stress tensor P P in classical physics, the force dF dF (vector) acting on an infinitesimal area ds ds (vector) equals dF=P⋅ds dF=P⋅ds

How to introduce stress tensor on manifolds?

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I want to understand the type of stress tensor P $\mathbf{P}$ in classical physics. Usually in physics it is said that the force dF $\text{d}\mathit{F}$ (vector) acting on an infinitesimal area ds $\text{d}\mathit{s}$ (vector) equals dF=P⋅ds $\text{d}\mathit{F}=\mathbf{P}\cdot \text{d}\mathit{s}$ where ⋅ $\cdot$ is a "scalar product". How can it be rigourised? I guess directed area can be ⋆s $\star s$ where s $s$ is a 2-form, but can I avoid using ⋆ $\star$ by employing the volume form for example? The force should be 1-form. How is the power of surface forces is written? Usually it is given by dAdt =∫ S v⋅dF $$\frac{dA}{dt}={\int }_S\mathit{v}\cdot \text{d}\mathit{F}$$ v $\mathit{v}$ being the speed of the surface of the deformed body. What would be the corresponding local form, that is the power density of surface forces? UPDATE 1 If it helps, I found a whole appendix "The Classical Cauchy Stress Tensor and Equations of Motion" in the book "The Geometry of Physics: An Introduction" by Theodore Frankel. Particularly it says The Cauchy stress should be a vector-valued pseudo-(n−1) $(n-1)$ -form. However currently I don't know what does it mean. Further development in the book is rather obscure and I'm afraid of that "pseudo". If a thing called "pseudo-something" I would prefer it stated as "actual another thing". UPDATE 2 Stress tensor can also be viewed as a (molecular) flux of momentum. Then the equation for balance of momentum would be the Newton's second law. Probably this approach would be more fruitful, analogues can be made with the flux of density. differential-geometry classical-mechanics mathematical-physics share|cite|improve this question edited Aug 9 '12 at 18:03 asked Aug 5 '12 at 15:36 Yrogirg 1,457828

1   My guess: P $P$ is a 1 $1$ -form valued 2 $2$ -form. Surface force f $f$ is also a 1 $1$ -form valued 2 $2$ -form, and power density is the 2 $2$ -form that results from contracting f $f$ with the surface velocity. – timur Aug 8 '12 at 0:18 @timur what is 1-form valued 2-form? Is it ⋆P $\star P$ ? – Yrogirg Aug 8 '12 at 4:00 No, but as I said it is just a guess. I am curious why do you think it be star P? – timur Aug 8 '12 at 5:34 @timur I just don't know what is "1-form valued 2-form", I was guessing. Btw, see my update to the answer. – Yrogirg Aug 8 '12 at 7:26 2   A "foo valued" 2-form is, roughly speaking, something that, when combined with a bivector (or an ordered pair of tangent vectors), yields a "foo". The kind of n-form you're used to is a "scalar-valued" n-form. – Hurkyl Aug 8 '12 at 7:29  |  show 9 more comments

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I found a paper supporting my comment that P $P$ is a 1-form valued 2-form, that surface force f is also a 1-form valued 2-form, and power density is the 2-form that results from contracting f with the surface velocity. The paper is R. Segev and L. Falach. Velocities, stresses and vector bundle valued chains. J. Elast. 105:187-206, 2011. share|cite|improve this answer answered Aug 16 '12 at 3:31 timur 9,9051231

Given an inviscid fluid with a 0-form p $p$ for preassure, how would you make a stress tensor for it? – Yrogirg Aug 21 '12 at 12:47 @Yrogirg: Pressure is a 3-form (If you have a 0-form then just take its Hodge dual). The stress tensor s $s$ corresponding to the 3-form p $p$ is the following: Given a vector field X $X$ , the contraction s(X) $s(X)$ , which is supposed to be a 2-form, is given by i X p $i_{X}p$ . – timur Aug 23 '12 at 1:39 add a comment |

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It is important to distinguish between covariant and contravariant indices of a tensor. Differential forms are totally antisymmetric covariant tensor fields. So a 2-form has 2 covariant indices, and when you swap them, the sign changes. Contravariant indices are written as upper indices and covariant indices as lower indices. You can raise and lower indices by use of a metric. Now, the stress tensor has one covariant index and one contravariant index. When you lower the contravariant index, you get a symmetric tensor field, not a differential form. In local coordinates, you simply have a matrix associated to every point, say P(x ⃗ ) $\mathbf{P}(\overrightarrow{x})$ . The easiest way to understand what the stress tensor does is to imagine the effect of infinitesimal deformations inside the body, described by a vector field, say v ⃗ (x ⃗ ) $\overrightarrow{v}(\overrightarrow{x})$ . The actual displacement at x ⃗  $\overrightarrow{x}$ could be written as v ⃗ (x ⃗ )dr $\overrightarrow{v}(\overrightarrow{x})dr$ . Now, the Energy density released by this displacement is dE=P j i v i ;j  dr $dE=P_i^j{v}_{;j}^{i}dr$ , or, if you take v ⃗ (x ⃗ ) $\overrightarrow{v}(\overrightarrow{x})$ as a velocity, P j i v i ;j  $P_i^j{v}_{;j}^i$ will simply be the power density. The semicolon indicates the covariant derivative. You can compute it by taking local coordiantes such that at x ⃗  $\overrightarrow{x}$ the metric is the Euclidean metric and all derivates of the metric are zero. In such local coordinates, P j i v i ;j =tr(PJ v ⃗  ) $P_i^j{v}_{;j}^i=\mathrm{t}\mathrm{r}({\mathbf{P}\mathbf{J}}_{\overrightarrow{v}})$ , with J v ⃗   ${\mathbf{J}}_{\overrightarrow{v}}$ the Jacobi matrix of v ⃗  $\overrightarrow{v}$ . Edit: What I am saying is that you cannot use differential forms alone. They are special tensors, but you need more general tensors. The stress tensor is a vector-valued 1-form (which, in 3 dimensions, is equivalent to a vector-valued 2-form, by Hodge duality, which gives a little more weight to the surface interpretation you formulated above). A vector is a contravariant 1-tensor, a 1-form is a covariant 1-tensor. Using the metric, you can transform one into the other, so you could even write the stress tensor as a 1-form-valued 1-form (or (n−1) $(n-1)$ -form, in n $n$ dimensions), but that seems not very physical to me. share|cite|improve this answer edited Aug 9 '12 at 22:57 answered Aug 9 '12 at 13:23 Hendrik Lönngren 912

2   You can go from one to another using the metric, but I think the ideal formulation should be (or at least should try to be) independent of metric. An example I have in mind is Maxwell's electrodynamics, where the electric and magnetic fields are naturally 1- and 2-forms, and the metric enters only through the laws. – timur Aug 9 '12 at 23:55 Hendrik, I prefer coordinate-free presentation since coordinates seem "not very physical to me". – Yrogirg Aug 10 '12 at 4:21 @timur: What we need at any rate is a connection. We can make the stress tensor independent of the metric by multiplying it with the volume form: P j i ω klm  $P_i^j{\omega }_{klm}$ – we get a (vector-valued 1-form)-valued 3-form. We cannot make it independent of the connection though. – Hendrik Lönngren Aug 10 '12 at 11:32 @Yrogirg: The covariant derivative is coordinate-free, I just gave a method to compute it. You could write tr(P∇v) $\mathrm{t}\mathrm{r}(\mathbf{P}\mathrm{\nabla }v)$ instead for the power density, which does not depend on coordinates. – Hendrik Lönngren Aug 10 '12 at 11:38 and then what is P $\mathbf{P}$ in a coordinate-free view? What is the space it belongs to? – Yrogirg Aug 10 '12 at 11:54 负比热出现的原因在于体系的能量将不再是一个广延量 负比热出现的原因在于体系的能 量将不再是一个广延量，子系统之间的相互 作用必须予以考虑，这是天体、原子核以及 纳米团簇等负比热体系的共同特点 這是 http://www.ccast.ac.cn/workshop/cond-2007/wenzhang/raoj.ppt 的 HTML 檔。 G o o g l e 在網路漫遊時會自動將檔案轉換成 HTML 網頁。     Negative specific heat, phase transition and particles spilling from a potential well J.Rao, Q.H.Liu and T.G.Liu 引     言   纳米团簇中负比热的成功观测[1]使得有 限体系中的统计物理日益成为一个研究热点。 通常认为，负比热出现的原因在于体系的能 量将不再是一个广延量，子系统之间的相互 作用必须予以考虑，这是天体、原子核以及 纳米团簇等负比热体系的共同特点。  W.Thirring在一篇论文[2]中指出在某些体系中尽管 各态历经情况下的微正则比热为正，但其中某些非各 态历经成分却可能出现负比热。该文中W.Thirring专 门提出了一个简单的模型，本文称为“noninteracting- particle-in-the-well”模型。这种模型中，  个粒子被限 制在一个大盒子里，盒子中央有一个势阱。体系能量 升高时，粒子有可能从阱内溢出，使得动能降低，实 现负比热。 W.Thirring的模型为实现负比热指出了一 条新的途径。    Thirring在其论文里假定，体系中每一个粒子的能量分别守恒，粒子间不交换能量，因此是一个非各态历经的体系。本文在Thirring工作的基础上更进一步，考虑一个各态历经的体系，也就是仅系统的总能量守恒，但粒子间可以交换能量。通过 计算，我们发现该体系严格可解，在有限粒子的情况下该模型有可能出现微正则的负比热。最后，我们进一步讨论了这种负比热与一级相变的联系。 MODEL AND ITS MICROCANONICAL ENSEMBLE TREATMENT  在“noninteracting-particle-in-the-well”模型中,我 们假定  个无相互作用的粒子在势场   中运动。                                 （1） 其中：                                                               （2） 给定总能量 ，可求得微正则条件下相空间的 总体积：                                                            （3） 其中：         ,            为第 个粒子的能量。       系统的微正则平均动能： （4） 阱内粒子数的平均值为：   （5） 以 为能量单位，每个粒子的平均动能为： 体系的平均能量：               曲线即为体系的caloric曲线，而比热 则可作如下定义： 通过直接计算可求得平均动能的表达式：           （6）     代表势阱体积占总体积的比例，函数 势阱内的平均粒子数为：                                                                                                                                                                       （7） 平均动能的表达式,(6)可作如下理解： 对某一特定的位形，比如 个粒子位于阱内，那 么它对位形空间积分的贡献为： 微正则分布给出动量空间中的热力学状态数正比于： 因此，对于该位形相空间中代表点数目正比于： SINGLE PARTICLE NSH       只有一个粒子时，其能量达到势阱高度即可从阱内溢出；而当粒子数很多时，由于能均分的作用，粒子平均能量达到势阱高度时，并不能使全部粒子从阱内溢出。势阱的体积比越小和(或)粒子数越多，离子越容易从阱内溢出。   当体积比    时,(6)式和(7)式将只剩下    一项，因此：   这正是理想气体的情形。       当体积比       时，仅有与       成正比的项对(6)式和(7)式有贡献。这时我们得到规则锯齿形结构的caloric曲线以及呈梯状变化的平均粒子数(阱内)曲线。具体表达如下： (8)     Fig.1中最下端的锯齿形曲线为          时的caloric曲线，当       时为一个周期函数，其周期为           。   从图上我们可以看出，每一个粒子精确的吸收能量  ，一个接一个地溢出势阱，当粒子全部溢出时，体系释放出全部动能且处于能量       的状态。动能下降同时总能量上升意味着单粒子的负比热。   在热力学极限     ，caloric曲线得变化周期趋于零，          。(8)式化为：     当           时，(6)式和(7)式不能进一步解析地简化。Fig.1-Fig.3显示了数值计算结果。Fig.1显示了粒子数     时，不同体积比                   下的caloric曲线, Fig.2和Fig.3则显示了相同体积比下体系粒子数不同时的caloric曲线。   由图中的caloric曲线可以看出，粒子数越大，锯齿波的振幅越小。因此，对一定的体积比，大粒子数体系的caloric曲线将呈平滑趋势。     Fig.1中的六条曲线显示了该模型中caloric曲线在各种条件下的基本特征，容易看出这些曲线限制在    和   两个极端之间。随着体积比的增大，总能量越来越趋向于在每一个粒子间均分，而且在    附近，caloric曲线通常有一个锯齿形结构。   能量均分在阱内和阱外粒子之间同时发生。一方面尽管阱内粒子受到扰动，但仍然是一个个(而不是一群群)溢出阱外；另一方面，阱外粒子一旦获得了足够的能量，粒子从阱内溢出后，体系并不出现负比热现象。    显然，粒子从阱内的溢出可以分为三种类型：ⅰ体积比很大且趋于1，这时没有负比热发生，caloric曲线呈单调增长；ⅱ体积比很小且趋于0，这时，整个能量增长的过程都伴随着负比热；ⅲ 给定粒子数  ，存在某个体积比  值的区间，在低能区域体系存在负比热现象，随着能量的升高负比热现象趋于消失。 Dense/Dilute Particle state and Phase Transition  当平均能量  为正数且足够大时所有粒子可 看作均匀分布在体积  内，粒子数密度为     。 当平均能量取一个中间值时，阱内外两种不 同密度的状态将共存。我们可以将阱外的粒子 看作是一种气相，而阱内粒子则可看成是一种 凝聚相。因此，当粒子数有限时，两相共存意 味着负比热的一种相变解释。 定义两个密度 阱内粒子数密度： 阱外粒子数密度： 一级相变可以重新定义为两相密度差： 在相变过程中，两相可以共存。  在这样一种定义下，发生相变的过程 中并不一定伴随着负比热现象。在热力 学极限情况下，仅仅在体积比取无穷小 量时，这种相变才可能重现为通常的一 级相变。由Fig.3可以看出，这时，相变 潜热对应着caloric曲线中的平坦部分， 粒子数       ，体积比        。 讨论： (1)凝聚相不能进一步区分成固相和液相，这是因 为：ⅰ)该模型不能肯定阱内是否存在某种有序 结构；ⅱ)对给定的粒子数和体积比，凝聚相仅仅在 密度方面不同；    (2)由于粒子间不存在相互作用，这种有限体系和 其他有限体系相比在相变方面有重大不同。包括ⅰ) 体系的负比热完全来自单粒子效应，ⅱ)负比热并 不是发生相变的必要条件。    (3)我们试图通过多种定义来概括相变，发现上述 定义是自洽且物理上合理的。   “noninteracting-particle-in-the-well”模型的 微正则统计力学已经建立。既然粒子间没有相 互作用，体系必然为气相。密度和气相不同的 另一相位于阱内。相变因此可以定义为高密度 态和气态两相共存。尽管粒子的溢出通常伴随 着负比热，但负比热并不是相变的一个先决条 件。 Conclusion 参考文献： 1 M. 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