Arbitrary Loading Histories Research Articles

Amplitude dependence of filler-reinforced rubber: Experiments, constitutive modelling and FEM – Implementation

The focus of the present paper is the experimental investigation, the constitutive representation and the numerical simulation of the amplitude dependence of filler-reinforced elastomers. A standard way to investigate the dynamic properties of viscoelastic materials is via the dynamic modulus which is obtained from stress signals due to harmonic strain excitations. Based on comprehensive experimental data, an amplitude-dependent constitutive model of finite viscoelasticity is developed. The model is based on a modified Maxwell chain with process-dependent viscosities which depend on additional internal state variables. The evaluation of this thermodynamically consistent model is possible in both the time domain, via stress-time signals, and in the frequency domain, via the dynamic modulus. This property is very profitable for the parameter identification process. The implementation of the constitutive model into the commercial finite element code ANSYS with the user-programmable feature (UPF) USERMAT for large deformations in updated Lagrange formulation is presented. This implementation allows simulating the time-dependent behaviour of rubber components under arbitrary transient loading histories. Due to physical and geometrical nonlinearities, these simulations are not possible in the frequency domain. But, transient FEM computations of large loading histories are sometimes not possible in an acceptable time. In the context of the parameter identification the fundamental ideas are presented, how this problem has been solved. Transient FEM simulations of real rubber components are also shown to visualize the properties of the model in the context of the transient material behaviour.

FEM simulation of viscous properties for granular materials considering the loading rate effect

The deformation and strength characteristics of sandy soils as a kind of granular materials are very complex. The experimental results show that when the strain rate suddenly changes in monotonic loading (ML) case, the stress–strain curve of sandy soils changes sharply and then gradually converges into the original inferred one that would be obtained by continuous ML at constant strain rate after having exhibited clear yielding. Similar behaviors are also observed when ML is restarted at a constant strain rate following a creep loading or stress relaxation stage. An elasto-viscoplastic constitutive model for granular materials is developed, which consists of three components. One of the most important features of the model is that it can take into account the effects of loading rate due to viscous properties on the stress–strain behavior. The stress ratio-axial strain–time relations from four drained plain strain compression (PSC) tests on the saturated Toyoura sand are successfully simulated by the finite element method (FEM) code incorporating the proposed constitutive model. It is shown that the FEM code can simulate the viscous behaviors of sand accurately under arbitrary loading history.

Effects of loading rate on viscoplastic properties of polymer geosynthetics and its constitutive modeling

AbstractOn the basis of the special tensile test results under various loading histories, the rate‐dependent behaviors of three polymer geosynthetics due to their viscous properties have been investigated. All the investigated polymer geosynthetics show significant loading rate effects, creep deformation, and stress relaxation. Except for the polyester geogrid showing the combined viscosity, all the investigated polymer geosynthetics exhibit the isotach viscosity. An elasto‐viscoplastic constitutive model described in a nonlinear three‐component model framework is developed to simulate the rate‐dependent behaviors of polymer geosynthetics. The developed constitutive model is verified by comparing its simulated results with the experimental data of polymer geosynthetics presented in this study and those available from the literature. The comparison indicates that the developed model can reasonably interpret the rate‐dependent behaviors of polymer geosynthetics under arbitrary loading histories, including the step‐changed strain rate loading, creep, and stress relaxation applied during otherwise monotonic loading (ML). POLYM. ENG. SCI., 2010. © 2009 Society of Plastics Engineers

Damage Evolution in Composites with a Homogenization-based Continuum Damage Mechanics Model

This paper develops a 3D homogenization-based continuum damage mechanics (HCDM) model for fiber reinforced composites undergoing micromechanical damage. Micromechanical damage in the representative volume element (RVE) is explicitly incorporated in the form of fiber—matrix interfacial debonding. The model uses the evolving principal damage coordinate system as its reference in order to represent the anisotropic coefficients. This is necessary for retaining accuracy with nonproportional loading. The material constitutive law involves a fourth order orthotropic tensor with stiffness characterized as macroscopic internal variable. Damage in 3D composites is accounted for through functional forms of the fourth order damage tensor in terms of macroscopic strain components. The HCDM model parameters are calibrated by using homogenized micromechanical (HMM) solutions for the RVE for a few strain histories. The proposed model is validated by comparing the CDM results with HMM response of single and multiple fiber RVEs subjected to arbitrary loading history. Finally the HCDM model is incorporated in a macroscopic finite element code to conduct damage analysis in a structure. The effect of different microstructures on the macroscopic damage progression is examined through this study.

Homogenization Based 3D Continuum Damage Mechanics Model for Composites Undergoing Microstructural Debonding

This paper develops a microscopic homogenization based continuum damage mechanics (HCDM) model framework for fiber reinforced composites undergoing interfacial debonding. It is an advancement over the 2D HCDM model developed by Raghavan and Ghosh (2005, “A Continuum Damage Mechanics Model for Unidirectional Composites Undergoing Interfacial Debonding,” Mech. Mater., 37(9), pp. 955–979), which does not yield accurate results for nonproportional loading histories. The present paper overcomes this shortcoming through the introduction of a principal damage coordinate system (PDCS) in the HCDM representation, which evolves with loading history. The material behavior is represented as a continuum constitutive law involving a fourth order orthotropic tensor with stiffness characterized as a macroscopic internal variable. The current work also extends the model of Raghavan and Ghosh to incorporate damage in 3D composites through functional forms of the fourth order damage tensor in terms of macroscopic strain components. The model is calibrated by homogenizing the micromechanical response of the representative volume element (RVE) for a few strain histories. This parametric representation can significantly enhance the computational efficiency of the model by avoiding the cumbersome strain space interpolations. The proposed model is validated by comparing the CDM results with homogenized micromechanical response of single and multiple fiber RVEs subjected to arbitrary loading history.

Effect of Combined Loading Due to Bending and Internal Pressure on Pipe Flaw Evaluation Criteria

Considering a rational maintenance rule of Light Water Reactor piping, reliable flaw evaluation criteria are essential to determine how a detected flaw is detrimental to continuous plant operation. Ductile fracture is one of the dominanant failure modes to be considered for carbon steel piping, and can be analyzed by the elastic-plastic fracture mechanics. Some analytical efforts have been provided as flaw evaluation criteria using load correction factors such like the Z-factors in the JSME codes on fitness-for-service for nuclear power plants or the ASME boiler and pressure vessel code section XI. The present correction factors were conventionally determined taken conservatism and simplicity into account, however, the effect of internal pressure which would be an important factor under an actual plant condition was not adequately considered. Recently, a J-estimation scheme, “LBB. ENGC” for ductile fracture analysis of circumferentially through-wall-cracked pipes subject-ed to combined loading was newly developed to have a better prediction with more realistic manner. This method is explicitly incorporated the contribution of both bending and tension due to internal pressure by means of the scheme compatible with an arbitrary combined loading history. In this paper, the effect of internal pressure on the flaw evaluation criteria was investigated using the new J-estimation scheme. A correction factor based on the new J-estimation scheme was compared with the present correction factors, and the predictability of the current flaw evaluation criteria was quantitatively evaluated in consideration of internal pressure.

Continuum approach to high-cycle fatigue modeling

A continuum approach to modeling of high-cycle fatigue is proposed. The approach takes a very simple and appealing form and is based on the concepts of a moving endurance surface in the stress space and an evolving damage variable. If the stress state is inside the endurance surface, no damage development occurs whereas damage may develop if the stress state is outside the surface. The endurance surface may move in the stress space as a function of the stress history. The theory is formulated within a proper continuum mechanics framework and it treats uniaxial and multiaxial stress states in a unified manner for arbitrary load histories. A unique feature of the theory is that it allows for the concepts of fatigue limits and accumulation of damage during the load history and thereby avoids cycle-counting techniques. Predictions of the theory are compared with experimental data and the agreement turns out to be close.

Effect of Combined Loading Due to Bending and Internal Pressure on Pipe Flaw Evaluation Criteria

Considering a rational maintenance rule of Light Water Reactor piping, reliable flaw evaluation criteria are essential to determine how a detected flaw is detrimental to continuous plant operation. Ductile fracture is one of the dominanant failure modes to be considered for carbon steel piping, and can be analyzed by the elastic-plastic fracture mechanics. Some analytical efforts have been provided as flaw evaluation criteria using load correction factors such like the Z-factors in the JSME codes on fitness-for-service for nuclear power plants or the ASME boiler and pressure vessel code section XI. The present correction factors were conventionally determined taken conservatism and simplicity into account, however, the effect of internal pressure which would be an important factor under an actual plant condition was not adequately considered. Recently, a J-estimation scheme, “LBB. ENGC” for ductile fracture analysis of circumferentially through-wall-cracked pipes subject-ed to combined loading was newly developed to have a better prediction with more realistic manner. This method is explicitly incorporated the contribution of both bending and tension due to internal pressure by means of the scheme compatible with an arbitrary combined loading history. In this paper, the effect of internal pressure on the flaw evaluation criteria was investigated using the new J-estimation scheme. A correction factor based on the new J-estimation scheme was compared with the present correction factors, and the predictability of the current flaw evaluation criteria was quantitatively evaluated in consideration of internal pressure.

Fatigue reliability analysis of composites based on residual strength

The fatigue reliability of Gr/PEEK composites has been investigated in the present study. In the experimental investigation, constant amplitude loading, multiple-level loading and random loading have been considered step by step during the fatigue tests. In particular, a residual strength degradation model was presented for predicting the fatigue life of composites subjected to constant amplitude fatigue loading. The accumulation of fatigue damage was determined, using the residual strength degradation model. The residual strength distribution after an arbitrary fatigue cycle can be represented by a two-parameter Weibull distribution function. A fatigue reliability analysis method for arbitrary load history was thus developed through the Weibull distribution function. Good correlation between theory and experiment was obtained for constant amplitude loading, two-level loading, three-level loading and random loading. The result of the present study can be used as a basis for the reliability design of mechanical components made of Gr/PEEK composites.

Multiaxial fatigue of rubber: Part I: equivalence criteria and theoretical aspects

ABSTRACTThis paper investigates commonly used approaches for fatigue crack nucleation analysis in rubber, including maximum principal strain (or stretch), strain energy density and octahedral shear strain criteria. The ability of these traditional equivalence criteria, as well as a recent equivalence criterion (the cracking energy density) to predict multiaxial fatigue behaviour is explored. Theoretical considerations are also introduced relating to the applicability of various fatigue life analysis approaches. These include the scalar nature of traditional equivalence criteria, robustness of the criteria investigated for a wide range of multiaxial loadings, effects of crack closure and applications to non‐proportional multiaxial loadings. It is shown that the notion of a stress or strain amplitude tensor used for the analysis of multiaxial loading in metals is not appropriate in the analysis of rubber due to nonlinearity associated with finite strains and near incompressibility. Taken together, these considerations illustrate that traditional criteria are not sufficiently consistent or complete to permit confident analysis of arbitrary multiaxial loading histories, and that an analysis approach specific to the failure plane is needed. Of the three traditional criteria, maximum principal strain is shown to match most closely to the cracking energy density criterion, in terms of a failure locus in principal stretch space.

A Four-Parameter Iwan Model for Lap-Type Joints

The constitutive behavior of mechanical joints is largely responsible for the energy dissipation and vibration damping in built-up structures. For reasons arising from the dramatically different length scales associated with those dissipative mechanisms and the length scales characteristic of the overall structure, this physics cannot be captured through direct numerical simulation (DNS) of the contact mechanics within a structural dynamics analysis. The difficulties of DNS manifest themselves either in terms of Courant times that are orders of magnitude smaller than that necessary for structural dynamics analysis or as intractable conditioning problems. The only practical method for accommodating the nonlinear nature of joint mechanisms within structural dynamic analysis is through constitutive models employing degrees of freedom natural to the scale of structural dynamics. In this way, development of constitutive models for joint response is a prerequisite for a predictive structural dynamics capability. A four-parameter model, built on a framework developed by Iwan, is used to reproduce the qualitative and quantitative properties of lap-type joints. In the development presented here, the parameters are deduced by matching joint stiffness under low load, the force necessary to initiate macroslip, and experimental values of energy dissipation in harmonic loading. All the necessary experiments can be performed on real hardware or virtually via fine-resolution, nonlinear quasistatic finite elements. The resulting constitutive model can then be used to predict the force/displacement results from arbitrary load histories.

CONCRETE-FILLED TUBULAR COLUMNS: PART 1 — CROSS-SECTION ANALYSIS

A fiber-based approach is adopted to study the cross-sectional behavior of concrete-filled steel tubes subjected to monotonic and cyclic loading. Constitutive behavior of the cross-section is formulated by explicitly dividing the section into fibers, and uniaxial stress–strain rules for steel and concrete are defined. Interaction curves, Moment–Thrust–Curvature (M–P–φ) relations and Moment–Thrust–Strain (M–P–ε) relations are developed for circular and box-shaped in-filled sections based on fiber analysis. A fiber model is also developed to describe the nonlinear response of concrete-filled steel tubes subjected to cyclic loading. M–P–φ relations are developed for arbitrary loading histories using this simple yet reliable approach. Results are compared with the numerical and experimental data available in published literature, and the accuracy of the proposed model is thus ascertained.

Non-linear rheology of tension structural element under single and variable loading history Part I: Theoretical derivations

The present paper concerns the macroscopic overall description of rheologic properties for steel wire and synthetic fibre cables under variable loading actions according to non-linear creep and/or relaxation theory. The general constitutive equations of non-linear creep and/or relaxation of tension elements - cables under one-step and the variable stress or strain inputs using the product and two types of additive approximations of the kernel functions are presented in the paper. The derived non-linear constitutive equations describe a non-linear rheologic behaviour of the cables for a variable stress or strain history using the kernel functions determined only by one-step - constant creep or relaxation tests. The developed constitutive equations enable to simulate and to predict in a general way non-linear rheologic behaviour of the cables under an arbitrary loading or straining history. The derived constitutive equations can be used for the various tension structural elements with the non-linear rheologic properties under uniaxial variable stressing or straining.

Analysis of a Rocking and Walking Punch—Part II: General Numerical Solution Using Quadratic Programming

AbstractThis paper pursues a numerical approach to the solution of the contact problem of a rigid punch on an incompressible half-plane, subjected to a shearing force together with a normal force which may be offset from the centerline of the punch. A piecewise-linear representation of the shear tractions is employed and quadratic programming techniques are used to solve the problem. This method enables an arbitrary load history to be followed. Results are presented which show excellent agreement with other solutions to the special cases of monotonic and steady-state cyclic loading. It is shown that the traction distribution reaches a steady-state cycle after only a few cycles of loading. The existence of an interesting stick-slip regime, where a central zone of slip is bordered by two stick regions is highlighted.

The Payne effect in finite viscoelasticity: constitutive modelling based on fractional derivatives and intrinsic time scales

As we know from experimental testing, the stiffness behaviour of carbon black-filled elastomers under dynamic deformations is weakly dependent on the frequency of deformation but strongly dependent on the amplitude. Increasing strain amplitudes lead to a decrease in the dynamic stiffness, which is known as the Payne effect. In this essay, we develop a constitutive approach of finite viscoelasticity to represent the Payne effect in the context of continuum mechanics. The starting point for the constitutive model resulting from this development is the theory of finite linear viscoelasticity for incompressible materials, where the free energy is assumed to be a linear functional of the relative Piola strain tensor. Motivated by the weak frequency dependence of the dynamic stiffness of reinforced rubber, the memory kernel of the free energy functional is of the Mittag Leffler type. We demonstrate that the model is compatible with the Second Law of Thermodynamics and equal to a fractional differential equation between the overstress of the Second Piola Kirchhoff type and the Piola strain tensor. In order to represent the dependence of the dynamic stiffness on the amplitude of strain, we replace the physical time by an intrinsic time variable. The temporal evolution of the intrinsic time is driven by an internal variable, which is a measure for the current state of the material's microstructure. The material constants of the model are estimated using a stochastic identification algorithm of the Monte Carlo type. We demonstrate that the constitutive approach pursued here represents the combined frequency and amplitude dependence of filler-reinforced rubber. In comparison with the micromechanical Kraus model developed for sinusoidal strains, the theory set out in this essay allows the representation of the stress response under arbitrary loading histories.

The Effect of Scale and Criticality in Rock Slope Stability

Summary The apparent shear strength of rock discontinuities is considerably smaller than that of small scale samples. At the same time, the sliding behavior is characterized, in situ, by marked instabilities, with the typical features of Critical Phenomena. Contact mechanics permits to calculate normal and tangential forces at any point, and to follow the stick-slip transition for arbitrary loading histories. On the other hand, the above aspects are not captured by the classical theories, including those based on roughness indices. We argue that the multiscale topology of contact domains plays a fundamental role in determining the behavior of rock joints. In particular, experiments and numerical simulations show that these domains are lacunar sets with fractal dimension smaller than 2.0. This provides peculiar scaling of normal and tangential pressures at the interface, and the consequent size-dependence of the apparent friction coefficient. Moreover, we implement Renormalization Group to determine the critical point (e.g. the critical shear force) when rock sliding occurs. We show that the critical force is less than the one predicted by the classical Coulomb’s theory, and that it depends on the specimen size and on the topology of the interface. The same reasoning can be extended to other phenomena, e.g., to the rupture of brittle materials.

Bonding of a viscoelastic periodic rough surface to a rigid layer

AbstractA theoretical analysis is performed for the study of the bonding of a viscoelastic rough surface to a rigid substrate. The mechanics of contact and adhesion are studied with the Dugdale–Barenblatt model for surface interaction. Exact solutions are obtained for arbitrary surface profiles and loading histories. Detailed solutions are given for a power‐law viscoelastic material. This solution is used to determine the time for the self‐bonding of surfaces (solid sintering under zero load). The time to self‐bonding is shown to be extremely sensitive to the aspect ratio of the asperities. A closed form expression is derived for the time needed to achieve full contact when the surfaces are compressed with a load that increases linearly with time. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 545–561, 2002; DOI 10.1002/polb.10113

Vibration isolation and design of automotive test benches

The design of two automotive test benches for fatigue and comfort assessment of cars is illustrated. The vibration isolation of such test benches, designed to operate under arbitrary load histories in the frequency range 0–200 Hz, has been obtained with a fibre-reinforced concrete mass equipped with a dual suspension system. Because the imposed load histories are arbitrary, the eventuality that the resilient isolation mass of the bench may undergo resonant condition under harmonic excitation must be duly taken into account to exclude any fatigue damage under such extreme conditions. For this reason the research involved both the theoretical definition of the suspension system and the study of optimum concrete mixtures. Amongst the mixes considered, the two best, with polypropylene and steel fibre reinforcement respectively, were selected for the fashioning of two model benches, each of revealed very satisfactory behaviour when subsequently tested for fatigue under resonant conditions. The experimental behaviour of the real full-scale bench, constructed according to the guidelines stemming from the research, fully complies with the theoretical predictions, substantiating the methodological approach.

Prediction of the forming limit diagram on the basis of the damage criterion under non-proportional loading

Based on the theory of damage mechanics, a recently developed anisotropic plastic damage model for the prediction of forming limit diagrams (FLDs) is extended to take into account the effect of rotation of the principal damage coordinates on the deformation and damage behaviour. With the aid of the damage plastic potential, the damage evolution equations are established. A damage criterion for localized necking under arbitrary loading history is proposed. The model is employed to predict the FLDs of AL6111-T4 alloy. The predicted results agree well with those determined experimentally.

Modeling and compensation of hysteresis in piezoceramic actuators for vibration control

It has been established that the Maxwell resistive capacitor (MRC) hysteresis model accurately represents hysteretic behavior in the dielectric domain of PZT-based piezoceramics. Methods for MRC and inverse MRC online model identification have been developed along with an MRC-based framework for calculating continuous hysteretic energy loss for arbitrary loading histories. Building on these developments, an adaptive feedforward compensation technique for attenuating the effect of hysteresis in vibration control applications is proposed and evaluated experimentally via studies on a 1–3 piezoceramic composite. Also, indirect electrical excitation is compared to direct voltage drive of PZT devices for reduced hysteretic behavior in vibration control applications. [Work supported by the Office of Naval Research.]