Connecting the macro and microstrain responses in technical porous ceramics. Part II: microcracking

Following previous study on non-microcracked porous ceramics (SiC and alumina), we studied the micro and macrostrain response of honeycomb porous microcracked ceramics under applied uniaxial compressive stress. Cordierites of different porosities were compared. Both macroscopic and microscopic strains were measured, by extensometry and neutron diffraction, respectively. Lattice strains were determined using a single diffraction peak (steady-state neutron source) in both the axial and the transverse sample directions. Complementarily, we measured the macroscopic Young’s modulus of these materials as a function of temperature, at zero load, using high-temperature laser ultrasound spectroscopy. This allowed having a non-microcracked reference state for all the materials investigated. Confirming our previous study, we observed that macrostrain relaxation occurs at constant load, which is not observed in non-microcracked compounds, such as SiC. This relaxation effect increases as a function of porosity. Moreover, we generally observed a linear dependence of the diffraction modulus on porosity. However, for low and very high applied stress, the lattice strain behavior versus stress seems to be influenced by microcracking and shows considerable strain release, as already observed in other porous microcracked ceramics. We extended to microcracked porous ceramics (cordierite) the macro to microstrain and stress relations previously developed for non-microcracked ceramics, making use of the integrity factor (IF) model. Using the whole set of data available, the IF could also be calculated as a function of applied stress. It was confirmed that highly porous microcracked materials have great potential to become stiffer and more connected.     

Experimental evaluation of the elastic limit of carbon‐fibre reinforced epoxy composites under a large range of strain rate and temperature conditions

The mechanical behaviour of organic matrix composite materials such as T700GC/M21 carbon fibre reinforced polymer (CFRP) is generally considered by the industry as being orthotropic elastic for the sizing of aeronautical structures under normal isothermal “static” flight loads. During the aircraft lifetime, it may be exposed to severe loading conditions at various temperatures. However, the mechanical behaviour of CFRP is known to exhibit a linear behaviour or a non‐linear behaviour according to the types of loads that are considered creep or extreme conditions. The observed non‐linearity can be commonly attributed to several physical phenomena such as non‐linear viscosity, plasticity, or damage. In the literature, different models can be found that are based on three components: a first elastic reversible behaviour, a second non‐linear behaviour, and a failure criterion. An important issue is to understand and characterize the transition between the elastic reversible behaviour and the non‐linear behaviour. To answer this question, the present paper describes an experimental methodology that permits to evaluate this transition thanks to raw experimental data, and its application to a range of constant but different strain rate and temperature tests performed on the T700GC/M21 CFRP material.     

Multiaxial notch fatigue life prediction based on pseudo stress correction and finite element analysis under variable amplitude loading

A new computational methodology is proposed for fatigue life prediction of notched components subjected to variable amplitude multiaxial loading. In the proposed methodology, an estimation method of non‐proportionality factor (F) proposed by authors in the case of constant amplitude multiaxial loading is extended and applied to variable amplitude multiaxial loading by using Wang‐Brown's reversal counting approach. The pseudo stress correction method integrated with linear elastic finite element analysis is utilized to calculate the local elastic‐plastic stress and strain responses at the notch root. For whole local strain history, the plane with weight‐averaged maximum shear strain range is defined as the critical plane in this study. Based on the defined critical plane, a multiaxial fatigue damage model combined with Miner's linear cumulative damage law is used to predict fatigue life. The experimentally obtained fatigue data for 7050‐T7451 aluminium alloy notched shaft specimens under constant and variable amplitude multiaxial loadings are used to verify the proposed methodology and equivalent strain‐based methodology. The results show that the proposed methodology is superior to equivalent strain‐based methodology.     

Strain-life approach in thermo-mechanical fatigue evaluation of complex structures

This paper is a contribution to strain‐life approach evaluation of thermo‐mechanically loaded structures. It takes into consideration the uncoupling of stress and damage evaluation and has the option of importing non‐linear or linear stress results from finite element analysis (FEA). The multiaxiality is considered with the signed von Mises method. In the developed Damage Calculation Program (DCP) local temperature‐stress‐strain behaviour is modelled with an operator of the Prandtl type and damage is estimated by use of the strain‐life approach and Skelton's energy criterion. Material data were obtained from standard isothermal strain‐controlled low cycle fatigue (LCF) tests, with linear parameter interpolation or piecewise cubic Hermite interpolation being used to estimate values at unmeasured temperature points. The model is shown with examples of constant temperature loading and random force‐temperature history. Additional research was done regarding the temperature dependency of the K_p used in the Neuber approximate formula for stress‐strain estimation from linear FEA results. The proposed model enables computationally fast thermo‐mechanical fatigue (TMF) damage estimations for random load and temperature histories.     

A unified mathematical programming formulation of strain driven and interior point algorithms for shakedown and limit analysis

A mathematical programming formulation of strain‐driven path‐following strategies to perform shakedown and limit analysis for perfectly elastoplastic materials in an FEM context is presented. From the optimization point of view, standard arc‐length strain‐driven elastoplastic analyses, recently extended to shakedown, are identified as particular decomposition strategies used to solve a proximal point algorithm applied to the static shakedown theorem that is then solved by means of a convergent sequence of safe states. The mathematical programming approach allows: a direct comparison with other non‐linear programming methods, simpler convergence proofs and duality to be exploited. Owing to the unified approach in terms of total stresses, the strain‐driven algorithms become more effective and less non‐linear with respect to a self‐equilibrated stress formulation and easier to implement in the existing codes performing elastoplastic analysis. The elastic domain is represented avoiding any linearization of the yield function so improving both the accuracy and the performance. Better results are obtained using two different finite elements, one with a good behavior in the elastic range and the other suitable for performing elastoplastic analysis. The proposed formulation is compared with a specialized implementation of the primal–dual interior point method suitable to solve the problems at hand. Copyright © 2011 John Wiley & Sons, Ltd.     

Stress states and nonlinear mechanical behavior associated with nanoindentation testing of polymers

Interest in determining material properties on the nanoscale has promoted use of nanoindentation testing as a measurement technique. Classical elasticity solution of indentation geometry has provided values of the mechanical properties for linear elastic materials. Recent attempts to apply this test technique to polymers have given indications of time dependent response in the early relaxation period. There is corresponding interest in the possibility of obtaining their nonlinear viscoelastic behavior. As a preliminary to analytical study providing a basis for such testing, the first part of this paper examines the initial stress and strain condition in the vicinity of the indenter. Data from recent tests on poly(vinyl acetate) material at load levels typical of current testing indicate that stress magnitudes in the nonlinear and possibly plastic-like range are present near the specimen surface. The second part of this study pursues the examination of how the heavily nonlinear region may be characterized for polymers in analogy with the treatment utilized for metals and other elastic-plastic materials. As an example, analysis of data on PVAc indicates that its behavior in nanoindentation should in several respects correspond to materials exhibiting a relatively low value of the ratio of elastic modulus to yield stress.     

Shear correction for Mindlin type plate and shell elements

In this paper, two shear correction methods, which can guarantee good accuracy, suitable for different materials and easy to implement, are proposed for the first‐order theory based plate and shell elements. The general variational principles based on the thermodynamic potentials are first derived, to provide a uniform frame for the different material non‐linearity. Then, the two shear correction methods are derived from the obtained general variational principles using the assumed strain and assumed stress methods, respectively. The shear correction factors (SCF) for homogeneous elastic materials are evaluated using the two methods, and the same result of 0.8 is obtained. The proposed shear correction method is then applied to an elastic‐linear hardening plastic deep beam for verification, and the conformable result is obtained as compared with the plane strain result. Copyright © 2006 John Wiley & Sons, Ltd.     

Effects of approximations in analyses of beams of open thin‐walled cross‐section—part II: 3‐D non‐linear behaviour

In a companion paper, the effects of approximations in the flexural‐torsional stability analysis of beams was studied, and it was shown that a second‐order rotation matrix was sufficiently accurate for a flexural‐torsional stability analysis. However, the second‐order rotation matrix is not necessarily accurate in formulating finite element model for a 3‐D non‐linear analysis of thin‐walled beams of open cross‐section. The approximations in the second‐order rotation matrix may introduce ‘self‐straining’ due to superimposed rigid‐body motions, which may lead to physically incorrect predictions of the 3‐D non‐linear behaviour of beams. In a 3‐D non‐linear elastic–plastic analysis, numerical integration over the cross‐section is usually used to check the yield criterion and to calculate the stress increments, the stress resultants, the elastic–plastic stress–strain matrix and the tangent modulus matrix. A scheme of the arrangement of sampling points over the cross‐section that is not consistent with the strain distributions may lead to incorrect predictions of the 3‐D non‐linear elastic–plastic behaviour of beams. This paper investigates the effects of approximations on the 3‐D non‐linear analysis of beams. It is found that a finite element model for 3‐D non‐linear analysis based on the second‐order rotation matrix leads to over‐stiff predictions of the flexural‐torsional buckling and postbuckling response and to an overestimate of the maximum load‐carrying capacities of beams in some cases. To perform a correct 3‐D non‐linear analysis of beams, an accurate model of the rotations must be used. A scheme of the arrangement of sampling points over the cross‐section that is consistent with both the longitudinal normal and shear strain distributions is needed to predict the correct 3‐D non‐linear elastic–plastic behaviour of beams. Copyright © 2001 John Wiley & Sons, Ltd.     

A two‐dimensional ordinary,state‐based peridynamic model for linearly elastic solids

Peridynamics is a non‐local mechanics theory that uses integral equations to include discontinuities directly in the constitutive equations. A three‐dimensional, state‐based peridynamics model has been developed previously for linearly elastic solids with a customizable Poisson's ratio. For plane stress and plane strain conditions, however, a two‐dimensional model is more efficient computationally. Here, such a two‐dimensional state‐based peridynamics model is presented. For verification, a 2D rectangular plate with a round hole in the middle is simulated under constant tensile stress. Dynamic relaxation and energy minimization methods are used to find the steady‐state solution. The model shows m‐convergence and δ‐convergence behaviors when m increases and δ decreases. Simulation results show a close quantitative matching of the displacement and stress obtained from the 2D peridynamics and a finite element model used for comparison. Copyright © 2014 John Wiley & Sons, Ltd.     

Simulation and experimental studies of the repaired composite laminate

Firstly the compress experiment is undertaken to investigate the efficiency of repaired panels in this paper, and then modeling of the mechanical behavior of the repaired composite panel under compressive static load is conducted by using of the finite element method. The effect of geometric non‐linearity on the stress‐strain response is considered in the numeric analysis. Fatherly, the user material subroutine (UMAT) is integrated with the ABAQUS package with the geometric non‐linearity effect for studying the damage initiation and its progression in the composite structure, and quadrilateral, linear, thick shell elements (S8R) are adopted. Finally, the predicted strain distribution, damage evolution and strength of the laminate are compared with the test results.     

Crack growth in an orthotropic medium

Abstract

The elastostatic problem of an orthotropic body having a central inclined crack and subjected at infinity to a uniform biaxial load is considered. It is assumed that the crack line does not coincide with an axis of elastic symmetry of the body. The problem must be considered as one of general orthotropy, due in particular to the fact that the elastic coefficients of the material change with rotation of the reference system. The stress fields at the crack tip are reported and the presence of the non‐singular terms underlined. The Strain Energy Density Theory is extended to orthotropic materials. It is assumed that the Critical Strain Energy Density Factor has a polar variation. The crack initiation is determined via minimization of the ratio of the strain energy density over the material critical strain energy density, pointing out the effects of orthotropy and load biaxiality. The effects of the non‐singular terms on crack growth for different orthotropic materials is also studied, underling the relation between orthotropy and non‐singular terms.     

Classification of metallic alloys for fatigue damage accumulation: A conservative model under strain control for 304 stainless steels

Fully reversed uniaxial tests performed under total strain and stress control on 304 stainless steels specimens show that, under strain control the fatigue damage for High–Low (H–L) cycling is more significant than that using Miner’s rule, but under stress control opposite results are obtained. This has been attributed to opposite effects of pre-hardening under strain and stress control. Classical non linear damage accumulation models are not able to take into account this difference in sequence effect. Smith–Watson–Topper (SWT) and Fatemi–Socie (FS) criterion combined to linear damage accumulation can take into account this difference in sequence effect through the presence of maximum stress. However these models require an elastic–plastic constitutive law which is difficult to propose due to the presence of high cycle secondary hardening observed on 304 stainless steel. A conservative model for damage accumulation under variable amplitude strain control loading is thus proposed, which does not require a constitutive law. Linear damage accumulation is used, while sequence effect is taken into account using the elastic–plastic memory effect through cyclic strain–stress curves (CSSC) with pre-hardening. This modeling classifies metallic alloys in two groups for damage accumulation, with a stable (independent to pre-hardening) CSSC as for aluminum alloys and with an unstable (dependent to pre-hardening) one as for austenitic stainless steels. For the former case the modeling is identical to Miner’s rule. The modeling is approved based on a large number of tests on 304 stainless steel and is compared with SWT and FS models. In presence of mean stress the modeling permits in a qualitative way to explain the fact that tensile mean stresses in constant amplitude strain control tests are more detrimental than for constant amplitude stress control tests. Moreover it is shown that the SWT model is not always able to predict accurately the fatigue life in presence of a mean stress. Finally, it is concluded that for a 304 stainless steel, in order to take into account the mean stress in fatigue life, the mean stress effect has to be decomposed into two parts: maximum and “intrinsic” mean stress effects.     

A finite element constitutive update scheme for anisotropic,viscoelastic solids exhibiting non‐linearity of the Schapery type

This study presents a new scheme for performing integration point constitutive updates for anisotropic, small strain, non‐linear viscoelasticity, within the context of implicit, non‐linear finite element structural analysis. While the basic scheme has been presented earlier by the authors for linear viscoelasticity, the present work illustrates the generality of the underlying fundamentals by extending to Schapery's non‐linear model. The method features a judicious choice of state variables, a stable backward Euler integration step, and a consistent tangent operator. Its greatest strength lies in ready incorporation into existing FEM codes. Numerical examples involving homogeneous stress states such as uniaxial extension and simple shear, and non‐uniform stress states such as a beam under tip load, were carried out by incorporating the present scheme into a general purpose FEM package. Excellent agreement with analytical results is observed. Copyright © 1999 John Wiley & Sons, Ltd.     

Optimized Dynamic Acousto-elasticity Applied to Fatigue Damage and Stress Corrosion Cracking

The dynamic acousto-elasticity (DAE) technique uniquely provides the elastic (speed of sound and attenuation) behavior over a dynamic strain cycle. This technique has been applied successfully to highly nonlinear materials such as rock samples, where nonlinear elastic sources are present throughout the material. DAE has shown different nonlinear elastic behavior in tension and compression as well as early-time memory effects (i.e. fast and slow dynamics) that cannot be observed with conventional dynamic techniques (e.g. resonance or wave mixing measurements). The main objective of the present study is to evaluate if the DAE technique is also sensitive to (1) fatigue damage and (2) a localized stress corrosion crack. A secondary objective is to adapt the DAE experimental setup to perform measurements in smaller specimens (thickness of few cm). Several samples (intact aluminium, fatigued aluminium and steel with a stress corrosion crack) were investigated. Using signal processing not normally applied to DAE, we are able to measure the nonlinear elastic response of intact aluminium, distinguish the intact from the fatigued aluminium sample and localize different nonlinear features in the stress corrosion cracked steel sample.     

A relationship between non-exponential stress relaxation and delayed elasticity in the viscoelastic process in amorphous solids: Illustration on a chalcogenide glass

Inorganic glasses are viscoelastic materials since they exhibit, below as well as above their glass transition temperature, a viscoelastic deformation under stress, which can be decomposed into a sum of an elastic part, an inelastic (or viscous) part and a delayed elastic part. The delayed elastic part is responsible for the non-linear primary creep stage observed during creep tests. During a stress relaxation test, the strain, imposed, is initially fully elastic, but is transformed, as the stress relaxes, into an inelastic and a delayed elastic strains. For linear viscoelastic materials, if the stress relaxation function can be fitted by a stretched exponential function, the evolution of each part of the strain can be predicted using the Boltzmann superposition principle. We develop here the equations of these evolutions, and we illustrate their accuracy by comparing them with experimental evolutions measured on GeSe₉ glass fibers. We illustrate also, by simple equations, the relationship between any kind of relaxation function based on additive contribution of different relaxation processes and the delayed elastic contribution to stress relaxation: the delayed elasticity is directly correlated to the dispersion of relaxations times of the processes involved during relaxation.     

A fatigue damage indicator parameter for P91 chromium‐molybdenum alloy steel and fatigue pre‐damaged P54T carbon steel

Two grades of structural steel were subjected to fully reversible, constant stress amplitude cyclic loading. The local strain response of the material was measured and recorded during the test, with the applied testing technique enabling the monitoring of hysteresis loop variation for the narrowest cross‐section of the hourglass specimen. Changes in hysteresis loop width, representing the local inelastic response of the material, were recorded in order to monitor the density of structural imperfections. Material ratcheting behaviour was observed as changes in the mean strain for selected load cycles. Ratcheting was attributed to local deformation of the material in the vicinity of imperfections such as voids or inclusions, as well as deformation induced by the propagation of microcracks. Definitions of a damage indicator parameter and damage parameter were proposed. The fatigue behaviour of the two investigated grades of steel was finally illustrated in the form of damage curves for different stress amplitudes and for undamaged and fatigue pre‐damaged material.     

Continuous fatigue damage prediction of a rubber fibre composite structure using multiaxial energy‐based approach

This paper details an advanced method of continuous fatigue damage prediction of rubber fibre composite structures. A novel multiaxial energy‐based approach incorporating a mean stress correction is presented and also used to predict the fatigue life of a commercial vehicle air spring. The variations of elastic strain and complementary energies are joined to form the energy damage parameter. Material parameter α is introduced to adapt for any observed mean stress effect as well as being able to reproduce the well‐known Smith‐Watson‐Topper criterion. Since integration to calculate the energies is simplified, the approach can be employed regardless of the complexity of the thermo‐mechanical load history. Several numerical simulations and experimental tests were performed in order to obtain the required stress‐strain tensors and the corresponding fatigue lives, respectively. In simulations, the rubber material of the air spring was simulated as nonlinear elastic. The mean stress parameter α , which controls the influence of the mean stress on fatigue life, was adjusted with respect to those energy life curves obtained experimentally. The predicted fatigue life and the location of failure are in very good agreement with experimental observations.     

An accelerated FFT algorithm for thermoelastic and non‐linear composites

A fast numerical algorithm to compute the local and overall responses of non‐linear composite materials is developed. This alternative formulation allows us to improve the convergence of the existing method of Moulinec and Suquet (e.g. Comput. Meth. Appl. Mech. Eng. 1998; 157 (1–2):69–94). In the present method, a non‐linear elastic (or conducting) material is replaced by infinitely many locally linear thermoelastic materials with moduli that depend on the values of the local fields. This makes it possible to use the advantages of an algorithm developed by Eyre and Milton (Eur. Phys. J. Appl. Phys. 1999; 6 (1):41–47), which has faster convergence. The method is applied to compute the local fields as well as the effective response of non‐linear conducting and elastic periodic composites. Copyright © 2008 John Wiley & Sons, Ltd.     

Enhanced assessment of ageing phenomena in steel structures based on materials data and non‐destructive testing

The increasing amount of ageing civil steel infrastructure requests an enhanced assessment of this infrastructure in terms of determining its residual fatigue life in a more realistic way than this has been done in the past. Often the relevant materials data for cyclic loading of such an ageing infrastructure is not available and its retrieval turns out to be relatively cumbersome bearing the urgency in data availability and continuous cost pressure in mind. This article addresses different approaches and techniques on how materials data for cyclic loading can be obtained at a fraction of the effort compared to state‐of‐the‐art techniques, considering load increase tests, non‐destructive testing techniques and finally even a stepped bar specimen allowing a complete set of materials data (stress‐strain behaviour and stress‐ and strain‐life curve) to be obtained with a single specimen in the end only. Options for ’digitizing’ materials data evaluation are discussed and some prospect on application of those novel approaches and techniques in damage accumulation assessments on real steel infrastructure is provided.     

Mean stress and plasticity effect prediction on notch fatigue and crack growth threshold,combining the theory of critical distances and multiaxial fatigue criteria

In the present work, we propose a robust calibration of some bi‐parametric multiaxial fatigue criteria applied in conjunction with the theory of critical distances (TCD). This is based on least‐square fitting fatigue data generated using plain and sharp‐notched specimens tested at two different load ratios and allows for the estimation of the critical distance according to the point and line method formulation of TCD. It is shown that this combination permits to incorporate the mean stress effect into the fatigue strength calculation, which is not accounted for in the classical formulation of TCD based on the range of the maximum principal stress. It is also shown that for those materials exhibiting a low fatigue‐strength‐to‐yield‐stress ratio σ_fl,R = ?1/σ_YS, such as 7075‐T6 (σ_fl,R = ?1/σ_YS = 0.30), satisfactorily accurate predictions are obtained assuming a linear‐elastic stress distribution, even at the tip of sharp notches and cracks. Conversely, for any materials characterized by higher values of this ratio, as quenched and tempered 42CrMo4 (σ_fl,R = ?1/σ_YS = 0.54), it is recommended to consider the stabilized elastic‐plastic stress/strain distribution, also for plain and blunt‐notched samples and even in the high cycle fatigue regime still with the application of the TCD.