Modeling the Stress Strain Relationships and Predicting Failure Probabilities for Graphite Core Components

Modeling the Stress Strain Relationships and Predicting FailureProbabilities for Graphite Core Components Request for Pre-Application RPA Submitted to: Center for Advanced Energy Studie

Modeling the Stress Strain Relationships and Predicting Failure

Probabilities for Graphite Core Components

Request for Pre-Application (RPA)

Submitted to:

Center for Advanced Energy Studies (CAES)

sponsored by the Department of Energy (DOE)

Office of Nuclear Energy (NE)

Technical Work Scope Identifier No G4A-1

Gen IV Materials (NGNP)

Principle Investigators:

Stephen F Duffy PhD, PE, F.ASCE Professor and Chair, Civil & Environmental Engineering Director, CSU University Transportation Center

Modeling the Stress Strain Relationships and Predicting Failure

Probabilities for Graphite Core Components

Summary (Abstract)

In order to assess how close the component is to a failure state accurate stress states are required, both the elastic and inelastic components One also needs to know the strain state to assess deformations in order to ascertain serviceability issues relating to failure, e.g., do we have too much shrinkage for the core to work

The design engineer must be equipped with ability to compute the current stress state andthe current strain state given a load history But you also need a failure model that reflects the variability of the material (at least in tensile regimes) and possible anisotropy And it would be nice to assess the damage exposure to radiation has on the mechanical properties of the material and how that damage will impact the stress state and strain state But assessing stress and strain

as opposed to failure predictions are distinct modeling efforts

Having the capability to accurately predict stress and strain states in graphite core is critical to the Gen IV initiative Having accurate stress states facilitates assessment of graphite component failure probabilities In addition, accurately assessing strain states is paramount to designing the geometry of core subcomponents The effort described in this pre-proposal

sketches a framework aimed at developing the mechanistic modeling capabilities needed for implementation of the next generation of commercially available nuclear reactors

To mechanistically assess graphite core components the following analytical tools are required:Task #1 Thermoelastic constitutive (stress-strain) relationship that exhibits different behavior intension 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 fordifferent 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 asfollows where an individual graduate student is assigned to each task

Year 1 $ 50,250.00 Tuition and Stipend

$144,992.50 Estimated Total $ 434,977.50 Three year total

in this pre-application

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

W

* 0

Alternatively, the area above the  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

 



A d

In the multiaxial case this relationship is expressed as

The strain energy density W and the complementary energy density are functions of

strain, ij, and stress ij, respectively It is evident that the two are related through the following relationship

j i j i

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

 k l

j i j

An elastic material defined by this expression is termed a Cauchy elastic material Note that F ij

is a second order tensor operator (i.e., the result is a second order tensor) that is a function of a second order tensor (ij)

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.,

j i

j i j

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.,

j i

j i j

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), theconcepts 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  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 i j Thus W can be

j i j

i j

i j

i

W W

1 1

I I

I I

i j

i j

where

1 1

Note that 's are not tensor quantities in the same sense that the s are not tensor quantities in the same sense that the I 's are not tensor quantities in the same sense that the 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

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

i 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 repeatedfirings 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 E 0 as the Young's are not tensor quantities in the same sense that the s modulus of a

virgin material, and E as the current value of Young's are not tensor quantities in the same sense that the 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  can be defined as

where  is known as continuity The damage parameter  ranges from 0 (E = E 0, the undamaged

state) to 1 (E = 0) where the material has totally lost the ability to sustain an applied load

Consequently the continuity parameter  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 A 0 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

2

1

C

C dt

 ( 1 /( 1 ))

2 0

where C 1 and C 2 are material constants, 0 is the applied uniaxial stress (constant over time)

From this, an expression for a time to failure (t f) can be obtained by noting that

1

2 0



C C

which leads to the simplification of  as follows

)) 1 /(

1 ( 2

0 100 200 300 400 500 600 700 800 900

Figure 1 Rising stress as a function of time and damage – power law damage rate

Note that the power law expression describing the evolution of damage has very little effect during a good portion of the life of the material However, as enough damage is accumulated a somewhat rapid increase in deterioration occurs This behavior is easily influenced by the choice

in the model parameters C 1 and C 2 Furthermore the power law formulation of the damage rate issomewhat generic, but there are some arguments in the literature that this formulation represents the physics of damage accumulation in metals undergoing creep damage The author does not advocate this formulation for damage arising from surface erosion kinetics (see discussions in a later section for a more appropriate rate law) It has been introduced here to facilitate the

presentation of concepts