Reliability Analysis for NuclearGraphite
Doctoral Research Proposal
Prepared by:
Chengfeng Xiao;
Department of Civil and Environmental Engineering
ClevelandStateUniversity
Cleveland, Ohio, 44115-2214
Tel: 216-255-8939
E mail:
Dr. Stephen F. Duffy, Ph.D., P.E.
Professor, Department of Civil and Environmental Engineering
ClevelandStateUniversity
Cleveland, Ohio44115-2214
Tel: 216 687-3874
Email:
TABLE OF CONTENTS
1. Introduction
2. Failure criteria isotropic models of brittle materials
2.1Integra basis
2.2New Integra basis
2.3One-Parameter model: Von Mises yield criterion
2.4Two-Parameter model: Drucker-Prager criterion
2.5Three-Parameter model: William-Warnke criterion
3. Develop the isotropic model to anisotropic model (transverse isotropic)
3.1Anisotropic model of Drucker-Prager
3.2New model with five invariants
4. Proposed Research
4.1Correct and develop the new model and estimation of parameters
4.2Compare new model’s result with experiment data
4.3Reliability model based on the new model
4.4Application
5.Reference
- Introduction
With the development of society, more and more energy is required. Nuclear power plants generate sustainable, economical, safe and reliable energy. Figure 1 shows the evolution of reactors since their introduction in the 1950s.The core of Gen IV reactor, for example Pebble Bed Reactor, is conducted by large amounts of graphite components, including replaceable fuel blocks, replaceable and permanent moderator blocks and core support posts.During reactor operation the graphite components of the core are subjected to complex stress states arising from structural loads, thermal gradients, neutron irradiation damage, and seismic events. Of particular concern is the potential for crack formation and even rupture in individual blocks. Therefore, failure theories that predict reliability of graphite components are desired.
Graphite is ananisotropic material. The compressive stress is not equal to the tensile stress and the stress-strain relationship depends on the orientation of the material. But graphite does not act in a truly brittle manner. It has a biaxial softening where failure stresses in a biaxial stress state are lower than the uniaxial failure stresses. These properties make it difficult to set up an accurate failure model.Deterministic and probabilisticapproaches are the main research direction to predict graphite behavior.
Traditional deterministic failure criteria models includevon Mises criterion, Coulumb-Mohr Criterion and so on. Griffith energy balance criterion for fracture states provides an isotropic brittle continuum. Batdorf and Crose represented the first attempt at extending fracture mechanics to reliability analysis in a consistent and rational manner. For discrete particle-toughened materials, such as graphite, failure remains a stochastic process. So some researchers developed probabilistic approaches. Weibull firstly introduced a good method for quantifying variability in fracture strength and the size effect in brittle material, which is based on Weakest Link Theory. Barnett and Freudenthal proposed Principle of Independent Action. Dr. Duffy et al. presented an array of failure models to predict reliability of ceramic components that have isotropic, transversely isotropic, or orthotropic material symmetry.
The purpose of the paper is exploring one general model of failure in convenience mathematical. Through the application of invariant theory and Cayley-Hamilton theorem, consideration of transverse isotropic material direction, an integrity basis with finite number of invariants is developed. Using the particular vector space, different kinds of failure criterion model are rebuilt, from simple Von Mises criteria, Drucker-Prager criteria to complex William-Warnke, from isotropic cases to anisotropic ones. The strength parameters defining the new general failure criteria model are assumed to be random variables, thereby transforming the deterministic design into a probabilistic criterion. The process will realized by Fortran or Matlabprogram and Monte Carlo method.The results willbe extensively compared with the experimental data from Oar Ridge National Laboratory.The model will also use the output of the finite element analysis(Ansys) to computer element-by-element reliability. On the basis of the weakest link concept the component survivability is simply the produce of individual element reliabilities with Weibull distribution.
- Failure criteria function of brittle Materials
The strength of brittle materials, such as concrete and graphite, under multiaxial stresses is a function of the state of stress and cannot be predicted by liminations of simple tensile, compressive, and shearing stresses independently of each other.
is orthogonal unit vector that represent the local principal material direction, and represents the Cauchy stress tensor. For isotropic materials, is equal to 0.
2.1 Integra basis
If is expressed as a polynomial in all possible traces and products of traces of and, which are divided into infinite groups as following
Group 1:,,,
Group 2:,,,
Group 3:,,,
Group 4:,,,
By the Cayley-Hamilton theorem, the tensor will satisfy its own characteristic polynomial
Where
Multiplying equation by gives
Taking the trace yields
Or
It shows that is a function of,and. According to Induction method,
So the group1 just remain , , are irreduced.
For group 2
We can conclude. So these invariants are also eliminated.
For the group 3
Multiplies then
The same reason as the first group, the third group just remain, .
For the fourth group
,
So the fourth group is eliminated.
Above all, the integrity basis for the scalar function include five invariants as following
, ,, , .
2.2 New integrity basis
A failure criterion of isotropic materials based upon state of stress must be an invariant function of the state of stress, i.e., independent of the choice of the coordinate system by which stress is defined. One method of representing such a function is to use the principal stresses, i.e.,
to indicate the general functional form of the failure criterion. In general case of a multiaxial state of stress, the approach to establishing a failure function is difficult to pursue. It is also difficult to supply both a geometrical and a physical explanation of failure on this basis. So a new available basis should be explored.
or
which is a cubic equation forwith three real roots
Since the principal stresses cannot depend on the choice of coordinate axes, the quantities cannot change if the coordinate system is refined; hence they are referred to as invariants of the stress tensor.
The stress tensor can be expressed as the sum of a purely hydrostatic stress and a deviation from the hydrostatic state
Assume
Then we get the new basis
(or)is the first invariant of stress tensor and represents a purely hydrostatic pressure;(or)is the second invariant of deviator tensor and represents a purely hydrostatic pressure.
(or) is the third invariant of deviator tensor and representsthe angle in deviator plane.
For sake of understanding, all different kinds of failure models are projected on three planes: the principle stress plane(), the deviator plane through origin normal to the hydrostatic line(plane) and meridian plane.
Based on the three projected graphics on different planes, it is clear to image the failure surface in two dimensions space.
2.3One-Parameter model: Von Mises yield criterion
The octahedral shearing stress is a convenient alternative choice to the maximum shearing stress as the key variable for causing yield of materials which are pressure-independent. This is known as Von Mises yield criterion, dating from 1913.
or
Run the experiment
1) Uniaxial tensile stress:
;
Substitutive these invariants into failure function, then solve the equation and get
So the failure function is
In von Mises model that term() disappears means hydrostatic pressure has no effective on the failure. It is very simple in mathematical, but the brittle materials, such as graphite, are dependent on pressure. So we need to expand von Mises to pressure-dependent model. It is Drucker-Prager.
2.4Two-Parameter model: Drucker-Prager criterion
One failure surface yield criterion model is a simple modification of the von Mises yield criterion in the form
or
run the experiment tests:
1) Uniaxial tensile stress:
;
Substitutive invariants into the failure function, then solve the equation and get
2) Uniaxial compressive stress:
;
Substitutive invariants into the failure function, then solve the equation and get
Solve the equations and get the parameters
The failure surface in principal-stress space is clearly a right-circle cone whose meridian and cross section on -plane are shown in the figures. The Drucker-Prager Model has two shortcomings: 1)ordepends linearly on; 2) and independence on the angle of
The meridian lines have been experimentally shown to be curved, and the trace of the failure surface on deviatoric section is noncircular but depends on the angle of similarity. As a first step to generalize the Drucker-Prager surface, two approaches can be taken: (1) to assume a parabolic dependence of (or)on(or) but to keep the deviatoric section independent of, that is , to retain the circular cross section, and (2) to retain the linear relation but let the deviatoric sections exhibit dependence, i.e., have a noncircular cross section. This will lead to three-parameter model.
2.5Three-Parameter model: William-Warnke criterion
or
or
Run the isotropic biaxial tests
1) Uniaxial tensile stress:
;
2) Uniaxial compressive stress:
;
3) Equal biaxial compressive stress:
;
Solve the equations, we will get formulations.is elliptical approximation, so
Figure
3Develop the isotropic model to anisotropic model (transverse isotropic)
3.1Anisotropic model of Drucker-Prager
Model explaining, material strong direction
Run 4 tests:
Isotropic plane
1) Uniaxial tensile stress:
;
2) Uniaxial compressive stress:
;
Strong direction
4) Uniaxial tensile stress:
;
5) Uniaxial compressive stress:
.
Figure
Analysis the limination of this model
3.2New model with five invariants
Run 5 tests:
Isotropic plane
1) Uniaxial tensile stress:
;
2) Uniaxial compressive stress:
;
3) Equal biaxial compressive stress:
;
Strong direction
4) Uniaxial tensile stress:
;
5) Uniaxial compressive stress:
.
Use equations to make Matrixes as the following:
Solve
or
If,, , , then
4. Proposed Research
4.5Correct and develop the new model
Derivate the formulation ofand, then obtain the slope of meridian line
W.F. Chen equation (5.30)
According to relationshipand, we get a new equation
In order to find the maximum of, assume, run the following test
Then the invariants are calculated
Substitute invariants into failure function
We get
Expanding and combining the function, we get
Failure surface
Example
If,,,, then
Strong direction
- Case and
- Case and
- Case and
which matches the Matlab program figures.
In order to find the maximum of, assume, run the following test
Then the invariants are calculated
Substitute invariants into failure function
Then
Expanding and combining the function
Failure surface
Example
If,, , , then
Strong direction
and , which matches the Matlab program figures.
So the slope of meridian line is
So the failure function is
Examples
Strong direction:,
1) Case and
2) Case and
3) Case and
The following figure is proposed one we need to explore.
4.2Correct and develop the new model and estimation of parameters
4.3Compare new model’s result with experiment data
4.4Reliability model based on the new model
4.5Application
5. Reference