cosas

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nafems4.md

@@ -217,7 +217,7 @@ $$
 For the boundary conditions of the particular problem that the radial stress should be equal to the negative of the internal pressure at $r=a$ and null at $r=b$, the axial displacement has the particular solution
 divert(0)
 
-#### Displacements
+#### Displacements {#sec:u}
 
 Remember that when any solid body is subject to external forces, it has to react in such a way to satisfy the equilibrium conditions. The way solids do this is by deforming a little bit in such a way that the whole body acts as a compressed (or elongated) spring balancing the load.
 Starting from [@fig:timoshenko-eq] and after some mathematics (detailed shown in reference\ [@pipe-linearized]), which is what most of us have already done in college, we can find that the displacement field has the following analytical solution:
@@ -245,7 +245,7 @@ What does [@eq:ur] mean? Well, that overall the whole pipe expands a little bit
 
 That is how an infinite pipe withstands internal pressure. And that is what we are taught in college, which is actually true by the way!
 
-#### Stresses
+#### Stresses {#sec:stresses}
 
 As the solid is deformed, that is to say that different parts are relatively displaced one from another, strains and stresses appear. When seen from a cylindrical coordinate system, the stress tensor (recall [@sec:tensor]) has these features.
 
@@ -400,9 +400,9 @@ Since we already agreed there is no way to obtain analytical expressions for the
 For instance, [@fig:ur] shows how some results obtained with the finite element method using different number and order of elements compare to the analytical solution from the previous section. The bullets correspond to the nodal values of the radial displacements and stresses obtained with the finite element method [@pipe-linearized]. The more number of nodes employed, the more accurate the results are---at the expense of an increase of time and computational effort needed to solve the problem.
 
 :::: {#fig:pipe-linearized}
-![$u_r(r)$ [@eq:ur].](ur.svg){#fig:ur width=90%}
+![$u_r(r)$ from [@sec:u].](ur.svg){#fig:ur width=90%}
 
-![$\sigma_r(r)$ [@eq:sigmar].](sigmar.svg){#fig:sigmar width=90%}
+![$\sigma_r(r)$ from [@sec:stresses].](sigmar.svg){#fig:sigmar width=90%}
 
 Comparison between analytical and FEA results (ref.\ [@pipe-linearized]).
 ::::
@@ -412,13 +412,13 @@ As a nuclear engineer, I learned (theoretically in college but practically after
 Therefore, it is worth to take some time to think about what we need to do, what our choices are to build finite-element [models](https://www.seamplex.com/blog/say-modeling-not-simulation.html) and which one is the most convenient in terms of costs and efficiency. First of all, we need to define which transients are going to be taken into account. For the current imaginary case study, we define that the piping system from [@fig:cad-figure] will be subject to the the four simple (and again imaginary) time histories for the internal pressure\ $p$ and the fluid temperature\ $T$ as a function of time shown in [@fig:pt].^[Actual real piping systems might be subject to dozens of more complex transients.]
 
 ::::: {#fig:pt}
-![Transient #1: heating from cooldown state.](pt-1.svg){#fig:pt1 width=95%} 
+![Transient #1: heating from cooldown state (200 cycles).](pt-1.svg){#fig:pt1 width=95%} 
 
-![Transient #2: cooling from hot pressurised state.](pt-2.svg){#fig:pt2 width=95%} 
+![Transient #2: cooling from hot pressurised state (200 cycles).](pt-2.svg){#fig:pt2 width=95%} 
 
-![Transient #3: cooldown from full power to zero in hot condition.](pt-3.svg){#fig:pt3 width=95%} 
+![Transient #3: cooldown from full power to zero in hot condition (100 cycles).](pt-3.svg){#fig:pt3 width=95%} 
 
-![Transient #4: power reduction and turbine trip.](pt-4.svg){#fig:pt4 width=95%} 
+![Transient #4: power reduction and turbine trip (100 cycles).](pt-4.svg){#fig:pt4 width=95%} 
 
 The four (imaginary) transient operational conditions for the case study.
 :::::
@@ -459,7 +459,7 @@ Remember the main issue of the fatigue analysis in these systems is to analyse w
 On the one hand, a reasonable number of nodes^[Remember it is the number of degrees of freedom that defines the problem size, which in the finite element method is given by the number of nodes and not by the number of elements. Conversely, if we used finite volumes, it would be given by the number of elements and not by the number of nodes. The two meshes below have the same number of nodes but the one on the right has more nodes and will thus give far more accurate results.\newline![](distorted.png){width=100%}] in order to get a decent grid is around a couple of thousand for each piping system under study (@fig:mech). On the other hand, solving dozens of transient heat transfer problems during a few thousands of seconds over a couple hundred of thousands of nodes might take more time and storage space to hold the results than we might have.
 
 ::: {#fig:mech}
-![Full view of 200k nodes spanning $\approx$ 100k 2nd-order tetrahedra.](mech-view1.png){width=95% #fig:mech-msh}
+![Full view of 200k nodes spanning $\approx$ 100k 2nd-order tetrahedra.](mech-view1-annotated.svg){width=95% #fig:mech-msh}
 
 ![Detail around the material interface where the mesh is locally refined.](mech-view2.png){width=95% #fig:mech-zoom}
 
@@ -477,15 +477,24 @@ We can then merge this idea by Asimov with an adapted version of the [Saint-Vena
  2. assume the part of the full system which is not contained in this reduced mesh has an uniform (though not constant in time) temperature as if the reduced model was extended in each direction.
 
 
-Instead of solving the transient heat-conduction problem on the full 200k-nodes mesh of [@fig:cad-figure], we solve in on a reduced model consisting of half the valve body a small length of the pipes at both the valve inlet and outlet as shown in [@fig:valve-temp], which has 30k nodes 100k first-order tetrahedra. Once the temperature distribution\ $\hat{T}(\mathbf{x},t)$ for each time is obtained in the reduced mesh, the actual temperature distribution\ $T(\mathbf{x},t)$ is computed by an algebraic generalisation of $\hat{T}(\mathbf{x},t)$ in the full mesh of [@fig:mech]. Those locations which are not covered by the reduced model are generalised with a time-dependent uniform temperature which is the mean value of the temperature at the inlet and outlet of the reduced mesh from [@fig:valve-temp].
+Instead of solving the transient heat-conduction problem on the full 200k-nodes mesh of [@fig:cad-figure], we solve in on a reduced model consisting of half the valve body a small length of the pipes at both the valve inlet and outlet as shown in [@fig:valve-mesh], which has 30k nodes. Once the temperature distribution\ $\hat{T}(\mathbf{x},t)$ for each time is obtained in the reduced mesh ([@fig:valve-temp]), the actual temperature distribution\ $T(\mathbf{x},t)$ is computed by an algebraic generalisation of $\hat{T}(\mathbf{x},t)$ in the full mesh of [@fig:mech]. Those locations which are not covered by the reduced model are generalised with a time-dependent uniform temperature which is the mean value of the temperature at the inlet and outlet of the reduced mesh, marked as “end faces” in\ [@fig:valve-temp]. It can also be seen there that since the nozzle made of carbon steel has a larger thermal conductivity the temperature is almost uniform whilst there is a non-trivial temperature distribution within then adjacent stainless steel valve body. Indeed, [@fig:temp-zoom] shows how the temperature at SCLs\ #2 and\ #4 evolve as a function of time during the ramp of transient #1. It is this difference one of the main reasons for the need to perform fatigue analysis of critical piping systems in a nuclear power plant.
 
-![Transient temperature distribution in a reduced mesh around the valve.](temp-valve-commented.svg){#fig:valve-temp width=100%}
+::::: {#fig:valve}
+![Mesh with 30k nodes and 120k first-order tetrahedra.](thermal-mesh.png){#fig:valve-mesh width=100%}
+
+![Transient temperature distribution around the valve.](temp-valve-commented.svg){#fig:valve-temp width=100%}
+
+Reduced mesh around the valve including the carbon-steel nozzle.
+::::: 
+
+![Temperature evolution at the SCLs for transient #1.](temp-zoom.svg){#fig:temp-zoom width=100%}
 
 
 Note that there is no need to have a one-to-one correspondence between the elements from the reduced mesh with the elements from the original one. Actually, the reduced mesh contains first-order elements whilst the former has second-order elements. Also the grid density is different, yet both of them are locally refined around the material interface. Nevertheless, the finite-element solver [Fino](https://www.seamplex.com/fino)---used to solve both the heat and the mechanical problems---allows to read functions of space and time defined over one mesh and continuously evaluate and use them into another one even if the two grids have different elements, orders or even dimensions.
 
 
 
+
 ### Seismic loads {#sec:seismic}
 
 
@@ -531,7 +540,7 @@ Since the computation of the loads that a certain earthquake gives rise to would
 ![$f_3 \approx 50$\ [Hz](https://www.seamplex.com/docs/nafems4/mode3.webm)](mode3.0000.png){width=48%}\  
 ![$f_4 \approx 70$\ [Hz](https://www.seamplex.com/docs/nafems4/mode5.webm)](mode4.0000.png){width=48%}
 
-![$f_5 \approx 90$\ [Hz](https://www.seamplex.com/docs/nafems4/mode5.webm)](mode5.0000.png){width=48%}\  
+![$f_5 \approx 85$\ [Hz](https://www.seamplex.com/docs/nafems4/mode5.webm)](mode5.0000.png){width=48%}\  
 ![$f_6 \approx 90$\ [Hz](https://www.seamplex.com/docs/nafems4/mode6.webm)](mode6.0000.png){width=48%}
 
 First natural oscillation modes. Videos available online ([@sec:online]).
@@ -1031,7 +1040,7 @@ Primary stresses at the internal points of two SCLs for the “break” step.
 
   1. The material properties are temperature-dependent (we use data from [ASME\ II](https://en.wikipedia.org/wiki/ASME_Boiler_and_Pressure_Vessel_Code#ASME_BPVC_Section_II_-_Materials) part\ D).
   2. Thermal expansion in the pipes is taken into account. The reference temperature (i.e. the temperature at which there is no expansion) is\ 20ºC that coincides with ASME’s decision of the reference temperature for the mean thermal expansion coefficients in section\ II part\ D.
-  3. The temperature distribution\ $T(x,y,t,z)$ for bullets 1 & 2 is the generalisation of the temperature computed in the reduced-model of [@fig:valve-temp] to the full mesh of [@fig:mech] as explained in\ [@sec:thermal].
+  3. The temperature distribution\ $T(x,y,t,z)$ for bullets 1 & 2 is the generalisation of the temperature computed in the reduced-model of [@fig:valve] to the full mesh of [@fig:mech] as explained in\ [@sec:thermal].
   4. The internal faces of the pipes are subject to an uniform pressure\ $p(t)$ given by the definition of the transients from [@fig:pt].
   5. There are mechanical supports throughout the piping system. Depending on the type of the support (i.e. vertical, lateral, axial, full, etc.) one or more degrees of freedom (i.e. displacements in $x$, $v$ and/or $z$) are fixed to zero. The ends of the CAD models are chosen always to have axially-null displacements. [@Fig:mech] shows the location and nature of the supports.
   6. The earthquake-equivalent volumetric force\ $\mathbf{f}(x,y,z)$ is only be applied at the time\ $t$ where the maximum stresses is expected to occur.
@@ -1054,7 +1063,7 @@ $$ \sigma = E(T) \cdot \epsilon $$
 
 What changes with temperature is the slope of\ $\sigma$ with respect to\ $\epsilon$ (think and imagine!), but the relationship between them is _still linear_.
 
-On the other hand, we have a non-trivial temperature distribution\ $T(\mathbf{x}, t)$ 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 and obtaining something like [@fig:temp-valve]). Let us now forget about the time, as after all we are solving a quasi-static elastic problem. Now you can trust me or ask a FEM teacher, but the continuous displacement formulation can be loosely written as
+On the other hand, we have a non-trivial temperature distribution\ $T(\mathbf{x}, t)$ 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 and obtaining something like [@fig:valve-temp]). Let us now forget about the time, as after all we are solving a quasi-static elastic problem. Now you can trust me or ask a FEM teacher, but the continuous displacement formulation can be loosely written as
 
 $$ K\big[E\left(T(\mathbf{x})\right), \mathbf{x}\big] \cdot \mathbf{u}(\mathbf{x}) = \mathbf{b}(\mathbf{x})$$
 
@@ -1074,7 +1083,7 @@ To recapitulate, the steps discussed so far include
  2. defining the number and details of each operational transient to be included into the analysis ([@fig:pt])
  3. defining the number and locations of the stress classification lines ([@fig:scls])
  4. creating a mesh for the main domain refining locally around the material interfaces ([@fig:mech])
- 5. computing a heat conduction (“bake”) transient problem with temperatures as a function of time from the operational transients in a simple domain using temperature-dependent thermal conduction coefficients ([@fig:valve-temp])
+ 5. computing a heat conduction (“bake”) transient problem with temperatures as a function of time from the operational transients in a simple domain using temperature-dependent thermal conduction coefficients ([@fig:valve])
  6. performing a modal analysis (“shake”) on the main domain to obtain the main oscillation frequencies and modes ([@fig:modes])
  7. obtaining a distributed force statically-equivalent to the earthquake load
  8. solving a quasi-static linear elastic problem for different reading the temperature $T(\vec{x},t)$ computed in step\ 4 taking into account
@@ -1088,29 +1097,28 @@ To recapitulate, the steps discussed so far include
 
 A pretty nice list of steps, which definitely I would not have been able to tackle when I was in college. Would you?
 
-divert(-1)
 
 ## Cumulative usage factors {#sec:usage}
 
-Strictly speaking, finite elements are not needed anymore at this point of the analysis. But even though we are (or want to be) FEM experts, we have to understand that if the objective of a work is to evaluate fatigue (or fracture mechanics or whatever), finite elements are just a mean and not and end. If we just mastered FEM and nothing else, our field of work would be highly reduced. We need to use all of our computational knowledge to perform actually engineering tasks and to be able to tell our bosses and clients whether the pipe would fail or not. This tip is induced in college but it is definitely reinforced afterwards when working with actual clients and bosses.
+Strictly speaking, finite elements are not needed anymore at this point of the analysis. But even though we are (or want to be) FEM experts, we have to understand that if the objective of a work is to evaluate fatigue (or fracture mechanics or whatever), finite elements are just a mean and not and end. If we just mastered FEM and nothing else, our field of work would be narrow and bounded. We need to use all of our computational knowledge to perform actual engineering tasks and to be able to tell our bosses and/or clients whether the pipe will fail or not. This important hint is indirectly induced in college but it is definitely reinforced afterwards when working with actual clients and bosses.
 
-Another comment I would like to add is that I had to learn fatigue practically from scratch when faced with this problem for the first time in my engineering career. I remembered some basics from college (like the general introduction from [@sec:fatigue]), but I lacked the skills to perform a real computation by myself. Luckily there still exist books, there are a lot of interesting online resources (not to mention Wikipedia) and, even more luckily, there are plenty of other fellow engineers that are more than eager to help you. My second tip is: when faced to a new challenging problem, read, learn and ask for guidance to real people to see if you read and learned it right.
+Another comment I would like to add is that I had to learn fatigue practically from scratch when faced with this problem for the first time in my engineering career. I remembered some basics from college (like the general introduction from [@sec:fatigue]), but I lacked the skills to perform a real computation by myself. Luckily there still exist books, there are a lot of interesting online resources (not to mention Wikipedia) and, even more luckily, there are plenty of other fellow engineers that are more than eager to help you. My second hint in this section is: when faced to a new challenging problem, read, learn and ask for guidance to real people to see if you got what you read right.^[I managed to take a graduate-level university course on material fatigue after having worked in this field learning stuff on my own, which configures the third hint in a row: do take official college courses if you can.]
 
 \medskip
 
-Back and distantly, in\ [@sec:case] we said that people noticed there were some environmental factors that affected the fatigue resistance of materials. The basic ASME approach does not take care of these factors, and it is regarded as fatigue “in air.” We are interested in taking them into account, so we follow the US Nuclear Regulatory Commission guidelines to evaluate fatigue “in water.”
+Back and distantly, in\ [@sec:case] we said that people noticed there were some environmental factors that affected the fatigue resistance of materials. The basic ASME approach does not take care of these factors, and it is regarded as fatigue “in air.” We are interested in taking them into account, so we follow the US\ Nuclear Regulatory Commission guidelines to evaluate fatigue “in water” [@nrc].
 
 ### In air (ASME’s basic approach) {#sec:in-air}
 
-We already said in\ [@sec:fatigue] that the stress-life fatigue assessment method gives the limit number\ $N$ of cycles that a certain mechanical part can withstand when subject to a certain periodic load of stress amplitude\ $S_\text{alt}$. If the actual number of cycles\ $n$ the load is applied is smaller than the limit\ $N$, then the part is fatigue-resistant. In our case study there is a mixture of several periodic loads, each one expected to occur a certain number of times. ASME’s way to evaluate the resistance is to break up the stress history into partial stress amplitudes\ $S_{\text{alt},j}$ between a “peak” and a “valley” and to compute individual usage factors\ $U_j$ for the\ $j$-th amplitude (which does not need to coincide with one of the \ $k$ transient loads) as
+We already said in\ [@sec:fatigue] that the stress-life fatigue assessment method gives the limit number\ $N$ of cycles that a certain mechanical part can withstand when subject to a certain periodic load of stress amplitude\ $S_\text{alt}$. If the actual number of cycles\ $n$ the load is applied is smaller than the limit\ $N$, then the part is fatigue-resistant. In our case study there is a mixture of several periodic loads, each one expected to occur a certain number of times. ASME’s way to evaluate the resistance is first to build a juxtaposed stress history from all transients under consideration and then to break it up into partial stress amplitudes\ $S_{\text{alt},j}$ between a “valley” and a “peak.” Each valley-peak pair\ $j$ is assigned an individual usage factor\ $U_j$ as
 
 $$U_j = \frac{n_j}{N_j}$$
 
-The overall cumulative usage factor is then the algebraic sum of the partial contributions, a.k.a. [Miner’s rule](https://en.wikipedia.org/wiki/Fatigue_%28material%29%23Miner%27s_rule) as learned in college:
+Under the assumption that [Miner’s rule](https://en.wikipedia.org/wiki/Fatigue_%28material%29%23Miner%27s_rule) (a.k.a. Palmgreen’s rule) holds, the overall cumulative usage factor is then the algebraic sum of the partial contributions [@schijve]:
 
 $$\text{CUF} = U_1 + U_2 + \dots + U_j + \dots$$
 
-When\ $\text{CUF} < 1$, then the part under analysis can withstand the proposed cyclic operation. Now, if the extrema of the partial stress amplitude correspond to different transients, then the following note in ASME III’s NB-3224(5) should be followed:
+When\ $\text{CUF} < 1$, the part under analysis can withstand the proposed cyclic operation. Now, if the valley of the partial stress amplitude corresponds to one transient and the peak to another one, then the following note in ASME III’s NB-3224(5) should be followed:
 
 
 > In determining $n_1$, $n_2$, $n_3$, $\dots$, $n_j$ consideration shall be given to the superposition of cycles of various origins which produce a total stress difference range greater than the stress difference ranges of the individual cycles. For example, if one type of stress cycle produces 1,000 cycles of a stress difference variation from zero to +60,000\ psi and another type of stress cycle produces 10,000 cycles of a stress difference variation from zero to −50,000\ psi, the two types of cycle to be considered are defined by the following parameters:
@@ -1119,7 +1127,12 @@ When\ $\text{CUF} < 1$, then the part under analysis can withstand the proposed
 > (b) for type 2 cycle, $n_2 =$ 9,000 and $S_{\text{alt},2} = (50,000 + 0)/2$.
 
 
-This cryptic paragraph can be better explained by using a clearer example. To avoid using actual sensitive data from a real power plant, let us use the same test case used by both the [US Nuclear Regulatory Commission](https://en.wikipedia.org/wiki/Nuclear_Regulatory_Commission) (in its report NUREG/CR-6909) and the [Electric Power Institute](https://en.wikipedia.org/wiki/Electric_Power_Research_Institute) (report 1025823) called “EAF (Environmentally-Assisted Fatigue) Sample Problem 2-Rev.\ 2 (10/21/2011)”.
+This cryptic paragraph is a clear example of stuff that cannot be learned at college. No matter how good your university is, there is no way to cover all theories and methodologies which a mechanical engineer could need in his or her professional life.
+
+
+dnl -------------------------------------------------------
+
+using a clearer example. To avoid using actual sensitive data from a real power plant, let us use the same test case used by both the [US Nuclear Regulatory Commission](https://en.wikipedia.org/wiki/Nuclear_Regulatory_Commission) (in its report NUREG/CR-6909) and the [Electric Power Institute](https://en.wikipedia.org/wiki/Electric_Power_Research_Institute) (report 1025823) called “EAF (Environmentally-Assisted Fatigue) Sample Problem 2-Rev.\ 2 (10/21/2011)”.
 
 
 ![A low-alloy steel vessel nozzle (blue) welded to a stainless steel pipe (grey)](axi-inches-3d.png){#fig:axi-inches-3d width=35%}
@@ -1252,37 +1265,41 @@ About your favourite FEM program, ask yourself these two questions:
 
 And finally, make sure that at the end of the journey from college theory to an actual engineering problem your conscience is clear knowing that there exists a report with your signature on it. That is why we all went to college in the first place.
 
+divert(0)
+
 ### Online stuff {#sec:online}
 
 Here is a list of sub-problems and stuff to play with.
 
  * The pendulum-swing video from [@sec:intro]
-    - <https://youtu.be/Q-lKK4A2OzA>
- * “On convergence of linearized stresses in an infinite pipe computed using the finite element method” from [@sec:infinite-pipe;@sec:infinite-pipe;@sec:infinite-pipe-fem]
-    - <https://www.seamplex.com/fino/doc/pipe-linearized/> 
-dnl divert(-1)    
+   - <https://youtu.be/Q-lKK4A2OzA>
+ * “On convergence of linearized stresses in an infinite pipe computed using the finite element method” from [@sec:infinite-pipe;@sec:infinite-pipe]
+   - <https://www.seamplex.com/fino/doc/pipe-linearized/> 
+divert(-1)    
  * The three cubes from [@sec:linearity]
-    - case A: pure normal loads (<https://caeplex.com/p/d8fe>)
-    - case B: pure shear loads (<https://caeplex.com/p/b494>)
-    - case C: the combination of A & B (<https://caeplex.com/p/9899>)
-dnl divert(0)
- * The animations of natural oscillations in [@fig:modes]
-    - <https://www.seamplex.com/docs/nafems4/mode1.webm>
-    - <https://www.seamplex.com/docs/nafems4/mode2.webm>
-    - ...
- * The animations of natural oscillations in [@fig:case-mode]
-    - <https://www.seamplex.com/docs/nafems4/case-mode1.webm>
-    - <https://www.seamplex.com/docs/nafems4/case-mode2.webm>
-    - ...
+   - case A: pure normal loads (<https://caeplex.com/p/d8fe>)
+   - case B: pure shear loads (<https://caeplex.com/p/b494>)
+   - case C: the combination of A & B (<https://caeplex.com/p/9899>)
  * The parametric tee repository from [@sec:parametric]
    - <https://github.com/seamplex/tee>
  * The environmental fatigue sample problem repository from [@sec:in-air;@sec:in-water]
    - <https://github.com/seamplex/cufen>
+divert(0)    
+ * The videos of the thermal transients in\ [@fig:valve]
+   - <https://www.seamplex.com/docs/nafems4/temp-1.mpg>
+   - <https://www.seamplex.com/docs/nafems4/temp-1.mpg>
+   - <https://www.seamplex.com/docs/nafems4/temp-1.mpg>
+  
+ * The animations of natural oscillations in [@fig:modes]
+   - <https://www.seamplex.com/docs/nafems4/mode1.webm>
+   - <https://www.seamplex.com/docs/nafems4/mode2.webm>
+   - ...
+ * A ready-to-play-with CAEplex case with the modal problem
+   - <https://caeplex.com/project/results.php?id=42180c3>
 
 See <https://www.seamplex.com/nafems> for new material, updated links and the full version of this case with many more details about the case and the associated mathematics.
 
 
 dnl # Concluding Remarks
 
-divert(0)
 # References

references.bib

@@ -122,7 +122,7 @@
  hyphenation = "english", 
 }
 
-@report{pipe-linearized,
+@techreport{pipe-linearized,
  author = "Jeremy Theler",
  institution = "Seamplex",
  title = "On convergence of linearized stresses in an infinite pipe computed using the finite element method",
@@ -163,3 +163,20 @@
  langid = "english",
  language = "english",
 }
+
+@techreport{nrc,
+ author = "Omesh Chopra and Gary L. Stevens",
+ title = "Effect of {LWR} Water Environments on the Fatigue Life of Reactor Materials",
+ year = "2018",
+ url = "https://www.nrc.gov/reading-rm/doc-collections/nuregs/contract/cr6909/r1/",
+ number = "NUREG/CR-6909",
+ version = "1",
+ year = "2018",
+ institution = "US NRC",
+ langid = "english",
+ language = "english",
+}
+ 
+ 
+# https://www.epri.com/research/products/1025823
+# 

temp-zoom.svg

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