linearity

nafems4.md

@@ -967,9 +967,14 @@ 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.
 
-geometrical
-material
+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:
 
+ 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. 
 
 # Fatigue
 
@@ -1004,3 +1009,5 @@ Back in college, we all learned how to solve engineering problems. But there is
  * remember that welded materials with different thermal expansion coefficients may lead to fatigue under cyclic temperature changes
  * if you have time, try to get out of your comfort zone and do more than what other expect from you (like parametric computations)
  * clone the [parametric tee repository](https://bitbucket.org/seamplex/tee), understand how the figures from\ [@sec:parametric] where built and expand them to cover “we might go on...” bullets
+ * think thermal-mechanical plus eartquakes as “bake, break and shake” problems
+ * the elastic problem is still linear