OpenLLM-France/wikimedia huggingface dataset:
# Elasticity/Hertz contact
## The Hertz problem : rigid cylindrical punch

* Hertz indentation
* The contact length $a$ depends on the load $F\,$.
* There is no singularity at $x=\pm a$.
* The radius of the cylinder ($R\,$) is large.
We have,
    ${\frac {d^{2}u_{0}}{dx^{2}}}=-{\frac {1}{R}}$
Hence,
    $u_{0}=C_{0}-{\frac {x^{2}}{2R}}=C_{0}-{\frac {a^{2}\cos(2\phi )}{4R}}-{\frac {a^{2}}{4R}}$
and
    ${\frac {du_{0}}{d\phi }}=-{\frac {a^{2}\sin(2\phi )}{2R}}$
Therefore,
    $u_{1}=0~;~~u_{2}={\frac {a^{2}}{2R}}~;~~u_{n}=0~(n>2)$
and
    $p_{0}=-{\frac {F}{\pi a}}~;~~p_{1}=0~;~~p_{2}={\frac {2\mu a}{R(\kappa +1)}}~;~~p_{n}=0~(n>2)$
Plug back into the expression for $p(\theta )$ to get
    $p(\theta )=\left(-{\frac {F}{\pi a}}+{\frac {2\mu a}{R(\kappa +1)}}\cos(2\theta )\right)/\sin \theta$
This expression is singular at $\theta =0$ and $\theta =\pi$, unless we choose
    ${\frac {F}{\pi a}}={\frac {2\mu a}{R(\kappa +1)}}\Rightarrow a={\sqrt {\frac {F(\kappa +1)R}{2\pi \mu }}}$
Plugging $a$ into the equation for $p(\theta )$,
    $p(\theta )=-{\frac {2F\sin \theta }{\pi a}}\Rightarrow p(x)=-{\frac {2F{\sqrt {a^{2}-x^{2}}}}{\pi a^{2}}}$

### Two deformable cylinders

If instead of the half-plane we have an cylinder; and instead of the rigid cylinder we have a deformable cylinder, then a similar approach can be used to obtain the contact length $a\,$
    $a={\sqrt {{\frac {FR_{1}R_{2}}{2\pi (R_{1}+R_{2})}}\left({\frac {\kappa _{1}+1}{\mu _{1}}}+{\frac {\kappa _{2}+1}{\mu _{2}}}\right)}}$
and the force distribution $p$
    $p(x)=-{\frac {2F{\sqrt {a^{2}-x^{2}}}}{\pi a^{2}}}$
# Elasticity/Hu-Washizu principle
## Hu-Washizu Variational Principle

In this case, the admissible states are not required to meet any of the field equations or boundary conditions.
Let ${\mathcal {A}}$ denote the set of all admissible states and let ${\mathcal {W}}$ be a functional on ${\mathcal {A}}$ defined by
    ${\mathcal {W}}[s]=\int _{\mathcal {B}}U({\boldsymbol {\varepsilon }})-\int _{\mathcal {B}}{\boldsymbol {\sigma }}:{\boldsymbol {\varepsilon }}~dV-\int _{\mathcal {B}}({\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}+\mathbf {f} )\bullet \mathbf {u} ~dV+\int _{\partial {\mathcal {B}}^{u}}\mathbf {t} \bullet {\widehat {\mathbf {u} }}~dA+\int _{\partial {\mathcal {B}}^{t}}(\mathbf {t} -{\widehat {\mathbf {t} }})\bullet \mathbf {u} ~dA$
for every $s=[\mathbf {u} ,{\boldsymbol {\varepsilon }},{\boldsymbol {\sigma }}]\in {\mathcal {A}}$.
Then,
    $\delta {\mathcal {W}}[s]=0$
at an admissible state $s$ if and only if $s$ is a solution of the mixed problem.
# Elasticity/Kinematics example 4
## Example 4

Given:
Displacement field $\mathbf {u} =\kappa X_{2}{\widehat {\mathbf {e} }}_{1}+\kappa X_{1}{\widehat {\mathbf {e} }}_{2}$.
Find:
* The Lagrangian Green strain tensor ${\boldsymbol {E}}\,$.
* The infinitesimal strain tensor ${\boldsymbol {\varepsilon }}\,$.
* The infintesimal rotation tensor ${\boldsymbol {\omega }}\,$.
* The infinitesimal rotation vector ${\boldsymbol {\theta }}\,$.
* The exact longitudinal strain in the reference material direction $\mathbf {e} _{1}\,$.
* The approximate longitudinal strain in the direction $\mathbf {e} _{1}\,$ based on the infinitesimal strain tensor ${\boldsymbol {\varepsilon }}\,$.

### Solution

The Maple output of the computations are shown below:
    $u:=\left[\!k\,{X_{2}},\,k\,{X_{1}},\,0\!\right]$
    $x:=\left[\!k\,{X_{2}}+{X_{1}},\,k\,{X_{1}}+{X_{2}},\,{X_{3}}\!\right]$
    $F:={\begin{bmatrix}1&k&0\\k&1&0\\0&0&1\end{bmatrix}}$
    $C:={\begin{bmatrix}1+k^{2}&2\,k&0\\2\,k&1+k^{2}&0\\0&0&1\end{bmatrix}}$
    $E:={\begin{bmatrix}{\frac {k^{2}}{2}}&k&0\\[2ex]k&{\frac {k^{2}}{2}}&0\\[2ex]0&0&0\end{bmatrix}}$
    $gradu:={\begin{bmatrix}0&k&0\\k&0&0\\0&0&0\end{bmatrix}}$
    $\varepsilon :={\begin{bmatrix}0&k&0\\k&0&0\\0&0&0\end{bmatrix}}$
    $\omega :={\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}}$
    ${\mathit {stretch1}}:={\sqrt {1+k^{2}}}$
    ${\mathit {longStrain1}}:={\sqrt {1+k^{2}}}-1$
    ${\mathit {approxLongStrain1}}:=0$
The geometrical difference between the large strain and small strain cases can be observed by looking at the figures from the previous examples.
# Elasticity/Homogeneous and inhomogeneous displacements
## Homogeneous and inhomogeneous displacements
### Homogeneous Displacement Field

A displacement field $\textstyle \mathbf {u} (\mathbf {X} )$ is called homogeneous if
    $\mathbf {u} (\mathbf {X} )=\mathbf {u} _{0}+{\boldsymbol {A}}\bullet [\mathbf {X} -\mathbf {X} _{0}]$
where $\textstyle \mathbf {X} _{0},\mathbf {u} _{0},{\boldsymbol {A}}$ are independent of $\textstyle \mathbf {X}$.

#### Pure Strain

If $\textstyle \mathbf {u} _{0}=0$ and $\textstyle {\boldsymbol {A}}={\boldsymbol {\varepsilon }}$, then $\textstyle \mathbf {u}$ is called a pure strain from $\textstyle \mathbf {X} _{0}$, i.e.,
    $\mathbf {u} (\mathbf {X} )={\boldsymbol {\varepsilon }}\bullet [\mathbf {X} -\mathbf {X} _{0}]$

### Inhomogeneous Displacement Field

Any displacement field that does not satisfy the condition of homogeneity is inhomogenous. Most deformations in engineering materials lead to inhomogeneous displacements.
# Elasticity/Kinematics example 1
## Example 1

Take a unit cube of material. Rotate it 90 degrees in the clockwise direction around the z-axis. Calculate the strains. Discuss your results - their accuracy and the reasons for your conclusions.

### Solution

The strains are related to displacements by
    $\epsilon _{xx}={\frac {\partial u}{\partial x}};~\epsilon _{yy}={\frac {\partial v}{\partial y}};~\epsilon _{zz}={\frac {\partial w}{\partial z}};~\gamma _{xy}={\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial x}};~\gamma _{yz}={\frac {\partial v}{\partial z}}+{\frac {\partial w}{\partial y}};~\gamma _{zx}={\frac {\partial w}{\partial x}}+{\frac {\partial u}{\partial z}}$
Let us consider rotation about the center of the cube. Since the problem concerns a pure rotation, a cylindrical co-ordinate system is appropriate. This problem also provides us a easy way of trying out Maple. Here are the steps that you can follow to find the strains at a point in the cube.
r := sqrt(x^2+y^2);
    $r:={\sqrt {x^{2}+y^{2}}}$
theta := arctan(y/x);
    $\theta :=arctan({\frac {y}{x}})$
x1 := r*cos(theta);
    $x1:={\frac {\sqrt {x^{2}+y^{2}}}{\sqrt {1+{\frac {y^{2}}{x^{2}}}}}}$
y1 := r*sin(theta);
    $y1:={\frac {{\sqrt {x^{2}+y^{2}}}\,y}{x\,{\sqrt {1+{\frac {y^{2}}{x^{2}}}}}}}$
x2 := r*cos(theta+Pi/2);
    $x2:=-{\frac {{\sqrt {x^{2}+y^{2}}}\,y}{x\,{\sqrt {1+{\frac {y^{2}}{x^{2}}}}}}}$
y2 := r*sin(theta+Pi/2);
    $y2:={\frac {\sqrt {x^{2}+y^{2}}}{\sqrt {1+{\frac {y^{2}}{x^{2}}}}}}$
u := x2 - x1;
    $u:=-{\frac {{\sqrt {x^{2}+y^{2}}}\,y}{x\,{\sqrt {1+{\frac {y^{2}}{x^{2}}}}}}}-{\frac {\sqrt {x^{2}+y^{2}}}{\sqrt {1+{\frac {y^{2}}{x^{2}}}}}}$
v := y2 - y1;
    $v:={\frac {\sqrt {x^{2}+y^{2}}}{\sqrt {1+{\frac {y^{2}}{x^{2}}}}}}-{\frac {{\sqrt {x^{2}+y^{2}}}\,y}{x\,{\sqrt {1+{\frac {y^{2}}{x^{2}}}}}}}$
epsx := simplify(diff(u,x));
    $epsx:=-{\frac {\sqrt {x^{2}+y^{2}}}{x\,{\sqrt {\frac {x^{2}+y^{2}}{x^{2}}}}}}$
epsy := simplify(diff(v,y));
    $epsy:=-{\frac {\sqrt {x^{2}+y^{2}}}{x\,{\sqrt {\frac {x^{2}+y^{2}}{x^{2}}}}}}$
gamxy := simplify(diff(u,y) + diff(v,x));
    $gamxy:=0$
From the above Maple calculation, and noting that there is no motion in the $z$ direction, the strains in the cube are
    $\epsilon _{xx}=-1;~\epsilon _{yy}=-1;\epsilon _{zz}=0;\gamma _{xy}=0;\gamma _{yz}=0;\gamma _{zx}=0$
A pure rigid body rotation should not result in any non-zero strains.
Therefore, the measure of strain we have used is not appropriate for large rigid body motions.
# English-Chinese/After his tenth beer

After his tenth beer, he blacked out and we couldn't wake him up.
在他第十次喝啤酒之后，他昏了过去，我们无法将他叫醒。
# English-Chinese/After the new president was sworn in

After the new president was sworn in, there was an inaugural ball.
在新总统宣誓就职后，有一个就职舞会。
# English-Chinese/Aging is an individual's change over time

Aging is an individual's change over time.
老化是个体随时间的变化。
# English-Chinese/Air

Air is the Earth's atmosphere. Air around us is a mixture of many gases and dust particles. It is the clear gas in which living things live and breathe. It has an indefinite shape and volume. It has no color or smell. It has mass and weight, because it is matter. The weight of air creates atmospheric pressure. There is no air in outer space.
空气 是 地球的 大气。我们 周围的 空气 是 许多 气体 和 尘埃颗粒的 混合物。它是 生物 存活 和 呼吸的 清澈的 气体。它 具有 不确定的 形状 和 体积。它 没有 颜色 或 气味。它 有 质量 和 重量，因为 它是 物质。空气的 重量 会 产生 大气压力。外太空 没有 空气。
# English-Chinese/After a serious head injury

After a serious head injury, he stayed in the hospital for two days.
头部严重受伤后，他在医院住了两天。
# English-Chinese/Age is the most common cause of cataracts

Age is the most common cause of cataracts. As time goes by, the lens protein will degenerate, and diseases such as diabetes and high blood pressure will accelerate this process.
年齡是白內障最常見的原因。 隨著時間的推移，晶狀體蛋白會退化，糖尿病和高血壓等疾病會加速這一過程。
# English-Chinese/Agreement

An agreement is a promise or act on part of offeree indicating a willingness to be bound by terms and conditions contained in an offer.
协议是对提议方的的承诺或行动，表明愿意接受提议中包含的条款和条件的约束。
# English-Chinese/Air is a mixture of gases

Air is a mixture of gases. It has no color or odor (smell). We breathe air to live.
空气是气体的混合物。它没有颜色或气味。 我们呼吸空气来生存。
# English-Chinese/African Union

The African Union (AU) is an international organization consisting of fifty-five African member countries. Founded in Durban in July 2002, the AU was formed as a replacement to the Organization of African Unity (OAU) and the African Economic Community (AEC).
非盟（AU）是 一 个 由 五十五 个 非洲 成员国 组成 的 国际 组织。非盟 于 2002年7月 成立于 Durban，成为 非洲统一组织（非统组织）和 非洲经济共同体（AEC）的 替代 组织。
# English-Chinese/Aging

Ageing (British English) or aging (American English) is the process of becoming older. It represents the accumulation of changes in a person over time. In humans, ageing is several physical and psychological changes.
衰老 是 变老 的 过程。它 代表了 一 个 人 随 时间 变化 的 积累。在 人类 中，衰老 是 身体 和 心理 的 一些 变化。
# English-Chinese/Wheat is classified according to grain properties for market needs

Wheat is classified according to grain properties for market needs. Wheat buyers use these to decide which wheat to buy, as each class has special uses, and producers use them to decide which classes of wheat will be most profitable to cultivate.
根据市场需求，小麦根据谷物属性进行分类。 小麦买家使用这些来决定购买哪种小麦，因为每个类别都有特殊用途，并且生产者使用它们来决定哪种小麦最有利于种植。
# English-Chinese/Wheat is grown on more land area than any other food crop

Wheat is grown on more land area than any other food crop.
小麦种植面积比其他任何粮食作物都要多。
# English-Chinese/Wheat is 13% water, 71% carbohydrates, and 1.5% fat. Its 13% protein content is mostly gluten (75-80% of the protein in wheat)

Wheat is 13% water, 71% carbohydrates, and 1.5% fat. Its 13% protein content is mostly gluten (75-80% of the protein in wheat).
小麦含有13％的水，71％的碳水化合物和1.5％的脂肪。 其13％的蛋白质含量主要是麸质（小麦中蛋白质的75-80％）。
# English-Chinese/Wheat is a significant food for human nutrition

Consumed worldwide by billions of people, wheat is a significant food for human nutrition, particularly in the least developed countries where wheat products are primary foods.
小麦在全球范围内被数十亿人消费，是人类营养的重要食品，特别是在小麦产品是主要食品的最不发达国家。
# English-Chinese/Wheat bran has also been used for tanning leather since at least the 16th century

Wheat bran has also been used for tanning leather since at least the 16th century.
至少从16世纪开始，麦麸也被用于制革。
# English-Chinese/Wheat is an important source of carbohydrates

Wheat is an important source of carbohydrates. Globally, it is the leading source of vegetal protein in human food, having a protein content of about 13%, which is relatively high compared to other major cereals but relatively low in protein quality for supplying essential amino acids.
小麦是碳水化合物的重要来源。 在全球范围内，它是人类食物中植物蛋白的主要来源，其蛋白质含量约为13％，与其他主要谷物相比相对较高，但蛋白质质量相对较低，不足以供应必需氨基酸。
# English-Chinese/Wheat bran is useful as feed for poultry and other livestock, as part of a balanced ration with other inputs

Wheat bran is useful as feed for poultry and other livestock, as part of a balanced ration with other inputs.
麦麸可用作家禽和其他牲畜的饲料，作为与其他投入的平衡日粮的一部分。
# English-Chinese/Wheat gluten proteins are poor in lysine

Because the proteins present in the wheat endosperm (gluten proteins) are particularly poor in lysine, white flours are more deficient in lysine compared with whole grains.
由于小麦胚乳（麸质蛋白）中存在的蛋白质在赖氨酸中特别差，因此与全谷物相比，白色面粉在赖氨酸中更缺乏。
# English-Chinese/What will your life amount to?

What will your life amount to? Will you become successful?
你的生命将达到什么目的？你会成功吗？
# English-Chinese/Wheat

Wheat is a grass widely cultivated for its seed, a cereal grain which is a worldwide staple food.
小麦是广泛种植的草种子，是世界常见的谷类主食。
# English basics

English Basics Hello, this is a project to make flash cards for english learners. The plan is to have a series of flash cards with some text, a sound file and a selection of images, the english learner reads the text listens to the sound file and clicks on the appropriate image. If correct they move to the next card if incorrect the can try again. The flash cards can be grouped in increasing order of difficulty and by topic.

## Food
### Milk

* Question 1:

### Bread

* Question 2:

### Cheese

* Question 3:

### Rice

* Question 4:

### Potatoes

* Question 5:

### Corn

* Question 6:
* Corn

### Fish

* Question 7:
* Fish

### Chicken

* Question 8:
* Chicken

### Lamb

* Question 9:

### Apple

* Question 10:

### Pear

* Question 11:

### Orange

* Question 12:
* Orange

### Mango

* Question 13:

### Banana

* Question 14:

### Sugar

* Question 15:
# English basics/101 q1

A man
# English basics/101 q1 clicked24

Try again
A man
# English basics/101 q1 clicked3

Try again!
A man
# English basics/101 q1 clicked32

Try again!
A man
# English basics/101 q1 clicked2

Try again!
A man
# Introduction to Algorithms/Flow Control

This is a lesson in in the course, Introduction to Algorithms, which is a part of The School of Computer Science
* Objective
* Flow Control
* Our current toast making algorithm works well if you are making toast for yourself, but what if you have a few friends around for breakfast and you need to make them toast as well? The algorithm clearly needs to be improved so that we can make toast for everyone. Also, simply waiting for the toasted is very boring, we could be doing something else while we wait. The third problem with making toast for friends is that they are fussy, what if some of them want butter whilest others prefer margarine, or even just plain toast
* Algorithm for making buttered toast (4.0)
* LET A = a loaf of bread
* LET T = a toaster
* LET P = a plate
* LET B = some butter
* LET M = some margarine
* FOR every friend (F) eating toast{
* LET S = cut slice from A
* move S to T
* turn on T
* WHILE T is not finished{
* talk to F
* move S to P
* IF F likes B{
* LET X = B
* }ELSEIF F likes M{
* LET X = M
* }ELSE{
* LET X = NOTHING
* spread S with X
* move P to F
* As can be seen, to resolve the three problems above, I had to implement the three main types of flow control, a For Loop, a While Loop and an If Statement. It should be obvious from the example what each of these does. I also involved a new variable X, which unlike the variables I have used before changes its value as the algorithm progresses.
* Assignments
Completion status: this resource is a stub, which means that pretty much nothing has been done yet.
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# Introduction to Asian American Studies: Literature, Culture, and Historical Experience
## Course Description

An interdisciplinary subject that draws on literature, history, anthropology, film, and cultural studies to examine the experiences of Asian Americans in U.S. society. Covers the first wave of Asian immigration in the nineteenth century, the rise of anti-Asian movements, the experiences of Asian Americans during World War II, the emergence of the Asian American movement in the 1960s, and the new wave of "post-1965" Asian immigration. Examines the role these historical experiences played in the formation of Asian American ethnicity, and explores how these experiences informed Asian American literature and culture. Addresses key societal issues such as racial stereotyping, media racism, affirmative action issues, the glass ceiling, the "model minority" syndrome, and anti-Asian harassment or violence.
Introduction to Asian American Studies: Literature, Culture, and Historical Experience
# Introduction to Algorithms/Subalgorithms

This is a lesson in in the course, Introduction to Algorithms, which is a part of The School of Computer Science
* Objective
* The objective is to teach you all the necessary stuff that is usually taught in a university level course.
* Subalgorithm
* Algorithms are often broken down into smaller chunks, called Subalgorithm. This is both so that they are easier to read, and also because then parts of the algorithm can be reused. What follows is the above algorithm in a more formal manner. The text in italics below are the names of the Subalgorithm that are being called.
* Algorithm for making buttered toast (2.0)
* get a loaf of bread.
* cut slice from the loaf of bread.
* move the slice of bread to the toaster.
* turn on the toaster.
* wait for the toaster to finish.
* move the slice of toast to a plate.
* spread the slice of toast with butter.
* turn off the toaster.
* Assignments
* Subalgorithm for applying mascara
* grab the mascara from the cabinet
* unscrew the lid
* dab any excess product on rim
* use the mascara wand and apply to lashes in upward motion
* repeat this step until you get the desired results
* apply to bottom lashes
* screw the lid back on
* place mascara back in cabinet
Completion status: this resource is a stub, which means that pretty much nothing has been done yet.
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# Introduction to Algorithms/Efficiency

This is a lesson in in the course, Introduction to Algorithms, which is a part of The School of Computer Science
* Objective
* Algorithm efficiency refers to making the steps of the algorithm concise and eliminating redundancy. This could include eliminating steps altogether by combining or deleting particular ones, or to refining a step to make it run with less code or run time, etc.
* Assignments
Completion status: this resource is a stub, which means that pretty much nothing has been done yet.
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# Introduction to Biomembranes
## Biomembrane Transport

A biomembrane is a structure present in all cells, being either the external plasma membrane or internal membranes (e.g. endoplasmic reticulum, Golgi apparatus). Membranes exist in order to allow the cell to control and change its internal environment, and to protect important organelles from environmental damage or attack from pathogens. All Membranes consist of at least three substances:
* Lipids
* Proteins
* Carbohydrates (Glyco-Proteins, -lipids, sterols for structure)
The ratio between lipid and proteins depends on the function of the cell, some examples of different cell membrane compositions are shown in the table below.
| Membrane | % By weight | % By weight | % By weight |
| | Protein | Lipid | Carbohydrate |
| Myelin | 18 | 79 | 3 |
| Human erythtocyte | 49 | 43 | 8 |
| Mouse liver | 44 | 52 | 4 |
| Ameba | 54 | 42 | 4 |
| Chloroplast spinach lamellae | 70 | 30 | 0 |
| Halobacterium purple membrane | 75 | 25 | 0 |
| Mitochondrial inner membrane | 76 | 24 | 0 |

### Membranes as Lipid Bilayers

Membranes are made of a lipid by layer of phospholipids which are made up of a hydrophilic "polar" heads(water) and hydrophobic fatty acid tails (oil). Phospholipids will spontaneously form lipid bilayers in aqueous solution; tails aggregate and exclude water from a core, the structure is stabilised by tail-tail/ head-head interactions. The polar head region's charge is variable and fatty acid (R) chains vary in length from C$_{12}$ to C$_{22}$ as well as in their degree of unsaturation (number of double bonds). Each C=C bond produces a rigid kink in the the phospholipid tail. In 1972 Singer and Nicholson proposed the "Fluid Mosaic" model suggesting that the membrane itself is a two dimensional sea of mobile lipid in which proteins diffuse or 'float'.

### Mechanisms by which substances cross the membrane

* Diffusion
** For hydrophobic molecules, occurs through the phospholipid bilayer down a concentration gradient (i.e. from a region of high concentration of the substance to a region of low concentration)
** For hydrophillic molecule diffusion, see Transport
* Osmosis (net water flow)
** As water is hydrophillic, it cannot pass through the hydrophobic cell membrane, and so requires special hydrophillic proteins called aquaporins in order to assist their transport.
* Transport (Passive or Active)
** Transport via proteins. This can be passive (i.e. down a concentration gradient, no energy input) or active (against a concentration gradient, requiring energy input)
# Introduction to Algorithms/Variables

This is a lesson in in the course, Introduction to Algorithms, which is a part of The School of Computer Science
* Objective
* Variables
* In the above example it is fairly simple to follow the piece of toast through the algorithm, but if you imagine that the algorithm could become more complex, and thus the progress of the toast much harder to follow. In order that, at any point in the algorithm, it is clear what is being talked about, names are often given to things. For example
* Algorithm for making buttered toast (3.0)
* LET A = a loaf of bread
* LET T = a toaster
* LET P = a plate
* LET B = some butter
* LET S = cut slice from A
* get A
* move the S to T
* turn on T
* wait for T
* move the S to P
* spread S with B
* Assignments
Completion status: this resource is a stub, which means that pretty much nothing has been done yet.
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# Elasticity/Principal stresses
## Principal Stresses in Two and Three Dimensions

The principal stresses are the components of the stress tensor when the basis is changed in such a way that the shear stress components become zero. To find the principal stresses in two dimensions, we have to find the angle $\textstyle \theta$ at which $\textstyle \sigma _{12}^{'}=0$. This angle is given by
$\theta ={\cfrac {1}{2}}\tan ^{-1}\left({\frac {2\sigma _{12}}{\sigma _{11}-\sigma _{22}}}\right)$
Plugging $\textstyle \theta$ into the transformation equations for stress we get,
    ${\begin{aligned}\sigma _{1}&={\frac {\sigma _{11}+\sigma _{22}}{2}}+{\sqrt {\left({\frac {\sigma _{11}-\sigma _{22}}{2}}\right)^{2}+\sigma _{12}^{2}}}\end{aligned}}$
Where are the shear tractions usually zero in a body?
The principal stresses in three dimensions are a bit more tedious to calculate. They are given by,
    ${\begin{aligned}\sigma _{1}&={\frac {I_{1}}{3}}+{\frac {2}{3}}\left({\sqrt {I_{1}^{2}-3I_{2}}}\right)\cos \phi \\\sigma _{2}&={\frac {I_{1}}{3}}+{\frac {2}{3}}\left({\sqrt {I_{1}^{2}-3I_{2}}}\right)\cos \left(\phi -{\frac {2\pi }{3}}\right)\\\sigma _{3}&={\frac {I_{1}}{3}}+{\frac {2}{3}}\left({\sqrt {I_{1}^{2}-3I_{2}}}\right)\cos \left(\phi -{\frac {4\pi }{3}}\right)\end{aligned}}$
where,
    ${\begin{aligned}\phi &={\cfrac {1}{3}}\cos ^{-1}\left({\frac {2I_{1}^{3}-9I_{1}I_{2}+27I_{3}}{2(I_{1}^{2}-3I_{2})^{3/2}}}\right)\\I_{1}&=\sigma _{11}+\sigma _{22}+\sigma _{33}\\I_{2}&=\sigma _{11}\sigma _{22}+\sigma _{22}\sigma _{33}+\sigma _{33}\sigma _{11}-\sigma _{12}^{2}-\sigma _{23}^{2}-\sigma _{31}^{2}\\I_{3}&=\sigma _{11}\sigma _{22}\sigma _{33}-\sigma _{11}\sigma _{23}^{2}-\sigma _{22}\sigma _{31}^{2}-\sigma _{33}\sigma _{12}^{2}+2\sigma _{12}\sigma _{23}\sigma _{31}\end{aligned}}$
The quantities $\textstyle I_{1},I_{2},I_{3}$ are the stress invariants.
Note: Be careful while implementing above relations in a solver, as the value of:
    ${\frac {2I_{1}^{3}-9I_{1}I_{2}+27I_{3}}{2(I_{1}^{2}-3I_{2})^{3/2}}}$
can be out of range of $\cos ^{-1}$, which is (-1, 1).

## Related Content

Introduction to Elasticity

## External Links

* Online Calculator for Principal Stress and Principal Plane