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

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., we have too much shrinkage for the core to work The design engineer must be equipped with ability to compute the current stress state and the 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 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 $ 50,250.00 $ 30,000.00 $ 5,790.00 $ 1,605.00 $ 47,347.50 $ 10,000.00 $144,992.50 Year Tuition and Stipend Faculty Salary Fringes Faculty Fringes Students Indirects Travel Estimated Total $ 50,250.00 $ 30,000.00 $ 5,790.00 $ 1,605.00 $ 47,347.50 $ 10,000.00 $144,992.50 Year Tuition and Stipend Faculty Salary Fringes Faculty Fringes Students Indirects Travel Estimated Total $ $ $ $ $ $ Year Tuition and Stipend Faculty Salary Fringes Faculty Fringes Students Indirects Travel 50,250.00 30,000.00 5,790.00 1,605.00 47,347.50 10,000.00 $144,992.50 Estimated Total $ 434,977.50 Three year total Contributions by CSU 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 using sound 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 also convenient to derive constitutive models both for the elastic and the inelastic response of the material The first proposed contribution of the effort sketched out 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   A   A  W  A  d   d dW O    d  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 A W   A  d In the multiaxial case, the strain energy density is the sum of the contributions of all the stress components, i.e.,  *i j W  *i j  i j d i j 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  A  d In the multiaxial case this relationship is expressed as  *i j  *i j   i j d i j 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 W  i j  i j 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  i j F i j  k l  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 (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.,  W  i j   i j ij 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.,   i 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), 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  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 expressed as W W  I , I , I  where I  kk I   km mk I   km kn mn With  W  i j   i j ij then  W I1  W I  W I  i j   I1   i j I   i j I   i j Carrying out the implied differentiation, this expression can be restated as  i j  1 i j  2 i j  3 i k  j k where W  1 I W  2 I and W  3 I Note that '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 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 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 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  can be defined as    E E0    (Eq 1) where  is known as continuity The damage parameter  ranges from (E = E0, the undamaged state) to (E = 0) where the material has totally lost the ability to sustain an applied load Consequently the continuity parameter  ranges from (undamaged) to (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 crosssectional area at some point in time after a constant stress has been applied As the material damages under load A  A0 (Eq 2) and the macroscopic (or volume averaged) damage associated with this tensile specimen is  A0  A A0  (Eq 3) or, alternatively in terms of continuity A (Eq 4) A0 With this definition of damage the concept of an "effective" stress (a term often used in continuum damage mechanics literature) or current stress can be introduced Once again, assume a tensile test specimen A load 0 is applied and held constant over time Define current stress by the expression    P A  PA     A0  A   (Eq 5) 0  As the material damages under a constant load the current stress is initially equal to the applied stress, but approaches infinity (i.e., a singularity) in the limit as continuity decreases to zero The next step is defining how the damage parameter evolves with time One commonly used evolutionary law for the continuity function is a simple power law that has been used for metals undergoing creep damage The damage rate is expressed as d dt     C1    C2 (Eq 6) which yields the following expression for the continuity parameter  1  C1 ( ) C2 (C  1) t  (1 /( C 1)) (Eq 7) where C1 and C2 are material constants, 0 is the applied uniaxial stress (constant over time) From this, an expression for a time to failure (tf) can be obtained by noting that t t f (Eq 8)  0 (Eq 9) when This results in the following expression tf  C2 C1 ( ) (C  1) (Eq 10) which leads to the simplification of  as follows    1   t  t  f     (1 /( C 1)) (Eq 11) Utilizing equations 5, 10 and 11, a plot of effective stress versus time can be developed, as depicted in Figure Figure 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 C1 and C2 Furthermore the power law formulation of the damage rate is somewhat 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 Develop elastic constitutive model with anisotropic features that is coupled to a continuum damage mechanics model Will capture strain softening exhibited by graphite material Adopt an existing viscoplastic model that allows for different creep behavior in tension and compression and characterize the modeling parameters in an appropriate fashion Implement both the elastic constitutive model and the viscoplastic constitutive model into the COMSOL finite element analysis software Evolve an existing interactive reliability model based on Weibull analysis principles for materials with different behavior in tension and compression to accommodate anisotropic, stochastic failure behavior Relevance One of the nice things about phenomenological models is that you not have to stipulate what the phenomena is a priori Yes I am sure graphite does not exhibit viscoplastic behavior in the sense that metals exhibit it, but the behavior of graphite is non linear especially in the compressive regimes And the material must exhibit some sort of history dependence in that previous load histories are reflected in current behavior Graphites also appear to me to be linear elastic in the tensile regimes I believe both the linear and nonlinear behavior can be captured by the same phenomenological model by simply characterizing the modeling parameters in an appropriate fashion In monolithic ceramics, creep deformation and creep rupture are generally controlled by the physical and chemical properties of the grain and grain boundaries These microstructural phenomena, including the nucleation, growth, and coalescence of cavities, lead eventually to macrocracks which propagate and ultimately lead to component fracture Much work continues in determining the exact mechanisms responsible for the observed creep deformation and rupture behavior of ceramics An in-depth discussion of the micromechanical aspects of creep of ceramics is beyond the scope of this work Here attention will focus on the macromechanical phenomena (specifically, creep deformation) that can be accounted for in the proposed model Creep rupture is not considered in the model, although incorporating damage mechanics concepts into the present theory could yield a workable creep rupture model This task is reserved for a future enhancement Under isothermal conditions, a typical creep test entails abruptly applying a constant stress (or load) to a test specimen The material response in this type of test is conveniently described in terms of strain vs time In a typical creep strain vs time plot, the material response usually (but not always) exhibits three distinct regions of behavior The first, denoted primary creep, is identified as the initial period during which the strain rate diminishes with time Following the primary creep response, the strain rate stabilizes and remains constant for a (relatively) long period of time This region is denoted as secondary, or steady-state, creep In the third region, the strain rate accelerates, leading to specimen or component fracture (failure) or rupture The response in this region is denoted tertiary creep Many types of monolithic ceramics typically don’t exhibit tertiary creep behavior Therefore, many constitutive equations formulated for these material systems appropriately describe only the primary and secondary creep regions This is the case with the proposed model However, the tertiary creep response (including creep rupture) can be added to the current model by incorporating continuum damage mechanics concepts This task is reserved for a future enhancement CONSTITUTIVE THEORY The complete theory was derived in detail in the previous manuscript by the authors (Janosik and Duffy, 1998) For brevity, the derivation of the constitutive theory will only be summarized here The theory is derived from a scalar dissipative potential function attributed to Robinson (1978), which is identified here as  Under isothermal conditions, this function is dependent upon the applied Cauchy stress (ij) and internal state variable (aij), i.e.,       ,    ij ij  K 2  1 n  R m      F dF +  H  G dG    (12) In this formulation, ij and ij represent second rank Cartesian tensors Indicial notation is used with the convention that repeated indices imply summation The parameters K, , R, H, n, and m are material constants related to viscoplasticity The authors realize that several of these quantitites identified as material constants in the theory may be strongly temperature-dependent in a non-isothermal environment However, for simplicity, the present work is restricted to isothermal conditions A paper by Robinson and Swindeman (1982) provides the approach by which an extension can be made to non-isothermal environments This task is reserved for a future enhancement to the current model Specific macroscopic behavior (e.g., different behavior in tension and compression) is readily embedded in the model through the use of invariant theory A three-parameter yield criterion originally proposed by Willam and Warnke (1975) for concrete serves as the threshold flow criterion, F, for the model F  ~ ~ ~ I1 , J , J   c  ~    J2   r  ~       ~ I 3  1/ (13) c The functional form of the scalar state function, G, has similar mathematical formulation, following the framework of previously proposed constitutive models based on Robinson’s (1978) viscoplastic law For frame indifference, the scalar functions F and G (and hence ) must be form invariant under all proper orthogonal transformations Combining the principal stress invariants (, , for the scalar function F, and , , for the scalar function G) in an intelligent fashion allows the model to account for the viscoplastic response to general multiaxial loading conditions encountered when utilizing advanced monolithic ceramics in practical engineering applications Specifically, the proposed formulation permits the constitutive model to exhibit a sensitivity to the hydrostatic component of stress, and allows different behavior in tension and compression The additional parameters , c, and r() shown in equation (2) are presented in detail in the previous article (Janosik and Duffy, 1998) and will not be discussed herein With the functions F and G (and hence ) completely defined, the flow law (i.e., the inelastic strain rate) and the evolutionary law are derived from the potential function The flow law is derived from the potential function  by taking the partial derivative with respect to the applied stress, i.e.,  i j =   i j (14) Here, the inelastic strain rate vector is the gradient (directed outward normal) to level surfaces of , similar to the structure encountered in classical plasticity The evolutionary law is similarly derived from the flow potential The rate of change of the internal stress with deformation history is expressed as  i j = -h   i j (15) where h is a scalar hardening function of the inelastic state variable (i.e., the internal stress) only The complete multiaxial statement of the viscoplastic constitutive theory was presented in the previous manuscript by the authors The remainder of this paper will focus on applying the theory to account for asymmetric tensile and compressive behavior, as well as sensitivity to hydrostatic stress exhibited by advanced monolithic ceramics References “Miniature High Frequency Focused Ultrasonic Transducers for Minimally Invasive Imaging Procedures,” Fleischman, A., Modi, R., Nair, A., Talman, J., Lockwood, G., and Roy, S., Sensors and Actuators A, Volume 103, 2003, pp 76-82 “Miniature high frequency focused ultrasonic transducers for minimally invasive procedures,” presented originally at the 15th IEEE MEMS conference in Las Vegas, USA, January 2002 Timoshenko, S., Woinosky-Kreiger, S., Theory of Plate and Shells, McGraw Hill Classic Textbook Reissue, 1987 Eaton, W., Bitsie, F., Smith, J., Plummer, D., “A New Analytical Solution fro Diaphragm Deflection and its Application to a Surface-Micromachined Pressure Sensor,” Sandia National Laboratories, 1999 Altenbach, H., “An Alternative Determination of Transverse Shear Stiffness for Sandwich and Laminated Plates,” International Journal of Solids and Structures, Vol 37, 3502-3520, June 2000 Principal Investigator/Program Director (Last, First, Middle): Duffy, Stephen F BIOGRAPHICAL SKETCH NAME POSITION TITLE Duffy, Stephen F Professor Civil Engineering – CSU eRA COMMONS USER NAME duffysf EDUCATION/TRAINING (Begin with baccalaureate or other initial professional education, such as nursing, and include postdoctoral training.) INSTITUTION AND LOCATION DEGREE (if applicable) YEAR(s) FIELD OF STUDY University of Akron University of Akron University of Akron BSCE MSCE PhD 1978 1981 1987 Civil Engineering Geotechnical Engineering Engineering Mechanics A Positions and Honors Positions and Employment 1985 – Professor & Chair of Civil & Environmental Engineering, Cleveland State University Professional Memberships 1989 - Member, American Society of Testing Materials (ASTM) 1995 - Chair, ASTM Subcommittee C 28.02 Design and Evaluation of Advanced Ceramics 1997 - Member, American Ceramic Society 1978 - Member, American Society of Civil Engineers (ASCE) 1987 - Member, Board of Directors, Cleveland Section American Society of Civil Engineers 2002 - Secretary, Transportation Security Committee, American Society of Civil Engineers Honors 1994 ASTM Award of Appreciation - presented by ASTM Committee C-28 for Advanced Ceramics 1995 ASTM C-28 Advanced Ceramics Award 1995 ASME Award of Appreciation B Selected peer-reviewed publications (in chronological order) Books & Book Chapters "Design Practices for Whisker Toughened Ceramic Components," S.F Duffy and A.F Saleeb, in Engineered Materials Handbook Volume 4: Ceramics and Glasses, Samuel J Snyder tech ed., ASM International, pp 733 740, 1991 "Structural Design Methodologies for Ceramic Based Material Systems," S.F Duffy, A Chulya, and J.P Gyekenyesi, Chapter 14 in Flight Vehicle Materials, Structures and Dynamics Technologies Assessment and Future Directions, eds A.K Noor and S.L Venneri, The American Society of Mechanical Engineers, pp 265 285, 1992 (also published as NASA TM 103097) 3 Life Prediction Methodologies and Data for Ceramic Materials, ASTM STP 1201, C.R Brinkman and S.F Duffy eds., American Society for Testing and Materials, Philadelphia, 1994 "High Temperature Life Prediction of Monolithic Silicon Carbide Heat Exchanger Tubes," J.B Sandifer, M.J Edwards, T.S Brown, and S.F Duffy, in Life Prediction Methodologies and Data for Ceramic Materials, ASTM STP 1201, C.R Brinkman and S.F Duffy eds., American Society for Testing and Materials, Philadelphia, pp 373-389, 1994 "A Methodology to Predict Creep Life for Advanced Ceramics Using Continuum Damage Mechanics Concepts," T J Chuang and S.F Duffy, in Life Prediction Methodologies and Data for Ceramic Materials, ASTM STP 1201, C.R Brinkman and S.F Duffy eds., American Society for Testing and Materials, Philadelphia, pp 207-227, 1994 "Reliability and Life Prediction of Ceramic Composite Structures at Elevated Temperatures," S.F Duffy and J.P Gyekenyesi, in High Temperature Mechanical Behavior of Ceramic Composites, S.V Nair and K Jakus, eds., Butterworth-Heineman, Boston, pp 471-515, 1995 "Issues Relating to Reliability Based Component Design," S.F Duffy, in Handbook on Discontinuously Reinforced Ceramic Matrix Composites, K Bowman, S.K El Rahaiby, and J Wachtman eds., DoD Ceramics Information Analysis Center Purdue University, pp 310-355, 1995 "Design with Brittle Materials," S.F Duffy and L.A Janosik, in Engineered Materials Handbook: Volume 20 Material Selection and Design, G Dieter, volume chair, ASM International, pp 622-638, 1997 "Life Prediction of Structural Components," S.F Duffy, L.A Janosik, A.A Wereszczak, B Schenk, A Suzuki, J Lamon, and D.J Thomas, in Progress in Ceramic Gas Turbine Development: Volume Ceramic Gas Turbine Component Development and Characterization, M van Roode, M.K Ferber, and D.W Richerson, volume chairs, ASME Press, pp 553-606, 2003 Refereed Journal Articles "A Viscoplastic Constitutive Theory for Transversely Isotropic Materials," D.N Robinson, S.F Duffy and J.R Ellis, in Third Symposium on Nonlinear Relations for High Temperature Applications, NASA CP 10010, Akron, Oh., pp 25, June, 1986 "A Viscoplastic Constitutive Theory for Metal Matrix Composites at High Temperature," D.N Robinson, S.F Duffy and J.R Ellis, in 1987 Pressure Vessels and Piping Conference, PVP Vol 123, San Diego, Ca., pp 49 56, June, 1987 (also published as NASA CR-17953) "A Continuum Deformation Theory for Metal Matrix Composites at High Temperature," D.N Robinson and S.F Duffy, Journal of Engineering Mechanics, Vol 116, No 4, pp 832 844, April, 1990 "A Unified Inelastic Constitutive Theory for Sintered Powder Metals," S.F Duffy, Mechanics of Materials, Vol 7, No 3, pp 245 254, December, 1988 "Noninteractive Macroscopic Statistical Failure Theory for Whisker Reinforced Ceramic Composites," S.F Duffy and S.M Arnold, Journal of Composite Materials, Vol 24, No 3, pp 293 308, March, 1990 "Noninteractive Macroscopic Reliability Model for Ceramic Matrix Composites with Orthotropic Material Symmetry," S.F Duffy and J.M Manderscheid, Journal of Engineering for Gas Turbines and Power, Vol 112, No 4, pp 507 511, October, 1990 (also published as NASA TM 101414) "Analysis of Whisker Toughened Ceramic Components A Design Engineer's Viewpoint," S.F Duffy, J.M Manderscheid, and J.L Palko, Ceramic Bulletin, Vol 68, No 12, pp 2078 2083, December, 1989 (also published as NASA TM 102333) "Extension of a Noninteractive Reliability Mod for Ceramic Matrix Composites," S.F Duffy, R.C Wetherhold, and L.K Jain, Transactions of the ASME Journal of Engineering for Gas Turbines and Power, Vol 115, No 1, pp 205 207, January, 1993 (also published as NASA CR 185267) 9 "Structural Reliability Analysis of Laminated CMC Components," S.F Duffy, J.L Palko, and J.P Gyekenyesi, Transactions of the ASME Journal of Engineering for Gas Turbines and Power, Vol 115, No 1, pp 103 108, January, 1993 (also published as NASA TM 103685) 10 "Reliability Analysis of Structural Components Fabricated from Ceramic Materials Using a Three Parameter Weibull Distribution," S.F Duffy, L.M Powers, and A Starlinger, Transactions of the ASME Journal of Engineering for Gas Turbines and Power, Vol 115, No 1, pp 109 116, January, 1993 (also published as NASA TM 105370) 11 "Analysis of Whisker Toughened CMC Structural Components Using an Interactive Reliability Model," S.F Duffy and J.L Palko, AIAA Journal, Vol 32, No 5, pp 1043-1048, May, 1994 (also published as NASA CR 190755) 12 "Reliability Analysis of Laminated Ceramic Matrix Composites Using Shell Subelement Techniques," A Starlinger, D.J Thomas, S.F Duffy, and J.P Gyekenyesi, AIAA Journal, Vol 31, No 11, pp 21812183, November, 1993 (also published as NASA TM 105413) 13 "Reporting Strength Data and Estimating Weibull Distribution Parameters for Advanced Ceramics," S.F Duffy, G.D Quinn and C.A Johnson, ASTM Standard Practice C 1239 94 14 "Parameter Estimation Techniques Based on Optimizing Goodness of Fit Statistics for Structural Reliability," A Starlinger, S.F Duffy, and J.L Palko, Proceedings of the 10th Biennial Conference on Reliability, Stress Analysis, and Failure Prevention, DE-Vol 55, R.J Schaller ed., ASME, Albuquerque, NM, pp 65-77, September, 1993 15 "Composites Research at NASA Lewis Research Center," S.R Levine, S.F Duffy, A Vary, M.V Nathal, R.V Miner, S.M Arnold, M.G Castelli, D.A Hopkins, and M.A Meador, Composites Engineering, Vol 4, No 8, pp 787-810 16 "Trends in the Design and Analysis of Components Fabricated From CFCCs," S.F Duffy, J.L Palko, J.B Sandifer, C.L DeBellis, M.J Edwards, and D.L Hindman, Transactions of the ASME Journal of Engineering for Gas Turbines and Power, Vol 119, No 1, pp 6, January, 1997 17 "An Overview of Engineering Concepts and Current Design Algorithms for Probabilistic Structural Analysis," S.F Duffy, J Hu, and D.A Hopkins, in Proceedings of the 1995 Design Engineering Technical Conferences - Volume 2, DE-Vol 83, Boston, Massachusetts, pp 3-16, September, 1995 18 "A Viscoplastic Constitutive Theory for Monolithic Ceramics - I," L.A Janosik and S.F Duffy, Transactions of the ASME Journal of Engineering for Gas Turbines and Power, Vol 120, No 1, pp 155 161, January, 1998 19 "Weibull Analysis Effective Volume and Effective Area for a Ceramic C – Ring Test Specimen," S.F Duffy, E.H Baker, A.A Wereszczak and J.J Swab, ASTM Journal of Testing and Evaluation, Vol 33, No 4, pp 233-238, July 2005 Industry Standards "Standard Practice for Reporting Uniaxial Strength Data and Estimating Weibull Distribution Parameters for Advanced Ceramics," S.F Duffy, G Quinn, and C Johnson, ASTM Designation: C 1239, (1995, 2000) "Fine Ceramics (Advanced Ceramics, Advanced Technical Ceramics) - Weibull Statistics for Strength Data," S.F Duffy, ISO Designation: FDIS 20501, (2003) Industry Reports "Final Report: Modifications to NIKE2D and NIKE3D to Accommodate Weibull Analytical Methods," J.L Palko, E.H Baker, and S.F Duffy, submitted to the Lawrence Livermore National Laboratory, Livermore, California, September 2001 "Evaluation of Effective Volume & Effective Area for Four Point Bend Sectored Tube Test Specimen," S.F Duffy and E.H Baker, report submitted to the Army Research Laboratory, Aberdeen, Maryland, September 2002 3 "Ceramic Armor Reliability Analysis and Evaluation of Circular, Hexagonal, and Square Geometries," S.F Duffy, report submitted to the Army Research Laboratory, Aberdeen, Maryland, February 2003 "Reliability Analysis Of Ceramic Gun Barrel," S.F Duffy, report submitted to the Army Research Laboratory, Aberdeen, Maryland, August 2003 "Forensic Engineering Report: Reinforced Concrete Slabs Clearon Building #311 South Charleston, West Virginia," S.F Duffy, submitted to Dechert LLP, October 2003 "Effective Volume/Area & Confidence Bounds For 120 mm Ceramic Gun Barrel," S.F Duffy and E.H Baker, report submitted to the Army Research Laboratory, Aberdeen, Maryland, May 2004 "Estimates Of The Geometry/Load History Parameters Associated With Time Dependent Reliability Analysis: C-Ring And The Sectored Flex Bar Test Specimens," S.F Duffy, report submitted to the Army Research Laboratory, Aberdeen, Maryland, May 2004 ”Burst Tube Test Analysis," S.F Duffy and E.H Baker, report submitted to the Army Research Laboratory, Aberdeen, Maryland, January 2006 "Evaluation of Effective Volume & Effective Area for O – Ring Test Specimen, "S.F Duffy and E.H Baker, report submitted to the Army Research Laboratory, Aberdeen, Maryland, April 2005 10 "Size Effects in Ceramic Materials: Computational Issues Associated With Parameter Estimations," S.F Duffy and E.H Baker, Army Research Laboratory Report #ARL-CR-0560, April 2005 11 "Analysis and Design of a Nanoporous Membrane for a Drug Delivery System," E.H Baker, S.F Duffy, and J.L Palko, report submitted to the Cleveland Clinic, Cleveland, Ohio, October 2005 12 "Designing Advanced Ceramic Parts That Last," ANSYS Solutions Magazine, Vol 7, Issue 1, pp 912, January 2006 C Research Support "Reliability Analysis of Micro-Electro-Mechanical Systems (MEMS)" Role: Co-Investigator Agency: NASA Agency Research No.: C-72118-T Time Period: October 1, 2002 through September 30, 2005 Provide design and analysis relative to ceramic materials in support of a team developing a nanoporous membrane for drug delivery "Ceramic Gun Barrel Technology" Role: Co-Investigator Agency: Army Research Laboratory Agency Research No.: W911QX04P0502 Time Period: April 23, 2002 May 31, 2005 Includes finite element modeling and study of, c-ring specimens, sectored flexure bars, the internal pressure burst test specimen and the o-ring specimen Additionally, upgrades made to WeibPar and CARES in order to accommodate the anticipated large finite element models of the ceramic gun barrel "Reliability Analysis of Microturbine Components" Role: Principle Investigator Agency: Oak Ridge National Laboratory Agency Research No.: 4000021849, Time Period: April 1, 2002 March 31, 2006 Update and enhance various software algorithms (ANSCARES, CARES, and WeibPar) that are provided to DER industrial partners Provide technical support (e.g., theoretical development and modeling advice) to DER industry partners Support interfacing the ORNL software algorithm IRASoft with CARES and WeibPar "A Short Course on Weibull Analysis and Parameter Estimation" Role: Principle Investigator Agency: Oak Ridge National Laboratory Agency Research No.: 4000036595 Time Period: March 10, 2003 through March 12, 2002 Short course provided information on the following elements: ceramic reliability background and theory; Parameter estimation background and theory; an overview of the analytical approach to ceramic analysis and the life or probability of failure estimation; interpreting and processing the ceramic test data and using the test data with CARES; utilizing CARES, explanation of the control input parameters; post processing, interpreting CARES results; hands-on examples "Curriculum Development: Work Zone Safety" Role: Principal Investigator Agency: Department of Education Agency Research No.: P116Z040299 Time Period: June 1, 2004 August 31, 2005 Funding supported three goals: the establishment of a safety curriculum at the university level; the establishment of a training curriculum for construction personnel; and the creation of a summer intern research program at the undergraduate level Budget The budget consists of supporting two graduate students for three consecutive semesters (fall, spring and summer under tuition grants (TGs) The other major component of the budget is the purchase of a license from the ANSYS corporation The Fenn College of Engineering at Cleveland State University has a site license for MATLAB $45,434.7 $ 2,000.00 $12,000.0 $59,434.7 Student Tuition Grants Travel ANSYS license Total .. .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. .. 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... 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