Scalar Damage Parameter Research Articles

Deformation and damage of nuclear power station fuel elements under cyclic loading

Deformation and damage of nuclear power station fuel element shells under irradiation and cyclic loading due vibrations are studied. Constitutive equations include dependencies for a creep-damage equation with a scalar damage parameter, as well as terms for thermal and irradiation creep strains, elastic, thermal, and swelling strains. The acceleration of the creep-damage process due to cyclic variation of internal pressure is considered with a dynamic creep model, for which constitutive equations are derived using the method of asymptotic expansions and averaging over a period of cyclic loading. Stress and strain states in the fuel element shell are determined by use of an in-house finite element method creep-damage code for shells of revolution. Results show the essential variation in the initially symmetric fuel element shell form, as well as the acceleration of creep-damage processes due to the cyclic pressure.

On the Choice of Forming Modes and Estimation of Residual Service Life by Kinetic Equations with the Scalar Damage Parameter

On the Choice of Forming Modes and Estimation of Residual Service Life by Kinetic Equations with the Scalar Damage Parameter

Physical Biology of Axonal Damage.

Excessive physical impacts to the head have direct implications on the structural integrity at the axonal level. Increasing evidence suggests that tau, an intrinsically disordered protein that stabilizes axonal microtubules, plays a critical role in the physical biology of axonal injury. However, the precise mechanisms of axonal damage remain incompletely understood. Here we propose a biophysical model of the axon to correlate the dynamic behavior of individual tau proteins under external physical forces to the evolution of axonal damage. To propagate damage across the scales, we adopt a consistent three-step strategy: First, we characterize the axonal response to external stretches and stretch rates for varying tau crosslink bond strengths using a discrete axonal damage model. Then, for each combination of stretch rates and bond strengths, we average the axonal force-stretch response of n = 10 discrete simulations, from which we derive and calibrate a homogenized constitutive model. Finally, we embed this homogenized model into a continuum axonal damage model of [1-d]-type in which d is a scalar damage parameter that is driven by the axonal stretch and stretch rate. We demonstrate that axonal damage emerges naturally from the interplay of physical forces and biological crosslinking. Our study reveals an emergent feature of the crosslink dynamics: With increasing loading rate, the axonal failure stretch increases, but axonal damage evolves earlier in time. For a wide range of physical stretch rates, from 0.1 to 10 /s, and biological bond strengths, from 1 to 100 pN, our model predicts a relatively narrow window of critical damage stretch thresholds, from 1.01 to 1.30, which agrees well with experimental observations. Our biophysical damage model can help explain the development and progression of axonal damage across the scales and will provide useful guidelines to identify critical damage level thresholds in response to excessive physical forces.

Delayed fracture of plates under creep condition in unsteady complex stress state in the presence of aggressive medium

• The decision of diffusion equation is based on the introduction of a diffusion front. • The time to fracture of the plate is determined using a fractional linear creep model. • The times to fracture while using scalar and vector damage parameters are compared. Delayed fracture of bended rectangular plate is researched under creep condition in unsteady complex stress state considering the influence of the ambient medium. Using Rabotnov's kinetic theory, time to fracture of such plate is determined during sequential bending in different planes. Piecewise constant dependence of the bending moment direction and level on time is considered. The time to fracture of the plate is determined using a linear fractional creep model. The effect of the ambient medium on the creep and the creep rupture of the plate is attributed to diffusive penetration of the ambient medium elements into the material of the plate. An approximate method based on the introduction of a diffusion front is used to evaluate the diffusion process rate. The ambient medium effect is taken into consideration by introducing the function of cumulative average concentration into the constitutive and kinetic linear fractional equations. The times to fracture for the scalar and vector damage parameters are compared.

Creep of supersonic aircraft panels under the periodic action of airstream pressure

Irreversible deformation of supersonic aircraft panel under the aerodynamic heating and periodical action of airstream pressure is regarded. The distribution for airstream pressure which is obtained from the solution of aerodynamics problem, was simplified and presented as a cycle with triangular form of pressure increasing-decreasing in a part of period with definite constant static component. The mathematical statement of creep problem for thin shells of revolution loaded by normal pressure having constant and periodically varying components is presented. The theory of finite deflections is used. Equations are derived for the case of conical shell. The constitutive equations, which are obtained by use the method of asymptotic expansions of creep strain and scalar damage parameter with subsequent averaging in a period of stress varying, are used. The form of these equations is similar to Norton law - Rabotnov-Kachanov evolution equation with multiplayers represent the influence of periodic loading in each equation. Numerical simulation was performed by use of programming complex for thin shells of revolution creep calculations. It uses the Finite Element Method jointly with finite differences method for time integration. The example of numerical solution of the problem of deformation of heated cylindrical panel made from duralumin alloy and subjected to periodical loading from airstream puffs is presented. The distributions of stress-strain state components were obtained. The numerical results include distribution of normal deflection in the panel after 1000 seconds of flight as well as the von Mises stress relaxation curve. The analysis of obtained data shows the sufficient increasing of normal deflections as well as stress redistributions. The necessity of consideration of current deformed state of the panel in the problem of analysis of airstream flow is discussed.

High temperature creep of steel 09G2S under non-stationary loading

Experimental analysis of deformation of steel 09G2S under creep conditions has been carried out to determine the optimal modes of high temperature forming of steel shells. The efficient temperature range in terms of saving of the plasticity resource was defined. The material parameters including scalar damage parameters have been determined by taking into account the description of the deformation of a material under non-stationary loading conditions. A satisfactory description of the unsteady creep processes of the material 09G2S in the temperature range from 700 to 800°C has been received.

Damage and fracture: Classical continuum theories

This paper reports the research results on the continuum theory of damage which goes back to the works of Kachanov, Gurson, and Rabotnov. In these models, internal variables that generally have different mathematical structure are explicitly introduced to constitutive relations. The internal variables describe a non-oriented (using scalar damage parameters) or oriented (using different-order tensors) damage distribution in the material. Then, a fracture criterion is introduced based on mechanical or thermodynamic considerations. Models of this type are still most frequently used in the structural analysis of strength of some materials (e.g., composites). Since damage nucleation and growth are closely related to strain localization, consideration is given to formulations and methods for analyzing the stability of inelastic deformation processes. Much attention is given to the effect of the finite element mesh on simulation results, to solution algorithms for such problems, and to the possibilities of using non-local constitutive models. The studies that use gradient models are also included, because damage formation is associated with sharp spatial variations of kinematic and/or dynamic characteristics which must be described by non-classical constitutive relations (gradient, non-local, micromorphic continuum).

Prediction of creep rupture in 2.25Cr–1Mo notched bars

This paper re-examines the creep life methodology based on the continuum damage mechanics (CDM) of the Kachanov and Rabotnov theory. Uniaxial creep and multiaxial creep rupture formulations are presented taking into account the primary creep effect. The scalar damage parameter is computed up to time-to-rupture as a function of time and stress. The methodology implemented is based on the uniaxial time-to-rupture obtained experimentally. The times-to-rupture for bars with different notches are calculated. It is demonstrated that the use of the damage parameter is vital to indicate the critical damage location where failure occurs. Results are compared to those obtained experimentally. It is shown that the primary creep inclusion has a significant effect on the damage distribution zone.

Non-linear incremental fatigue damage calculation for multiaxial non-proportional histories

Most fatigue models must somehow identify and count individual load events before quantifying the damage induced by each one of them, making multiaxial fatigue damage calculations under non-proportional variable amplitude loadings a challenging and laborious task in practical applications. Moreover, to apply such models it is usually necessary to use semi-empirical methods to evaluate the non-proportionality of the load path of each event, through path-equivalent ranges obtained e.g. using a convex enclosure or the MOI (Moment Of Inertia) method. To avoid this burden, a new approach called the Incremental Fatigue Damage methodology is proposed in this work to continuously accumulate multiaxial fatigue damage under actual service loads, without requiring path-equivalent range estimators or rainflow counters. This new approach is not based on questionable Continuum Damage Mechanics concepts, or on the integration of some unrealistic scalar damage parameter based on elastoplastic work. Instead, inspired on multiaxial plasticity procedures, a framework of nested damage surfaces is introduced, allowing the calculation of fatigue damage even for general 6D multiaxial load histories. In this way, fatigue damage itself can be continuously integrated along the load path, considering damage parameters adopted by traditional fatigue models and following reliable procedures well tested in engineering practice. The proposed approach is experimentally validated by non-proportional tension-torsion tests on tubular 316L stainless steel specimens.

Modeling Steel Concrete Bond Strength Reduction Due to Corrosion

Reinforcement corrosion in reinforced concrete (RC) structures induces concrete cracking and leads to a reduction in both steel cross section and steel-concrete bond strength. In this paper, a new model is proposed based on a scalar damage parameter Dc that depends on the loss of cross-sectional area of steel due to corrosion. The model predictions are plotted against experimental data from the literature based on both accelerated and natural corrosion processes and compared to other existing empirical models. The proposed new model significantly improves the prediction of the bond strength of corroding steel bars and is valid for all diameters ranging between 10 and 20 mm (0.39 and 0.79 in.). Moreover, two additional parameters have been implemented in the model, allowing for the consideration of natural (localized) corrosion and the presence of stirrups. A stirrups confinement coefficient st and a pit penetration depth coefficient a have been successfully calibrated using experimental results from the literature.

Numerical analysis of the in-plane behaviour of three-leaf stone masonry panels consolidated with grout injection

This paper presents the calibration procedures of a numerical model based on the results of uniaxial and shear-compression tests on three-leaf stone masonry panels in scale 1:1 and 2:3, both in original and injected conditions. The stone masonry panels were simulated considering a macro-scale approach, where a nonlinear continuum damage model with distinct scalar damage parameters for tension and compression, implemented in the finite element software Cast3 M, was used to simulate the walls behaviour. The main goal of this paper is, first to calibrate, based on the obtained experimental results and using a phenomenological calibration strategy, a single set of parameters that represent a macroscopic constitutive law, by type of masonry, which is able to describe the different experimental tests. Afterwards, a parametric study was performed using the calibrated material laws, with the scope of assessing the influence of vertical pre-compression, slenderness and material strength on failure mechanisms and on different performance parameters such as the maximum load and displacement capacity (drift at ultimate state and drift at maximum load), for each type of masonry condition.

Finite Element Analysis of Influence of Non-homogenous Temperature Field on Designed Lifetime of Spatial Structural Elements under Creep Conditions

Abstract The techniques of modeling of continual fracture process for spatial bodies under long-term static force loading condition in non-homogenous temperature field are presented. The scalar damage parameter is used to describe the material continual fracture process. A stress-strain problem solution made with semianalytic finite element method (SFEM). Results of lifetime determination of responsible parts are presented.

Technique for selecting the functions of the constitutive equations of creep and long-term strength with one scalar damage parameter

The strain-strength characteristics of aerostructures made of hardening materials under uniaxial tension in creep conditions are determined. The problem is reduced to a system of ordinary differential equations of the kinetic theory of creep with one scalar damage parameter. The approximate solutions of the problem are obtained with the help of the implicit Euler method and of the arc length method in combination with the explicit methods of the Runge–Kutta family for cylindrical St.45 steel samples and 3V titanium alloy plates.

Методика выбора функций определяющих уравнений ползучести и длительной прочности с одним скалярным параметром поврежденности

Методика выбора функций определяющих уравнений ползучести и длительной прочности с одним скалярным параметром поврежденности

Creep fracture of plates with variable bending moments in the presence of an aggressive medium

The dissipated fracture of a rectangular plate with bending under conditions of a time-varying plane stress state taking account of the effect of an aggressive medium is investigated. A method for determining the time up to the fracture of such a plate when it is successively bent in different planes is proposed. Using Rabotnov's kinetic theory, the times up to fracture are compared in the case of scalar and vector damage parameters using a power creep model. The deviations of the sum of the partial times from unity are investigated in the case of piece-wise constant relations between the level of the bending moment and time. An analogy with the results of experiments on the creep strength of rods with a piecewise-constant tensile stress is shown.

Evolution of damage and fracture in two families of Ni–Cu–Mo sinter-hardened steels with various initial porosities

The damage evolution of two families of industrial sinter-hardened low alloy steels with different density levels was studied by means of mechanical tests coupled with microstructural observations. Several differences between the families were highlighted through the characterization of their microstructure, hardness and porosity. The presence of nickel rich austenite was revealed in the first family and larger pores were found in the latter. This work shows that those metallurgical characteristics markedly influence the behavior in tension and the damage evolution during mechanical loading. The ultimate tensile strength and the elongation at fracture are up to twice as high for the first family than for the second one which points out the beneficial role of the austenite as well as the detrimental role of larger pores to the mechanical properties. A fractography analysis showing mixed fracture modes supports these results. Two stages in the evolution of damage were highlighted by the evolution of the Young's modulus during loading–unloading tests up to fracture of specimens. As plastic deformation increases, a competition takes place between the damage growth in the neighborhood of the pores and plasticity mechanisms inside the metallic matrix. The evolution law linking a scalar damage parameter to the deformation was finally identified considering only plasticity mechanisms thanks to a previous continuum damage mechanics model developed within the framework of thermodynamics.

Round-robin tests on damage evaluation of concrete based on the concept of acoustic emission rates

Making out a draft of a recommendation practice on damage evaluation of concrete by acoustic emission (AE) technique was postponed in RILEM TC 212-ACD (AE and related NDE techniques for crack detection and damage evaluation in concrete). A follow-up RILEM committee TC 239-MCM (On-Site Measurement of Concrete and Masonry Structures by Visualized NDT) was approved in March, 2010. In this committee, therefore, round-robin tests have been conducted and results are under examination for preparing the recommendation. Damage estimation of concrete by the compression test of core samples is developed, by applying AE measurement and the scalar damage parameter in damage mechanics. The procedure is named damage estimation of concrete by acoustic emission technique (DeCAT), which is based on estimating the intact modulus of elasticity in concrete from AE database. In addition to available data already published, results of the round-robin tests conducted by three groups are discussed.

Comparison of continuous and discontinuous constitutive models to simulate concrete behaviour under mixed‐mode failure conditions

SummaryThe paper presents detailed FE simulation results of concrete elements under mixed‐mode failure conditions according to the so‐called shear‐tension test by Nooru‐Mohamed, characterized by curved cracks. A continuous and discontinuous numerical two‐dimensional approach was used. In order to describe the concrete's behaviour within continuum mechanics, two different constitutive models were used. First, an elasto‐plastic model with isotropic hardening and softening was assumed. In a compression regime, a Drucker–Prager criterion with a non‐associated flow rule was used. In turn, in a tensile regime, a Rankine criterion with an associated flow rule was adopted. Second, an isotropic damage constitutive model was applied with a single scalar damage parameter and different definitions of the equivalent strain. Both constitutive laws were enriched by a characteristic length of micro‐structure to capture properly strain localization. As an alternative approach, the extended finite element method was used. Our results were compared with the experimental ones and with results of other FE simulations reported in the literature. Copyright © 2015 John Wiley & Sons, Ltd.

Development Length in Reinforced Concrete Structures Exposed to Steel Corrosion: A Correction Factor for AS3600 Provisions

In reinforced concrete structures, reinforcement corrosion induces concrete cracking and leads to a reduction in both the steel cross-section and the steel-concrete bond strength and ultimately affects the development length. In this paper, a new correction factor is proposed to calculate the development length of reinforced concrete structures prone to corrosion. A scalar bond damage parameter is introduced to relate bond strength reduction to corrosion. The new model agrees well with all experimental results found in the literature. The bond damage parameter is further used as a correction factor modifying the AS3600 provisions for the development length of reinforcement in reinforced concrete structures located in a saline environment. The relationship between steel-concrete bond damage and corrosion induced concrete cracking is also discussed.

A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model

A one-dimensional displacement second-gradient damage continuum theory has been already presented within the framework of a variational approach. Damage is associated with strain concentration. Therefore, not only non-local effects of displacement second-gradient modelling should be considered in a comprehensive model, but also any plastic effects. The aim of this paper is therefore to extend such a model to plasticity. The action is intended to depend not only with respect to first and second gradient of displacement field and to a scalar damage field, but also to further two internal variables, i.e. the accumulated plastic tension and the accumulated plastic compression. A constitutive prescription on the stiffness is given in terms of the scalar damage parameter in a usual way, i.e. as in many other works, it is prescribed to decrease as far as the damage increases. On the other hand, the microstructural material length (i.e. the square of the constitutive function in front of the squared displacement second-gradient term in the action functional) is prescribed to increase as far as the damage increases, being this last assumption connected to the interpretation that a damage state induces a microstructure in the continuum and that such a microstructure is more important as far as the damage increases. Initial damage threshold and yield stresses are naturally introduced in the action in front of linear terms, respectively, of damage and plastic internal variables. The hardening matrix is also introduced in a natural way as the coefficient matrix in front of the quadratic terms of the two plastic internal variables. At a given value of damage and plastic parameters, the behaviour is referred to second-gradient linear elastic material. However, the damage and plastic evolutions make the model not only nonlinear, but also inelastic. The second principle of thermodynamics is considered by assuming that the scalar damage and plastic parameters do not decrease their values in the process of deformation, and this implies a dissipation for the elastic strain energy. A novel result of this investigation, where displacement second-gradient and plastic effects are combined, is that the distributed and concentrated external double forces do not make work on the displacement gradient but only to its elastic part and this means that the displacement gradient cannot be prescribed, at the border, independently of the plastic internal variables.