Multiaxial Constitutive Model Research Articles

Behavior of high performance fiber reinforced cement composites under multi-axial compressive loading

Experiments were conducted to better understand the behavior of strain hardening, high performance fiber reinforced cement composites (HPFRCC) when subjected to uniaxial, biaxial, and triaxial compression. The experimental parameters were: type of fiber, fiber volume fraction, and loading path. Two types of commercially available fibers, namely high-strength hooked steel fiber and ultra high molecular weight polyethylene fiber, with volume fractions ranging from 1.0% to 2.0%, were used in a 55-MPa mortar matrix. The selected loading paths consisted of uniaxial compression and tension, equal biaxial compression, and triaxial compression with two levels of lateral compression. The test results revealed that the inclusion of short fibers can significantly increase both strength and ductility under uniaxial and biaxial loading paths, but that the role of volume fraction is rather small for the range of fiber volume contents considered. The results also showed that the confining effect introduced by the fibers becomes minor in triaxial compression tests, where there is relatively high external confining pressure. The experimental information documented herein can serve as input for the development of multiaxial constitutive models for HPFRCCs.

Stochastic damage model for concrete based on energy equivalent strain

Starting with the framework of conventional elastoplastic damage mechanics, a class of stochastic damage constitutive model is derived based on the concept of energy equivalent strain. The stochastic damage model derived from the parallel element model is adopted to develop the uniaxial damage evolution function. Based on the expressions of damage energy release rates (DERRs) conjugated to the damage variables thermodynamically, the concept and its tensor formulations of energy equivalent strain is proposed to bridge the gap between the uniaxial and the multiaxial constitutive models. Furthermore, a simplified coupling model is proposed to consider the evolution of plastic strain. And the analytical expressions of the constitutive model in 2-D are established from the abstract tensor expression. Several numerical simulations are presented against the biaxial loading test results of concrete, demonstrating that the proposed models can reflect the salient features for concrete under uniaxial and biaxial loading conditions.

A multi-axial constitutive model for metal matrix composites

Metal matrix composites (MMCs) generally do not follow the classical plasticity theory, even though the matrix metals do deform plastically. A tension–compression yield asymmetry is typically observed in MMCs. For particulate-reinforced MMCs, this non-classical response is mainly due to the variation of damage evolution with loading modes. In this paper, a viscoplastic multi-axial constitutive model for plastic deformation of MMCs is constructed using the Mises–Schleicher yield criterion. The subsequent plastic flow is characterized by an associated and decomposed flow rule considering effects from both deviatoric and hydrostatic stresses. This model is capable of describing the multi-axial yield and flow behavior of MMCs by using simulated or measured asymmetric tensile and compressive stress–strain responses as input. As an example, the influence of damage evolution in terms of interfacial debonding in MMCs (obtained from FEM simulations) is incorporated through the different tensile and compressive stress–strain behaviors. Applying this model to predict the torsion and the pressure-dependant tensile responses of some commonly used MMCs provides good agreement with experimental data.

Nonlinear constitutive models for FRP composites using artificial neural networks

This paper presents a new approach to generate nonlinear and multi-axial constitutive models for fiber reinforced polymeric (FRP) composites using artificial neural networks (ANNs). The new nonlinear ANN constitutive models are complete and have been integrated with displacement-based FE software for the nonlinear analysis of composite structures. The proposed ANN constitutive models are trained with experimental data obtained from off-axis tension/compression and pure shear (Arcan) tests. The proposed ANN constitutive model is generated for plane–stress states with assumed functional response in some parts of the multi-axial stress space with no experimental data. The ability of the trained ANN models to predict material response is examined directly and through FE analysis of a notched composite plate. The experimental part of this study involved coupon testing of thick-section pultruded FRP E-glass/polyester material. Nonlinear response was pronounced including in the fiber direction due to the relatively low overall fiber volume fraction (FVF). Notched composite plates were also tested to verify the FE, with ANN material models, to predict general non-homogeneous responses at the structural level.

Constitutive model of rock under static-dynamic coupling loading and experimental investigation

The importance of study on constitutive model of statically loaded rock experiencing dynamic load is set forth, and the studying methods on dynamic constitutive model are classified according to the current studying status. By way of combining statistic damage model and viscoelastic model, uni-axial and multi-axial constitutive models of statically loaded rock experiencing dynamic load (static-dynamic coupling constitutive model) under intermediate strain rate are established. The verification experiment on 2D constitutive model under different static stress and dynamic stress with different frequencies is designed and performed. It is found that there is a good agreement between the experimental stress-strain curves and the theoretical stress-strain curves.

Modeling of Anisotropic Deformation in Superplastic Sheet Metal Stretching

Currently available models describing superplastic deformation are mostly based on uniaxial tensile test data and assume isotropic behavior, thus leading to limited predictive capabilities of material deformation and failure. In this work we present a multi-axial microstructure-based constitutive model that describes the anisotropic superplastic deformation within the continuum theory of viscoplasticity with internal variables. The model accounts for microstructural evolution and employs a generalized anisotropic dynamic yield function. The anisotropic yield function can describe the evolution of the initial state of anisotropy through the evolution of unit vectors defining the direction of anisotropy during deformation. The generalized model is then reduced to the plane stress condition to simulate sheet metal stretching in superplastic blow forming using pressurized gas. Different ratios of biaxial stretching were investigated, including the case simulating the uniaxial loading condition, where the model successfully captured the uniaxial experimental data. The model is also used to develop a new forming pressure profile that accounts for anisotropy and microstructural evolution.

In-plane shear testing of thick-section pultruded FRP composites using a modified Arcan fixture

A modified Arcan fixture with butterfly specimen geometry is designed to measure the in-plane shear response of thick-section pultruded FRP composites. The objective of the proposed testing method is to determine both the material shear stiffness and its non-linear stress–strain response up to ultimate stress. The tested pultruded specimens include two alternating layers in the form of a unidirectional glass roving and continuous filament mat layers. Finite element models for the butterfly specimen are generated to examine the effects of the notch radius and material orthotropy on the uniformity and distribution of stresses in the gage area. Butterfly geometry with a blunted notch and roving orientation parallel to the applied load is found to have a uniform shear stress in the gage section. The axial-shear response is measured under different biaxial stress states by varying the angle of the applied load. The tested non-linear shear stress–strain responses compare favorably to results previously obtained from off-axis compression tests used to calibrate a multi-axial constitutive model for this material. Results from strain gage and Infrared thermography measurements provide confirmation for the effectiveness of the fixture and the specimen geometry.

Multiaxial constitutive model accounting for the strength-differential in Inconel 718

The nickel-base alloy Inconel 718 exhibits a strength-differential, that is, a different plastic flow behavior in uniaxial tension and uniaxial compression. A phenomenological viscoplastic model founded on thermodynamics has been extended for material behavior that deviates from classical metal plasticity by including all three stress invariants in the threshold function. The model can predict plastic flow in isotropic materials with or without a flow stress asymmetry as well as with or without pressure dependence. Viscoplastic material parameters have been fit to pure shear, uniaxial tension, and uniaxial compression experimental results at 650°°C. Threshold function material parameters have been fit to the strength-differential. Four classes of threshold functions have been considered and nonproportional loading of hollow tubes, such as shear strain followed by axial strain, has been used to select the most applicable class of threshold function for the multiaxial model as applied to Inconel 718 at 650 °C. These nonproportional load paths containing corners provide a rigorous test of a plasticity model, whether it is time-dependent or not. A J 2 J 3 class model, where J 2 and J 3 are the second and third effective deviatoric stress invariants, was found to agree the best with the experimental results.

Nonlinear constitutive models for pultruded FRP composites

Three nonlinear multi-axial constitutive models for pultruded fiber reinforced plastic composites are proposed and examined in this study. The first two are macromechanical models that idealize the entire composite material as homogeneous orthotropic under plane stress conditions. The third is a new three-dimensional (3D) micromechanical model where the fiber and matrix responses are explicitly recognized and the nonlinear behavior is expressed at the matrix level. The pultruded composite material system considered in this study consists of two alternating layers of roving and continuous filament mat. The two layers have E-glass/vinylester fiber/matrix constituents. Coupon tests were performed for calibration and verification of the proposed models. Nonlinear response is calibrated using V-notch tests, to generate the axial-shear stress–strain, and uniaxial transverse tests. Off-axis coupons were cut with different roving orientations in order to generate in-plane multi-axial stress states. The nonlinear axial stress–strain curves of the off-axis tests are compared with the predicted curves from the three proposed models. Good agreement is shown for all off-axis angles when comparing the experimental stress–strain curves with those predicted by the 3D micromodel. The nonlinear curves predicted by the two nonlinear orthotropic models are also in good agreement with the experimental results but not for all off-axis orientations. All three models can be easily integrated within a finite element code for the general nonlinear analysis of pultruded composite structures using layered shell or 3D type elements.

A Constitutive Model Taking Account of Non-Symmetric Creep Behavior in Tension and Compression.

A multiaxial constitutive model that distinguishes between creep-damage behaviors of a class of polycrystalline metallic materials under tensile and compressive stress states is developed from phenomenological points of view. The formulation is achieved by modifying the coupled creep-damage model based on the von Mises stress criterion. The effective stresses of the von Mises type that control the rates of creep and damage are scaled, respectively, to show asymmetry in tension and compression. For this purpose, anisotropic scaling parameters are introduced which depend on the hydrostatic stress and the third invariant of the stress deviator. Two kinds of modified version of the coupled creep-damage model are presented which consider kinematic and isotropic hardening of materials, respectively. These new models describe the primary, secondary and tertiary creep behavior with asymmetry in tension and compression. Numerical simulations of creep behavior under constant and variable stress conditions are also carried out.

Time-Dependent Strain Distribution in Polymers under Complex Stress History: Experimental Study by Moire Fringe Method and Finite Element Predictions

A combination of experimental and computational methods are developed, for predictions of nonlinear viscoelastic creep of polymers in sheet form, subjected to inhomogeneous stress states and stress history. A recently proposed multiaxial constitutive model for glassy polymers was implemented in a commercial finite element (FE) package. The model was tested by means of creep experiments on PMMA at elevated temperature, using specimens with a central circular hole. The experiments were performed using a tensile creep machine, and the geometric Moire fringe method was employed for measuring strain distribution. The results obtained from the experiment and FE analysis were compared.

Time-Dependent Strain Distribution in Polymers under Complex Stress History: Experimental Study by Moire Fringe Method and Finite Element Predictions

A combination of experimental and computational methods are developed, for predictions of nonlinear viscoelastic creep of polymers in sheet form, subjected to inhomogeneous stress states and stress history. A recently proposed multiaxial constitutive model for glassy polymers was implemented in a commercial finite element (FE) package. The model was tested by means of creep experiments on PMMA at elevated temperature, using specimens with a central circular hole. The experiments were performed using a tensile creep machine, and the geometric Moire fringe method was employed for measuring strain distribution. The results obtained from the experiment and FE analysis were compared.

Multiaxial Experiments and Constitutive Modeling of Superplasticity

In the first part of the paper, multiaxial experiments of 5083Al alloy are carried out at 833K at constant strain-rates to elucidate the characteristic features of superplastic deformation under general forming conditions. Monotonic tension, compression and torsion tests were performed by using solid circular cylindrical specimens together with thin-walled tubular specimens. The values of the equivalent strain-rate of von Mises type are specified as 2 × 10 -3 , 1 × 10 -3 , 5 × 10 -4 and 2 × 10 -4 s -1 . The significant strain-rate dependence of flow stress and the material softening due to cavity growth are observed under the tension tests. The results of the tension and compression tests show that the material hardens under tension while it softens under compression. The comparison between tension and torsion tests shows that the equivalent flow stress of von Mises type under torsion is much smaller than that of the tension at the same equivalent strain-rate of von Mises type. Based on these observations, a constitutive model of superplasticity is then formulated based on the finite deformation theory. The rate of deformation tensor is divided into the sum of the elastic and inelastic parts, and the former part is represented by the isotropic linear elastic law. The inelastic part is modeled based on the invariant theory by employing the steady-state creep law of hyperbolic sine type and by taking account of the effects of yielding, grain and cavity growth. The rate of drag stress is assumed to be the sum of the terms due to the grain and cavity growth. The evolution equation of the grain growth is formulated by taking account of static growth and strain-induced growth. The evolution equation of the cavity volume fraction is represented by the sum of the two terms due to cavity diffusion and viscoplastic deformation. Comparison of the results of the proposed model with those of the experiments shows that the proposed model gives satisfactory predictions.

History-dependent coupled growth of creep damage under variable stress conditions

A multiaxial constitutive model is presented for the description of the history-dependent inelastic and damage behavior of polycrystalline metallic materials in the high-temperature range under variable loading conditions. A particular emphasis is placed on a new formulation for the history-dependent material degradation which is based on a coupling between hardening and damage. The total damage is given by the sum of damage components which have different history dependence. The associated continuum damage variables are defined through their evolution equations coupled with hardening parameters involved with a kinematic hardening model. The non-classical coupling between hardening and damage is incorporated into the modified kinematic hardening model which has. been developed for describing the complicated hardening and softening behavior under repeated stress changes. Numerical simulations for different material parameters show that the proposed damage-coupled constitutive model can predict an acceleration as well as a deceleration of creep damage growth due to stress variations.

Inelastic Response of Off-Axis MMC Lamina

Damage evolution and failure characteristics were investigated for a 10 deg off-axis SiC/It unidirectional metal matrix composite (MMC) lamina subjected to monotonic loading. A replica technique was used to monitor sequential damage evolution under monotonic loading for the 10 deg MMC lamina. The replicas were then examined under a Scanning Electron Microscope (SEM). Characteristic debonding along the fiber—a dominant damage mechanism, matrix plasticity in the form of slip bands and fiber cracks were observed at various strain levels and were rationalized based on the state of stress in the off-axis lamina. This investigation provides an insight into the off-axis material response for MMC lamina and associated damage mechanisms and can guide mechanism-based multiaxial constitutive model and failure criteria development.

Phenomenological Multiaxial Constitutive Model for Shape Memory Alloys and Its Application. 2nd Report, Micromechanical Analysis for Hysteretic Behavior of TiNi-SMA Embedded Unidirectional Composite Laminae.

Hysteretic behavior of unidirectional TiNi SMA/Epoxy composites subjected to a single isothemal loading and unloading cycle has been analyzed using a three-dimensional micromechanical unitcell model. The multiaxial constitutive equations developed in the previous study were applied to the description of the stress-induced rhombohedral and martensitic transformations of the SMA fibers embedded in the epoxy matrix. It was clearly observed that the global behavior of the SMA composites was governed by the pseudoelastic property of the SMA fibers. The hysteretic responses of SMA composites were significantly influenced by the SMA fiber volume fraction. For a high volume fraction of the SMA fibers, the rhombohedral transformation of the embedded SMA fibers characterized a pseudoelasticity-like behavior of SMA composites. This predication indicates a feature unique to the multiaxial SMA constitutive model developed in the previous report.

Coupled Inelasticity and Damage Model for Metal Matrix Composites

Multiaxial constitutive models to describe fully coupled inelastic deformation and damage of unidirectional fiber-reinforced metal matrix composites are developed from a phenomenological point of view. A damage-coupled kinematic hardening model for transversely isotropic materials is first formulated in the invariant form on the basis of the internal variable approach. An isotropic hardening model coupled with damage is then constructed by assuming a particular representation of the kinematic hardening variable. The extent of composite damage is described using a scalar variable, and a different damage growth rate depending on the orientation of composite is considered through a fourth-rank anisotropic tensor. The rates of damage and hardening are affected by the magnitudes of the relevant internal variables, respectively, so that a full coupling between inelasticity and damage can be predicted. An analytical expression for the creep rupture time under a constant stress condition can be derived from the isotropic hardening model. This expression of the creep life depends on both the time when the minimum creep rate is attained and the degree of damage at that instant. Numerical simulations have been performed to elucidate the characteristics of the present formulation.

Analysis of primary creep of S2 fresh-water and saline ice

A simple multiaxial constitutive model for primary creep of S2 fresh-water and saline ice is proposed. This model is applied to available primary creep data for fresh-water, sea, and laboratory-grown saline ice. Results of these comparisons are used to examine microscopic and macroscopic aspects of creep of S2 ice. Also the proposed model provides dimensionless representations that can be useful for design of creep tests and presentation of creep data.

形状記憶合金の現象論的多軸構成モデルとその応用 : 第1報, 多軸表現の比較とTiNi-SMAモデルへの一般化 (<小特集>インテリジェント材料・流体システム)

A multiaxial constitutive model to describe the pseudoelastic and shape-memory behavior for TiNi shape memory alloys has been developed from a phenomenological point of view. The rhombohedral and martensitic phase transformations of TiNi shape memory alloys have been taken into account. A generalized expression for the transformation strain range which depends on applied stress and current phase volume fraction has been proposed to obtain closed hysteresis loops for closed cyclic loading paths. The predictive capability of the present model for the hysteretic behavior of TiNi shape memory alloys has been demonstrated through simulations of the stress-strain curves under isothermal uniaxial and multiaxial loading-unloading cycles. A comparison between existing constitutive models for shape memory alloys has been also presented.

Multiaxial constitutive modeling of aircraft engine materials

Based on a newly developed theory (Lu and Weng, Acta Mech., in press) the high temperature behavior of an aircraft engine material is studied under combined stress state. Both monotonic and cyclic deformations are examined to uncover its stress-strain response, as well as its cyclic hardening and strain-ratchetting characteristics. Under a biaxial loading it is disclosed that tensile cyclic hardening is greatly magnified with a superimposed lateral tension, whereas the strain-ratchetting process is led to an enhanced, unsettling state with a superimposed lateral compression. The biaxial transient and steady-state creep strains have also been calculated. The results suggest that while a superimposed lateral tension will inhibit the creep deformation, a lateral compression can greatly promote the inelastic flow. To reflect the practical service conditions of an aircraft engine, the theory is further applied to examine the effect of loading frequency on the development of inelastic strains under concurrent thermal and mechanical loading. It is found that a more frequently flying aircraft will have a greater accumulation of creep strains and, consequently, a greater possibility of material damage in its engine components over the same total flying time.