Modeling the Stress Strain Relationships and Predicting Failure Probabilities for Graphite Core Components
Technical Work Scope Identifier No. G4A-1
Gen IV Materials (NGNP)
Principle Investigator:
Stephen F. Duffy PhD, PE, F.ASCE
Professor and Chair, Civil & Environmental Engineering
Summary
In order to assess how close a nuclear core component is to failing accurate stress states are required, both the elastic and inelastic components. The design engineer must be able to compute the current stress state and the current strain state given a load history. The engineer also needs to know the strain state to assess deformations in order to ascertain serviceability issues relating to failure, e.g., has too much shrinkage taken place for the core to function properly. Work on developing inelastic constitutive models is proposed here that will yield the requisite stress-strain information necessary for graphite component design.
Failure probabilities (as opposed to safety factors) are required in order to capture the variability in failure strength (at least in tensile regimes). The current stress state is used to predict the probability of failure. Stochastic failure models are proposed in this work that can accommodate possible material anisotropy. This work also proposes to assess material damage due to radiation exposure, i.e., develop a predictive capability on how the mechanical properties degrade with radiation exposure. It is noted that assessing stress and strain as opposed to failure predictions are distinct work efforts.
Contribution and Relevance
Having the capability to accurately predict stress and strain states in graphite core is critical to the Gen IV initiative. As Eason et al. (2008) point out constitutive models for nuclear graphite material is largely empirically based, especially when the effects of irradiation are accounted for. Historically the work of Greenstreet et al. (1973) and Batdorf (1975) attempt to capture stress and strain states for graphite materials using phenomenological models. Indeed Onat (1970) points out that materials used in elevated temperature applications have stress states that are dependent on past thermal and mechanical load histories. This requires the use of state variables in order to capture the non-linear behavior exhibited by graphite, especially for compressive stress regimes. The use of potential functions is convenient to derive constitutive models both for the elastic and the inelastic response of the material.
Having accurate stress states that include elastic and inelastic stress components facilitates the assessment of graphite component failure probabilities. In addition, accurately assessing strain states is paramount to designing the geometry of core subcomponents. The tasks outlined below provides a framework aimed at developing the mechanistic modeling capabilities needed for implementation of the next generation of commercially available nuclear reactors:
Task #1 Thermoelastic constitutive (stress-strain) relationship that exhibits different behavior in tension and compression and allows for material anisotropy (transverse isotropy is alluded to in the literature). This capability is available in the COMSOL finite element analysis software.
The appropriate temperature dependent equations for the various elastic material constants must be identified by finding those relationships in the literature, or developing spline functions that fit published (or unpublished) material data.
Damage mechanics model that accounts for radiation damage. Radiation damage must initially be manifested through a change in elastic material constants. A scalar state variable is initially proposed, but a tensorial damage state variable will be examined to account for directional damage.
Task #2 An inelastic, time dependent constitutive model developed by Janosik and Duffy will be incorporated into COMSOL. This multiaxial constitutive model allows for creep and different stress-strain behavior in tension and compression. If the inelastic behavior of graphite exhibits anisotropic behavior the model will be extended to account for transversely isotropic time dependent responses.
If the inelastic behavior of graphite material exposed to radiation is affected, then damage mechanics can be coupled with the inelastic constitutive model in order to capture phenomenon such as strain softening. Damage mechanics can also be utilized with an inelastic constitutive model to capture tertiary creep. Both phenomenon, i.e., strain softening and tertiary creep will be modeled using a scalar state variable.
Task #3 Incorporate into the CARES algorithm an interactive reliability model that accounts for different behavior in tension and compression analogous to the Burchell reliability model. Importance sampling techniques will be utilized to compute the probability failure within a continuum element (finite element). The model will be integrated into the CARES algorithm for use with COMSOL.
Since the interactive reliability model is phenomenologically based, tensorial invariants used by Duffy et al. ( ) will be incorporated into the model to account for anisotropic failure behavior.
Budget
Note that for the most part the tasks identified above can move ahead in parallel and can be funded for the most part independently of one another. Hence an estimated budget is provided as follows where an individual graduate student is assigned to each task.
Year 1, Year 2 and Year 3$ 50,250.00 / Tuition and Stipend
$ 30,000.00 / Summer Salary
$ 5,790.00 / Fringes Faculty
$ 1,605.00 / Fringes Students
$ 47,347.50 / Indirects
$ 20,000.00 / COMSOL (license and training)
$ 10,000.00 / Travel
$164,992.50 / Estimated Total per year
The work effort will span three years
References
Contributions by CSU
Nonlinear Isotropic Elastic Constitutive Relationship
An elastic material is characterized by its total reversibility. In the uniaxial case (see the figure below) this implies that the stress-strain curve will retrace itself during unloading, i.e., loading will follow OA along the stress-strain curve, and unloading will follow path AO. Thus completion of load cycle OAO will leave the material in its original configuration.
This type of reversibility implies that the mechanical work done by external loading is regained when the load is applied slowly and removed slowly. Thus any work may be considered as being stored in the deformed body as strain energy.
In the uniaxial case, the strain energy stored per unit volume of the material, or strain energy density, W, is represented by the area under the stress-strain curve and the strain axis. This quantity is expressed as
In the multiaxial case, the strain energy density is the sum of the contributions of all the stress components, i.e.,
Alternatively, the area above the s-e curve in the figure above represents the complementary energy density (or the complementary energy per unit volume). For the uniaxial case this quantity is expressed as
In the multiaxial case this relationship is expressed as
The strain energy density W and the complementary energy density W are functions of strain, eij, and stress sij, respectively. It is evident that the two are related through the following relationship
At this point we can employ two approaches to describe the reversible elastic behavior under consideration in this section. The first approach assumes that there exists a one-to-one relationship between stress and strain that can be expressed as
An elastic material defined by this expression is termed a Cauchy elastic material. Note that Fij is a second order tensor operator (i.e., the result is a second order tensor) that is a function of a second order tensor (eij).
Alternatively, one can assume that the components of the stress tensor are obtained from the derivatives of the strain energy density function with respect to the components of the strain tensor, i.e.,
or that the components of the strain tensor are obtained from the derivatives of the complementary strain energy density function with respect to the components of the stress tensor, i.e.,
A material whose elastic stress-strain relationship is described by taking derivatives in the manner indicated above is referred to as a Green elastic material.
It is this potential-normality structure embodied in a Green elastic material relationship that provides a consistent framework. According to the stability postulate of Drucker (1959), the concepts of normality and convexity are important requirements which must be imposed on the development of any stress-strain relationship. The convexity of the elastic strain-energy surface assures stable material behavior, i.e., positive dissipation of elastic work, a concept based on thermodynamic principles. Constitutive relationships developed on the basis of these requirements assure that the elastic boundary-value problem is well posed, and solutions obtained are unique.
Nonlinear Isotropic Elastic Constitutive Relationships based on W and W
For an isotropic elastic material we found that the strain energy density function W can be expressed in terms of any three independent invariants of the strain tensor ei j. Thus W can be expressed as
where
With
then
Carrying out the implied differentiation, this expression can be restated as
where
and
Note that a's are not tensor quantities in the same sense that the I 's are not tensor quantities. It should also be noted that the choice of the three independent strain invariants appearing in the expressions above is arbitrary. Deviatoric strain invariants (which are functionally dependent on e i j) can be used. However, according to the Cayley-Hamilton theorem, the three invariants chosen above span the space that defines the function W (see a previous section of your notes for the proof).
Yet, there is no a priori reason for requiring all three invariants to appear in the functional dependence for W. Nor is there any a priori reason to stipulate the powers of the invariants as they appear in the polynomial function for W (i.e., linear, quadratic, cubic, square root , cube root, etc.). Nor is there any a priori reason to specify ahead of time that the function W must be a polynomial in terms of the invariants. We could just as easily specify a rational form for W, or a hyperbolic form for W. The possibilities are endless. What we do is allow the experimental evidence to guide us in our choice for the functional form of the strain energy density function. This a recurring theme throughout the study of constitutive relationships.
Thus it may be advantageous to expand W as a polynomial function of only two invariants, or even one invariant. In this case we would be constructing W in what the algebraist would call a reduced subspace. Note the quadratic and the zero order strain terms in the expression above for si j. If we wanted a linear relationship between stress and strain, one could easily suppress the dependence of W on the first and third invariants of strain, and obtain a linear formulation directly.
Ideally, any theory that predicts the behavior of a material should incorporate parameters that are relevant to its microstructure (grain size, void spacing, etc.) and the physics/chemistry associated with the application. However, this requires a determination of volume averaged effects of microstructural phenomena reflecting nucleation, growth, and coalescence of microdefects that in many instances interact. This approach is difficult even under strongly simplifying assumptions. In this respect, Leckie (1981) points out that the difference between the materials scientist and the engineer is one of scale. He notes the materials scientist is interested in mechanisms of deformation and failure at the microstructural level and the engineer focuses on these issues at the component level. Thus the former designs the material and the latter designs the component. Here, we adopt the engineer’s viewpoint and note from the outset that continuum damage mechanics does not focus attention on microstructural events.
However, in keeping with the philosophy of the above discussion that a design life protocol should in some respect reflect the physics/chemistry of deterioration at the microstructure, then the thermally activated process(es) that drives erosion deterioration in gun barrels should also be captured by design methods in some fashion. Thus there must be an attempt to bridge design issues at the micro and macro levels. Although this methodology is by no means complete or comprehensive, the author wishes to sketch a framework that points to how one can include a thermo-chemical activated damage process into a design protocol that may lead to the ability of predicting barrel life using stochastic principles. This logical first approach may provide a practical model for erosion damage which macroscopically captures changes in microstructure induced by erosive ballistic processes. As noted in the summary, this approach lends itself to element "death" approaches found in some finite element algorithms. Thus one can "teach" elements in a gun barrel finite element analysis (FEA) to evolve and die based on suitable damage rate models. In this fashion the loss in rifling that occurs after repeated firings of a gun barrel can be modeled given a suitable rate of change in damage locally.
Damage Definition and the Concept of an Effective Stress
In this section the concept of a damage parameter is developed that captures the essence of a material undergoing a process that consumes its ability to sustain applied loads. A simple and elegant method of representing damage is associating a damage parameter with the loss of stiffness in a material undergoing a degradation process. Define E0 as the Young's modulus of a virgin material, and E as the current value of Young's modulus in a material subjected to a damage process, e.g., creep fatigue, chemical erosion, cyclic fatigue, or recession. Stiffness decreases with damage and is easy to assess in a test specimen. The damage parameter w can be defined as
(Eq 1)
where f is known as continuity. The damage parameter w ranges from 0 (E = E0, the undamaged state) to 1 (E = 0) where the material has totally lost the ability to sustain an applied load. Consequently the continuity parameter f ranges from 1 (undamaged) to 0 (material can not sustain load).
If we assume failure is the direct result of the evolution and accumulation of microdefects, i.e., the typical defect size is on the order of the average grain size of the material, then use of fracture mechanics principles becomes somewhat cumbersome in order to determine the life of the material. In addition, the damage process is a thermodynamic process, and the author notes that either damage parameter defined above can serve as a "state variable" in an engineering mechanics model. Thus let A0 represent the cross-sectional area of a test specimen subjected to a tensile load in the undamaged or reference state. Denote A as the current cross-sectional area at some point in time after a constant stress has been applied. As the material damages under load