Neural network modelling for damage behaviour of composites using full-field strain measurements

This paper presents a neural network modelling method for damage behaviour of composite materials in conjunction with full-field strain measurements. The proposed method utilises the overall structural response of a laminate composite specimen to develop the constitutive model of a single ply unidirectional laminate. Based on an energy principle, a performance function for training the neural networks is derived in terms of the applied external work and the induced strain energy. This allows the proposed method to develop the neural networks without the presence of stress information that is not necessarily obtainable in experiments with non-uniform deformation. The use of neural networks also enables the proposed method to model the damage behaviour without the constraints on the parameter space, such that a more representative model is developed for the actual material behaviour. An example of tailoring the proposed method to model the in-plane shear damage behaviour of a carbon fibre reinforced plastic (CFRP) laminate is demonstrated as well as its numerical validation. The practical application of the proposed method to multi-axial damage-related nonlinear behaviour of composite is presented using the experimental data obtained from a tensile test with an open-hole specimen.     

Constitutive equation for transient creep of cellulose nitrate under hydrostatic pressure

The behaviour of polymers is influenced by the hydrostatic pressure during the unloading creep period as well as during loading creep. Moreover, the behaviour during the unloading creep is also influenced by the value of the unloading strain at the final point of the loading process. In this paper, a constitutive equation for transient creep in the unloading process is proposed by using the same postulates as in the loading process of the previous paper, which includes the effect of the hydrostatic pressure and the effect of the unloading strain. The proposed unloading creep equation is in good agreement with the actual creep data on cellulose nitrate.     

考虑加/卸载速率影响的Ti-Ni形状记忆合金简化本构模型

系统研究了Ti-Ni形状记忆合金丝(SMA)应力-应变曲线、特征点应力、耗能能力、等效阻尼比随材料直径、应变幅值、加载速率、加载循环次数的变化规律;针对SMA唯象Brinson本构模型无法描述SMA动态力学性能的缺点,结合前述试验结果,提出了一种可考虑加/卸载速率影响的SMA简化本构模型。应用该模型对试验用SMA丝进行模拟,所得应力-应变曲线各特征点平均误差仅为3%,结果表明:所建立的速率相关SMA简化本构模型可较为精确地描述SMA在应力诱发相变过程中的超弹性力学行为,同时可反映加/卸载速率和应变幅值等主要因素对其动力本构模型的影响;该模型结构形式简单,具有较好的工程应用前景。     

混凝土坝地震动力损伤分析

基于塑性损伤本构理论,将损伤变量作为内变量,在Drucker-Prager本构模型中引入损伤变量,考虑材料损伤引起的材料劲度的退化,基于非关联流动法则计算材料的塑性应变,根据材料的有效塑性应变计算损伤量,考虑到张开裂缝闭合时材料弹性劲度的恢复,推导了考虑塑性损伤的混凝土动态本构关系,并给出了内变量的计算步骤和动力方程的迭代格式。最后利用建立的动态本构模型对Koyna重力坝进行了非线性地震响应时程分析,并给出了关键时刻坝体最大受拉损伤分布,结果表明在坝颈和坝基处出现了较大的损伤,坝颈处的损伤最终形成由下游向上游的开裂破坏,这与该坝的实际震害较为一致。     

Autoprogressive training of neural network constitutive models

A new method, termed autoprogressive training, for training neural networks to learn complex stress–strain behaviour of materials using global load–deflection response measured in a structural test is described. The richness of the constitutive information that is generally implicitly contained in the results of structural tests may in many cases make it possible to train a neural network material model from only a small number of such tests, thus overcoming one of the perceived limitations of a neural network approach to modelling of material behaviour; namely, that a voluminous amount of material test data is required. The method uses the partially-trained neural network in a central way in an iterative non-linear finite element analysis of the test specimen in order to extract approximate, but gradually improving, stress–strain information with which to train the neural network. An example is presented in which a simple neural network constitutive model of a T300/976 graphite/epoxy unidirectional lamina is trained, using the load–deflection response recorded during a destructive compressive test of a [(±45)₆]_S laminated structural plate containing an open hole. The results of a subsequent forward analysis are also presented, in which the trained material model is used to simulate the response of a compressively loaded [(±30)₆]_S structural laminate containing an open hole. Avenues for further improvement of the neural network model are also suggested. The proposed autoprogressive algorithm appears to have wide application in the general area of Non-Destructive Evaluation (NDE) and damage detection. Most NDE experiments can be viewed as structural tests and the proposed methodology can be used to determine certain damage indices, similar to the way in which constitutive models are determined. © 1998 John Wiley & Sons, Ltd.     

混凝土循环加卸载动态损伤特性研究

利用大型多功能动静力三轴仪对混凝土试件进行了5种应变速率下的动态循环加卸载压缩试验。对混凝土的物理力学参数的变化规律进行了统计分析。结果表明:峰值应力和弹性模量随加载速率的提高而增大,但峰值应变随加载速率的变化表现出较大的离散性。在此基础上,进一步研究了混凝土在不同加载速率下的刚度退化规律。最后,选用基于Weibull统计理论的分段式动态损伤本构模型对试验数据进行拟合。经验证,此模型能够较好的模拟混凝土材料的本构特性。     

三轴加载条件下混凝土的神经网络本构模型

神经网络由大量并行处理单元构成,适合于描述多影响因素的非线性复杂因果规律,为研究材料本构特性提供了一条崭新的途径。利用 BP 网络的模拟能力来代替传统的方法,建立了一个三轴加载情况下混凝土的神经网络本构模型,用于描述混凝土在侧压力恒定轴向单调加载条件下的本构关系。从模型对训练和检验样本的模拟结果可以看出,这个经过训练的含有双隐层的神经网络本构模型具有很高的学习精度和良好的泛化能力,适合在结构工程问题中应用。     

Large-Deformation Constitutive Theories for Structural Composites: Rate-Dependent Concepts and Effect of Microstructure

Abstract:  Polymer-based composite materials are widely used in applications subjected to a variety of loading types, including shock and impact loading in the range of hundreds of strain per second. The behaviour of composite laminates loaded at those rates is typically nonlinear and may involve rather large strains to failure. In the present study, the large-deformation characteristics and constitutive representations of structural composites were investigated as functions of strain rate and temperature. A plain-weave vinyl ester composite material was selected for the study. Tensile tests of off-axis coupon specimens were conducted over several orders of strain rates and limited change of temperatures. A three-parameter constitutive model was proposed to model the large-deformation stress–strain relationship. The constitutive model was then used to predict the material response at different strain rates. The model predictions were verified by a different set of tests. The basic concepts and methodologies involved in reducing such data to constitutive equations that can be used in commercial computational codes to enable structural analysis in the presence of large-strain progressive damage under dynamic loading is discussed.     

Numerical implementation of a shape memory alloy thermomechanical constitutive model using return mapping algorithms

Previous studies by the authors and their co‐workers show that the structure of equations representing shape Memory Alloy (SMA) constitutive behaviour can be very similar to those of rate‐independent plasticity models. For example, the Boyd–Lagoudas polynomial hardening model has a stress‐elastic strain constitutive relation that includes the transformation strain as an internal state variable, a transformation function determining the onset of phase transformation, and an evolution equation for the transformation strain. Such a structure allows techniques used in rate‐independent elastoplastic behaviour to be directly applicable to SMAs. In this paper, a comprehensive study on the numerical implementation of SMA thermomechanical constitutive response using return mapping (elastic predictor‐transformation corrector) algorithms is presented. The closest point projection return mapping algorithm which is an implicit scheme is given special attention together with the convex cutting plane return mapping algorithm, an explicit scheme already presented in an earlier work. The closest point algorithm involves relatively large number of tensorial operations than the cutting plane algorithm besides the evaluation of the gradient of the transformation tensor in the flow rule and the inversion of the algorithmic tangent tensor. A unified thermomechanical constitutive model, which does not take into account reorientation of martensitic variants but unifies several of the existing SMA constitutive models, is used for implementation. Remarks on numerical accuracy of both algorithms are given, and it is concluded that both algorithms are applicable for this class of SMA constitutive models and preference can only be given based on the computational cost. Copyright © 2000 John Wiley & Sons, Ltd.     

A phenomenological constitutive modelling of polyethylene foam under multiple impact conditions

This paper extends the knowledge into the mechanical behaviour characterizations and constitutive modelling of polyethylene (PE) foam under multiple loading and unloading. The mechanical properties of PE foam subjected to single loading cases can be obtained by uniaxial compressive tests at quasi‐static and dynamic states. And the multiple loading and unloading behaviours of the foam can be revealed by consecutive drop tests. The major objective of this research is to propose a phenomenological model consists of shape function and modulus function, which can be predicted compressive response of PE foam for single loading cases. The constitutive models of foamed PE under multiple loading and unloading conditions are established by both using hyperbolic function, where the relations between coefficients and residual strain are introduced. And then, experiments are conducted to validate the proposed model by comparing the constitutive models proposed in this paper and those predicting by finite element software ABAQUS with those by experiments, showing that the proposed models are more accurate for predicting acceleration‐times curves of multiple drop scenarios.     

损伤材料的动力响应特性

研究了工程材料在动力载荷下损伤演化的计算模型。提出了一般材料在各向异性损伤状态下的两种动力损伤模型。第一种以有效应力的等效值的幂函数为基础 ,第二种以损伤应变能释放率为基础。通过数值分析研究了损伤结构元件的动力响应及损伤材料的动力特性。说明了结构元件中损伤发展的分析方法和它们的有限元程序的执行过程。该研究表明 :损伤结构的频谱下移 ,损伤材料的阻尼比变高 ,响应的振幅明显增加 ,损伤结构可能发生由于损伤发展引起的共振。     

Modeling constitutive relationship of Ti40 alloy using artificial neural network

Constitutive relationship equation reflects the highly non-linear relationship of flow stress as function of strain, strain rate and temperature. It is a necessary mathematical model that describes basic information of materials deformation and finite element simulation. In this paper, based on the experimental data obtained from Gleeble-1500 Thermal Simulator, the constitutive relationship model for Ti40 alloy has been developed using back propagation (BP) neural network. The predicted flow stress values were compared with the experimental values. It was found that the absolute relative error between predicted and experimental data is less than 8.0%, which shows that predicted flow stress by artificial neural network (ANN) model is in good agreement with experimental results. Moreover, the ANN model could describe the whole deforming process better, indicating that the present model can provide a convenient and effective way to establish the constitutive relationship for Ti40 alloy.     

Phenomenological formulation of viscoplastic constitutive equation for polyethylene by taking into account strain recovery during unloading

A viscoplastic constitutive equation for polyethylene that properly describes significant strain recovery during unloading was proposed. The constitutive equation was formulated by combining the kinematic hardening creep theory of Malinin and Khadjinsky with the nonlinear kinematic hardening rule of Armstrong and Frederick. In order to describe the strain recovery, the nonlinear kinematic hardening rule was modified. First, a loading surface was defined in a viscoplastic strain space. A loading–unloading criterion was then introduced using the loading surface. Moreover, a new parameter was defined by the relationship between the loading surface and the current state of the viscoplastic strain, and the evolution equation of back stress was modified using this parameter, which has some value only during unloading. Experimental results for polyethylene were simulated by using the modified constitutive equations, and cyclic inelastic deformation in both uniaxial and biaxial states of stress was predicted. Finally, the validity of the above-described modification was verified, and the features of the constitutive equation and the deformation were discussed.     

复合固体推进剂黏弹性微裂纹损伤本构模型

为建立复合固体推进剂的损伤本构模型,在介观尺度上视其为微裂纹损伤,选取微裂纹密度为损伤内变量。在Abdel-Tawab本构方程的基础上,基于微裂纹均匀化理论,推导了损伤映射张量的一般形式。该张量通常具有非完全对称性,其物理意义是将真实应力空间中各向异性材料的多轴加载映射为等效应力空间中各向同性材料的更为复杂的多轴加载。其次,基于黏弹性动态裂纹扩展模型和裂纹扩展阻力曲线的概念,建立了损伤内变量的演化方程。该演化方程仅含4个物理意义明确的细观参数,并且参数的取值规律与宏观应力曲线的变化规律相一致。数值结果表明,建立的模型能够有效反映材料损伤的应变率、温度依赖性及各向异性特征,并且具有一定的蠕变损伤预测能力。     

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。     

A bounding surface plasticity model for cyclic loading of granular soils

A constitutive model for describing the stress–strain behaviour of granular soils subjected to cyclic loading is presented. The model is formulated using bounding surface theory within a critical state framework. A single set of material parameters is introduced for the complete characterization of the constitutive model. The shape of the bounding surface is based on experimental observations of undrained stress paths for loose samples. A mapping rule which passes through stress reversal points is introduced to depict the stress–strain behaviour during unloading and reloading. The effect of particle crushing is considered through a modified critical state line. Essential features of the model are validated using several experimental data from the literature. Both drained and undrained loading conditions are considered. The characteristic features of behaviour in granular soils subjected to cyclic loading are captured. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

A viscoelastic continuum damage model and its application to uniaxial behavior of asphalt concrete

An existing viscoelastic constitutive model which accounts for the effects of rate-dependent damage growth is described and applied successfully to characterize the uniaxial stress, constant strain rate behavior of asphalt concrete. The special case of an elastic continuum damage model with multiaxial loading, which is based upon thermodynamics of irreversible processes with internal state variables, is first reviewed and then it is shown how this model has been extended to a corresponding viscoelastic damage model through the use of an elastic-viscoelastic correspondence principle. The general mathematical model is next specialized to uniaxial loading. A rate-type evolution law, similar in form to a crack growth law for a viscoelastic medium, is adopted for describing the damage growth within the body. Results from laboratory tests of uniaxial specimens under axial tension at different strain rates are then shown to be consistent with the theory. The discussion of data analysis describes the specific procedure used here to obtain the material parameters in the constitutive model for uniaxial loading and how the method may be generalized for multiaxial loading.     

The fracture process in quasi‐brittle materials simulated using a lattice dynamical model

The damage process in quasi‐brittle materials is characterized by the evolution of a micro‐crack field, followed by the joining of micro‐cracks, stress localization and crack instability. In network models, masses are lumped at nodal points which are interconnected by one‐dimensional elements with a bilinear constitutive relation, considering the energy consistency during the simulated process. In order to replicate the material imperfections, to render a realistic behaviour in damage localization, the model has not only random elastic and rupture properties, but also a geometric perturbation. In the present paper 2D plates with different levels of brittleness are simulated. The numerical results are presented in terms of global stress vs strain diagram, final network configuration, energy balance during the process and as geometric damage evolution. Therefore, the predictive potential of the lattice discrete element model to capture fracture processes in quasi‐brittle materials is demonstrated.     

A viscoelastic constitutive model with nonlinear evolutionary internal variables

Summary Based on the internal variable theory, a viscoelastic constitutive model of a highly deformable continuous medium is proposed. A set of second rank tensorial internal state variables corresponding to Biot's strain is introduced, and a nonlinear evolution law for these internal variables is suggested. The proposed model may be considered as an extension of the network theory of rubber elasticity to take the viscous effects into account. In order to verify the validity of the present model, uniaxial tension tests for HDPE are carried out at different strain rates. The prediction of the present model shows a good agreement with the experimental data. Finally, a discussion of the present constitutive model is given. It is found that the present constitutive model is more flexible to describe the strain rate sensitivity of polymeric materials in a wide range of strain rates.