Nonlocalized Fatigue Damage of Metals and Alloys. Part 2. Interrelation between Fatigue and Inelasticity

The paper considers interrelation between the regularities in the fatigue fracture and inelastic deformation taking into account the material properties, stress gradient and concentration, specimen size, load cycle asymmetry, complex stress state, temperature, loading frequency and conditions, scatter of the test results. Analysis of the data from the literature and results of original investigations has shown that the characteristics of inelasticity, which define nonlocalized fatigue damage, can be used in studies of general regularities of the high-cycle fatigue fracture. Specific results of such analysis are presented. __________ Translated from Problemy Prochnosti, No. 5, pp. 5 – 29, September – October, 2005.     

EFFICIENT AND ACCURATE EXPLICIT INTEGRATION ALGORITHMS WITH APPLICATION TO VISCOPLASTIC MODELS

Several explicit integration algorithms with self-adaptive time integration strategies are developed and investigated for efficiency and accuracy. These algorithms involve the Runge–Kutta second order, the lower Runge–Kutta method of orders one and two, and the exponential integration method. The algorithms are applied to viscoplastic models put forth by Freed and Verrilli and Bodner and Partom for thermal/mechanical loadings (including tensile, relaxation, and cyclic loadings). The large amount of computations performed showed that, for comparable accuracy, the efficiency of an integration algorithm depends significantly on the type of application (loading). However, in general, for the aforementioned loadings and viscoplastic models, the exponential integration algorithm with the proposed self-adaptive time integration strategy worked more (or comparably) efficiently and accurately than the other integration algorithms. Using this strategy for integrating viscoplastic models may lead to considerable saving in computer time (better efficiency) without adversely affecting the accuracy of the results. This conclusion should encourage the utilization of viscoplastic models in the stress analysis and design of structural components.     

A time integration algorithm for a 3D constitutive model for SMAs including permanent inelasticity and degradation effects

Components based on shape‐memory alloys are often subjected to several loading cycles that result in substantial alteration of material behavior. In such a framework, accurate models, as well as robust and efficient numerical approaches, become essential to allow for the simulation of complex devices. The present paper focuses on the numerical simulation of quasi‐static problems involving shape‐memory alloy structures or components subjected to multiple loading‐unloading cycles. A novel state‐update procedure for a three‐dimensional phenomenological model able to describe the saturation of permanent inelasticity, including degradation effects, is proposed here. The algorithm, being of the predictor‐corrector type and relying on an incremental energy minimization approach, is based on elastic checks, closed‐form solutions of polynomial equations, and nonlinear scalar equations solved through a combination of Newton‐Raphson and bisection methods. This allows for an easy implementation of model equations and to avoid the use of regularization parameters for the treatment of nonsmooth functions. Numerical results assess the good performances of the proposed approach in predicting both pseudoelastic and shape‐memory material behavior under cyclic loading as well as algorithm robustness.     

Calculation of thermal stress field with non-linear surface heat-transfer coefficient during quenching

The heat conduction equation in a coupled thermo-inelastic body, was derived by means of the irreversible thermodynamics and the measured data of the inner dissipation for metal. The coupled coefficients, which are dependent on temperature and inelastic strain rate, can be determined by the measured data of inner dissipation for metal. The functional of a coupled thermal inelastic problem was derived by means of nonlinear functional analysis theory. Finally, it is proved that the critical points of the functional are the solutions of coupled thermal inelastic equations.     

On the numerical implications of multiplicative inelasticity with an anisotropic elastic constitutive law

The statement that theories of inelasticity at finite strains have arrived at a high level of development is only true in conjunction with isotropic material behaviour. From both points of view (theoretical and computational), the extension to anisotropic material behaviour seems to be a complicated task. The statement is especially true when the multiplicative decomposition of the deformation gradient is considered a basis for the formulation. Of special interest are questions related to the mathematical form of the stored energy function or, equivalently, of the constitutive relation for the material stress tensor as the thermodynamical force. This paper deals with the above issues. The anisotropic formulation is accomplished using the notion of structural tensors. Here we suggest that the privileged directions of the material should be transformed in a specific way under the action of the inelastic part of the deformation gradient. The inelastic behaviour is assumed to be governed by evolution equations of the unified type. Numerically, we deal with the full multiplicative structure of the theory. The numerical treatment is developed in full detail. Expressions concerning the local iteration as well as the tangent operator are derived. Various numerical examples with applications to shells are presented which demonstrate the influence of anisotropy and the applicability of the theory. Copyright © 2003 John Wiley & Sons, Ltd.     

薄壁钢结构的材料和几何非线性等参样条有限条法的研究进展

给出了等参样条有限条法(ISFSM)用于分析开孔薄壁钢结构的材料非弹性和几何非线性的研究进展。简要介绍了ISFSM理论。提出了运动学方程、应力-位移关系和假定的本构关系。通过增量平衡条件,推导出正切刚度矩阵。讨论了板带连续和边界条件的要求。特别强调了塑性理论和求解速率方程的方法,以及相关的欧拉后退法和材料模量一致的假定。开孔薄壁钢结构的非弹性后屈曲性能分析的准确性和有效性,证实了目前的等参样条有限条法的有效性。     

Tensile cracks in polycrystalline ice under transient creep: Part II — Numerical simulations

This paper discusses predictions of a numerical model presented in the companion paper (Nanthikesan and Shyam Sunder, 1995) to analyze tensile cracks in polycrystalline ice undergoing transient creep. The numerical model is based on the internal state variable constitutive theory of transient creep in ice developed by Shyam Sunder and Wu (1989a,b, 1990). The finite element model uses the boundary layer approach of Rice (1968), in conjunction with a mid-point crack-tip element and reduced integration, to simulate the asymptotic stress and deformation fields in the vicinity of the crack tip, including incompressible creep deformations.

The problem of a stationary, traction-free, tensile (mode I) crack is analyzed to predict the size, shape and time evolution of the creep-dominated fracture process zone surrounding the crack-tip. The numerical simulations quantify the effects of transient creep, material strain hardening, fabric anisotropy, loading rate, temperature, and finite fracture test-specimen boundary on the development of the creep zone. A range of stress-intensity rates from 1 to 100 kPa s⁻¹ and temperatures from −5° to −25°C is considered in the simulations.

The results from a comprehensive numerical simulation study show that: (i) transient creep increases the creep zone size by more than an order of magnitude over that for a power-law creeping material, i.e., about 40 times for the isotropic, equiaxed granular ice tested by Jacka (1984); (ii) material strain hardening significantly affects the creep zone size, i.e., the creep zone for the transversely-isotropic columnar-grained ice tested by Sinha (1978), with the crack loaded in the plane of isotropy, is about 4 times smaller than that for the granular isotropic ice; (iii) fabric anisotropy increases the size of the creep zone by a factor of at least two for cracks in the transversely-isotropic, columnar-grained ice loaded in the plane of isotropy; (iv) the Riedel and Rice (1980) equation, which was derived for an isotropic power-law creeping material subjected to a suddenly applied constant stress-intensity, overestimates the creep zone size by a factor of 4.2 for a constant stress-intensity rate loading; and (v) as the crack size increases, linear elastic fracture mechanics becomes increasingly applicable at lower loading rates and higher temperatures.     

A logarithmic‐exponential backward‐Euler‐based split of the flow rule for anisotropic inelastic behaviour at small elastic strain

A basic aspect of modern algorithmic formulations for large‐deformation hyperelastic‐based isotropic inelastic material models is the exponential backward‐Euler form of the algorithmic flow rule in the context of the multiplicative decomposition of the deformation gradient. Advantages of this approach in the isotropic context include the exact algorithmic fulfilment of inelastic incompressibility. The purpose of this short work is to show that such an algorithm can be formulated for anisotropic inelastic models as well under assumption of small elastic strain, i.e. for metals. In particular, the current approach works for both phenomenological anisotropy as well as for crystal plasticity. The major difference between the current and previous approaches lies in the fact that the elastic rotation is reduced algorithmically to a dependent internal variable, resulting in a smaller internal variable system. Copyright © 2006 John Wiley & Sons, Ltd.     

Material and geometric nonlinear isoparametric spline finite strip analysis of perforated thin-walled steel structures—Analytical developments

This paper presents the analytical developments of the application of the Isoparametric Spline Finite Strip Method (ISFSM) to the material inelastic and geometric nonlinear analysis of perforated thin-walled steel structures. The general theory of the ISFSM is briefly introduced. The formulations of the kinematics, strain–displacement and constitutive assumptions are presented, and the tangential stiffness matrix is derived by applying the incremental equilibrium condition. The requirements for strip continuity and boundary conditions are also discussed. In particular, the plasticity theory and the methods to integrate the ‘rate equations’ are emphasized, and the related ‘backward Euler return method’ and use of a ‘consistent material modulus’ are highlighted. The present isoparametric spline finite strip analysis is verified against a number of analyses of perforated and non-perforated plates and plate assemblages, as described in the companion paper (Yao and Rasmussen, submitted for publication) [1], demonstrating its accuracy and efficiency for the predictions of the inelastic post-buckling behavior of perforated thin-walled steel structures.