Constitutive modeling and prediction of hot deformation flow stress under dynamic recrystallization conditions

Simple modeling approaches based on the Hollomon equation, the Johnson–Cook equation, and the Arrhenius constitutive equation with strain-dependent material’s constants were used for modeling and prediction of flow stress for the single-peak dynamic recrystallization (DRX) flow curves of a stainless steel alloy. It was shown that the representation of a master normalized stress–normalized strain flow curve by simple constitutive analysis is successful in modeling of high temperature flow curves, in which the coupled effect of temperature and strain rate in the form of the Zener–Hollomon parameter is considered through incorporation of the peak stress and the peak strain into the formula. Moreover, the Johnson–Cook equation failed to appropriately predict the hot flow stress, which was ascribed to its inability in representation of both strain hardening and work softening stages and also to its completely uncoupled nature, i.e. dealing separately with the strain, strain rate, and temperature effects. It was also shown that the change in the microstructure of the material at a given strain for different deformation conditions during high-temperature deformation is responsible for the failure of the conventional strain compensation approach that is based on the Arrhenius equation. Subsequently, a simplified approach was proposed, in which by correct implementation of the hyperbolic sine law, significantly better consistency with the experiments were obtained. Moreover, good prediction abilities were achieved by implementation of a proposed physically-based approach for strain compensation, which accounts for the dependence of Young’s modulus and the self-diffusion coefficient on temperature and sets the theoretical values in Garofalo’s type constitutive equation based on the operating deformation mechanism. It was concluded that for flow stress modeling by the strain compensation techniques, the deformation activation energy should not be considered as a function of strain.     

Constitutive equation for a class of non-simple elastic materials

Summary Initial yield is the upper limit of the purely elastic deformation behaviour of an elasticplastic solid. Thus the choice of the constitutive equation describing the purely elastic deformation behaviour determines the initial yield function. The constitutive equation of a simple elastic material is only compatible with von Mises yield criterion, a conclusion which applies also to the classical infinitesimal theory. A more general form of constitutive equation for an elastic material is formulated by way of the concept of a stress loading function, the proposed constitutive equation being quadratic in the stress. The two loading coefficients associated with the stress loading function are assumed to be deriveable from a generalised isotropic yield criterion which is now assumed to hold over the entire range of deformation, and in this context is referred to as the stress intensity function. The proposed constitutive equation has the same representation in terms of the left Cauchy-Green deformation tensor as that for a simple elastic material. Using the Cayley-Hamilton theorem, this representation is rearranged and expressed in terms of a measure of finite strain which is defined to be one quarter of the difference between the left Cauchy-Green deformation tensor and its inverse. In this way the strain properties of the proposed constitutive equation are formulated by way of the concept of a strain response function. The three response coefficients associated with the strain response function are assumed to be deriveable from a generalised, isotropic, strain intensity function. The predictions of the proposed constitutive equation are considered in the context of the combined stressing of a thin sheet of incompressible material. In this way, it is shown that the proposed constitutive equation is not limited in the same way as the constitutive equation of a simple elastic material.     

Constitutive equation for rubber-like solids

Summary It has been found convenient to refer the deformation to the reference configuration of the body, the resulting constitutive equation being expressed in terms of the right Cauchy-Green deformation tensor, and a stress tensor which is also concerned solely with the reference configuration of the body. This particular measure of stress is referred to as the second referential stress tensor. The ratio, principal Cauchy stresses/principal stretches, identifies the proper numbers of the second referential stress tensor as the components of the force per unit underformed area. Using the Cayley-Hamilton theorem, the proposed constitutive equation is rearranged and expressed in terms of a quantity which is defined to be one quarter of the difference between the right Cauchy-Green deformation tensor and its inverse. In this way the (effective) strain properties of the proposed constitutive equation are formulated by way of the concept of a strain response function. The three response coefficients associated with the strain response function are assumed to be derivable from a strain intensity function. Two types of strain intensity function are considered. One type is characterised by being continuously differentiable, the other being characterised by the existence of vertices, but is piece-wise linear and continuous. The proposed constitutive equation is used to formulate an effective stress-effective strain relation. This effective stress-effective strain relation is used to correlate the individual stress-strain relations characteristic of various simple modes of deformation to give a single stress-strain relation. Correlation between the predictions of the proposed constitutive equation and the results of experiment proceeds by way of the concept of a generalised shear modulus.With 3 Figures     

Multiscale Nonlinear Constitutive Modeling of Carbon Nanostructures Based on Interatomic Potentials

Continuum-based modeling of nanostructures is an efficient and suitable method to study the behavior of these structures when the deformation can be considered homogeneous. This paper is concerned about multiscale nonlinear tensorial constitutive modeling of carbon nanostructures based on the interatomic potentials. The proposed constitutive model is a tensorial equation relating the second Piola-Kirchhoff stress tensor to Green-Lagrange strain tensor. For carbon nanotubes, some modifications are made on the planar representative volume element (RVE) to account for the curved atomic structure resulting a non-planar RVE. Using the proposed constitutive model, the elastic behavior of the graphene sheet and carbon nanotube are studied.     

复合材料蠕变本构关系及实验测定 Ⅱ实验测定

在前一部分,本文得到复合材料蠕变的本构关系,在此基础上,本文进一步分析了复合材料蠕变本构关系的具体形式,实验测得了长纤维增强复合材料在蠕变、恢复两个阶段的应变,以用来确定本构关系中的待定参数,考虑到本构关系为复杂的非线性方程,本文提出了用离散变量和最小二乘法联合的方法确定参数,进而拟合蠕变本构关系的理论公式,分离出了蠕变过程中的弹性变形、粘弹性变形和粘塑性变形,对本构关系中的几个参函数,本文根据有限的实验数据拟合了其函数.     

复合材料蠕变本构关系及实验测定Ⅱ实验测定

在前一部分,本文得到复合材料蠕变的本构关系,在此基础上,本文进一步分析了复合材料蠕变本构关系的具体形式,实验测得了长纤维增强复合材料在蠕变、恢复两个阶段的应变,以用来确定本构关系中的待定参数,考虑到本构关系为复杂的非线性方程,本文提出了用离散变量和最小二乘法联合的方法确定参数,进而拟合蠕变本构关系的理论公式,分离出了蠕变过程中的弹性变形、粘弹性变形和粘塑性变形,对本构关系中的几个参函数,本文根据有限的实验数据拟合了其函数。      

Hyper-elastic constitutive equations of conjugate stresses and strain tensors for the Seth-Hill strain measures

The use of hypo-elastic constitutive equations for large strains in nonlinear finite element applications usually requires special considerations. For example, the strain does not tend to zero upon unloading in some elastic loading-unloading closed cycles. Furthermore, these equations are based on objective material time rate tensors, which require incrementally objective algorithms for numerical applications and integration. Hyper-elastic constitutive equations on the other hand do not require such considerations. However, their behaviour for large elastic strains is important and may differ in tension and compression. In the present work, Hyper-elastic constitutive equations for the Seth-Hill strains and their conjugate stresses are explored as a natural generalisation of Hook’s law for finite elastic deformations. Based on the uniaxial and simple shear tests, the response of the material for different constitutive equations is examined. Together with an objective rate model, the effect of different constitutive laws on Cauchy stress components is compared. It is shown that the constitutive equation based on logarithmic strain and its conjugate stress gives results closer to that of the rate model. In addition, the use of Biot stress-strain pairs for a bar element results in an elastic spring which obeys the Hook’s law even for large deformations and has the same behaviour in both tension and compression. The effect of the constitutive equation on the volume change of the material has also been considered here.     

Nonlinear constitutive equations for transient creep of viscoelastic polymers including the effect of hydrostatic pressure

The behaviour of polymers is known to be significantly influenced by the hydrostatic pressure in creep deformation or elastic-plastic deformation. The effect of the third stress invariant on the nonlinear viscoelastic deformation is much smaller than that of the hydrostatic pressure. In this paper, a constitutive equation for transient creep is proposed, which includes the effect of the hydrostatic pressure on the yield function. The creep and plastic strains or the creep strain rate converge to zero with increasing hydrostatic pressure. The proposed constitutive equation is in good agreement with the actual creep data of cellulose nitrate and cellulose acetate, under various combinations of superimposed tensile and hydrostatic loadings.     

Thermomechanical analysis,modelling and simulation of the forming of pre-impregnated thermoplastics composites

In this paper, the viscoelastic behaviours of pre-impregnated (prepreg) thermoplastic composites are analysed using the bias-extension test. A new constitutive model is proposed in order to simulate the forming of thermoplastic composite prepregs at the macroscopic scale. The model is based on a continuous approach. Hyperelastic behaviours are associated with dry reinforcements. Four principal deformation modes, all considered independent, define the hyperelastic potential: the elongation in warp direction, the elongation in weft direction, the in-plane shear strain and the bending contribution. Experience shows that viscoelastic behaviour is mainly associated with the in-plane shear deformation. A non-linear visco-hyperelastic model based on the generalisation of Maxwell rheological model is considered for this type of deformation mode. The finite element simulation of a stamping case using this model is introduced. The influence of temperature on the forming stage and the performance of the model are evaluated.     

A numerical study of hypoelastic and hyperelastic large strain viscoplastic Perzyna type models

For the case of metals with large viscoplastic strains, it is necessary to define appropriate constitutive models in order to obtain reliable results from the simulations. In this paper, two large strain viscoplastic Perzyna type models are considered. The first constitutive model has been proposed by Ponthot, and the elastic response is based on hypoelasticity. In this case, the kinematics of the constitutive model is based on the additive decomposition of the rate deformation tensor. The second constitutive model has been proposed by García Garino et al., and the elasticresponse is based on hyperelasticity. In this case, the kinematics of the constitutive model is based on the multiplicativedecomposition of the deformation gradient tensor. In both cases, the resultant numerical models have been implemented in updated Lagrangian formulation. In this work, global and local numerical results of the mechanical response of both constitutive models are analyzed and discussed. To this end, numerical experiments are performed and different parameters of the constitutive models are tested in order to study the sensitivity of the resultantalgorithms. In particular, the evolution of the reaction forces, the effective plastic strain, the deformed shapes and the sensitivity of the numerical results to the finite element mesh discretization have been compared and analyzed. The obtained results show that both models have a very good agreement and represent very well the characteristic of the viscoplastic constitutive model.     

On the selection of constitutive equation for modeling the friction stir processes of twin roll cast wrought AZ31B

Proper numerical modeling of the Friction Stir Processes (FSPs) requires the identification of a suitable constitutive equation which accurately describes the stress-strain material behavior under an applicable range of strains, strain rates, and temperatures. While some such equations may be perfectly suitable to simulate processes characterized by low (or high) strains and temperatures, FSPs are widely recognized for their relatively moderate ranges of such state variables. In this work, a number of constitutive equations for describing flow stress in metals were screened for their suitability for modeling Friction Stir Processes of twin roll cast (TRC) wrought magnesium Mg–AL–Zn (AZ31B) alloy. Considered were 4 different reported variations of the popular Johnson–Cook equation and one Sellars–Tegart equation along with their literature–reported coefficients for fitting AZ31B stress–strain behavior. In addition, 6 variations of the (rarely used in FSPs simulations) Zerilli–Armstrong equation were also considered along with their literature–reported coefficients. The screening assessment was based on how well the considered constitutive equations fit experimental tensile stress–strain data of twin roll cast wrought AZ31B. Goodness of fit and residual sum of squares were the two statistical criteria utilized in the quantitative assessment whereas a ‘visual ’ measure was used as a qualitative measure. Initial screening resulted in the selection of one best fitting constitutive equation representing one of each of the Johnson–Cook, Sellars–Tegart, and Zerilli–Armstrong equations. An HCP–specific Zerilli–Armstrong constitutive equation (dubbed here as ZA₆ ) was found to have the best quantitative and qualitative fit results with an R² value of 0.967 compared to values of 0.934 and 0.826 for the Johnson–Cook and Sellars–Tegart constitutive equations, respectively. Additionally, a 3D thermo–mechanically coupled FEM model was built in DEFORM 3D to simulate the experimental tensile test from which the experimental load–deflection data was obtained. The three ‘finalist ’ equations were fed into the FEM simulations and were compared based on the 1) simulations’ running times and 2) goodness of fit of the simulation results to the experimental load–deflection data. It was found that the ZA₆ constitutive equation exhibited favorable run times even when contrasted against the simpler mathematical form of the Sellars–Tegart equation. On average, the ZA₆ equation showed improvements in solution time by 5.4% as compared with the Johnson–Cook equation and almost identical solution time (0.9% increase) with that of the ST equation. This result indicates that the proposed equation is not numerically expensive and can be safely adopted in such FEM simulations. Based on the favorable running times and goodness of fit, it was concluded that the HCP–specific Zerilli–Armstrong constitutive equation ZA₆ holds an advantage over all other considered equations and was, therefore, selected as most suitable for the numerical modeling of FSP of twin roll cast AZ31B.     

Visco-hyperelastic constitutive law for modeling of foam’s behavior

This paper proposes a new visco-hyperelastic constitutive law for modeling the finite-deformation strain rate-dependent behavior of foams as compressible elastomers. The proposed model is based on a phenomenological Zener model, which consists of a hyperelastic equilibrium spring and a Maxwell element parallel to it. The hyperelastic equilibrium spring describes the steady state response. The Maxwell element, which captures the rate-dependency behavior, consists of a nonlinear viscous damper connected in series to a hyperelastic intermediate spring. The nonlinear damper controls the rate-dependency of the Maxwell element. Some strain energy potential functions are proposed for the two hyperelastic springs. compressibility effect in strain energy is described by entering the third invariant of deformation gradient tensor into strain energy functions. A history integral method has been used to develop a constitutive equation for modeling the behavior of the foams. The applied history integral method is based on the Kaye–BKZ theory. The material constant parameters, appeared in the formulation, have been determined with the aid of available uniaxial tensile experimental tests for a specific material.     

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.     

Multi-Scale Modeling the Mechanical Properties of Biaxial Weft Knitted Fabrics for Composite Applications

In this paper a multi-scale numerical model for simulating the mechanical behavior of biaxial weft knitted fabrics produced based on 1×1 rib structure is presented. Fabrics were produced on a modern flat knitting machine using polyester as stitch yarns and nylon as straight yarns. A macro constitutive equation was presented to model the fabric mechanical behavior as a continuum material. User defined material subroutines were provided to implement the constitutive behavior in Abaqus software. The constitutive equation needs remarkable tensile tests on the fabric as the inputs. To solve this drawbacks meso scale modeling of the fabric was used to predict stress–strain curves of the fabric in three different directions (course, wale and 45°). In these simulations only the yarn properties are needed. To evaluate the accuracy of the proposed macro and meso models, fabric tensile behavior in 22.5 and 67.5° directions were simulated by the calibrated macro model and compared with experimental results. Spherical deformation was also simulated by the multi scale model and compared with experimental results. The results showed that the multi-scale modeling can successfully predict the tensile and spherical deformation of the biaxial weft knitted fabric with least required experiments. This model will be useful for composite applications.     

A Lagrangian model for hardening behaviour of materials at finite deformation based on the right plastic stretch tensor

In this paper a modified multiplicative decomposition of the right stretch tensor is proposed and used for finite deformation elastoplastic analysis of hardening materials. The total symmetric right stretch tensor is decomposed into a symmetric elastic stretch tensor and a non-symmetric plastic deformation tensor. The plastic deformation tensor is further decomposed into an orthogonal transformation and a symmetric plastic stretch tensor. This plastic stretch tensor and its corresponding Hencky’s plastic strain measure are then used for the evolution of the plastic internal variables. Furthermore, a new evolution equation for the back stress tensor is introduced based on the Hencky plastic strain. The proposed constitutive model is integrated on the Lagrangian axis of the plastic stretch tensor and does not make reference to any objective rate of stress. The classic problem of simple shear is solved using the proposed model. Results obtained for the problem of simple shear are identical to those of the self-consistent Eulerian rate model based on the logarithmic rate of stress. Furthermore, extension of the proposed model to the mixed nonlinear isotropic/kinematic hardening behaviour is presented. The model is used to predict the nonlinear hardening behaviour of SUS 304 stainless steel under fixed end finite torsional loading. Results obtained are in good agreement with the available experimental results reported for this material under fixed end finite torsional loading.     

二维编织C/SiC复合材料非线性损伤本构模型与应用

基于二维编织C/SiC复合材料的基本力学性能试验, 建立了该材料的宏观正交各向异性非线性损伤本构模型。模型中以可检测的应变作为变量, 采用形式简单的函数分别描述了单轴拉伸和剪切加载下的材料损伤演变下的应力-应变关系, 以及卸载状态的刚度变化规律。同时, 考虑了材料的单边效应以及拉压应力状态转换时的损伤钝化行为。将此本构模型编写成UMAT子程序并引入ABAQUS有限元软件, 可以完整描述该材料的加载非线性和卸载线性的应力-应变关系特征, 及其加卸载历史。通过对带孔板的拉伸模拟, 孔边应变分布与试验结果吻合较好, 验证了本构模型的有效性。