Micromechanical modelling of cyclic plasticity incorporating damage

In this paper a numerical investigation on the possibility to simulate and predict cyclic plastic response incorporating damage has been performed. To this purpose, unit cell and continuum approaches based on porous metal plasticity and continuum damage mechanics (CDM) have been considered. In particular, the porous metal plasticity model of Leblond, Perrin and Devaux (LPD model) and the CDM model developed by Pirondi and Bonora were used. Finite element (FE) simulations were performed for each approach with different degrees of triaxiality and the results are analyzed and compared.     

Micromechanical modelling for effect of inherent anisotropy on cyclic behaviour of sand

The inherent anisotropy more or less exists in sand when preparing samples in laboratory or taking from field. The purpose of this paper is to model cyclic behaviour of sand by means of a micromechanical approach considering inherent anisotropy. The micromechanical stress–strain model developed in an earlier study by Chang and Hicher (2005) is enhanced to account for the stress reversal on a contact plane and the density state-dependent dilatancy. The enhanced model is first examined by simulating typical drained and undrained cyclic tests in conventional triaxial conditions. The model is then used to simulate drained cyclic triaxial tests under constant p′ on Toyoura sand with different initial void ratios and different levels of p′, and undrained triaxial tests on dense and loose Nevada sand. The applicability of the present model is evaluated through comparisons between the predicted and the measured results. The evolution of local stresses and local strains at inter-particle planes due to externally applied load are discussed. All simulations have demonstrated that the proposed micromechanical approach is capable of modelling the cyclic behaviour of sand with inherent and induced anisotropy.     

Frictional Hertzian contact problems under cyclic loading using static reduction

In this study we investigate an axisymmetric Hertzian contact problem of a rigid sphere pressing into an elastic half-space under cyclic loading. A numerical solution is sought to obtain a steady state, which demands a large amount of computer memory and computing speed. To achieve a tractable problem, the current numerical model uses a “static reduction” technique, which employs only a contact stiffness matrix rather than the entire stiffness of the problem and is more accurate than the approach used by most finite element codes. Investigation of the tendency of contact behavior in the transient and steady states confirms that a steady state exists, showing converged energy dissipation. The dependence of dissipation on load amplitude shows a power law of load amplitude less than 3, which may explain some deviations in the experimental findings.     

Computational modelling of the stress-softening phenomenon of rubber-like materials under cyclic loading

When a rubber specimen is subjected to cyclic loading, not only non-linear behaviour but also damage-induced stress-softening phenomena (the Mullins effect) have been observed. Applications of a continuum damage mechanics model and Ogden and Roxburgh's pseudo-elastic model to describe the Mullins effect in elastomers have been considered. Both models together with Gao's elastic law were implemented to describe the mechanical behaviour of rubber-like materials including the stress-softening phenomenon. Two sets of experimental data (a simple tension test and a simple tension and pure shear test) are used to validate the constitutive models. Model parameters are estimated via an inverse technique. Computational results show that both constitutive models together with Gao's elastic law can describe the typical Mullins effect. From engineering point of view, the pseudo-elastic model has the advantages that (i) the model is simple and practical, since it considers that the stress-softening function is only activated on unloading or reloading paths, (ii) the model with a slight modification of the damage variable is very stable in finite element calculations, and (iii) the numerical results agree very well with experimental data in both simple tension and pure shear deformation. Two applications illustrate the capability of combining the pseudo-elastic model with Gao's elastic law in describing the Mullins effect. It is emphasized that both models are applicable to multiaxial states of stress and strain because both models are energy-based and not strain-based.     

Review of brittle fracture criteria in case of static and cyclic mixed mode loading

The paper gives in the first part in pressed form a survey of brittle fracture criteria using a reference intensity factor in case of static mixed mode loading. Criteria (expressed in terms of different quantities such as stress, deformation and strain energy) usually refer to a parameter that is characteristic of the material response at fracture. Criteria include information on two basic hypotheses (crack propagation direction and unstable crack growth). In the second part a generalized method is suggested for application of cyclic reference intensity factor in case of cyclic mixed mode loading. Three basic hypotheses describe crack growth direction, stable crack growth steps and unstable crack growth.     

Micromechanical modelling of shear banding in single crystals

Plastic deformation of a single crystal in double-conjugate single slip was considered. A micromechanical model based on dislocation density evolution was used. The mobile dislocation densities were assumed to relax rapidly to quasi-steady state values dictated by the current values of the slowly varying immobile dislocation densities (“adiabatic” approximation). In this approximation, a uniform time (or strain) dependent solution of the set of the governing equations was obtained. Linear analysis of stability of the uniform plastic flow with respect to band-shaped perturbations was carried out. The correspondence of the instability condition obtained with the results of previous, more macroscopic analyses was examined. Numerical examples illustrating the onset of instability were considered.     

Micromechanical modelling of remanent properties of morphotropic PZT

The present paper investigates the capability of micromechanical material models to predict the ferroelectric behaviour of morphotropic PZT ceramics in a rate-independent approximation based on realistic microscopic material parameters. Starting point is a three-dimensional tetragonal model, which builds on the model of Pathak and McMeeking [2008. Three-dimensional finite element simulations of ferroelectric polycrystals under electrical and mechanical loading. Journal of the Mechanics and Physics of Solids 56, 663-683]. Volume fractions of the crystallographic variants represent the domain structure inside the grains. Interactions between the grains are taken into account by means of a representative volume element of the grain compound. A simplified set of realistic microscopic material parameters of the lattice in terms of Young's modulus, Poisson's ratio, dielectric constant, and spontaneous strain and polarisation is derived from experimental data and theoretical results given in the literature. The simulation of the macroscopic remanent polarisation and strain response due to two load cases shows explicitly that the tetragonal model is not capable to reproduce the behaviour of morphotropic PZT. Therefore, the model is extended by the rhombohedral phase, allowing a mixture of both phases with varying quantities inside the grains. A comparison of our results with experimental data shows a remarkably good agreement, revealing the capability of the extended model.     

Constitutive modelling of cyclic plasticity of metal particulate-reinforced brittle matrix composites under compression-compression loading

The Mori-Tanaka approach is used to modelling metal particulate-reinforced brittle matrix composites under cyclic compressive loading. The J₂-flow theory is considered as the relevant physical law of plastic flow in inclusions. Ratchetting of the composite is prevented by the strong constraint exerted by the matrix on the inclusions, even under the assumption of evanescent kinematic hardening. However, the weakening constraint power of the matrix caused by microfracture damage around inclusions is closely coupled with the plasticity of inclusion and leads to ratchetting even when the plastic deformation of inclusions is described by an isotropic hardening rule. A detailed parametric study has revealed that ratchetting is followed by either plastic or elastic shakedown, depending on the load amplitude, composite parameters and the mean length of microcracks.     

Study on the static properties of spiral springs under static loading

Applied Mathematics and Mechanics - Spiral springs have a wide range of applications in various fields. As a result of the complexity of friction, few theoretical analyses of spring belts under...     

Creep constitutive modeling of an aluminum alloy under multiaxial and cyclic loading

A constitutive model is developed to characterize creep response of polycrystalline metals. The model is based on the effective stress concept and back stress is utilized as an internal variable. A memory aspect is incorporated in the model to account for the previous maximal stress as a result of dislocation related micromechanisms. The model is used to predict the creep potential of an aluminum alloy under multiaxial and cyclic loading. The predicted results are compared with the available experimental results on an aluminum alloy (2618-T61). The associated six basic model parameters are evaluated by using an optimization technique, called ‘Box Algorithm’. These model parameters are then used to predict some unoptimized experimental data sets. The influence of multiaxial and cyclic loadings is investigated in detail for the selected aluminum alloy. Overall, excellent correlations are observed.     

Micromechanical modelling of skeletal muscles: from the single fibre to the whole muscle

The structure of a skeletal muscle is dominated by its hierarchical architecture in which thousands of muscle fibres are arranged within a connective tissue network. The single muscle fibre consists of many force-producing cells, known as sarcomeres, which contribute to the contraction of the whole muscle. There are a lot of questions concerning the optimisation of muscle strength and agility. To answer these questions, numerical testing tools, e.g. in the form of finite element models can be an adequate alternative to standard experimental investigations. The present approach is crucially based on the use of the finite element method. The material behaviour of the muscle is additively split into a so-called active and a passive part. To describe the passive part special unit cells consisting of one tetrahedral element and six truss elements have been derived. Embedded into these unit cells are non-linear truss elements which represent bundles of muscle fibres. Besides the representation of the material model, this contribution focuses on the application to anatomically based 3D problems, as the animal soleus muscle of the rat.     

Grip pressure distribution under static and dynamic loading

The dynamic response of a vibrating handarm system is strongly related to the grip force. While the relationship between total grip force and vibration characteristics of the hand-arm system has been extensively studied, no attempts have been made to investigate the distribution of grip pressure at the hand-handle interface. The local grip-pressure distribution may be more closely related to the finger blood flow, fatigue and loss of productivity than total grip force. In the present study, distribution of static and dynamic forces at a hand-handle interface is investigated using a grid of pressure sensors mounted on the handle. The pressure distribution is acquired for different values of static and dynamic grip forces in the range of 25–150 N. The dynamic measurements were conducted at various discrete frequencies in the 20–1000 Hz range with peak acceleration levels of 0.5 g, 1.0 g, 2.0 g and 3.0 g. The grip-pressure distribution under static loads revealed a concentration of high pressures near the tips of the index and middle fingers, and the base of the thumb. This concentration of high pressures shifted towards the middle of the fingers under dynamic loads, irrespective of grip force, excitation frequency and acceleration levels. These local pressure peaks may be related to impairment of blood flow to finger tips and the possible causation of vibration white finger. Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 8–11.     

Constitutive modelling of nonproportional cyclic plasticity

A multiplicative hardening function and a unified evolution rule of the hardening factors are proposed. The hardening factorf ₁ is introduced to describe cyclic hardening with respect to the plastic strain range, whilef ₂ andf ₃ describe, respectively, instantaneous and hereditary additional hardening with respect to the nonproportionality of the plastic strain path. Two material dependent memory parametersa ₁ anda ₃ are introduced to keep the memory of the largest cyclic and additional hardening in the previous plastic deformation history. Different hardening mechanisms are then embedded into a thermomechanically consistent constitutive equation through the hardening function. The constitutive response of 304 and 316 stainless steels subjected to biaxial nonproportional cyclic loading is analyzed and the proposed model is critically verified by comparing the results with experimental results obtained by Tanaka et al., and Ohashi et al. The project supported by National Natural Science Foundation of China     

Micromechanical modelling of dry and saturated cement paste: Porosity assessment using ultrasonic waves

This article assesses the relationship between porosity and ultrasonic parameters of cement paste. It includes theoretical assessment of ultrasonic wave velocities of cement paste materials. Theoretical micromechanical models describing cement paste as a two-phase composite were detailed. Mechanical (bulk and shear moduli) and ultrasonic (longitudinal and transverse velocities) properties were evaluated. They were then, compared to the experimental ultrasonic properties measured on dry and fully water saturated samples with varying porosity First, the obtained micromechanical results showed that the correlation between acoustic velocity and porosity yielded the expected values: longitudinal and transverse velocities decrease with porosity. Secondly, the dilute inclusion model was able to represent the acoustic parameter of the cement paste only at low porosity, up to 20%. The self-consistent model under-estimated the measured ultrasonic properties for almost all porosity ranges. The Mori–Tanaka and the Kuster–Toksöz models succeeded in describing the acoustic parameters in dry and saturated states when assuming spherical shaped pores.     

Delayed fracture of an isotropic viscoelastic material with a crack under cyclic loading

The delayed fracture of an isotropic viscoelastic plate is examined as a process involving the subcritical propagation of a straight normal-rupture crack during fatigue loading. Calculations are based on the modified {ie165-1} of fracture, it being assumed that the size of the prefracture zone ahead of the moving crack remains constant. This zone is also assumed to be small compared to the size of the crack itself. Solutions for a time-dependent crack length are given both for media which undergo quasi-viscous flow (an integral operator with an Abelian kernel is used) and for media whose creep curves have a horizontal asymptote (an integral operator with a kernel in the form of the fractional-exponential function of Yu. N. Rabotnov is used). S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 2, pp. 60–64, February, 1999.     

Failure of spot welds under in-plane static loading

Under in-plane loading conditions, two independent modes contribute to the failure of a spot weld: the in-plane shear mode and the in-plane rotational mode. In this work, the failures of both modes under large static load are examined individually. To study the combined failure of these two modes, two special test coupons are designed. The first coupon contains one spot weld. The second coupon contains five spot welds. Tests conducted in this work show that a very simple force-based failure criterion can be used to predict the failure of a spot weld under large in-plane combined static loads. Current multiaxial failure theory cannot explain this combined failure. This force-based spot weld failure criterion fits current automotive industry needs for body shell finite element application very well.     

Finite-deformation analysis of the crack-tip fields under cyclic loading

The plane-strain crack subjected to mode I cyclic loading under small scale yielding was analysed. The influence of the load range, load ratio and overload on the near-tip deformation-, stress- and strain-fields was studied. Although the near-tip zones of appreciable cyclic plastic flow for all loading regimes matched closely one another, when scaled with (ΔK/σ_Y)², the activities of plastic flow within them manifested dependence on K_max and K_min, as well as on overload. Cyclic trajectories of the crack-tip opening displacement (CTOD) converged to stable self-similar loops of the sizes proportional to ΔK² and positions in CTOD-K plane dependent on the maximum K along the whole loading route, including an overload. Computed near-tip deformation evidenced plastic crack advance, this way visualising of the Laird–Smith concept of fatigue cracking. This crack growth by blunting-resharpening accelerated with rising ΔK and was halted by an overload. Crack closure upon unloading had no place. The affinities were revealed between computed near-tip stress–strain variables and the experimental trends of the fatigue crack growth rate, such as its dependence on K_max and K_min (or ΔK and K_max), and retardation by overload. Thus, the effects of loading parameters on fatigue cracking, hitherto associated with crack closure, are attributable to the stress–strain fields in front of it as the direct drives of the key fatigue constituents – damage accumulation and bond breaking.