

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More formally, in continuum mechanics a body \(\mathcal B\) is a collection of elements which can be put into one-to-one correspondence with some region \(\mathcal R\) of Euclidean point space. An element \(p \in \mathcal B\) is called a particle (or material point). Thus, given a body \(\mathcal B\), there is necessarily a mapping \(\chi\) that takes particles \(p \in \mathcal B\) into their geometric locations \(y \in \mathcal R\) in three-dimensional Euclidean space: \(y = \chi(p) \quad \textrm{where} \quad p \in \mathcal B, \mathbf{y} \in \mathcal R.\) The mapping \(\chi\) is called a configuration of the body \(\mathcal B\); \(\mathbf y\) is the position occupied by the particle \(p\) in the configuration \(\chi\); and \(\mathcal R\) is the region occupied by the body in the configuration \(\chi\). Often, we write \(\mathcal R = \chi \left( \mathcal B \right)\).

Tags

#continuum-mechanics #reference-configuration

Question

Describe two reasons considering a reference configuration.

Answer

In order to identify a particle of a body, we must label the particles. The abstract particle label \(p\), while perfectly acceptable in principle and intuitively clear, is not convenient for carrying out calculations. It is more convenient to pick some arbitrary configuration of the body, say \(\chi_\textrm{ref}\) , and use the (unique) position \(x = \chi_\textrm{ref} (p)\) of a particle in that configuration to label it instead. Such a configuration \(\chi_\textrm{ref} (p)\) is called a reference configuration of the body. It simply provides a convenient way in which to label the particles of a body. The particles are now labeled by \(\mathbf x\) instead of \(p\).

A second reason for considering a reference configuration is the following: we can study the geometric characteristics of a configuration \(\chi\) by studying the geometric properties of the points occupying the region \(\mathcal R = \chi (\mathcal B)\). This is adequate for modeling certain materials (such as many fluids) where the behavior of the material depends only on the characteristics of the configuration *currently* occupied by the body. In describing most solids however one often needs to know the *changes* in geometric characteristics between one configuration and another configuration (e.g. the change in length, the change in angle etc.). In order to describe the change in a geometric quantity one must necessarily consider (at least) *two* configurations of the body: the configuration that one wishes to analyze, and a *reference configuration* relative to which the changes are to be measured.

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[unknown IMAGE 5525580942604]

Tags

#continuum-mechanics #has-images

Question

Draw a scheme showing a body, a reference configuration and another configuration, as well as the mapping between a particle in the body and the configurations.

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#continuum-mechanics #deformation

This induces a mapping \(\mathbf y = \hat{\mathbf y} (\mathbf x)\) from \(\mathcal R_\textrm{ref} \rightarrow \mathcal R\):

\(\mathbf y = \hat{\mathbf y} (\mathbf x) \stackrel{\textrm{def}}{=} \chi \left( \chi_\textrm{ref}^{-1} \left( \mathbf x\right) \right), \qquad \mathbf x \in \mathcal R_\textrm{ref}, \mathbf y \in \mathcal R;\)

\(\hat{\mathbf y}\) is called a *deformation* of the body from the reference confinguration \( \chi_\textrm{ref}\).

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Tags

#continuum-mechanics #deformation

Question

This induces a mapping \(\mathbf y = \hat{\mathbf y} (\mathbf x)\) from \(\mathcal R_\textrm{ref} \rightarrow \mathcal R\):

\(\mathbf y = \hat{\mathbf y} (\mathbf x) \stackrel{\textrm{def}}{=} \chi \left( \chi_\textrm{ref}^{-1} \left( \mathbf x\right) \right), \qquad \mathbf x \in \mathcal R_\textrm{ref}, \mathbf y \in \mathcal R;\)

\(\hat{\mathbf y}\) is called a *[...]* of the body from the reference confinguration \( \chi_\textrm{ref}\).

Answer

deformation

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el{\textrm{def}}{=} \chi \left( \chi_\textrm{ref}^{-1} \left( \mathbf x\right) \right), \qquad \mathbf x \in \mathcal R_\textrm{ref}, \mathbf y \in \mathcal R;\) \(\hat{\mathbf y}\) is called a <span>deformation of the body from the reference confinguration \( \chi_\textrm{ref}\). <span>

[unknown IMAGE 5525580942604]

Tags

#continuum-mechanics #has-images

Question

Describe some simplifications which arise when considering a single fixed reference configuration.

Answer

When working with a single fixed reference configuration, as we will most often do, one can dispense with talking about the body \(\mathcal B\), a configuration \(\chi\) and the particle \(p\), and work directly with the region \(\mathcal R_\textrm{ref}\), the deformation \(\mathbf y \left( \mathbf x \right)\) and the position \( \mathbf x\).

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Tags

#continuum-mechanics #has-images

Question

Discuss about material description, Eulerian or spatial description and Lagrangian or referential description.

Answer

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#continuum-mechanics #has-images

Question

Discuss about (differentiate) Cauchy and first Piola-Kirchhoff stress tensors.

Answer

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Tags

#continuum-mechanics #has-images

Question

Explain the reason for using Grad, grad, Div, div, Curl, and curl as operator names in continuum mechanics.

Answer

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#continuum-mechanics #has-images

Question

Derive the relationship between \(\operatorname{Grad} \theta\) and \(\operatorname{grad} \theta\).

Answer

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#continuum-mechanics #has-images

Question

Describe the motion of a body.

Answer

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#continuum-mechanics #has-images

Question

Define the velocity and acceleration of a particle in the context of continuum mechanics.

Answer

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