An upper-bound evaluation of interactive creep and growth in textured Zircaloy

Uniform strain-rate, upper-bounding constitutive modeling techniques are applied to the study of irradiation creep in textured hexagonal metals with a specific application to Zircaloy. A crystallite orientation distribution function is used to describe the distribution of grains in a polycrystal for use in strain and stress averaging. Stress dependent creep compliances are constructed for a system exhibiting slip on a discrete set of crystallographic slip systems. Superposition of creep and stress independent sources of internal strain, such as those produced by the irradiation growth mechanism, is studied. A model for irradiation growth is proposed which accounts for grain shape and size effects as well as non-uniform dislocation densities arising from variable fabrication history. The internal strains are shown to interact with those dependent on applied stress in a manner which alters the constitutive behavior.     

Application of an impact analysis code to reinforced concrete structures

A numerical constitutive model representing the behavior of concrete material is proposed in this paper. The stress-strain relations are kept in accordance with the updated information, such as stress, strains, strain rates in the principal directions of stress, crack states, yield states, rupture states. The algorithm of the constitutive model was implemented to the explicit impact analysis code DYNA3D. The experimental tests were also held, in which a 100 kg weight with 8 m/s velocity drops onto a reinforced concrete structure. The results of the DYNA3D analysis were compared with those of the tests and show a good agreement.     

One-dimensional propagation speed of an elastic-plastic-viscoplastic stress wave

A one-dimensional elastic-plastic-viscoplastic constitutive equation with the introduction of overstress and understress is proposed, and the propagation speed of an elastic-plastic-viscoplastic stress wave is derived using this equation. Employing the propagation theory of this wave, the strain-rate dependence of the stress, a quasi-static stress-strain curve, the high-strain-rate dependence of the stress, and the strain and strain-rate dependence of the speed can be expressed. It is also shown that the extent of the strain-rate dependence of the stress and the speed can be expressed by changing the viscoplastic coefficient n and the modulus k₂ in the elastic -plastic-viscoplastic constitutive equation and in the equation of the elastic -plastic-viscoplastic stress wave.     

Finite element formulation for thermal stress analysis of thin reactor structures

This paper describes the formulation of a finite-element procedure for the thermal stress analysis of thin wall reactor components. A general temperature-dependent constituent relationship is derived from a Gibbs potential function and a temperature-dependent yield surface. This form of constitutive relationship is applicable to problems of small strain. A similar form of a hypoelastic-plastic type is also developed for large strains. The variation of the yield surface with temperature is based upon a temperature-dependent, work-hardening model. The model uses a temperature-equivalent stress-plastic strain diagram which is generated from isothermal unaxial stress-strain data.The above constitutive relationships are incorporated into two computer codes and a previously developed numerical algorithm is used with minor modifications. A set of problems is presented validating the thermal analysis capability of the computer codes to a variety of problem types.     

单片式核燃料板轧制过程的数值模拟研究

针对UMo合金单片式核燃料板锆合金包壳材料应变率相关的力学本构关系,推导出其三维应力更新算法,相应地编写了定义其本构关系的VUMAT子程序并验证了程序的正确性;建立了对UMo合金单片式板元件的框架轧制过程进行计算模拟的有限元模型;利用显式动力有限元法,计算分析了复合坯内部的变形以及接触压强在轧制过程中的演化规律。研究结果表明,利用VUMAT用户材料子程序能方便正确地定义材料应变率相关的本构关系;燃料芯体与盖板之间的轧制接触压力随时间而演化,在靠近宽度方向的对称面处具有最大的接触压力。本研究为优化UMo合金单片式核燃料板的制造工艺参数提供了理论基础和计算手段。     

Dynamic strain-ageing of A203D nuclear structural steel

The present investigation deals with the dynamic strain-ageing behaviour of a nuclear structural steel, designated ASTM A203 grade D, in tempered martensitic and ferritic-pearlitic microstructural conditions. The serrated stress-strain curves, characteristic of this phenomenon have been observed in the temperature range 100–200°C, with nominal strain rates varying from $\text{1.33 × 10}{}^{\mathrm{?5}}$ to $\text{6.66 × 10}{}^{\mathrm{?4}}\text{/s}$. It has been noted that dynamic strain-ageing causes a sharp rise in ultimate strength and work-hardening rate, a marked decrease in ductility and a negative strain-rate sensitivity of the flow stress. In this temperature range, the yield stress also increases with increasing temperature but the rise in ultimate stress is much greater than the rise in yield stress. The temperature and strain-rate dependence of the onset of serrations yields an activation energy of 63 kj/mol (15 kcal/mol), which suggests that the process is controlled by interstitial diffusion, probably of nitrogen, in ferrite. It appears that microstructure does not have any strong influence on the changes in mechanical properties of this steel during dynamic strain-ageing.     

Application of dislocation dynamics to assess irradiation effect on materials with different grain sizes

Irradiation of materials by energetic particles produces defect clusters like vacancies, self-interstitial atoms and stacking-fault tetrahedra. These defect clusters form loops around existing dislocations, leading to their decoration and immobilization, which ultimately leads to radiation hardening in most of the materials. Effect of irradiation on material shear yield strength is analyzed using two-dimensional poly-crystal dislocation dynamics (DD) modelling. The plastic flow in the material is represented as collective behavior of a large number of edge dislocations distributed among many grains. The unit cell is assumed to have grains of hexagonal shape with uniform size. Grain boundaries are considered to be impenetrable to dislocations. The irradiation effects are modelled by taking all dislocations being locked by irradiation defects thus characterizing the fluence. When the total stress on the dislocations exceeds a critical stress value, they get unlocked and become free to move on their glide planes. Typical stress-strain curves for various critical values are obtained for irradiated Aluminium with different grain sizes, which reveal the effect of dislocation loops on increased yield stress as a function of both fluence and grain size. Critical locking stress is correlated to irradiation fluence by using single crystal yield stress values of irradiated Aluminium from DD analysis and corresponding experimentally available yield stress values.     

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.     

Relationship between ductile crack initiation and void volume fraction

An investigation on ductile crack initiation in structural steel has been made, based on the concept of Gurson's yield function for porous material. First, the condition of ductile crack initiation in the uniform stress field has been investigated. The condition of ductile crack initiation under various stress triaxiality obtained from the tests on axisymmetric notched tensile specimens is well expressed by the condition of constant void volume fraction analytically obtained from Gurson's model. This result means that the condition of constant void volume fraction may be used as the criterion of ductile crack initiation. Secondly, the behavior of void growth and ductile crack initiation in the area near the notch tip under mode I and mode II loading has been investigated. Under mode I loading, the increase in void volume fraction around the notch with an increase in applied load agrees well with the behavior of porous material predicted by the finite element analysis based on Gurson's yield function, and the ductile crack initiation can be predicted by the concept of critical void volume fraction as in the case of uniform stress-strain field given above. The same criterion is not applicable to the crack initiation under mode II loading and further study is needed.     

The viscoplasticity theory based on overstress applied to the analysis of beams under monotonic and cyclic creep loading

The viscoplasticity theory based on total strain and overstress can reproduce rate-dependent inelastic deformation without distinction between plastic and creep strain using two material functions. A viscosity function and an equilibrium stress-strain curve characterize rate-dependency and work hardening, respectively. The theory is used to analyze the creep and cyclic creep behavior of a beam subjected to a linearly increasing moment which is subsequently held constant.The analysis shows the existence of two possible states of equilibrium for creep deformation: termination of primary creep or secondary creep. They occur when the equilibrium stress-strain curve has positive or zero slope in the plastic range.The numerical experiments illustrate that the stress distribution at the end of the moment increase depends on the moment rate. The rate effects disappear with time when stress is redistributed. For practical purposes the equilibrium solution is obtained before 10⁷ s, when the material functions representing AISI Type 304 Stainless Steel at room temperature are used. The other equilibrium solution (secondary creep) is reached after primary creep when the constant moment is above the limiting equilibrium moment which corresponds to the plastic hinge moment of plasticity theory. The stress distribution during stationary creep is shown to be the solution corresponding to the Norton law of creep theory. The numerical experiments also illustrate the influence of various viscosity functions and equilibrium stress levels. A growth law for the equilibrium stress-strain curve is postulated and reversed loading as well as repeated loading are investigated.     

A constitutive model for structural analysis of fusion magnets

A constitutive model to simulate the elastic-plastic and slip actions of fusion magnets under operating or abnormal loads is outlined. To represent the elastic-plastic responses a unified material homogenization procedure based on the existing composite technology was applied to obtain an effective incremental stress-strain relationship for the heterogeneous, laminated magnets. The inter-layer slip behavior of the magnets was represented by a friction-type model from which slip deformation can be calculated for the situation where inter-layer shear stresses exceed the bonding strength of the adhesives. Numerical results of three example problems are presented to demonstrate the utility of the proposed model.     

A two-surface cyclic plasticity model consistent with fundamental stress-strain equations of the power-law type

A plasticity constitutive model which can describe primary features of stress-strain behaviors of cyclically loaded metallic materials under isothermal conditions is developed in the framework of the two-surface plasticity theory. A special plastic hardening modulus function, which is consistent with the power-law type stress-strain expressions frequently used for both monotonic and cyclic stable stress-strain curves, is introduced to represent cyclic hardening as well as saturated behaviors more realistically. The model is validated through its application to the simulation of uniaxial cyclic and biaxial behaviors of cyclic hardening materials. This model is then implemented into a general purpose finite element computer program and applied to the analysis of a notched plate subjected to cyclic loading.     

Constitutive relations for nuclear reactor core materials

A strain rate dependent constitutive equation is proposed which is capable of describing inelastic deformation behavior of anisotropic metals, such as Zircaloys, under complex loading conditions. The salient features of the constitutive equations are that they describe history dependent inelastic deformation behavior of anisotropic metals under three-dimensional stress states in the presence of fast neutron flux. It is shown that the general form of the constitutive relations is consistent with experimental observations made under both unirradiated and irradiated conditions. The utility of the model is demonstrated by examining the analytical results obtained for a segment of tubing undergoing different loading histories in a reactor.     

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.     

Some recent developments of the endochronic theory with applications

In this paper we presented a comprehensive review of recent developments of the endochronic theory of plasticity. The endochronic theory was first proposed by Valanis in 1971 with the aim of circumventing some of the difficulties associated with classical theories of plasticity, such as the concept of a yield surface and its motion in the stress space, criteria for unloading, hardening rules, etc. The theory is developed by using irreversible thermodynamics of internal variables. In the early version of the theory the intrinsic time was defined as a measure of length in strain space. This version is now called the simple endochronic theory. The derived constitutive equations have been applied with success to a number of problems of practical interest, e.g. cyclic response, cross hardening, cross ratcheting, etc. However, it has been shown by the first author that the simple theory leads to an unloading response which is not elastic at its onset. As a consequence infinitesimal hysteresis loops in the first quadrant of the stress-strain space are open, in disagreement with observed behavior in metals. Recently Valanis introduced a new intrinsic time scale which is a measure of length in the plastic strain space. As a result, a new model of endochronic theory has been developed which leads to closure of hysteresis loops in the small as well as the large. It is also shown that the new model predicts the existence of a yield surface. Hardening rules proposed in the classical theory of plasticity appear as special results.It is also shown that the constitutive equations derived from the new model are very powerful in their quantitative prediction of steady cyclic response under constant strain amplitude conditions. In the case of cyclic creep the theory predicts the dependence of cumulative axial creep on the plastic shear strain amplitude but that it overestimates the dependence of the latter on the number of cycles. At the present time we think that this difficulty cannot be overcome on the basis of an isotropic theory. This suggests a future area of investigation and application of the endochronic theory.     

Development of constitutive equations for nuclear reactor core materials

A set of strain rate dependent constitutive equations has been described which is capable of predicting deformation behavior of anisotropic metals under complex loading conditions with or without the presence of a neutron flux. The important feature of the constitutive equations is that they describe history dependent plastic deformation behavior of anisotropic metals under three-dimensional stress states. Since the analytical model accounts for the effect of prior deformation history at all times, it is capable of handling consecutive or simultaneous loading histories, such as post-irradiation loading, in-pile loading, etc. It is demonstrated that the general form of the constitutive relations is consistent with experimental observations made for Zircaloys under both unirradiated and irradiated conditions.     

Description of the flow behaviour of a high strength austenitic steel under biaxial loading by a constitutive equation

Uniaxial and biaxial tension-torsion tests were carried out on a high strength austenitic steel at room temperature in the strain rate range from 10⁻⁵ to 10² s⁻¹. This material shows a strong dependence of strength on strain rate. The yield loci of the combined tests are described by ellipses. The size and the shape of these ellipses are functions of strain and strain rate. The quasi-static and dynamic tension and tension-torsion behaviour of the austenitic steel is described by Perzyna's constitutive equation. There is good agreement between measured and calculated results if a yield criterion as a function of strain and strain rate, and a formula which contains the dependence of flow stress on strain rate based on thermal activation, are included.