Mechanical modeling of creep, swelling and damage under irradiation for polycrystalline metals

A constitutive equation of creep, swelling and damage under irradiation for polycrystalline metals applicable to structural analyses in multiaxial state of stress is developed. After reviewing microscopic mechanisms of irradiation creep and swelling, the relevant theories proposed so far from the view point of metallurgical physics and their applicability are discussed first. Then a constitutive model is developed by assuming that creep under irradiation can be decomposed into irradiation-affected thermal creep and irradiation-induced creep. By taking account of the Stress-Induced Preferential Absorption (SIPA) mechanism, the irradiation-induced creep is represented by an isotropic tensor function of order one and zero with respect to stress, which is, at the same time, the function of neutron flux and neutron fluence. The volumetric part of the irradiation-induced creep is identified with swelling. The irradiation-affected thermal creep is described by modifying Kachanov-Rabotnov theory for stress-controlled creep and creep damage by incorporating the effect of irradiation. Finally irradiation creep and swelling of 20% cold-worked type 316 stainless steel at elevated temperature are predicted by the proposed constitutive equations, and the numerical results are compared with the corresponding experimental results.     

The influence of cavitation damage upon high temperature creep under stationary and non-stationary loading conditions: Part III: Creep at steady increasing load and true stress

In this paper for ideally plastic materials the influence of high temperature cavitation damage upon creep at steady increasing loads is investigated. The damage function $\text{A(t)}$ enters a constitutive equation for plastíc flow through an effective stress σ_e. For given loading conditions the latter is derived from the solution of Hart's tensile test equation. In the present paper the case of time linear increase in load ($\text{F}$ = constant) and in true stress ($\text{/.s}$ = constant) is investigated. The creep equations for cavitating as well as for non-cavitating materials are derived and the volume change during creep at $\text{/.F}$ = constant are calculated.     

Mechanics and mechanisms of creep deformation and damage

In recent years, several important advances in the understanding of the mechanics and mechanisms of creep deformation, damage and fracture in polycrystalline alloys have been achieved through the synergistic efforts of materials scientists and applied mechanicians. Current understanding, while far from complete, nonetheless provides a physically-based framework for developing computationally tractable continuum constitutive relations which capture major features of the intrinsic mechanisms of creep damage and deformation. Such models, when applied to the analysis of scientifically and technologically relevant boundary value problems, provide a basis for a local approach to high temperature fracture. The first section of the paper reviews certain aspects of the phenomenology, mechanisms, and mathematical models of creep fracture. It is concluded that an important internal damage parameter influencing macroscopic tertiary creep in conventional polycrystalline materials is the density of grain boundary facet cracks. The development of a “damage” constitutive model can conceptually be broken into two parts. One aspect is to quantify the effect of an (instantaneous) state of damage on the mechanical behavior. In the second part of the paper, this topic is explored for the case of a (generally non-dilute) distribution of aligned facet cracks in a power law creeping matrix. The remaining aspect of the model is to provide evolution equations for the damage variable(s). In the third section of the paper, we develop a simplified model for the evolution of facet crack density under conditions of creep-constrained cavitation. A central feature of the model is the variability in cavity nucleation potency over the grain boundary population.     

Random cyclic stress–strain responses of a stainless steel pipe-weld metal II — a modeling

This paper pays special attention to an issue that there is a significant scatter of the stress–strain responses of a nuclear engineering material, 1Cr18Ni9Ti stainless steel pipe-weld metal. Efforts are made to reveal the random fatigue damage character by fracture surface observations and to model the random responses by introducing probability-based stress–strain curves of Ramberg–Osgood relation and its modified form. Results reveal that the fatigue damage is subjected to, 3-D interacting and involved microcracks. The three stages, namely microstructural short cracks (MSC), physical short cracks (PSC) and long cracks (LC) subdivided by Miller and de los Rios, can give a good characterization of the damage process. Both micro- and macro-behaviour of the material have the character of three stages. The 3-D effects are strong in the MSC stage, tend to a gradual decrease in the PSC stage, and then show saturation after going to the LC stage. Intrinsic causes of the random behaviour are the difference and evolution of the microstructural conditions ahead of the dominant crack tips. The ‘effectively short fatigue crack criterion’ introduced by Zhao et al. in observing the material surface short crack behaviour could facilitate an understanding of the mechanism of interaction and evolution. Based on the previous obtained appropriate assumed distribution, normal model, for the cyclic stress amplitude, the probability-based curves are approximated by the mean value and standard deviation cyclic stress–strain curves. Then, fatigue analysis at arbitrarily given reliability can be conveniently made according to the normal distribution function. To estimate these curves, a maximum likelihood method is developed. The analysis reveals that the curves could give a good modeling of the random responses of material.     

On the multiaxial behaviour of concrete exposed to high temperature

An attempt is made to formulate a multiaxial constitutive model for concrete in the temperature range up to 800°C. The proposed model can be characterized as isotropic, elastic-viscoplastic-plastic in the compression region. Brittle failure is assumed in the tensile region.The thermal strain increment is assumed to be a function of both temperature and the current stress tensor. This assumption implies that the thermal strain may have deviatoric components.The volumetric thermal strain is used as a scalar damage measure instead of temperature itself. The corresponding softening function is obtained from isothermal, uniaxial tests. Also the elastic properties are taken as functions of the volumetric thermal strain.The response of the model is illustrated and compared with experimental results.     

反应堆部件设计及寿命预测的有限元技术研究

为了能更好地设计、分析核反应堆部件以及预测其寿命,本文将Chaboche本构模型与损伤模型相结合,推导了Chaboche损伤本构模型。在弹性预测-径向返回和向后欧拉积分方法的基础上,研究了模型的隐式积分算法。将Chaboche损伤本构模型链接到ABAQUS有限元程序,并利用六面体单元模拟了某型反应堆部件材料的4种变形行为,计算结果与实验值吻合良好。     

Shear transfer constitutive model for pre-cracked RC plate subjected to combined axial and shear stress

This paper proposes the constitutive model for the shear transfer through the cracks of pre-cracked reinforced concrete (RC) plate subjected to combined axial and shear stress. The plate is a scale model of a shear wall of a nuclear power plant (NPP) building. Twelve plate specimens were initially cracked and then loaded to the failure point by increasing cyclic shear under constant axial stress. Tangential shear modulus, G_cr, values are estimated from the v–γ relationships observed in the test results and formulated to the constitutive model as the correlation function of the normal strain perpendicular to the crack plane, _cr, and shear strain, γ_cr, based on the smeared crack model concept. By incorporating the proposed model to a nonlinear FEM analysis program and comparing the analysis results with the test results, it is apparent that the program could be improved in its analytical accuracy. The proposed model will be useful for the nonlinear analysis of RC shear walls when the walls are exposed to simultaneous multi-directional load.     

Creep–fatigue interaction damage model and its application in modified 9Cr–1Mo steel

Creep–fatigue interaction damage evolution of the nuclear engineering materials modified 9Cr–1Mo steel is studied with Continuum Damage Mechanics (CDM) theory. Based on the Norton creep damage and fatigue dissipate potential theory, an effective stress controlled creep–fatigue interaction damage model has been developed in this paper, in which the creep and fatigue damage function are both considered as nonlinear variables. The damage evolution function consists of the stress amplitude and the range of mean strain. The damage parameters in the model have clear physical meaning and can be determined easily. A stress controlled creep–fatigue interaction experiment has been performed with the P91 steel to obtain the damage model. The experimental results indicated that the damage model is applicable to describe the damage evolution for creep–fatigue interaction.     

高温气冷堆舱室抗商用飞机撞击的耦合数值分析

本文建立了大型商用飞机撞击典型高温气冷堆核电站反应堆舱室的非线性有限元模型,计算中混凝土舱室直接采用工程用钢筋混凝土的损伤塑性本构模型,飞机结构采用Johnson-Cook本构模型。对飞机高速撞击高温气冷堆核电站反应堆舱室非线性撞击过程进行模拟计算,得出正面和侧面撞击条件下的撞击载荷曲线、撞击位移云图、反应堆舱室混凝土破坏情况等结果。评估表明,反应堆舱室结构在撞击条件下的整体损伤微小,可为保护内部关键设备提供重要的屏障功能。     

Numerical implementation of a transverse-isotropic inelastic, work-hardening constitutive model

This paper documents the numerical implementation of a transverse-isotropic inelastic, work-hardening plastic constitutive model. A brief review of the model is presented first to facilitate the understanding of its numerical implementation. This model is formulated in terms of ‘pseudo’ stress invariants, so that the incremental stress-strain relationship can be readily incorporated into existing finite-difference or finite-element computer codes. The anisotropic model reduces to its isotropic counterpart without any changes in the mathematical formulation or in the numerical implementation (algorithm) of the model.The input quantities to the algorithm are the initial stress components obtained at the end of the previous strain increment, and the new strain increments; the output quantities are the new values of the stress components. In the first step of the algorithm a set of elastic trial stresses computed, which are then tested with respect to the failure criteria. If they do not violate the failure criteria, the behavior of the material is truly elastic and these stresses are the correct stresses for the given strain increments. On the other hand, if the elastic trial stresses violate the failure criteria, they are corrected through an iteration scheme using an appropriate flow rule.A typical example of the model and its behavior in uniaxial strain and triaxial compression is presented.     

Thermo-mechanical constitutive equations for the description of shape memory effects in alloys

A class of alloys show ‘shape memory effects’ which make them applicable for many tasks. For them it is possible to remove imposed deformations nearly entirely by heating. By cooling the material again under constant loads one nearly obtains the old deformations again. It will be shown that this effect as well as others can be described in three dimensions by means of an extended classical theory of plasticity. Two temperature-dependent yield criteria are used which respond under different conditions. For the one-dimensional case the constitutive equations can be simulated by a rheological model. An algorithm makes the material functions applicable for engineering purposes. Numerical results are given for spacial bending of bars.     

Determination of internal state variables and constitutive modeling for Type 316 stainless steel

The purpose of this study is to develop an approach for unified constitutive modeling based on experimentally determined back stress and overstress. Back stress and overstress were experimentally determined for Type 316 stainless steel at 600°C, by analyzing the unloading curve for the stress-strain response of cyclic strain tests. The result has indicated that the cyclic strain hardening behavior is mainly caused by hardening in the back stress. A phenomenological unified constitutive model in which the back stress and drag stress are taken as the internal state variables is proposed, and has been shown that this model is able to simulate the cyclic inelastic behavior.     

How to use damage mechanics

The background of continuum damage mechanics is first presented in the framework of thermodynamics with some examples of constitutive equations for ductile damage, creep damage and fatigue damage. After the general scheme of structural calculations for macro-crack initiation, through non-coupled or coupled strain damage equations, some examples of “simple applications” are given: fracture limits of metal forming, surface initial damage in fatigue, creep fatigue interaction, and bifurcation of cracks.     

Application of endochronic theory in multi-dimensional stress states

The general endochronic theory and its specific constitutive equations for different stress states are studied. The determination of the material parameters, the stability and uniqueness of solution to an endochronic model are also examined. The endochronic material parameters can not be obtained by explicit means from simple tension test data. An approximate formula for these parameters is proposed. It is used to study the transverse behavior of a rod subjected to a uniaxial stress state and the behavior of a plate subjected to an in-plane biaxial stress state under static condition. Results for applications to dynamic problems are also reported. It is found that the endochronic model with simple relaxation function can be applied successfully to certain problems, although it may exhibit undesirable behavior for other problems. The flexibility in describing various material phenomena can be improved by using more realistic relaxation function and intrinsic time definition. This theory has great potential in practical applications.     

Design values of Inconel 617 for high temperature reactors operating at temperatures above 800°C

For extrapolation of the time-dependent stress values the following two methods are introduced:

• -extrapolation with constitutive equations

• -extrapolation with time-temperature parameters.

The first method is a system of coupled equations with creep and damaging parts (a physical-mathematical method), the second is an extrapolation method which combines three well-known time-temperature parameters (a statistical method). On the basis of the latter method time-dependent stress intensity values for a high temperature reactor material are derived and as an example the time safety against rupture is shown for a heat exchanger tube at a temperature of 950°C.     

Fracture mechanics in creep situation interpretation of results on a test specimen

Fracture mechanics in creep situation is a difficult challenge for the 1990s. In France, CEA Saclay has conducted experimental tests on compact tension (CT) specimens at 650°C in order to investigate crack initiation under creep situations. The constitutive material is the 316SPH austenitic stainless steel used for most LMFR structures.Numerical simulations using SYSTUS code and simplified method analysis were performed on one of the tests (CT specimen at 650°C under constant load) to compare some parameters (notch opening, initiation time) with experimental values. The material constitutive law was represented by the complete elasto-viscoplastic CHABOCHE model for computation. Owing to geometrical characteristics such as thickness, the situation of the CT specimen was likely to be intermediate between plane stress and plane strain assumptions. From C^* parameter, incubation time obtained using the R5 rule was conservative in comparison with the test result.The continuum damage model developed at Ecole des Mines de Paris has also been used to assess creep damage in the notch tip area. The crack initiation time has been deduced from critical damage at characteristic distance (X_c = 0.05 mm). Considering critical damage specifically, for a CT specimen (D_c = 0.05), initiation time obtained was higher than the test result.The results of this study will contribute to the development of a methodology for nocivity analysis of cracks in creep situation.     

Ductile fracture of A 508 Cl 3 steel in relation with inclusion content: The benefit of the local approach of fracture and continuum damage mechanics

A new methodology for ductile fracture analysis based on the local approach of fracture, through constitutive relations that take into account void growth and damage, has been applied to three heats of A 508 Cl 3 steel with different inclusion contents in the region of 10⁻³–10⁻⁴. The ductility of the three heats is well predicted by the ductile fracture model through its parameter f₀: the initial volume fraction of voids. The model, first calibrated with the simple notched tension test, gives a good prediction of crack initiation and growth in a precracked specimen. Finally the statistical aspects in ductile fracture are briefly discussed.     

Development of structural analysis program for non-linear elasticity by continuum damage mechanics

The brittle damage constitutive equation developed by Chow and Yang is used to simulate the non-linear elastic deformation behavior of graphite using finite element method (FEM). This model is achieved by introducing a damage surface that is similar to the yield function in the conventional theory of plasticity. A special form of damage surfaces is constructed to illustrate the application of the model. For verifying the FEM program including the Chow and Yang model, the predicted deformations by this model are compared with both the experimental ones in the graphite structural model and the calculated ones without the continuum damage mechanics.     

A dynamic model for element reliability

A dynamic model is developed for a system element reliability distribution over a generalized strength space. A differential equation is obtained describing the time-dependence of the reliability distribution function (RDF). The equation covers a wide class of power reactor system components which perform under intense stress conditions where a standard subdivision into a “burn-in” period, a “chance failures” range and a “wear-our” period is inapplicable.The hazard distribution function (HDF) over strength is introduced within the model and it is shown that a standard hazard rate is a strength-averaged failure intensity parameter with the RDF as a weighting function.It is shown that a well-known “bathtub” form of the hazard rate function corresponds to an analytical solution of the principal RDF transfer equation under some simplifying assumptions.