Bulge test method for measuring the hyperelastic parameters of soft membranes

Soft membranous materials widely exist in engineering and nature, and the determination of their constitutive parameters is of both scientific and engineering significance. In this paper, the bulge test method is extended to determine the hyperelastic parameters of soft membranes with or without initial stresses. Two extensively applied models—neo-Hookean model and Arruda–Boyce model—are employed to characterize the nonlinear behavior of the membrane under test. The hyperelastic parameters are then extracted from the pressure–deflection curve of the membrane recorded in the bulge tests. Our method is finally validated by both finite element simulations and uniaxial tension experiments. The proposed method can be used to evaluate various soft membranes and tissues and hold promise for numerous applications in such fields as biomedical engineering and bionic engineering.     

Hyperelastic modelling for mesoscopic analyses of composite reinforcements

A hyperelastic constitutive law is proposed to describe the mechanical behaviour of fibre bundles of woven composite reinforcements. The objective of this model is to compute the 3D geometry of the deformed woven unit cell. This geometry is important for permeability calculations and for the mechanical behaviour of the composite into service. The finite element models of a woven unit cell can also be used as virtual mechanical tests. The highlight of four deformation modes of the fibre bundle leads to definition of a strain energy potential from four specific invariants. The parameters of the hyperelastic constitutive law are identified in the case of a glass plain weave reinforcement thanks to uniaxial and equibiaxial tensile tests on the fibre bundle and on the whole reinforcement. This constitutive law is then validated in comparison to biaxial tension and in-plane shear tests.     

A thermovisco-hyperelastic constitutive model of HTPB propellant with damage at intermediate strain rates

The uniaxial compressive tests at different temperatures (223–298 K) and strain rates (\(0.40\mbox{--}63~\mbox{s}^{-1}\)) are reported to study the properties of hydroxyl-terminated polybutadiene (HTPB) propellant at intermediate strain rates, using a new INSTRON testing machine. The experimental results indicate that the compressive properties (mechanical properties and damage) of HTPB propellant are remarkably affected by temperature and strain rate and display significant nonlinear material behaviors at large strains under all the test conditions. Continuously decreasing temperature and increasing strain rate, the characteristics of stress-strain curves and damage for HTPB propellant are more complex and are significantly different from that at room temperature or at lower strain rates. A new constitutive model was developed to describe the compressive behaviors of HTPB propellant at room temperature and intermediate strain rates by simply coupling the effect of strain rate into the conventional hyperelastic model. Based on the compressive behaviors of HTPB propellant and the nonlinear viscoelastic constitutive theories, a new thermovisco-hyperelastic constitutive model with damage was proposed to predict the stress responses of the propellant at low temperatures and intermediate strain rates. In this new model, the damage is related to the viscoelastic properties of the propellant. Meanwhile, the effect of temperature on the hyperelastic properties, viscoelastic properties and damage are all considered by the macroscopical method. The constitutive parameters in the proposed constitutive models were identified by the genetic algorithm (GA)-based optimization method. By comparing the predicted and experimental results, it can be found that the developed constitutive models can correctly describe the uniaxial compressive behaviors of HTPB propellant at intermediate strain rates and different temperatures.     

Bayesian inference on a microstructural,hyperelastic model of tendon deformation

Microstructural models of soft-tissue deformation are important in applications including artificial tissue design and surgical planning. The basis of these models, and their advantage over their phenomenological counterparts, is that they incorporate parameters that are directly linked to the tissue’s microscale structure and constitutive behaviour and can therefore be used to predict the effects of structural changes to the tissue. Although studies have attempted to determine such parameters using diverse, state-of-the-art, experimental techniques, values ranging over several orders of magnitude have been reported, leading to uncertainty in the true parameter values and creating a need for models that can handle such uncertainty. We derive a new microstructural, hyperelastic model for transversely isotropic soft tissues and use it to model the mechanical behaviour of tendons. To account for parameter uncertainty, we employ a Bayesian approach and apply an adaptive Markov chain Monte Carlo algorithm to determine posterior probability distributions for the model parameters. The obtained posterior distributions are consistent with parameter measurements previously reported and enable us to quantify the uncertainty in their values for each tendon sample that was modelled. This approach could serve as a prototype for quantifying parameter uncertainty in other soft tissues.     

橡胶材料的非线性黏弹性本构方程

橡胶是一种非线性黏弹性材料,准确描述其非线性黏弹性力学响应的本构方程是橡胶材料及制品设计优化的关键。文中基于超弹性模型和并行流变模型(PRF)描述了橡胶材料的非线性黏弹性响应特征,重点探讨了由实验数据确定PRF本构方程材料参数的方法。首先通过单轴拉伸实验数据拟合得到超弹性模型,将应力松弛实验数据拟合得到表征材料线性黏弹性的模型-Prony级数,再将Prony级数转化为初始的PRF模型,进而不断优化得到PRF模型的准确材料参数,最后进行实验验证。结果表明,PRF模型计算的不同应变下的应力松弛数据与实验数据之间的误差仅为0.067%,PRF能准确地描述橡胶材料非线性响应的应力松弛行为。     

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.     

隔振橡胶本构建模研究

提出适合描述隔振橡胶在宽频振动时力学行为的本构模型。本构模型包含超弹性和粘弹性两个部分，超弹性部分表征橡胶材料的静态特性；非线性粘弹性部分描述橡胶材料在振动、冲击载荷下的动态响应。基于该本构模型，对橡胶材料在宽应变率范围内进行试验，九个材料参数通过高、低应变率下的试验数据拟合确定。模型预测结果与试验结果是相当吻合的。     

Introduction of imperfections in the cubic mesh of the truss‐like discrete element method

In the present version of the truss‐like discrete element method (DEM), masses are considered lumped at nodal points and interconnected by means of unidimensional elements with arbitrary constitutive relations. In previous studies of non‐homogeneous concrete cubic samples subjected to nominally uniaxial tension, it was verified that numerical predictions of fracture using DEM models are feasible and yield results that are consistent with the experimental evidence so far available, including the prediction of size and strain rate effects. In the DEM formulation, material failure under compression is assumed to occur by indirect tension. In previous simulations, it was verified that the response is satisfactorily modelled up to the peak load, when a sudden collapse usually occurs, characteristic of fragile behaviour. On the other hand, experimental stress versus displacement curves observed in small specimens subjected to compression typically present a softening branch, in part due to sliding with friction of the fractured parts of the specimens. A second deficiency of DEM models with a perfectly cubic mesh is that the best correlations with experimental results are obtained with material parameters that differ in tension and compression. This paper examines another cause of the excessively fragile behaviour of DEM predictions of the response of concrete elements subjected to nominally uniaxial compression, which is due to the regularity of the perfect cubic mesh, unable to capture nonlinear stability effects in the material. It is shown herein that the introduction of small perturbations of the DEM regular mesh significantly improves the predicting capability of the model and in addition allows adopting a unique set of material properties, which are independent of the nature of the loading.     

Permanent set and stress-softening constitutive equation applied to rubber-like materials and soft tissues

Many rubber-like materials present a phenomenon known as Mullins effect. It is characterized by a difference of behavior between the first and second loadings and by a permanent set after a first loading. Moreover, this phenomenon induces anisotropy in an initially isotropic material. A new constitutive equation is proposed in this paper. It relies on the decomposition of the macromolecular network into two parts: chains related together and chains related to fillers. The first part is modeled by a simple hyperelastic constitutive equation, whereas the second one is described by an evolution function introduced in the hyperelastic strain energy. It contributes to describe both the anisotropic stress softening and the permanent set. The model is finally extended to soft tissues’ mechanical behavior that present also stress softening but with an initially anisotropic behavior. The two models are successfully fitted and compared to experimental data.     

Calibrating constitutive models with full-field data via physics informed neural networks

The calibration of solid constitutive models with full-field experimental data is a long-standing challenge, especially in materials that undergo large deformations. In this paper, we propose a physics-informed deep-learning framework for the discovery of hyperelastic constitutive model parameterizations given full-field surface displacement data and global force-displacement data. Contrary to the majority of recent literature in this field, we work with the weak form of the governing equations rather than the strong form to impose physical constraints upon the neural network predictions. The approach presented in this paper is computationally efficient, suitable for irregular geometric domains, and readily ingests displacement data without the need for interpolation onto a computational grid. A selection of canonical hyperelastic material models suitable for different material classes is considered including the Neo–Hookean, Gent, and Blatz–Ko constitutive models as exemplars for general non-linear elastic behaviour, elastomer behaviour with finite strain lock-up, and compressible foam behaviour, respectively. We demonstrate that physics informed machine learning is an enabling technology and may shift the paradigm of how full-field experimental data are utilized to calibrate constitutive models under finite deformations.     

Analysis of the essential work of fracture method as applied to UHMWPE

The validity of the basic assumptions behind the method of essential work of fracture (EWF), as applied to ultra-high molecular weight polyethylene (UHMWPE), is evaluated using finite element modelling. To define a suitable model of constitutive behaviour, the mechanical properties of UHMWPE have been measured in both uniaxial tension and compression over a range of strain rates. The observed strain rate dependence of stress, including the observed differences in strain rate sensitivity between tension and compression, is interpreted in terms of a single Eyring process. The constitutive theory is constructed comprising an Eyring process and hyperelastic networks, the latter having responses symmetric with respect to tension and compression. This theory is implemented within a finite element scheme, and used to model fracture measurements made on the same material using double-edge notch tensile specimens. Calculations of the non-essential work and of the extent of the plastic zones are thus made possible. It is concluded that the specific non-essential work is essentially constant, but that the shape factor β, assumed constant in the conventional analysis, varies significantly with ligament length. The implication of this finding on the derived EWF value is evaluated and found to be slight.     

Mullins effect characterization of elastomers by multi-axial cyclic tests and optical experimental methods

Besides the typical hyperelastic behaviour, large elastic deformations with non-linear stress–strain behaviour, rubber-like materials may also exhibit some inelastic effects, like hysteresis and permanent set. One of them is a particular damage phenomenon called Mullins effect. This is visible when cyclic tension tests are performed with increasing values of deformation. Material is deformed up to a fixed strain value and then unloaded. When a second load is applied it is possible to observe a stress softening effect. In the present work uniaxial and equibiaxial tension tests have been carried out by a standard tensile machine and by an hydraulic bulge test experimental rig, respectively. In both tests optical methods have been used for strain measurement. Experimental data have been successively introduced in a numerical procedure that permitted to extract the best material parameters for two of the most known pseudo-elastic models [Ogden, R.W., Roxburgh, D.G., 1999. A pseudo-elastic model for the Mullins effect in filled rubber. Proceedings of the Royal Society London A 455, 2861–2877; Dorfmann, A., Ogden, R.W., 2004. A constitutive model for the Mullins effect with permanent set in particle-reinforced rubber. International Journal of Solids and Structures 41, 1855–1878] accounting for both stress-softening behaviour and residual strain.     

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.     

Constitutive modeling of woven composites considering asymmetric/anisotropic, rate dependent, and nonlinear behavior

The mechanical performance of woven composites was analyzed focusing on their nonlinear and rate dependent asymmetric/anisotropic deformation behavior. Three key characteristics were identified which are indispensable for realistically simulating the mechanical performance of woven composites: the asymmetric material behavior between tension and compression, its anisotropic and nonlinear evolution and rate dependency. To include all three characteristics into the nonlinear finite element analysis for woven composites, a phenomenological constitutive equation was developed based on an elasto-viscoplastic theory using the modified Drucker–Prager yield criterion and, in particular, developing the anisotropic nonlinear hardening law. A characterization method using both uniaxial tensile and compressive tests at different strain rates was proposed to determine the material properties for the constitutive equation. Then, the developed constitutive equation was incorporated into a finite element code and was validated by comparing the finite element simulation of the three points bending test with experiments.     

一种基于Seth应变张量的超弹性模型

为精确表征橡胶类材料在大变形范围内的力学行为,基于Seth应变张量不变量提出了一种适用于橡胶类材料的不可压缩各向同性超弹性模型。为考察其预测能力,分别利用Treloar经典试验数据和某型炭黑填充橡胶试验数据对该模型、Yeoh模型和二阶多项式模型进行了参数识别。结果表明,在同时使用单轴拉伸和等双轴拉伸试验数据情况下,相较于其他两种常用模型,该模型能够更准确地拟合两种橡胶材料的试验数据,并较好地预测纯剪切(或平面拉伸)试验数据。最后,分别基于前述三种超弹性模型对橡胶衬套进行了静刚度仿真计算和试验验证。结果表明,基于所提出的超弹性模型得到的径向刚度和轴向刚度仿真误差分别为6.61%和9.72%,显著小于基于其他两种模型得到的仿真误差。因此,提出的模型在一定误差范围内能够有效适用于橡胶产品的性能分析。该模型仅含4个材料参数,对不同的橡胶材料有较好地适用性,具有良好的工程应用价值。     

The effect of different forms of strain energy functions in hyperelasticity‐based crystal plasticity models on texture evolution and mechanical response of face‐centered cubic crystals

Micro‐mechanical and macro‐mechanical behavior of face‐centered cubic (FCC) crystals is investigated by using different forms of strain energy functions in hyperelastic material models in crystal plasticity finite element framework. A quadratic strain energy function with anisotropic elastic constants, a polyconvex strain energy function with invariants associated with the cubic symmetry, and a strain energy function from an inter‐atomic potential are considered in hyperelastic material models to describe the elastic deformation of FCC crystals. In our numerical experiments, the trajectories of {111} poles in the pole figure and the accumulated plastic slips of FCC coppers under uniaxial tension and simple shear depend on the choice of strain energy functions when the slip resistance of the slip systems is high. The ability of strain energy functions in this study to represent elastic lattice distortions in crystals varies with the amount of elastic deformation and the shape of deformed lattice. However, numerical results show that the change of macroscopic mechanical behavior of FCC coppers is not significant for the choice of strain energy functions, compared with the change of crystallographic texture evolution. Copyright © 2014 John Wiley & Sons, Ltd.     

A nonlinear fractional viscoelastic material model for polymers

In this contribution a test scheme based on tensile tests at different velocities, relaxation experiments and deformation controlled loading and unloading processes with intermediate relaxations has been used to experimentally characterize the nonlinear, inelastic material behavior. Based on the experimental observations a small strain nonlinear fractional viscoelastic material model is derived. In order to use the model within a finite element analysis, the constitutive equations have been generalized for the multiaxial case. The experimental test scheme and the fractional viscoelastic material model are subsequently applied to characterize and compute the mechanical behavior of the thermoplastic Polypropylene. After the identification of the material parameters several uniaxial and multiaxial simulations have been carried out and compared with experimental results.     

A polyconvex hyperelastic model for fiber-reinforced materials in application to soft tissues

In this paper a generalized anisotropic hyperelastic constitutive model for fiber-reinforced materials is proposed. Collagen fiber alignment in biological tissues is taken into account by means of structural tensors, where orthotropic and transversely isotropic material symmetries appear as special cases. The model is capable to describe the anisotropic stress response of soft tissues at large strains and is applied for example to different types of arteries. The proposed strain energy function is polyconvex and coercive. This guarantees the existence of a global minimizer of the total elastic energy, which is important in the context of a boundary value problem.     

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.