Thermoviscoplastic instabilities and double strain gradient approach in biaxial loading by effective instability analysis

In this article we study the influence of double strain gradient, reflecting microstructural inhomogeneities, on the instability regime of a thermoviscoplastic material caused by biaxial loading. A perturbation analysis proposed earlier by Dudzinski and Molinari [1991] is used. The gesults show the influence of the microstructural coefficient on the rate of growth of the instability for various values of strain hardening, strain rate sensitivity, and straining path. The role of optimal orientation is presented, and the cases of isothermal and anisothermal deformation are analysed. Our results are also compared with those predicted by the aforementioned analysis. Finally, a comparison of uniaxial and biaxial situations concerning the role of the microstructural parameter is presented.     

Adiabatic shearing of non-homogeneous thermoviscoplastic materials

In this work we present the role of material non-homogeneities, driven by position-dependent thermomechanical parameters on the emergence, evolution and localization of temperature and strain non-uniformities during the shearing of thermoviscoplastic materials of the rate type. We first present the solution of the quasistatic approximation and show the existence of two stages: the first is characterized by a possible “travelling” of non-uniformities, due to non-homogeneities, while in the second the non-uniformities localize and increase at regions with intense heterogeneities. Moreover, we show that, even under stability conditions, the non-uniformities increase at material regions with inhomogeneities, which affect both their spatial distribution and time evolution. The related strain non-uniformities in the form of shear banding, as well as the comparison with the non-uniformities caused by initial defects or temperature gradient, are presented. Finally, numerical results confirm the analytical findings.     

An inclined surface crack subject to biaxial loading

The elastic–plastic stress fields and mode mixity parameters for semi-elliptical surface cracks on biaxial loaded plates have been investigated using detailed three-dimensional finite element calculations. Different degrees of mode mixity are given by combinations of the far-field stress level, biaxial stress ratio and inclined crack angle. These analyses were performed for different surface flaw geometries to study the combined load biaxiality and mode mixity effects on the crack-front stress fields and the size and shape of the plastic zones. It is clear from considering the local stress distributions along the crack front that the elastic crack tip singularities have been derived for several particular cases of mixed mode biaxial loading. By theoretical analysis, the new formulae have been introduced for both the elastic and plastic mode-mixity parameters, accounting for ratios between the I/II, II/III and III/I modes. Particular attention was paid to the strong variations of the mode-mixity parameters along the semi-elliptical surface crack front. The mixed-mode behavior of the crack growth direction angle along the semi-elliptical crack front for different combinations of biaxial loading and inclination crack angles was also determined. It was done using methods based on the maximum tangential stress and the strain energy density criteria.     

On the surface instability of elastic half spaces

In this paper, we present some work on the surface instability of elastic half spaces. An analysis of surface instability of an incompressible half space under biaxial loading is summarized, and the critical condition for the onset of surface buckling is given. As an example in the case of compressible materials, the axisymmetric problem of surface instability for a half space made of a standard material is analyzed, and the dependence of buckling parameters on the material is revealed.     

Experimental Thermomechanical Analysis of Elastomers Under Uni- and Biaxial Tensile Stress State

The objective of the present work is to demonstrate an experimental methodology to determine the viscoelastic material behavior of elastomers independent of the mechanical loading conditions. For this purpose two model materials (elastic and damper formulation) were investigated. The experimental effort ranged from classical uniaxial dynamic thermomechanical analysis (DTMA) and monotonic loading to monotonic biaxial testing. The performed monotonic experiments were loading rate and temperature dependent. Before applying the experimental methodology, some fundamental presumptions had to be verified. First, the applicability of the well-known time-temperature superposition principle to the elastomers, and second the separation of thermomechanical loadings, in which the temperature effects on the mechanical behavior of the materials could be characterized by an appropriate shift factor function; if this function is fundamental, then it should be independent of the mechanical loading condition. The shift factor functions were determined for the investigated elastomers from the DTMA results and were applied for the monotonic uni- and biaxial loading. Two biaxial tests (bulge test and planar biaxial tension test) were performed to show a direct calculation method for the stresses from planar biaxial tests by utilizing FEA with a material model, whose parameters were defined by the bulge test results.     

Buckling of steel bars with Lüders bands

Experiments and simulations are presented for the study of interaction between material and structural instabilities that occur in mild steel bars under axial compression. The material instability consists of Lüders bands that nucleate and propagate along the specimens. The structural instability involves lateral deflections of the bar leading to collapse. The study includes an investigation of bars of several different lengths. The mechanical response in the experiments were monitored through measurements of axial load, axial and midspan lateral displacements, and full field imaging of a brittle coating showing the Lüders deformation. Interesting interactions exist between the localized deformation due to the material-level instabilities and the global collapse of the bars. Finite element simulations, using a constitutive model with a nonmonotonic stress–strain behavior, showed good agreement with the experiments and helped to explain the variety of collapse modes seen in the experiments. The symmetry of imperfections and/or loading misalignments have dramatic effects on the evolution of Lüders deformation and the eventual collapse mode. Certain imperfections lead to deformation modes that delay structural collapse.     

A plasticity model for pressure-dependent anisotropic cellular solids

The initial and subsequent yield surfaces for an anisotropic and pressure-dependent 2D stochastic cellular material, which represents solid foams, are investigated under biaxial loading using finite element analysis. Scalar measures of stress and strain, namely characteristic stress and characteristic strain, are used to describe the constitutive response of cellular material along various stress paths. The coupling between loading path and strain hardening is then investigated in characteristic stress–strain domain. The nature of the flow rule that best describes the plastic flow of cellular solid is also investigated. An incremental plasticity framework is proposed to describe the pressure-dependent plastic flow of 2D stochastic cellular solids. The proposed plasticity framework adopts the anisotropic and pressure-dependent yield function recently introduced by Alkhader and Vural [Alkhader M., Vural M., 2009a. An energy-based anisotropic yield criterion for cellular solids and validation by biaxial FE simulations. J. Mech. Phys. Solids 57(5), 871–890]. It has been shown that the proposed yield function can be simply calibrated using elastic constants and flow stresses under uniaixal loading. Comparison of stress fields predicted by continuum plasticity model to the ones obtained from FE analysis shows good agreement for the range of loading paths and strains investigated.     

Constitutive relation for rheological processes: Properties of theoretic creep curves and simulation of memory decay

We study the nonlinear stress-strain constitutive relation proposed earlier for describing one-dimensional isothermal rheological processes in the case of monotonous variation of the strain (in particular, viscoplasticity, creep, relaxation, plasticity, and superplasticity). This relation contains integral time operators of the strain and strain rate, which are the norms in the Lebesgue and Sobolev spaces equipped with special weight factors, one material function, and nine material parameters determined by the results of tests of the material for relaxation, creep, long-term strength, and constant-rate strain.We analytically inverse the constitutive relation and study the properties of the inverse operator. We derive the equation of creep curves corresponding to an arbitrary law of loading at the stage of passing from the zero stress to a given constant level. We study their dependence on the material parameters and the loading stage characteristics and find restrictions on the material parameters which ensure that the asymptotic behavior of the creep curves for large times is independent of the length of the loading stage and the specific law of stress variation during this stage, i.e., we find the conditions of the model memory decay in creep. Thus we have proved that the constitutive relation proposed above can adequately model both creep and the effect of the material memory decay.     

Analysis of Damage and Failure in Anisotropic Ductile Metals Based on Biaxial Experiments with the H-Specimen

Background

Dependence of strength and failure behavior of anisotropic ductile metals on loading direction and on stress state has been indicated by many experiments. To realistically predict safety and lifetime of structures these effects must be taken into account in material models and numerical analysis.

Objective

The influence of stress state and loading direction on damage and failure behavior of the anisotropic aluminum alloy EN AW-2017A is investigated.

Methods

New biaxial experiments and numerical simulations have been performed with the H-specimen under different load ratios. Digital image correlation shows evolution of strain fields and scanning electron microscopy is used to visualize failure modes on fracture surfaces. Corresponding numerical studies predict stress states to explain damage and fracture processes on the micro-scale.

Results

The stress state, the load ratio and the loading direction with respect to the principal axes of anisotropy affect the width and orientation of localized strain fields and the formation of damage mechanisms and fracture modes at the micro-level.

Conclusions

The enhanced experimental program with biaxial tests considering different loading directions and load ratios is suggested for characterization of anisotropic metals.

     

A finite deformation nonlinear thermo-elastic model that mimics plasticity during monotonic loading

We present and study a nonlinear thermo-elastic constitutive model that under monotonic loading closely reproduces the response seen in plasticity, showing the initial stiff elastic response, kneeing as if yielding, and then showing response resembling post-yield hardening. The proposed large deformation thermo-elastic response model is constructed based on four physically identifiable mechanical parameters, that are closely related to the parameters used to construct plasticity models, thermal expansion parameters and two thermodynamic parameters. The four mechanical parameters are the initial elastic shear and bulk moduli, the yield point in shear, the hardening slope in shear. The thermodynamic parameters are the heat capacity at a reference temperature and its rate of change with changes of temperature. The model can be considered an alternate to deformation plasticity models currently used and, as such, can be used as a lightweight substitute for plasticity modeling in certain analysis. Since the proposed model is thermodynamically based, not only thermal effects are integrated into the model, but also the stress is calculated in terms of the applied deformation, allowing the model to be integrated with other models when conducting numerical analysis. We study the response of the proposed model under simple shear, uniaxial extension, confined compression, partially-confined compression, and biaxial extension. We incorporate the elastic model into ABAQUS using its UMAT subroutine for solid elements and using UHYPER for shell elements. We compare the large deformation response from the proposed elastic model with J2-plasticity, and with plasticity and deformation plasticity models implemented in ABAQUS. The model in most cases compares very favorably to all such models. This comparison is done for both homogeneous and non-homogeneous problems including the case of a cantilever beam under tip loading. We show that for the problems that it applies to, the models run in approximately one tenth the computational time and with one tenth the number of iterations needed to conduct the analysis using the plasticity model in ABAQUS.     

酚醛层压材料的冲击力学行为及本构模型

为了研究酚醛层压材料的冲击力学行为并获得本构模型,利用万能试验机和整形修正的分离式霍普金森压杆(SHPB)装置,对材料试样进行了应变率范围为10-3~103 s-1的单轴压缩实验,得到了不同加载应变率下的应力应变曲线,对其在准静态、动态载荷下的压缩破坏机理进行了初步探讨。结果表明,酚醛层压材料具有较强的应变率效应,与准静态(1.67×10-3 s-1)时相比,在动态载荷(7×102 s-1)下,峰值应力增加了约10倍;破坏应变减少了约一半;在准静态和动态加载条件下试样力学性能的差异是由于纤维基体界面特性以及不同应变率下破坏模式的不同;采用朱-王-唐本构方程描述了酚醛层压材料力学行为,拟合得到了本构方程的系数,在加载过程中,理论计算值与实验结果吻合较好。     

Temperature dependent bulge test for elastomers

The bulge test is a particularly convenient testing method for characterizing elastomers under biaxial loading. In addition, it is convenient to utilize this test for validating material models in simulation due to the heterogeneous strain field induced during inflation. During the bulge test the strain field for elastomers covers uniaxial tension at the border to pure shear and equibiaxial tension at the pole. Elastomeric materials exhibit a hyperelastic material behavior, with a dependency on temperature and loading rate. The temperature effect on the mechanical behavior during biaxial loading is considered in the present study. A bulge test setup combined with a temperature chamber is developed in order to characterize this effect, and an exemplary temperature dependent characterization of a poly(norbornene) elastomer is performed with this setup. The equibiaxial stress–strain curves measured at 60 °C, 20 °C and −20 °C are presented.     

Post-buckling analysis of elastic honeycombs subject to in-plane biaxial compression

In this paper, employing the homogenization theory and the microscopic bifurcation condition established by the authors, we discuss which microscopic buckling mode grows in elastic honeycombs subject to in-plane biaxial compression. First, we focus on equi-biaxial compression, under which uniaxial, biaxial and flower-like modes may develop as a result of triple bifurcation. By forcing each of the three modes to develop, and by comparing the internal energies, we show that the flower-like mode grows steadily if macroscopic strain is controlled, while either the uniaxial or biaxial mode develops if macroscopic stress is controlled. Second, by analyzing several cases other than equi-biaxial compression, it is shown that a second bifurcation from either the uniaxial or biaxial mode to the flower-like mode, which is distorted, occurs under biaxial compression in a certain range of biaxial ratio under macroscopic strain control. Finally, the possibility of macroscopic instability under biaxial compression is discussed.     

Static analysis of simply supported functionally graded and layered magneto-electro-elastic plates

In this article, static analysis of functionally graded, anisotropic and linear magneto-electro-elastic plates have been carried out by semi-analytical finite element method. A series solution is assumed in the plane of the plate and finite element procedure is adopted across the thickness of the plate such a way that the three-dimensional character of the solution is preserved. The finite element model is derived based on constitutive equation of piezomagnetic material accounting for coupling between elasticity, electric and magnetic effect. The present finite element is modeled with displacement components, electric potential and magnetic potential as nodal degree of freedom. The other fields are calculated by post-computation through constitutive equation. The functionally graded material is assumed to be exponential in the thickness direction. The numerical results obtained by the present model are in good agreement with available functionally graded three-dimensional exact benchmark solutions given by Pan and Han [Pan, E., Han, F., in press. Green’s function for transversely isotropic piezoelectric functionally graded multilayered half spaces. Int. J. Solids Struct.]. Numerical study includes the influence of the different exponential factor, magneto-electro-elastic properties and effect of mechanical and electric type of loading on induced magneto-electro-elastic fields. In addition further study has been carried out on non-homogeneous transversely isotropic FGM magneto-electro-elastic plate available in the literature [Chen, W.Q., Lee, K.Y., Ding, H.J., 2005. On free vibration of non-homogeneous transversely isotropic magneto-electro-elastic plates].     

A parameter for measuring the magnitude of a change of strain path: Validation and comparison with experiments on low carbon steel

Various tension-tension and tension-shear strain sequential experiments have been performed on low carbon steel sheet along different material axes. Owing to the rapid plastic instability that occurs during the reloading in uniaxial tension of prestrained samples, the results are focussed on the evolution of the macroscopic reloading yield stress (back extrapolated stress). For a given prestrain amount, the reloading stress is a function of the magnitude of the strain path change. A parameter is proposed, which allows the comparison of different sequential loading tests: the scalar product of the unit tensors corresponding to the prestrain and to the subsequent strain modes, respectively. For low carbon steel, a single curve is obtained when the reloading stress, normalized by the stress along the monotonic strain path is plotted against this parameter whatever the combination of loading sequences and the material direction of the prestrain.     

多轴加载下混凝土细观破坏模拟的强度准则探讨

实际工程结构中混凝土材料大多处于双轴或三轴的复杂应力状态,已有的细观力学数值研究工作大多针对单轴加载问题,对于双轴或者三轴加载条件下混凝土破坏模拟的研究相对较少。复杂受力条件下的混凝土材料破坏模拟中,细观组分强度准则选取的合理与否将成为混凝土破坏模式及宏观力学性能数值研究准确和成功与否的关键。本文旨在探讨单轴强度准则,如最大拉应变准则在多轴加载条件下混凝土破坏过程研究中运用的合理性。鉴于此,首先在细观尺度上建立了混凝土试件的二维随机骨料模型,分别采用弹性损伤本构关系模型及塑性损伤本构关系模型来描述细观组分(即砂浆基质)的力学性能,对双轴加载条件下混凝土的细观破坏过程进行数值模拟,对比了单轴强度准则和多轴强度准则下混凝土试件破坏路径及宏观应力-应变关系的差异。数值结果表明,简单的单轴强度准则难以反映双轴加载下混凝土内部应力状态的复杂性,不宜采用单轴强度准则来描述多轴加载下混凝土的破坏行为。     

Stress distribution and wrinkling of webs under non-uniform and asymmetric edge loading

Non-uniform edge loading can cause unwanted wrinkling of a web. Here, Airy stress functions are obtained by a modified Fourier series method for webs under non-uniform and different edge loading on each side. The stress distributions of a web under non-uniform edge loading only on one side are subjected to the effect of Saint-Venant’s principle. For the web of small aspect ratio under non-uniform edge loading, the stress distribution converges to a uniform distribution or a linear distribution with both the resultant force and the resultant moment being equal to those of the applied loading. Conditions of wrinkling and the corresponding wrinkling modes of a web are found when the edge loadings are different on each side. The non-uniformities and shapes of the applied edge loadings result in different wrinkling conditions. Moreover, the non-uniformities together with the shapes of the applied edge loadings on each side affect the wrinkling conditions.     

Multiaxial and non-proportional loading responses,anisotropy and modeling of Ti–6Al–4V titanium alloy over wide ranges of strain rates and temperatures

Results from a series of multiaxial loading experiments on the Ti–6Al–4V titanium alloy are presented. Different loading conditions are applied in order to get the comprehensive response of the alloy. The strain rates are varied from the quasi-static to dynamic regimes and the corresponding material responses are obtained. The specimen is deformed to large strains in order to study the material behavior under finite deformation at various strain rates. Torsional Kolsky bar is used to achieve shear strain rates up to 1000 s⁻¹. Experiments are performed under non-proportional loading conditions as well as dynamic torsion followed by dynamic compression at various temperatures. The non-proportional loading experiments comprise of an initial uniaxial loading to a certain level of strain followed by biaxial loading, using a channel-type die at various rates of loadings. All the non-proportional experiments are carried out at room temperature. Experiments are also performed to investigate the anisotropic behavior of the alloy. An orthotropic yield criterion [proposed by Cazacu, O., Plunkett, B., Barlat, F., 2005. Orthotropic yield criterion for hexagonal closed packed metals. International Journal of Plasticity 22, 1171–1194.] for anisotropic hexagonal closed packed materials with strength differential is used to generate the yield surface. Based on the definition of the effective stress of this yield criterion, the observed material response for the different loading conditions under large deformation is modeled using the Khan–Huang–Liang (KHL) equation assuming isotropic hardening. The model constants used in the present study, were pre-determined from the extensive uniaxial experiments presented in the earlier paper [Khan, A.S., Suh, Y.S., Kazmi R., 2004. Quasi-static and dynamic loading responses and constitutive modeling of titanium alloys. International Journal of Plasticity 20, 2233–2248]. The model predictions are found to be extremely close to the observed material response.