Macroscopic Constitutive Model Research Articles

Axisymmetric plastic expansion of a cylindrical hole in isotropic metallic foam

By combining a finite element simulation with an analytical treatment, this paper provides quantitative information on the stress, strain and deformation states induced during the axisymmetric expansion of a cylindrical hole of an initial radius, located at the center of a block of closed-cell metallic foam of infinite size. Uniformly distributed radial loading is applied on the surface of the hole. A macroscopic phenomenological constitutive model of metal foam is first introduced, considering the initial and subsequent yielding surfaces in the space of the effective stress and hydrostatic stress. Isotropic hardening model is incorporated into the material property of the crushable foam. Two deformation stages are revealed from the numerical simulation, i.e. the initial yielding and then subsequent expansion of the hole accompanied with hardening. Preliminary analytical formulation is performed with respect to the pressure at the initial yielding and the size of the subsequent plastic deformation zone. It is found that after the onset of initial yielding, the maximum pressure is identical to that from the finite element analysis. The evolution of plastic zone during the expansion is discussed in terms of the results from the analytical and finite element studies. Furthermore, foam densification is observed from finite element analysis and a map is obtained showing the evolution of the three deforming zones, i.e. elastic, plastic and densification of the foam, when the applied pressure increases.

A macroscopic constitutive model of shape memory alloy considering plasticity

This paper presents a macroscopic constitutive model which is able to reproduce the thermo-mechanical behaviors of the super-elastic SMA undergoing plastic strain. A mechanical constitutive equation, which predicts the stress–strain response of the SMA undergoing plastic strain, is developed based on the expression of Gibbs free energy with plastic strain. A linear plastic constraint equation is supposed to describe the effect of plasticity on the phase transformation behaviors of SMA. A sine-type phase transformation equation is established to describe the phase transformation behaviors of the SMA undergoing plastic strain. The mechanical constitutive equation, plastic constraint equation, and phase transformation equation together compose the presented macroscopic constitutive model which reproduces the thermo-mechanical behaviors of the SMA undergoing plastic strain. Especially all material constants related to the presented macroscopic constitutive model can be determined through macroscopic experiments. Therefore it is easy to use this presented model for the practical applications of SMA. The mechanical behaviors of the supper-elastic SMA undergoing plastic strain and the effect of plasticity are numerically simulated by the presented macroscopic constitutive model. Results show that the presented macroscopic constitutive model can effectively reproduce the thermo-mechanical behaviors of the super-elastic SMA and express the effect of plasticity.

Hole expansion of twinning-induced plasticity steel

Fracture in the hole expansion test for a high-Mn twinning-induced plasticity (TWIP) steel sheet sample was investigated. The test results showed that cracking always occurred along the rolling direction (RD). The fracture behavior of the material was rationalized in terms of strain-rate sensitivity, yield surface shape and damage-producing MnS stringers aligned along the RD. In addition, according to the macroscopic constitutive modeling results, work hardening was shown to be almost isotropic for this material.

Theoretical Analysis of Functionally Graded Shape Memory Alloy Beam Subjected to Pure Bending

The functionally graded shape memory alloy(FG-SMA) is a new functional composite which possesses the excellent properties of both functionally graded material(FGM) and shape memory alloys(SMA).A macroscopic constitutive model of FG-SMA is established by using the mechanics of composite materials method and the existing SMA model.With this macro constitutive model,the mechanical behavior of a FG-SMA beam composed by elastic material A and SMA subjected to pure bending is investigated.The loading processes including elastic process and phase transformation process are discussed in detail and the analytical solutions are obtained.The distribution of stress on the cross section,the position of the neutral axis and the curvature-bending moment relation are discussed through numerical results.It can be learned that compared with common FGM,FG-SMA can decrease the maximum stress efficiently,preventing the destruction caused by oversize stress.This research can provide a base for the design and in-depth investigation of FG-SMA material.

Crack tip fields in rubber‐toughened polymers

AbstractNumerical simulations of the formation and the characteristic shape of plastic zones at crack tips in rubber‐toughened polymers are performed in order to get a better understanding of the underlying micromechanisms. Complementing previous work, the present study focuses on the contribution of distributed crazing. A macroscopic constitutive model therefore has been developed that describes the overall effect of distributed crazing as an anisotropic flow process. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Analysis of the rate-dependent coupled thermo-mechanical response of shape memory alloy bars and wires in tension

In this paper, the coupled thermo-mechanical response of shape memory alloy (SMA) bars and wires in tension is studied. By using the Gibbs free energy as the thermodynamic potential and choosing appropriate internal state variables, a three-dimensional phenomenological macroscopic constitutive model for polycrystalline SMAs is derived. Taking into account the effect of generated (absorbed) latent heat during the forward (inverse) martensitic phase transformation, the local form of the first law of thermodynamics is used to obtain the energy balance relation. The three-dimensional coupled relations for the energy balance in the presence of the internal heat flux and the constitutive equations are reduced to a one-dimensional problem. An explicit finite difference scheme is used to discretize the governing initial-boundary-value problem of bars and wires with circular cross-sections in tension. Considering several case studies for SMA wires and bars with different diameters, the effect of loading–unloading rate and different boundary conditions imposed by free and forced convections at the surface are studied. It is shown that the accuracy of assuming adiabatic or isothermal conditions in the tensile response of SMA bars strongly depends on the size and the ambient condition in addition to the rate dependency that has been known in the literature. The data of three experimental tests are used for validating the numerical results of the present formulation in predicting the stress–strain and temperature distribution for SMA bars and wires subjected to axial loading–unloading.

A Numerical Model for Powder Densification by SPS Technique

AbstractSpark plasma sintering (SPS) belongs to a class of sintering techniques that employs electric current to assist compaction. This technology seems very promising to obtain net‐shape components made of intermetallic alloys with a fine microstructure. However, the SPS process is difficult to stabilize because of density heterogeneities arising from non‐homogeneous temperature or stress in the powder. This motivates the development of a three‐dimensional finite element simulation in order to understand the distribution of current, temperature and porosity. The model couples three physical problems: electrical, thermal and mechanical. The numerical implementation is based on a monolithic formulation consisting in solving the different conservation equations on a single mesh including the specimen and the tooling. The general set of equations is described; the effect of the powder physical properties and of the geometry of the set up on the distribution of electrical current and temperature is discussed. A macroscopic Abouaf constitutive model is used to simulate powder densification; first results involving porosity evolution are presented.

A macroscopic damage‐plastic constitutive law for modeling quasi‐brittle fracture and ductile behavior of concrete

SUMMARYA new phenomenological macroscopic constitutive model for the numerical simulation of quasi‐brittle fracture and ductile concrete behavior, under general triaxial stress conditions, is presented. The model is particularly addressed to simulate a wide range of confinement stress states, as also, to capture the strong influence of the mean stress value in the concrete failure mechanisms.The model is based on a two‐surface damage‐plastic formulation. The mechanical behavior in different domains of the stress space is separately described by means of a quasi‐brittle or ductile material response: (i) For positive values of the mean stress (tensile states), an isotropic continuum damage model with strain softening is considered. In this context, and in order to avoid the Boundary Value Problem ill‐posedness induced by the softening law, a regularization technique based on the Continuum Strong Discontinuity Approach (CSDA) is adopted, which results equivalent to a damage model with embedded cohesive cracks providing anisotropic responses. (ii) A plastic model governs the material behavior when the mean stress is negative (confinement states). It is based on the classical plastic flow theory. In particular, a yield criterion similar to that of Willam and coauthors, which depends on the three stress invariants, is used. Additional features defining the plastic response are: an isotropic strain hardening law and a non‐associative flow rule.The paper presents the numerical implementation of the model using an efficient integration algorithm, namely, the Impl‐Ex scheme. Several widely known experimental tests (such as uniaxial, biaxial and triaxial tests) carried out on concrete specimens are used to calibrate and validate the performance of the proposed formulation. Finally, a classical 2D reinforced concrete beam example is analyzed in order to show the predictive capability of the model in structural analysis applications. Copyright © 2011 John Wiley & Sons, Ltd.

Homogenised elastic–viscoplastic simulation of plate–fin structures using duplex model

In this study, a duplex model is developed as a constitutive model for the homogenised elastic–viscoplastic behaviour of plate–fin structures. This model consists of plate and fin layers, which are individually ruled by different macroscopic constitutive models. An anisotropic, compressible power law equation that was derived in a recent study by the present authors is used to describe the homogenised viscoplastic behaviour of the fin layer. On the other hand, an isotropic, incompressible power law equation is used as the macroscopic viscoplastic constitutive equation of the plate layer. The duplex model developed is applied to an ultrafine plate–fin structure made of Hastelloy X. It is thus shown that the duplex model is more successful in simulating the homogenised viscoplastic behaviour under multiaxial loading than the corresponding simplex model in which plates and fins are non‐separately homogenised.

A combined analytical, numerical, and experimental study of shape-memory-alloy helical springs

In this paper, the pseudoelastic response of shape memory alloy (SMA) helical springs under axial force is studied both analytically and numerically. In the analytical solution two different approximations are considered. In the first approximation, both the curvature and pitch effects are assumed to be negligible. This is the case for helical springs with large ratios of mean coil radius to the cross sectional radius (spring index) and small pitch angles. Using this assumption, analysis of the helical spring is reduced to that of the pure torsion of a straight bar with circular cross section. A three-dimensional phenomenological macroscopic constitutive model for polycrystalline SMAs is reduced to the one-dimensional pure shear case and a closed-form solution for torsional response of SMA bars in loading and unloading is obtained. In the next step, the curvature effect is included and the SMA helical spring is analyzed using the exact solution presented for torsion of curved SMA bars. In this refined solution, the effect of the direct shear force is also considered. In the numerical analyses, the three-dimensional constitutive equations are implemented in a finite element method and using solid elements the loading–unloading of an SMA helical spring is simulated. Analytical and numerical results are compared and it is shown that the solution based on the SMA curved bar torsion gives an accurate stress analysis in the cross section of the helical SMA spring in addition to the global load–deflection response. All the results are compared with experimental data for a Nitinol helical spring. Several case studies are presented using the proposed analytical and numerical solutions and the effect of changing different parameters such as the material properties and temperature on the loading–unloading hysteretic response of SMA helical springs is studied. Finally, some practical recommendations are given for improving the performance of SMA helical springs used as energy dissipating devices, for example for seismic applications.

Multiscale Modelling Strategies for Heterogeneous Materials

This paper considers modelling strategies for the multiscale analysis of heterogeneous materials. Initial focus is on computational homogenization (e.g. [1-4]) which represents a powerful, albeit computationally intensive, technique for capturing the influence of an evolving microstrucuture on the overall macroscopic behaviour. First-order homogenization assumes that the representative volume element (RVE) is infinitely small compared to the macroscopic characteristic length scale and that there is clear separation of scales. Second-order schemes [2,4,5] are subsequently discussed as a possible solution. This approach introduces a material length scale into the macroscopic constitutive model and potentially enables geometric size effects to be captured. However, the size of the RVE now becomes an important parameter for the macroscopic response and there is no clear way of determining this length scale. Second-order computational homogenization is demonstrated in the context of an indentation problem. In situations where the RVE cannot be defined and there is no clear separation of scales, computational homogenization is not applicable and full resolution of the microstructure may be necessary. This paper summarises an efficient solution strategy for such situations following the work of Miehe and Bayreuther [6], with particular focus on fracturing heterogeneous materials, which utilises a geometric multi-grid preconditioner and the scale transition methodology from computational homogenization. This solution strategy is coupled with the use of the hybrid-Trefftz stress elements for modelling cohesive cracking [7]. The resulting modelling approach is demonstrated with the analysis of concrete dog-bone specimens where the different phases of matrix, aggregate and interfacial transition zone are explicitly modelled. Such an approach is still computationally intensive and so this paper also presents a novel hybrid multiscale approach that avoids the need for the finite element mesh to fully resolve all heterogeneities [8]. Classical micromechanics is utilised to enhance the approximation space in order to capture the effect of microstructural inclusions. In comparison to where the finite element mesh fully resolves the microstructure, this novel approach significantly reduces the number of degrees of freedom required while still accurately capturing the influence of the inclusions.

Multiscale constitutive modeling for plastic deformation of nanocrystalline materials

Macroscopic and microscopic constitutive modeling that can display large plastic deformation with shear band were presented for nanocrystalline materials subjected to uniaxial load over a wide strain rate range. The macroscopic model implemented with parameters microscopic meaning was established based on the theory of plastic dissipation energy. The microscopic model based on deformation mechanisms was composed of two parts: hardening and softening stages. In the hardening stage, the phase mixture model was used and a shear band deformation mechanism was proposed in the softening stage. Numerical simulations shown that the predications were in good agreement with experimental data. Finally, a parameter of normalized softening rate was proposed and its characteristics were evaluated quantitatively. It can be concluded that the failure strain could be prolonged when the normalized softening rate decrease through changing the softening path.

A generalized Hill’s lemma and micromechanically based macroscopic constitutive model for heterogeneous granular materials

Based on the Hill’s lemma for classical Cauchy continuum, a generalized Hill’s lemma for micro–macro homogenization modeling of gradient-enhanced Cosserat continuum is presented in the frame of the average-field theory. In this context not only the strain and stress tensors defined in classical Cosserat continuum but also their gradients are attributed to assigned micro-structural representative volume element (RVE), that leads to a higher-order macroscopic Cosserat continuum modeling and enables to incorporate the micro-structural size effects. The enhanced Hill–Mandel condition for gradient-enhanced Cosserat continuum is extracted as a corollary of the presented generalized Hill’s lemma. The derived admissible boundary conditions for the modeling are deduced to direct the proper presentation of boundary conditions to be prescribed on the RVE in order to ensure the satisfaction of the Hill–Mandel energy condition.With the link between the discrete particle assembly and its effective Cosserat continuum in an individual RVE, the boundary conditions prescribed on the RVE modeled as Cosserat continuum are transformed into those prescribed to the peripheral particles of the RVE modeled as the discrete particle assembly. The micromechanically based macroscopic constitutive model and corresponding rate forms of the macroscopic stress–strain relations taking into account the local microstructure and its evolution are formulated with neither need of specifying the macroscopic constitutive relation nor need of providing macroscopic material parameters.

Exact solutions for pure torsion of shape memory alloy circular bars

Shape memory alloy (SMA) structures are usually analyzed numerically; there are very few closed-form solutions in the literature. In this paper, we study the pure torsion of SMA bars with circular cross sections. First, a three-dimensional phenomenological macroscopic constitutive model for polycrystalline SMAs is reduced to the one-dimensional pure shear case and then a closed-form solution for torsional response of SMA bars in loading and unloading is obtained. Several case studies are presented in order to investigate the influence of different parameters, e.g. temperature and material properties on the torsional response of SMA circular bars.

Surface Energy Effect on Damage Evolution in a Viscoelastic Nanocomposite

In this article, the surface energy effect on damage evolution in a linear viscoelastic material filled with second phase nano-particles is studied. Because of the difference in mechanical properties between the matrix material and second phase particles, the progressive debonding of particle—matrix interface may occur under high stress triaxiality, and hence the damage evolution may be described by the nucleation and growth of these debonded voids. In order to study the surface energy effect on this kind of damage evolution, an energy criterion of interfacial debonding under spherically symmetrical loading is proposed, and an expression of the growth of debonded voids in a viscoelastic matrix material is derived, in which the effect of surface tension is also taken into account. Thus, a macroscopic constitutive model of the considered nanocomposite is presented. It is shown that the overall mechanical properties of such a nanocomposite are size dependent, due to the existence of the surface/interface energy effect.

1123 二層モデルによるプレートフィン構造体の弾粘塑性均質化シミュレーション(OS11. 材料の組織・強度に関するマルチスケールアナリシス(5),オーガナイズドセッション講演)

In this study, a two-layer model is developed for simulating the homogenized elastic-viscoplastic behavior of plate-fin structures. This model consists of plate and fin layers. which are individually ruled by macroscopic constitutive models. An anisotropic, compressible power-law equation that was derived in a recent study by the present authors is used to describe the homogenized viscoplastic behavior of fin layers. On the other hand, an isotropic. incompressible power-law equation is used as the macroscopic constitutive equation of plate layers. The two-layer model developed is applied to an ultra-fine plate-fin structure made of Hastelloy X at 900℃. It is thus shown that the two-layer model is more successful especially under combined loading than the corresponding one-layer model in which plates and fins are non-separately modeled.

112 二層モデルを用いた弾粘塑性プレートフィン構造体の均質化シミュレーション(OS1-3 微視構造を有する材料の変形と破壊 : シミュレーション)

In this study, a two-layer model is developed for simulating the homogenized elastic-viscoplastic behavior of plate-fm structures. This model consists of plate and fm layers, which are individually ruled by macroscopic constitutive models. An anisotropic, compressible power-law equation that was derived in a recent study by the present authors is used to describe the homogenized viscoplastic behavior of fm layers. On the other hand, an isotropic, incompressible power-law equation is used as the macroscopic constitutive equation of plate layers. The two-layer model developed is applied to an ultra-fme plate-fm structure made of Hastelloy X at 900℃. It is thus shown that the two-layer model is more successful especially under combined loading than the corresponding one-layer model in which plates and fms are non-separately modeled.

多凸性ひずみエネルギー関数を用いた異方性超弾性シェル要素の開発と衣服圧シミュレーションへの応用

The present study addresses the constitutive model of cloth using anisotropic hyperelastic materials and its implementation into a nonlinear finite shell element in order to simulate large deformation behavior of cloth. Present work focuses on a macroscopic continuum constitutive model able to capture the appropriate mechanical behavior of cloth, which is characterized by two families of yarns, i.e., warp and weft. Hyperelasticity allows the formulation of large deformation including the anisotropic effect by using a structural tensor which consists of fiber-directional unit vectors. In order to avoid non-physical behavior, the related strain-energy function must be polyconvex. The polyconvexity of a strain-energy function insures the existence of the global minimizers for the total elastic energy. In this paper, interaction effect of each fiber is accounted for using weight factors of the principal material directions in order for the accurate and robust modeling of clothing behavior. The present material model is implemented into a four-node shell element in Abaqus/Standard (S4R Type) via user-subroutine UMAT, and it is used to predict the outcome of uniaxial tensile tests and compute the clothing pressure applied to human body to design a form of cloth which better fits the human body.

608 一定内圧が作用する周期的微視構造物のマクロモデルの構築(OS8-2 工業材料の変形および特性評価2,OS8 工業材料の変形および特性評価)

608 一定内圧が作用する周期的微視構造物のマクロモデルの構築(OS8-2 工業材料の変形および特性評価2,OS8 工業材料の変形および特性評価)

A method of predicting macroscopic yield strength of polycrystalline metals subjected to plastic forming by micro–macro de-coupling scheme

We introduce a method of predicting the macroscopic yield strength of polycrystalline metals subjected to plastic forming, which hinges on the reliability of the micro–macro decoupling scheme for solving the corresponding two-scale boundary value problem. A polycrystalline aggregate composed of several crystal grains is adopted as a periodic microstructure, namely unit cell, and is regarded as a numerical specimen for numerical material tests (NMTs) based on the homogenization theory. The NMTs are conducted to identify the material parameters of the assumed approximate macroscopic constitutive model and followed by the de-coupled macro-scale analysis to simulate the macro-scale forming process. We then conduct the de-coupled micro-scale analysis by applying the resulting macroscopic deformation histories to the microstructures associated with the macroscopic points of interest in order to obtain the numerical specimens after plastic forming. Finally, the NMTs are conducted on the prepared specimens to evaluate the macroscopic post-forming yield strengths. We validate the proposed method by taking the three-step forming process as an example of macroscopic plastic forming. The validation of the method is made in comparison with the results with those obtained by the equivalent two-scale analysis with the coupling scheme. To demonstrate the capability and promise of the proposed method for practical applications, we also present an numerical example of the rolling process that causes the texture development in crystal grains.