eqns

nafems4.md

@@ -1,3 +1,5 @@
+$\renewcommand\vec[1]{\mathbf{#1}}$
+
 # Background and introduction
 
 First of all, please take this text as a written chat between you an me, i.e. an average engineer that have already taken the journey from college to performing actual engineering using finite element analysis and has something to say about it. Picture yourself in a coffee bar, talking and discussing concepts and ideas with me. Maybe needing to go to a blackboard (or notepad?). Even using a tablet to illustrate some three-dimensional results. But always as a chat between colleagues.
@@ -967,14 +969,20 @@ Finally we attempt to “break” the pipes successively solving many steady-sta
  9. The stress linearisation has to be performed individually for each principal stress\ $\sigma_1$, $\sigma_2$ and $\sigma_3$ to fulfill the requirements ASME\ III\ NB-3126 (see [@sec:in-air] below).  
  10. This “break” step is linear.
 
-Surely you’re joking, Mr.\ Theler! How can this extremely complex problem be linear? Well, let us see. First, there are two main kinds of non-linearities in FEM:
+[Surely you’re joking, Mr.\ Theler!](https://en.wikipedia.org/wiki/Surely_You're_Joking%2C_Mr._Feynman!) How can this extremely complex problem be linear? Well, let us see. First, there are two main kinds of non-linearities in FEM:
 
  1. Geometrical non-linearities
  2. Material non-linearities
 
 The first one is easy. Due to the fact that the pipes are made of steel, it is expected that the actual deformations are relatively small compared to the original dimensions. This leads to the fact that the mechanical rigidity (i.e. the stiffness matrix) does not change significantly when the loads are applied. Therefore, we can safely assume that the problem is geometrically linear.
 
-About material non-linearities, on the one hand we have the temperature-dependent issue. According to ASME\ II part\ D, what depends on temperature is the Young Modulus\ $E$. But the stress-strain relationship is still linear, what changes with temperature is the slope of\ $\sigma$ with respect to\ $\epsilon$ (think and imagine!). On the other hand, we have a given non-trivial temperature distribution\ $T(\vec{x}, t)$ within the pipes that is a snapshot of a transient heat conduction problem at a certain time\ $t$ (think and picture yourself taking photos of the temperature distribution changing in time). Let us now forget about the time, as after all we are solving a steady-state elastic problem. 
+About material non-linearities, on the one hand we have the temperature-dependent issue. According to ASME\ II part\ D, what depends on temperature\ $T$ is the Young Modulus\ $E$. But the stress-strain relationship is still linear, that is
+
+$$ \sigma = E(T) \cdot \epsilon $$
+
+What changes with temperature is the slope of\ $\sigma$ with respect to\ $\epsilon$ (think and imagine!). On the other hand, we have a given non-trivial temperature distribution\ $T(\vec{x}, t)$ within the pipes that is a snapshot of a transient heat conduction problem at a certain time\ $t$ (think and picture yourself taking photos of the temperature distribution changing in time). Let us now forget about the time, as after all we are solving a steady-state elastic problem.
+
+$$ K\big[E\left(T(\vec{x})\right), \vec{x}\big] \cdot \vec{u}(\vec{x}) = \vec{b}(\vec{x})$$
 
 # Fatigue