Department of Nuclear Engineering, University of California, Berkeley, CA 94720-1730
The following paper is a review of the basic mechanisms of creep. This was provided after a brief review of some of the basic mechanical behavior principles. Some examples of how creep affects nuclear power systems are suggested but not investigated in depth.
When designing any system there are a number of mechanical behavior of materials questions that must be answered. Among them are fracture, yield, fatigue, stress corrosion cracking, and creep, just to scratch the surface. Some of these considerations, like fracture and yield, are well recognized mechanical behavior problems. The other phenomena are more subtle and happen over longer time periods. Creep is the plastic deformation of a material that is subjected to a stress below its yield stress when that material is at a high homologous temperature. Homologous temperature refers to the ratio of a materials temperature to its melting temperature. The homologous temperatures involved in creep processes are greater than 1/3.
The temperatures in mainstream nuclear power applications such as Boiling and Pressurized Water Reactors (BWRs, PWRs) are very high as compared to their melting temperatures. In reactor systems such as Inertial Confinement Fusion, Liquid Metal Fast Breeder Reactors, and High Temperature Gas Cooled Reactors the designs allow for even higher temperatures. Due to the high temperatures involved, creep needs to be accounted for when designing nuclear power systems.
In order to understand how creep affects nuclear power systems its mechanisms and hence its behavior must first be understood. Before this can be done a review of basic mechanical behavior principles is warranted.
First a few definitions are in order. Stress is defined as the amount of force per unit area. Strain on the other hand is defined as the extension of a sample divided by its initial length. Macroscopically materials react to an applied stress in one of two ways: elasticity or plastically. Elastic behavior, as the name suggests, is the strain of a material that is not permanent and returns to zero when the applied stress is removed. A good example of elastic behavior is the behavior of a rubber band. On the other hand plastic behavior describes a material whose strain does not return to zero when the stress is removed. A good example of plastic behavior is the plastic behavior of taffy.
The stress at the point where a material's behavior changes form elastic to plastic is known as its yield stress. This point can be seen in figure 1 below where the line changes from linear to curved behavior. The linear behavior is characterized by Young's modulus. In this region strain is directly proportional to stress. In plastic deformation strain relates to stress in a power law form. This power law form is due to work or strain hardening.
Work hardening is a process by which the material grows stronger as it is deformed. This phenomena happens because microscopic defects in the material interact to make the strain of the material harder to achieve. At the maximum in figure 1 the amount of stress needed for further strain decreases. This stress is named the Ultimate Tensile Strength (UTS) and is the point where necking of the sample occurs. Necking is a decrease in the local area of sample due to a localization in stress.
When a material is plastically deformed too much the material pulls apart through a ductile fracture process. In this process voids start to form which further weaken the material and increase the local stress which eventually tears the material apart. This behavior is in direct contrast to a brittle fracture where the material shatters along preferred planes or sites in the material. The above described phenomena is shown below in figure 1.
Figure 1 Example of Stress-Strain curve
Note: The above graph shows the elastic region greatly exaggerated.
On a microscopic level the above macroscopic behavior can be explained. To start with most substances and virtually all metals are made of a regular three dimensional lattice of metal atoms. The lattice is commonly referred to as a crystal.
Periodically in crystals there exist imperfections in the regular ordering of the atoms in the lattice. There are zero, one, two, and three dimensional imperfections. Zero-dimensional or point defects are things such as atom vacancies, substitutional atoms, or interstitial atoms in the lattice. One-dimensional defects will be described later because they are most important to the creep process. One other type of imperfection in crystals is a two-dimensional defect. The most prominent of these defects are grain boundaries. In real materials the structure of the solid is only crystalline in tiny volumes called grains. These are surrounded by grain boundaries made up primarily of vacancies.
A three-dimensional defect is best exemplified by a precipitate. A precipitate in a matrix consists of a substance that has a different solid phase of composition than the rest of the lattice. Precipitates form because an alloy material is not soluble in the parent lattice.
Now the one-dimensional or line defect can be introduced. One type of line defect results from inserting an extra half plane of atoms into the crystal lattice as shown in the two dimensional picture below.
Figure 2 Atomic picture of dislocation.
This line defect creates a line of atoms into and out of the page called a dislocation line. It is these dislocation lines moving through the solid that cause a material to plastically deform under an applied shear stress. Dislocations because of their linear nature and the nature of the crystal structure of a lattice can only move in certain directions under applied stresses of given type. As can be visualized from figure 2 when a compressive stress field is applied to the crystal, an upward force will exerted on the extra half plane of atoms. This force results from the rule that like stresses repel and opposite stresses attract. Since the top of the crystal has extra atoms in it, the top part of the crystal is in compression. When a compressive stress is applied, a force is exerted on the half plane to relieve the compressive stress conflict. This effect is similar to squeezing a watermelon seed between ones fingertips until it pops out the side.
In a crystal for a dislocation to move like described above the atoms must be removed from the half plane of atoms. This requires molecular diffusion of atoms from the dislocation for the dislocation to move in the upward direction. Equivalently, vacancies must diffuse to the extra half plane. Diffusion is a thermally activated process. So, when a dislocation moves due to an applied normal stress the process is called thermally activated climb. Climb requires a whole line of atoms, into and out of the paper, to diffuse away from the dislocation.
As mentioned above work hardening happens because defects within the solid interact with one another. One specific interaction that happens to dislocations in dislocation jogging. When a dislocation becomes jogged by the intersection with another dislocation, those dislocations have regions containing the characteristics of the other dislocation and are hence jogged. That jog may impede the motion of that dislocation at low temperatures. It is this retardation of dislocation motion that gives rise to the strain hardening effect. At higher temperatures, that jog or dislocation may become mobile and climb in a direction perpendicular to the normal stress applied. As stated before climb is a diffusional process and depends strongly on the solid atoms ability to diffuse to or away from the dislocation.
Another reason that a material hardens is due to dislocation sources. When a material is stressed, various defects as mentioned above can produce dislocations when they interact with existing dislocations. With more dislocations present and moving in the material the more likely they will interact with each other or with other defects in the material so that they get pinned and hence strain harden the material.
With each of the above mentioned crystal defects is an associated strain energy since the perfect crystal lattice it being replaced by a non-lattice component. It is this strain energy that gives rise to the fact that like stesses repel and opposites attract. Whenever their is strain energy in the lattice it is thermodynamically favorable to remove that defect. When temperatures are high enough the kinetics of removing defects become favorable and the defect is removed releasing the strain energy. This process is called recovery.
With the basic definitions and mechanisms of material behavior explained the processes of creep can be examined. Creep can be subdivided into three categories primary, tertiary, and steady state creep. The qualitative behavior of the strain vs. time can be seen below.
Figure 3 Strain Vs. Time Creep Behavior.
Note: The abscissa in figure 3 has been exaggerated in the primary and tertiary regions.
As stated above creep is the plastic deformation of a material at a high homologous temperature below a material's yield stress. The effect of increasing the stress and temperature on the above curve is shown below.
Figure 4 Strain Vs. Time Creep Behavior Effect of Temperature and Stress.
As demonstrated above a material deforming by creep spends most of its time in the steady state region which is by far the dominant region when considering the effects of creep.
In the first region of the above graph is named the primary creep region. This region is characterized by the following equation.
Note: Where b is a constant.
Primary creep strain is usually less than one percent of the sum of the elastic, steady state, and primary strains. The mechanism in the primary region is the climb of dislocations that are not pinned in the matrix.
Since the amount of initial strain of a material is due to the number of dislocations initially present, the primary region is strongly dependent on the history of the material. If the material had been heavily worked before the creep test, there would have been many more dislocations present and the characteristics of the primary creep region would have been much different.
When the amount of strain is high creep fracture or rupture will occur. In the tertiary region the high strains will start to cause necking in the material just as in the uni-axial tensile test. This necking will cause an increase in the local stress of the component which further accelerates the strain. Eventually the material will pull apart in a ductile fracture around defects in the solid. These defects could be precipitates at high temperatures or grain boundaries at lower temperatures.
In any case the importance of the tertiary region to normal operation and creep design criteria is minimal. In the figure 3 above the time scale of the tertiary region is greatly expanded for the purpose of clarity. Considering the small amount of time in addition to the fact that the tertiary region develops a plastic instability similar to necking, operating in the tertiary region is not feasible. Therefore it is a conservative estimate to approximate the end of serviceable life of any component to coincide with the end of the steady state creep region. Only under accident conditions may the extra time in the tertiary region may be useful to consider.
The second region of the above graph is the steady state region. This region is so named because the strain rate is constant. In this region the rate of strain hardening by dislocations is balanced by the rate of recovery .
When the homologous temperature of a creep sample is between 0.3 and 0.7 of its melting temperature, the mechanism of creep is dislocation climb. This is the climb of dislocation jogs that impede the normal motion of the dislocation. This dislocation climb mechanism can be described by the following equation.
Note: Where H is the activation energy for diffusion . A is a constant , and sigma are the applied stresses.
When higher temperatures are applied and the temperature is above 0.7 of the melting temperature, a different mechanism takes place. In this region the creep is analogous to viscous flow. In this region the mechanism of the creep is diffusion of atoms from one place to another. This type of creep can be described by:
NOTE: Where dg is the diameter of the grains in the material, and H' is the diffusion activation energy. The above relation indicate that as grain size increases the creep rate decreases.
Two mechanisms of diffusional creep have been proposed. One mechanism by Coble proposes that the diffusion of atoms occurs along the grain boundaries of the material. Coble proposed this process because diffusion is easier along grain boundaries. This is due to the fact that vacancies are more common in the grain boundary than in the bulk. Since more vacancies are present, atoms will have more sites to jump to during diffusion which means the activation energy for diffusion will be less in the grain boundary than in the bulk. Another mechanism proposed by Herring and Nabarro suggests that the atomic diffusion takes place in the bulk, even though the activation energy for diffusion in the bulk is higher.
At higher temperatures in the homologous temperature region of 0.7 and above the bulk diffusion model dictates the creep performance. This is due to the larger number of diffusional paths in the bulk, as compared with the relatively few paths for diffusion along the grain boundaries. So, at high homologous temperatures in the 0.7 and above range the Herring-Nabarro model dominates due to the large number of paths for diffusion.
At lower temperatures, but still above homologous temperature of 0.7, the Coble model dominates the creep behavior. In this case the temperature is low enough that diffusion in the bulk is not activated due to its the relatively high activation energy. In the grain boundary diffusion is activated since the activation energy for diffusion along a grain boundary is lower.
In either case for these high temperature applications large grains are necessary to reduce the creep rate because they reduce the number of grain boundaries and increase the bulk diffusional distances.
Since nuclear power systems operate at such high temperatures the engineer must be conscious of creep when designing nuclear power plant systems. One might think that creep rupture would be the limiting design factor when applied to creep. Certainly a nuclear power system can not operate so that its components rupture due to creep. However, the more likely limiting factor in nuclear power systems is serviceable life of a component under high temperature conditions.
The high capital costs of nuclear power systems require that the life of the plant extend as long as possible. Therefore creep must be taken into account to ensure that a strain without rupture does not stop a system or the plant from functioning correctly. For example a turbine with extremely high tolerances my undergo creep which would cause the turbine to rub and therefore stop operating. Since high temperatures and stress are inevitable in nuclear power systems creep must be factored into serviceable life calculations. Otherwise systems must be designed to allow for the effects of creep and design for the amount of creep expected.
Another good example is the creep of the fuel cladding during operation. The creep of cladding over core life will change many characteristics of the fuel performance such as heat transfer and heat conduction characteristics. These factors must be accounted for when determining rod spacings and maximum heat flux safety factors.
During accident conditions creep rupture of components my become a viable concern. Under normal operating temperatures the time scale for creep is long and hence unnoticeable. Under extreme accident conditions creep may become a real time concern if the reactor can not be cooled. In a worst case scenario creep rupture of the reactor pressure vessel may occur under the most severe of accidents.
In any case since nuclear power systems are high temperatures systems, creep will always be a design concern.