A constitutive model for cavitation and cavity growth in rubber-like materials under arbitrary tri-axial loading

In this paper, we attempted to construct a constitutive model to deal with the phenomenon of cavitation and cavity growth in a rubber-like material subjected to an arbitrary tri-axial loading. To this end, we considered a spherical elementary representative volume in a general Rivlin’s incompressible material containing a central spherical cavity. The kinematics proposed by [Hou, H.S., Abeyaratne, R., 1992. Cavitation in elastic and elastic-plastic solids. J. Mech. Phys. Solids 40, 571–722] was adopted in order to construct an approximate but optimal field. In order to establish a suitable constitutive law for this class of materials, we utilized the homogenisation technique that permits us to calculate the average strain energy density of the volume. The cavity growth was considered through a physically realistic failure criterion. Combination of the constitutive law and the failure criterion enables us to describe correctly the global behaviour and the damage evolution of the material under tri-axial loading. It was shown that the present models can efficiently reproduce different stress states, varying from uniaxial to tri-axial tensions, observed in experimentations. Comparison between predicted results and experimental data proves that the proposed model is accurate and physically reasonable. Another advantage is that the proposed model does not need special identification work, the initial Rivlin’s law for the corresponding incompressible material is sufficient to form the new law for the compressible material resulted from cavitation procedure.     

A constitutive model dealing with damage due to cavity growth and the Mullins effect in rubber-like materials under triaxial loading

In this work, we attempted to describe the evolution of damage in rubber-like materials due to the Mullins effect and the cavity growth process. To this end we introduced two distinct internal variables into the constitutive laws; the first one essentially describes the Mullins damage and the second describes the cavity growth. The Mullins effect was considered as a continuous type of damage that can be modelled within the continuum damage theory. The cavity growth, being discontinuous at the microscopic scale, was also modelled by a continuous variable after a homogenization procedure. These analyses allow the establishment of a compressible constitutive law describing the strain-softening phenomena for rubber-like materials. In order to identify the material parameters and to verify the efficiency of the model, we carried out experimental studies involving uniaxial, biaxial, and hydrostatic tensions under monotonic and cyclic loading. Comparison between the model-predicted results and the experimental data shows that the present model can efficiently describe both the Mullins damage and the porosity evolution of rubber-like materials under triaxial monotonic or cyclic loading with a satisfactory accuracy. The proposed concept is simple and easy to apply to engineering calculations.     

Anisotropy of direction-based constitutive models for rubber-like materials

A study of direction-based models for the representation of isotropic and anisotropic hyperelastic behaviour of rubber-like materials is proposed. The interest in such models is sustained by their ability to account for the Mullins effect induced anisotropy. For such a purpose, the directional models should be initially isotropic and representative of the hyperelastic behaviour of rubber-like materials. Various models were defined according to different sets of directions. Models were tested in terms of their initial anisotropy and their ability to reproduce the classic full-network hyperelastic behaviour. Various models were proved to perform very well.     

Hyperelastic constitutive model for rubber-like materials based on the first Seth strain measures invariant

The mechanical behaviour of isotropic and incompressible vulcanized natural rubbers (NR's) and that of quasi-incompressible carbon black filled vulcanized natural rubbers (NR 70) are considered both theoretically and experimentally. We start by generalising the neo-Hookean model to derive an original form of the strain energy density function (W). W satisfies the hypothesis of the Valanis–Landel function, which allows reducing the number of needed experimental tests to identify the parameters of the model. In the present study, in the order to identity the analytical form of W, we undertake only simple tension tests. The two-dimensional field of in-plane homogeneous displacements is determined here using a home-developed image analysis cross-correlation technique. Our model is also identified using results taken from the literature in the case of (NR's) for different types of solicitations, including simple tension, equibiaxial tension and pure shear deformation. Comparison of numerical results with the experimental data indicates that the present model can characterise the hyperelastic behaviour of NR's and that of NR 70 for all the tested modes of deformation. Moreover, it seems to be valid over a wide range of deformation intervals.     

Computational modelling of the stress-softening phenomenon of rubber-like materials under cyclic loading

When a rubber specimen is subjected to cyclic loading, not only non-linear behaviour but also damage-induced stress-softening phenomena (the Mullins effect) have been observed. Applications of a continuum damage mechanics model and Ogden and Roxburgh's pseudo-elastic model to describe the Mullins effect in elastomers have been considered. Both models together with Gao's elastic law were implemented to describe the mechanical behaviour of rubber-like materials including the stress-softening phenomenon. Two sets of experimental data (a simple tension test and a simple tension and pure shear test) are used to validate the constitutive models. Model parameters are estimated via an inverse technique. Computational results show that both constitutive models together with Gao's elastic law can describe the typical Mullins effect. From engineering point of view, the pseudo-elastic model has the advantages that (i) the model is simple and practical, since it considers that the stress-softening function is only activated on unloading or reloading paths, (ii) the model with a slight modification of the damage variable is very stable in finite element calculations, and (iii) the numerical results agree very well with experimental data in both simple tension and pure shear deformation. Two applications illustrate the capability of combining the pseudo-elastic model with Gao's elastic law in describing the Mullins effect. It is emphasized that both models are applicable to multiaxial states of stress and strain because both models are energy-based and not strain-based.     

Growth of microcracks under monotonic loading

All-Union Scientific-Research Institute of Drilling Technology, Moscow. Translated from Prikladnaya Mekhanika, Vol. 24, No. 4, pp. 86–99, April, 1988.     

爆炸冲击下珊瑚砂动态本构模型

以珊瑚砂为主要覆盖域的岛礁在面临动力灾变时,确定岛礁工程抵抗极端冲击荷载的阈值至关重要,珊瑚砂的动态本构关系是防护工程设计的关键要素。本文中,根据SHPB实验和静态压缩实验的结果,提出了一种基于应变率强化规律确定珊瑚砂物态方程的方法,并确定了珊瑚砂动态本构模型的参数。分别基于流体弹塑性模型和Perzyna黏塑性帽盖模型,结合LS-DYNA有限元程序,通过对侵彻和爆炸的数值计算,验证了模型的适用性。基于建立的模型,对不同相对密实度的珊瑚砂开展了侵彻和爆炸数值计算,结果表明,密实度对爆炸波的衰减影响较大、对侵彻深度的影响较小。     

冲击载荷下猪后腿肌肉的横向同性本构模型

基于纤维增强复合材料连续介质力学理论及粘弹性理论,提出了猪后腿肌肉的率相关本构模型。 通过拟合以往研究中猪后腿肌肉的SHPB和SHTB实验应力应变曲线,确定了本构模型的相关参数。结果 表明:提出的本构模型既能描述猪后腿肌肉沿纤维方向的动态压缩力学性能又能描述其动态拉伸性能,理论 模型与实验模型有较好的一致性。该结果可为安全防护数值模拟提供一定的理论依据。     

A constitutive model and elastoplastic analysis of structures under cyclic loading

A constitutive model for cyclic plasticity is briefly outlined. Then the model is implemented in a finite element code to predict the response of cyclic loaded structural components such as a double-edge-notched plate, a grove bar and a nozzle in spherical shell. Comparision with results from other theories and experiments shows that the results obtained by using the present model are very satisfactory.The Project Supported by National Natural Science Foundation of China.     

Analytical solution to a notch tip field in rubber-like materials under tension

Two kinds of elastic laws have been adopted by Gao, 1990 and Gao, 1997 to analyze the deformation fields near a crack tip. One of them contains the response to volume change and shape change; the other contains the response to extension and compression. In this paper the two kinds of constitutive relations are examined by typical large deformation, and the restrictions on constitutive parameters are discussed.     

A nonlinear constitutive model for magnetostrictive materials

A general nonlinear constitutive model is proposed for magnetostrictive materials, based on the important physical fact that a nonlinear part of the elastic strain produced by a pre-stress is related to the magnetic domain rotation or movement and is responsible for the change of the maximum magnetostrictive strain with the pre-stress. To avoid the complicity of determining the tensor function describing the nonlinear elastic strain part, this paper proposes a simplified model by means of linearizing the nonlinear function. For the convenience of engineering applications, the expressions of the 3-D (bulk), 2-D (film) and 1-D (rod) models are, respectively, given for an isotropic material and their applicable ranges are also discussed. By comparison with the experimental data of a Terfenol-D rod, it is found that the proposed model can accurately predict the magnetostrictive strain curves in low, moderate and high magnetic field regions for various compressive pre-stress levels. The numerical simulation further illustrates that, for either magnetostrictive rods or thin films, the proposed model can effectively describe the effects of the pre-stress or residual stress on the magnetization and magnetostrictive strain curves, while none of the known models can capture all of them. Therefore, the proposed model enjoys higher precision and wider applicability than the previous models, especially in the region of the high field.The project supported by the National Natural Science Foundation of China (10132010 and 90405005)     

On the constitutive equations for cyclic plasticity under nonproportional loading

Experimental results on 316 stainless steel, at room temperature, under strain controled axial-torsion loadings, point out a large difference in strengthening between proportional and nonproportional loadings. The maximum difference is produced by 90° out of phase straining. In this paper we discuss the possibility of describing this additional hardening, without any new internal variable, by only modifying the expression for the yield criterion.     

A constitutive equation for filled rubber under cyclic loading

This paper describes experiments and the development of constitutive equations to predict the steady-state response of filled rubber under cyclic loading. An MTS servo-hydraulic machine was used to obtain the dynamic hysteresis curves for a filled rubber compound in uniaxial tension-compression. The material tests were performed with mean strains from −0.1 to 0.1, strain amplitudes ranging from 0.02 to 0.1, and strain rates between 0.01 and 10 s⁻¹. Temporary material set, the Payne effect and rate-dependence were observed from the experimental results. A hyper-viscoelastic constitutive model was developed to characterize the dynamic response of the rubber. A cornerstone of this constitutive modeling was to devise a scheme to evaluate material set and a finite strain, non-linear viscoelastic law from the test data. Predictions of the dynamic hysteresis curves using the proposed constitutive equation were found to be in good agreement with the uniaxial test results.     

Elastic beam on a rigid frictional foundation under monotonic and cyclic loading

An elastic beam resting on a frictional foundation and loaded by the concentrated force or moment applied at its tip is considered. The evolution of slip zones along the beam is discussed for both monotonie and cyclic loading. It is shown that an infinite number of slip zones develop and their propagation satisfies in some cases a self-similarity property. Transient hysteretic effects under cyclic loading are discussed. The closed form analytical solution is presented for the elastic friction model in the case of monotonie loading.     

Combining the logarithmic strain and the full-network model for a better understanding of the hyperelastic behavior of rubber-like materials

Suitably defined invariants of the logarithmic strain are shown to be more adequate than the usual invariants of the left Cauchy-Green tensor to define the type and intensity of a strain applied to hyperelastic rubber-like materials. Coupling these invariants with the macromolecular full-network model clarifies some features of the state of strain dependence of these materials. Finally, comparisons of the model with experimental data illustrate the efficiency of the full-network model and the dependence of the material parameters on the applied loading history.     

A thermomechanical constitutive model for phase transformations in silicon under pressure and contact loading conditions

Most of the technologically relevant abrasive machining techniques for silicon (Si) such as lapping, sawing and grinding are based on the interaction of the silicon surface with a hard particle or asperity. It has been long established that the governing deformation mechanism for Si under such contact loading conditions is stress induced phase transformation. The present work introduces a novel phenomenological constitutive model for phase transformations of silicon set up in a thermomechanical framework of broad applicability. Taking into account experimental observations as well as first principle and molecular dynamics calculations, it captures both the cd-Si → β-Si transition upon compression and the β-Si → a-Si transition upon rapid decompression, which are most relevant for indenter loading. The model was numerically implemented in analogy to incremental plasticity and successfully applied for finite-element (FE) simulations of nanoindentation.     

泡沫橡胶材料的超弹性本构模型

借助不可压橡胶类材料应变能函数,通过引入泡沫材料孔隙度,推导出了泡沫橡胶的本构方程。基于泡沫橡胶材料的单轴压缩实验结果,拟合确定了本构模型的参数,模型预测结果与实验结果吻合的比较好。     

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。