• Nuclear Engineering and Technology
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• v.55 no.1
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• pp.131-143
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• 2023

Constitutive equations in a nuclear reactor safety analysis code are mostly empirical correlations developed from experiments, which always accompany uncertainties. The accuracy of the code can be improved by modifying the constitutive equations fitting wider range of data with less uncertainty. Thus, the sensitivity of the code with respect to the constitutive equations is evaluated quantitatively in the paper to understand the room for improvement of the code. A new methodology is proposed which first starts by dividing the thermal hydraulic conditions into multiple sub-regimes using self-organizing map (SOM) clustering method. The sensitivity analysis is then conducted by multiplying an arbitrary set of coefficients to the constitutive equations for each sub-divided thermal-hydraulic regime with SOM to observe how the code accuracy varies. The randomly chosen multiplier coefficient represents the uncertainty of the constitutive equations. Furthermore, the set with the smallest error with the selected experimental data can be obtained and can provide insight which direction should the constitutive equations be modified to improve the code accuracy. The newly proposed method is applied to a steady-state experiment and a transient experiment to illustrate how the method can provide insight to the code developer.

• Korea-Australia Rheology Journal
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• v.14 no.1
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• pp.33-45
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• 2002

Recent results obtained for the port-pom model and the constitutive equations with time-strain separability are examined. The time-strain separability in viscoelastic systems Is not a rule derived from fundamental principles but merely a hypothesis based on experimental phenomena, stress relaxation at long times. The violation of separability in the short-time response just after a step strain is also well understood (Archer, 1999). In constitutive modeling, time-strain separability has been extensively employed because of its theoretical simplicity and practical convenience. Here we present a simple analysis that verifies this hypothesis inevitably incurs mathematical inconsistency in the viewpoint of stability. Employing an asymptotic analysis, we show that both differential and integral constitutive equations based on time-strain separability are either Hadamard-type unstable or dissipative unstable. The conclusion drawn in this study is shown to be applicable to the Doi-Edwards model (with independent alignment approximation). Hence, the Hadamardtype instability of the Doi-Edwards model results from the time-strain separability in its formulation, and its remedy may lie in the transition mechanism from Rouse to reptational relaxation supposed by Doi and Edwards. Recently in order to describe the complex rheological behavior of polymer melts with long side branches like low density polyethylene, new constitutive equations called the port-pom equations have been derived in the integral/differential form and also in the simplifled differential type by McLeish and carson on the basis of the reptation dynamics with simplifled branch structure taken into account. In this study mathematical stability analysis under short and high frequency wave disturbances has been performed for these constitutive equations. It is proved that the differential model is globally Hadamard stable, and the integral model seems stable, as long as the orientation tensor remains positive definite or the smooth strain history in the flow is previously given. However cautious attention has to be paid when one employs the simplified version of the constitutive equations without arm withdrawal, since neglecting the arm withdrawal immediately yields Hadamard instability. In the flow regime of creep shear flow where the applied constant shear stress exceeds the maximum achievable value in the steady flow curves, the constitutive equations exhibit severe instability that the solution possesses strong discontinuity at the moment of change of chain dynamics mechanisms.

• Proceedings of the Korean Society For Composite Materials Conference
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• 2004.10a
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• pp.147-150
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• 2004

In order to describe the mechanical behavior of highly anisotropic and asymmetric materials such as fiber­reinforced composites, the elastic-plastic constitutive equations were used here based on the recently developed yield criterion and hardening laws. As for the yield criterion, modified Drucker-Prager yield surface was used to represent the orthotropic and asymetric properties of composite materials, while the anisotropic evolution of back­stress was accounted for the hardening behavior. Experimental procedures to obtain the material parameters of the hardening laws and yield surface are presented for 3D Circular Braided Glass Fiber Reinforced Composites. For verification purpose, comparisons of finite element simulations using the elastic-plastic constitutive equations, anisotropic elastic constitutive equations and experiments were performed for the three point bending tests. The results of finite element simulations showed good agreements with experiments, especially for the elastic-plastic constitutive equations with yield criterion considering anisotropy as well as asymmetry and anisotropic back stress evolution rule.

• Structural Engineering and Mechanics
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• v.18 no.5
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• pp.565-589
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• 2004

The present paper concerns the macroscopic overall description of rheologic properties for steel wire and synthetic fibre cables under variable loading actions according to non-linear creep and/or relaxation theory. The general constitutive equations of non-linear creep and/or relaxation of tension elements - cables under one-step and the variable stress or strain inputs using the product and two types of additive approximations of the kernel functions are presented in the paper. The derived non-linear constitutive equations describe a non-linear rheologic behaviour of the cables for a variable stress or strain history using the kernel functions determined only by one-step - constant creep or relaxation tests. The developed constitutive equations enable to simulate and to predict in a general way non-linear rheologic behaviour of the cables under an arbitrary loading or straining history. The derived constitutive equations can be used for the various tension structural elements with the non-linear rheologic properties under uniaxial variable stressing or straining.

• Nuclear Engineering and Technology
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• v.56 no.8
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• pp.2999-3029
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• 2024

To improve the predictive capability of a nuclear thermal hydraulic safety analysis code by developing a better constitutive equation for individual phenomenon has been the general research direction until now. This paper proposes a new method to directly use complex experimental data obtained from integral effect test (IET) to improve constitutive models holistically and simultaneously. The method relies on the sensitivity of a simulation result of IET data to the multiple constitutive equations utilized during the simulation, and the sensitivity of individual model determines the direction of modification for the constitutive model. To develop a robust and generalized method, a clustering algorithm using an artificial neural network, sample space size determination using non-parametric statistics, and sampling method of Latin hypercube sampling are used in a combined manner. The value of the proposed methodology is demonstrated by applying the method to the ATLAS DSP-05 IET experiment. A sensitivity of each observation parameter to the constitutive models is analyzed. The new methodology suggested in the study can be used to improve the code prediction results of complex IET data by identifying the direction for constitutive equations to be modified.

• Transactions of the Korean Society of Mechanical Engineers
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• v.12 no.1
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• pp.64-71
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• 1988

A special set of constitutive equations is formulated to predict elastic-plastic strain hardening responses of porous metals. Including the effect of the material's strain hardening in the yield function, the constitutive equations are capable of showing no dip phenomena in uniaxial strain compression and prediction work-hardening response for plastically precyled porous metal. The proposed constitutive equations are compared with experimental data for porous tungsten.

• Transactions of Materials Processing
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• v.13 no.2
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• pp.160-167
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• 2004

In order to theoretically analyze the creep behavior of high Cr steel at $600^{\circ}C$, a unified elasto-viscoplastic constitutive model based on the consideration of dislocation density is proposed. A combination of a kinetic equation describing the mechanical response of a material at a given microstructure in terms of dislocation glide and evolution equations for internal variables characterizing the microstructure provides the constitutive equations of the model. Microstructural features of the material such as the grain size and spacing between second phase particles are directly implemented in the constitutive equations. The internal variables are associated with the total dislocation density in a simple model. The model has a modular structure and can be adjusted to describe a creep behavior using the material parameters obtained from uniaxial tensile tests.

• Transactions of Materials Processing
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• v.17 no.2
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• pp.97-101
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• 2008

Unique constitutive behavior of magnesium alloys as one of hexagonal close packed(hcp) metals has been implemented into the commercial finite element program ABAQUS. The constitutive equations can represent asymmetry in tension-compression yield stresses and flow curves. For the verification purpose, the springback of AZ31B magnesium alloy sheet was measured using the unconstrained cylindrical bending test proposed in Numisheet'2002 benchmark committee. Besides the developed constitutive models, the isotropic models based on tensile and compressive properties were also considered for comparison purpose. The predicted results by the finite element analysis and corresponding experiments showed enhanced prediction capability in springback analysis.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.24 no.5 s.176
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• pp.1075-1083
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• 2000

The prediction of the inelastic behavior of the structure is an essential part of reliability assessment procedure, because most of the failures are induced by the inelastic deformation, such as creep and plastic deformation. During decades, there has been much progress in understanding of the inelastic behavior of the materials and a lot of inelastic constitutive equations have been developed. These equations consist of the definition of inelastic strain and the evolution of the state variables introduced to quantify the irreversible processes occurred in the material. With respect to the definition of the inelastic strain, the inelastic constitutive models can be categorized into elastoplastic model, unified viscoplastic model and separated viscoplastic model and the different integration methods have been applied to each category. In the present investigation, the generalized integration method applicable for various types of constitutive equations is developed and implemented into ABAQUS by means of UMAT subroutine. The solution of the non-linear system of algebraic equations arising from time discretization with the generalized midpoint rule is determined using line-search technique in combination with Newton method. The strategy to control the time increment for the improvement of the accuracy of the numerical integration is proposed. Several numerical examples are considered to demonstrate the efficiency and applicability of the present method. The prediction of the inelastic behavior of the structure is an essential part of reliability assessment procedure, because most of the failures are induced by the inelastic deformation, such as creep and plastic deformation. During decades, there has been much progress in understanding of the inelastic behavior of the materials and a lot of inelastic constitutive equations have been developed. These equations consist of the definition of inelastic strain and the evolution of the state variables introduced to quantify the irreversible processes occurred in the material. With respect to the definition of the inelastic strain, the inelastic constitutive models can be categorized into elastoplastic model, unified viscoplastic model and separated viscoplastic model and the different integration methods have been applied to each category. In the present investigation, the generalized integration method applicable for various types of constitutive equations is developed and implemented into ABAQUS by means of UMAT subroutine. The solution of the non-linear system of algebraic equations arising from time discretization with the generalized midpoint rule is determined using line-search technique in combination with Newton method. The strategy to control the time increment for the improvement of the accuracy of the numerical integration is proposed. Several numerical examples are considered to demonstrate the efficiency and applicability of the present method.

• Computers and Concrete
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• v.25 no.2
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• pp.181-192
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• 2020

Nowadays, steel fiber reinforced concretes (SFRCs) are widely used in practical applications. Significant experimental research has thus been carried out to determine the constitutive equations that represent the behavior of SFRCs under multiaxial loadings. However, numerical modelling of SFRCs via FEM has been challenging due to the complexities of the implementation of these constitutive equations. In this study, following the literature, a plasticity model is constructed for the behavior of SFRCs that involves the Willam-Warnke failure surface with the relevant evolution laws and a non-associated flow rule for determining the plastic deformations. For the precise (yet rapid) integration of the constitutive equations, an explicit substepping scheme consisting of yield intersection and drift correction algorithms is employed and thus implemented in ABAQUS via UMAT. The FEM model includes various material parameters that are determined from the experimental data. Three sets of parameters are used in the numerical simulations. While the first set is from the experiments that are conducted in this study on SFRC specimens with various contents of steel fibers, the other two sets are from the experiments reported in the literature. The response of SFRCs under multiaxial compression obtained from various numerical simulations are compared with the experimental data. The good agreement between numerical results and the experimental data indicates that not only the adopted plasticity model represents the behavior of SFRCs very well but also the implemented integration scheme can be employed in practical applications of SFRCs.