Continuum Constitutive Model Research Articles

Non-ordinary state-based peridynamic simulation of elastoplastic deformation and dynamic cracking of polycrystal

The peridynamic correspondence material models provide a way to convert the classical continuum constitutive models into the framework of non-ordinary state-based peridynamics (NOSB PD) with inherent capabilities for describing long-range interactions and fracture. The present study introduces the modeling approach for crystal plasticity in the framework of NOSB PD with a penalty force stabilization technique. It also provides details for numerical implementation of explicit dynamic solver and implicit quasi-static nonlinear solver for many constitutive models. The quasi-static elastoplastic deformation and dynamic cracking examples illustrate that NOSB PD crystal plasticity can capture the fine shear bands that occur naturally in deforming polycrystalline aggregates and the elastoplastic dynamic cracking of polycrystals.

Shallow unloading deformation analysis on Baihetan super-high arch dam foundation

The Baihetan super-high arch dam is the largest hydropower station in the world under construction. Columnar jointed basalt (CJB) is widely distributed at the Baihetan dam foundation, and the many closely spaced joints pose potential risks on the unloading quality of the foundation surface. In this study, typical unloading cracking and relaxation features of CJB based on field survey and the acoustic wave test are revealed during foundation excavation. Considering the large-scale joints in category-I CJB, an equivalent continuum constitutive model was proposed for describing anisotropic deformation and the unloading relaxation modes and the mechanism of columnar joints are further analysed. Both monitoring and numerical results show that: 1) the unloading behaviours include shallow relaxation of columnar joints and combined rebound deformation of columnar joints and shear belts; 2) the relaxation range of columnar joints mainly occurs at the shallow foundation surface with the maximum depth of 3~4 m, and the relaxation degree is obviously exaggerated owing to the sliding of shear belts; 3) the unloading relaxation mode at the shallow foundation mainly belongs to rebound deformation and relaxation of oral joints. The tensile failure occurs in steep joints, and both tensile and shear failure occur in gentle joints. The proposed model can be applied to effectively simulate the unloading relaxation processes of dam foundation excavation.

Scratching of silicon surfaces

Nanoscale scratching of silicon surfaces is the elementary abrasive event for various machining techniques including fixed abrasive wire saw slicing, grinding, elliptical ultrasonic cutting and single-point diamond turning. The understanding of this process is essential for improving the surface quality and reducing sub-surface damage. Nanoscratching experiments are performed using a well characterized diamond tip geometry. The finite element method is employed in order to simulate the scratching process with a continuum constitutive model developed for phase transformation in silicon (Budnitzki, M., Kuna, M., 2016. Stress induced phase transitions in silicon. JMPS 95, 64–91). The required material parameters were determined from indentation experiments in a (111) single crystal Si wafer. The simulation results agree very well with data from scratch experiments without requiring additional calibration.

Localised failure mechanism as the basis for constitutive modelling of geomaterials

Localised failure of geomaterials in the form of cracks or shear bands always requires special attention in constitutive modelling of solids and structures. This is because the validity of classical constitutive models based on continuum mechanics is questionable once localised inelastic deformation has occurred. In such cases, due to the fact that the macro inelastic responses are mainly governed by the deformation and microstructural changes inside the localisation zone, internal variables, representing these microstructural changes, should be defined inside this zone. In this paper, the localised failure mechanism is identified and employed as an intrinsic characteristic upon which a constitutive model is based on at the first place, instead of being dealt with after developing the model using various regularisation techniques. As a result, inelastic responses of the model are correctly associated with the localisation bands, and not smeared out over the whole volume element as in classical continuum constitutive models. It is shown that this inbuilt localisation mechanism in a constitutive model can naturally capture important features of the material and possess intrinsic regularisation effects while minimising the use of additional phenomenological treatments, and also possessing intrinsic regularisation effects. The development of the proposed model is based on an additional kinematic enhancement to account for high gradient of deformation across the localisation band. This enrichment allows the introduction of an additional constitutive relationship for the localisation band, which is represented in the form of a cohesive-frictional model describing traction-displacement jump relationship across two sides of the localisation band. The model, formulated within a thermodynamically consistent approach, possesses constitutive responses of the bulk material and two localisation bands connected through internal equilibrium conditions. Its key characteristics are demonstrated and validated against experimental data from different types of geomaterials under different loading conditions at both material and structural levels.

Continuous constitutive model for bimodulus materials with meshless approach

One of the difficulties in dealing with bimodular materials including composites, rock and asphalt-mixture material is the discontinuity of Young's modulus when the principal stress changes sign, i.e. from a tensile stress state to a compressive stress state. According to the general elastic theory proposed by Ambartsumyan, there are two kinds of domains in which the coefficients of elasticity are constant. The discontinuity of Young's modulus causes divergence in the computational procedure. In order to overcome this difficulty, two continuous modes for bimodular materials are proposed in this paper. The non-linear equilibrium equations have been formulated with a continuous constitute equation of stress and strain. The meshless finite block method is successful in solving the nonlinear problems for bimodular materials. The numerical solutions of the meshless finite block method in a strong form are obtained using an iterative technique. The degree of accuracy and convergence of the proposed technique is demonstrated by directly comparing the achieved results with the finite element method and analytical solutions.

Small- and large-amplitude oscillatory rheometry with bead–spring dumbbells in Stokesian Dynamics to mimic viscoelasticity

In many areas of suspension mechanics, such as filled polymer fluids or household products such as toothpaste, the suspending fluid itself is inherently non-Newtonian and may exhibit viscoelastic properties. In this paper, we extend the Stokesian Dynamics formalism to incorporate a simple model of viscoelasticity by using small spheres as ‘beads’ in a bead–spring dumbbell (such as is found in the derivation of Oldroyd and FENE constitutive models for dilute polymer solutions). Various different spring laws are then tested in both small-amplitude and large-amplitude oscillatory shear, and their rheological behaviour is compared to continuum constitutive models.

Strain-injection and crack-path field techniques for 3D crack-propagation modelling in quasi-brittle materials

This paper presents a finite element approach for modelling three-dimensional crack propagation in quasi-brittle materials, based on the strain injection and the crack-path field techniques. These numerical techniques were already tested and validated by static and dynamic simulations in 2D classical benchmarks [Dias et al., in: Monograph CIMNE No-134. International Center for Numerical Methods in Engineering, Barcelona, (2012); Oliver et al. in Comput Methods Appl Mech Eng 274:289–348, (2014); Lloberas-Valls et al. in Comput Methods Appl Mech Eng 308:499–534, (2016)] and, also, for modelling tensile crack propagation in real concrete structures, like concrete gravity dams [Dias et al. in Eng Fract Mech 154:288–310, (2016)]. The main advantages of the methodology are the low computational cost and the independence of the results on the size and orientation of the finite element mesh. These advantages were highlighted in previous works by the authors and motivate the present extension to 3D cases. The proposed methodology is implemented in the finite element framework using continuum constitutive models equipped with strain softening and consists, essentially, in injecting the elements candidate to capture the cracks with some goal oriented strain modes for improving the performance of the injected elements for simulating propagating displacement discontinuities. The goal-oriented strain modes are introduced by resorting to mixed formulations and to the Continuum Strong Discontinuity Approach (CSDA), while the crack position inside the finite elements is retrieved by resorting to the crack-path field technique. Representative numerical simulations in 3D benchmarks show that the advantages of the methodology already pointed out in 2D are kept in 3D scenarios.

Spatiotemporal analysis of strain localization in dense granular materials

Predicting localized failure in granular materials is a problem of great interest from both the scientific and technological perspectives. The initiation and growth of strain localization have been studied using laboratory experiments and particle-based numerical simulations, and their findings have been preliminarily implemented into continuum constitutive models. In this study, we revisit strain localization in granular materials using the spatiotemporal data analysis technique, which has been extensively employed in data mining science and social science for the sake of different applications. A dense packing of granular material subjected to biaxial compression was simulated using the combined finite and discrete element method. The large amount of particle-level kinematical data, including the translational and rotational particle motion, fluctuating velocity, granular temperature, and local strain, are collected for subsequent spatiotemporal data analysis. The spatiotemporal data analysis provides a new perspective on the strain localization in dense granular materials and a rich body of new insights are presented for the first time. The spatial autocorrelation analysis results in Moran’s I values close to 1, and positive Z scores and statistically significant p values, which indicate that a dense granular system features clustered patterns during shearing. This finding proves yet again that dense granular materials have an inherent short range order. Eventually, correspondence between localized modes with different particle kinematics and spatial distributions of local Moran statistics and quadrant location map are investigated. By assuming shearing of dense granular materials is a first-order Markov chain process, the convergence and time homogeneity of this process are analyzed. Both the maximum likelihood and Pearson test statistics clearly demonstrate that shearing of dense granular materials is a time-homogenous Markov chain process.

Topology optimization of fusiform muscles with a maximum contraction.

Understanding the optimal designs in nature is critical in bionics. This paper presents a method for designing the configuration of fusiform muscle with a maximum contractile displacement based on topology optimization methods. A nearly incompressible continuum constitutive model of skeletal muscle is utilized. The contractile displacement from the relaxed state to the contracted state is regarded as the objective function. To handle the numerical difficulties that result from the existence of element density, an energy interpolation equation is employed, and a modification of the constitutive model of skeletal muscle is proposed. Several numerical examples are given to demonstrate the reasonability of the proposed method.

A method for establishing a continuous constitutive model of welded metals

Developing an accurate and continuous constitutive model across the weld bead and heat affected zone (HAZ) is an urgent challenge to address the inhomogeneous property distribution and its effects on plastic deformation of welded metals. In this study, such a universal method was proposed, in which the continuous variation in flow stress across these zones was characterized via the relationship among the flow stress, microhardness and weld shape. Through the method, the continuous constitutive model of welded metals can be expressed as a function of weld shape. Essentially, this continuous constitutive model eliminates the dependence of the accuracy on the partition of HAZ in the discrete constitutive model. Using the method, continuous constitutive models of a 2219 aluminum alloy welded plate (AAWP) and a QSTE340 welded tube (WT) were set up. These constitutive models were applied to the finite element (FE) modelings of longitudinal and transverse tensions of the 2219 AAWP and bending of the QSTE340 WT, respectively. The comparisons from simulation and experiment of both of the tensions show that the continuous constitutive model can accurately describe the deformation responses of the welded metals in simple loading processes. Comparison in prediction results of welded tube bending based on the continuous and discrete constitutive models shows that the continuous one presents higher prediction precision in complicated plastic deformation processes. Finally, the transverse tensile deformation behavior of the 2219 AAWP was obtained using the continuous constitutive model.

Prediction of muscle activation for an eye movement with finite element modeling

In this paper, a 3D finite element (FE) modeling is employed in order to predict extraocular muscles' activation and investigate force coordination in various motions of the eye orbit. A continuum constitutive hyperelastic model is employed for material description in dynamic modeling of the extraocular muscles (EOMs). Two significant features of this model are accurate mass modeling with FE method and stimulating EOMs for motion through muscle activation parameter. In order to validate the eye model, a forward dynamics simulation of the eye motion is carried out by variation of the muscle activation. Furthermore, to realize muscle activation prediction in various eye motions, two different tracking-based inverse controllers are proposed. The performance of these two inverse controllers is investigated according to their resulted muscle force magnitude and muscle force coordination. The simulation results are compared with the available experimental data and the well-known existing neurological laws. The comparison authenticates both the validation and the prediction results.

The Strong Discontinuity Approach as a limit case of strain localization in the implicit BEM formulation

The Implicit Formulation of the Boundary Element Method (BEM) is used to deal with physically non-linear 2D-problems in solids. An isotropic damage constitutive model, equipped with a strain softening rule, is applied to increasingly narrow bandwidths, determined by mesh refinement, showing the typical post-peak mesh-dependence behaviour. The Continuum Strong Discontinuity Approach (CSDA), characterized by the introduction of discontinuous jumps in the displacement field together with an equilibrium verification (using continuous constitutive models) on the discontinuity line, is also applied to the implicit formulation of the BEM. By a simple numerical example, it is shown that, beyond the mesh independence, the CSDA results represent the limit case of a zero localization bandwidth. Moreover, to demonstrate the accuracy of the CSDA, other three examples involving concrete fracture are presented and the obtained results are compared with experimental or analytical data available in the literature.

An exploration and further study of an enhanced Oldroyd model

The Oldroyd 6-constant constitutive model for polymeric liquids [J. G. Oldroyd, “On the formulation of rheological equations of state,” Proc. R. Soc. A 200, 523 (1950)] was shown in the work of Bird [“A modification of the Oldroyd model for rigid dumbbell suspensions with Brownian motion,” Z. Angew. Math. Phys. 23, 157 (1972)] to have insufficient flexibility even to describe all second-order time-dependent behaviors of the simple molecular model of rigid dumbbells in solution. Bird proposed an enhancement of the Oldroyd model that would remove this deficiency. The advantage of such a continuum constitutive model is that it is far easier to use in solving specific flow problems than the more physical, but far more cumbersome, molecular models. Thus, if this enhanced constitutive model is indeed able to replicate the results of the molecular models, it should be extremely useful over the applicable range. Here, we employ the enhanced constitutive model of Bird [Z. Angew. Math. Phys. 23, 157 (1972)] to solve four different flow problems, and we verify that our solutions coincide exactly with those of a molecular model: through third order for rectilinear steady shear flow and steady extensional flow, and for one steady non-rectilinear flow (eccentric disk rheometer flow); and through second order for one unsteady (i.e., oscillatory) shear flow, thus providing strong confirmation of the physical veracity and utility of this continuum constitutive model, at least for small and moderate shear or extension rates.

An industry‐oriented strategy for the finite element simulation of paperboard creasing and folding

The numerical simulation of paperboard creasing and folding processes is of increasing importance for the design and production of safe and reliable packaging systems. The extreme material anisotropy and the complexity of these processes require however simulation capabilities which are seldom available in commercial codes. Several approaches have been proposed in the literature over the years, in most cases making use of non‐linear material models developed ad hoc for this purpose. These models, some of which are very effective and accurate, are not in general available in commercial codes and are often based on the definition of a large number of parameters. In this paper, the possibility to obtain acceptable, first‐hand simulation results using only features already available in a commercial code is investigated. An advanced continuum constitutive model, recently presented in the literature, has been used as a reference for tuning the model and for assessing its accuracy. It is shown how standard features, usually available in state‐of‐the‐art commercial codes, can be employed to deal with the extreme material anisotropy, obtaining qualitatively good results in both the creasing and folding phases. The used standard model accounts for the extremely high anisotropy by means of embedded shell elements, playing the role of reinforcements in the fibre direction. The matrix is assumed to be isotropic and elastoplastic, with properties determined based on the behaviour in the thickness direction. The adopted plasticity model is a modified Drucker–Prager model with a cutoff on the tensile pressure side, available in the used commercial code. The procedure adopted for the identification of the small number of required material parameters is also discussed. Copyright © 2017 John Wiley & Sons, Ltd.

Hydrodynamic boundary layer of dilute emulsions of high-viscosity drops

This work investigates the viscous boundary layer in the flow of emulsions over a flat plate. A continuous constitutive model is employed to describe the macroscopic flow of dilute emulsions of high-viscosity drops that experience a regime of small surface deformation. The classical boundary layer hypothesis is applied to obtain a mathematical formulation for the flow within the emulsion boundary layer. The numerical results obtained with a stabilized Finite Element Method show that the dimensionless velocity profiles in the emulsion boundary layer are self-similar. Then, we perform an integral formulation of the emulsion boundary layer equations, which leads to a first-order, non-linear ordinary differential equation for the boundary layer thickness along the plate. We also investigate the existence of a pressure gradient in the cross-plate direction arising to balance elastic stresses in the emulsion boundary layer. The formulation developed here is used to study the effects of the capillary number of the flow on the boundary layer of such a complex fluid.

Application of vector form intrinsic finite element on integrated simulation of wind turbine

SummaryLarge wind turbine system is a periodic time‐varying system with many rigid‐flexible coupling bodies. It is difficult to deal with the singular stiffness matrix produced by the rotation of blades via the traditional finite element method. However, the vector form intrinsic finite element (VFIFE) method can effectively solve the geometric deformation of elastic continuum, the nonlinear or discrete constitutive model, the coupling motion continuum, and rigid body. In this study, a solver program of space beam element is developed by VFIFE method, and three typical examples are chosen to verify its accuracy. And then the integrated simulation of wind turbine system is established, and its dynamic response is analyzed. The natural frequencies of the turbine system, which are obtained by modal parameter identification, can agree well with the results obtained by traditional finite element method. The weighted amplitude wave superposition method and the proper orthogonal decomposition method as well as B‐spline surface interpolation are employed to obtain the wind time series of wind turbine under the normal operation condition. The wind‐induced dynamic response of wind turbine system is calculated by VFIFE method. The numerical results can reflect the periodic influence of gravity on the internal forces of blades and the interaction between blades and the tower.

Advances in Constitutive Modeling of Plasticity for Forming Applications

A succinct description of advanced constitutive models for applications to forming process simulations is provided. These models are continuum-based because they are more efficient in terms of computation time than microstructure–based models. However, they are so–called advanced because they are considered in many scientific studies but rather scarcely used in industrial applications. In addition, the relationship between these continuum constitutive models and multi-scale approaches based on crystal plasticity, dislocation dynamics and mechanics of multi-phase materials, such as advanced high strength steels, is substantiated.

A continuum constitutive model for mechanical behavior of 5052 resin epoxy containing various percentages of MWCNTs

In this article, a continuum-based constitutive model is developed to predict the mechanical behavior of 5052 resin epoxy reinforced by multiwalled carbon nanotubes (MWCNTs) based on experimentally generated data. For this purpose, MWCNTs/epoxy specimens with various percentages of functionalized and nonfunctionalized MWCNTs are prepared. The SEM graphs indicate that functionalization leads to a better bound between epoxy and MWCNTs and a higher level of dispersion. The specimens are then tested under standard ASTM D638-02 a procedure and their true plastic stress–strain curves are extracted. Investigations on experimentally generated data reveal that a wt% dependent equation which is obtained using any two series of these data can be successfully implemented for others. The equation is then implemented into a finite element software using a developed user material subroutine in which is utilized based on a particular solution algorithm. In order to verify the accuracy of the model some tensile as well as load-unload-reload tension tests are performed according to standard conditions and acceptable agreement between the numerical and experimental results are observed. Results also indicate that the proposed empirical model can precisely predict the stress–strain behavior of 5052 resin epoxy containing arbitrary wt% of MWCNTs in the range 0–1 wt%.

Local and nonlocal continuum modeling of inelastic periodic networks applied to stretching-dominated trusses

We present a nonlocal continuum model and its numerical implementation to describe the macroscale response of periodic discrete networks via second-order homogenization. The scale-bridging technique is applied to the specific example of stretching-dominated elastic and inelastic periodic truss networks. Experiments on small-scale truss structures have highlighted the importance of nodal connections on the effective stiffness and strength. Therefore, we describe the mechanics of trusses by accounting for the stretching of truss members and the deformation of nodes. For the representative 2D examples of lattices having square and triangular architectures and for example bar and nodal constitutive laws, we show that a simple continuum model based on affinely deforming a representative unit cell is sufficient to reproduce the nonlinear elastic behavior of discrete trusses. By contrast, localization that arises, e.g., from inelastic deformation requires a refined model. This is where the presented nonlocal continuum model is capable of accurately capturing details of localized deformation. We illustrate the performance of the model by comparing the results of example finite element simulations using the continuum constitutive model to discrete lattice calculations with elastic–plastic bars. Optimal performance is achieved when the representative unit cell of the continuum model agrees with the actual size of the discrete truss unit cell, which accounts for size effects even in regimes where a separation of scales between finite element size and unit cell size does not strictly apply.

Fibres and soils: A route towards modelling of root-soil systems

The addition of flexible fibres to granular, cohesionless soils, has a marked influence on the stress:strain and volumetric response. Experimental observations provide inspiration for the development of continuum models for the mechanical, pre-failure behaviour of these fibre/soil mixtures. Such generic models and the deduced mechanisms of response should be applicable to other combinations of soils and flexible fibres such as plant roots. Two features are particularly important: the distribution of the orientations of fibres (no method of preparation produces an isotropic distribution) and the allowance for the volume of void space not only occupied, but also influenced, by the presence of the fibres.A simple shear element is used as a quasi-one-dimensional demonstrator platform for the presentation of the continuum constitutive model. Such an element represents a familiar configuration in which phenomena, such as dilation and friction, can be directly observed. A basic constitutive model for sand is adapted to this simple shear element; the fibres are added as a separate component able to withstand tension but without flexural stiffness. As the soil-fibre mixture deforms, the straining of the soil generates stresses in favourably oriented fibres. The model is used to clarify some aspects of the response of the fibre-soil mixtures: the influence of fibres on the volumetric behaviour; the existence and nature of asymptotic states; and the stress–dilatancy relationship.