Rate Constitutive Equations Research Articles

A stepwise physics‐informed neural network for solving large deformation problems of hypoelastic materials

AbstractPhysics‐informed neural network (PINN) has been widely concerned for its higher computational accuracy compared with conventional neural network. The merit of PINN mainly comes from its ability to embed known physical laws or equations into data‐based neural networks. However, when dealing with the rate‐dependent nonlinear problems, such as elasto‐plasticity with loading and unloading and hypoelastic large deformation, the conventional PINN cannot obtain satisfactory results. In this article, a stepwise physics‐informed neural network (sPINN) is proposed to solve large deformation problems of hypoelastic materials. The whole process of sPINN can be divided into a series of time steps. In each time step, the rate constitutive equation expressed by Hughes‐Winget algorithm and momentum governing equation are incorporated into the loss function as physical constraints. The displacement and stress fields can be resolved by completing the training process of each time step. Three numerical examples are designed to validate the proposed method by comparing with the solutions of FEM. The results show that sPINN can accurately resolve the displacement and stress fields in path‐dependent large deformation problems. Furthermore, the performance of the sPINN on small data sets are also discussed, which illustrates that sPINN is more capable of predicting the global solution on small data sets as compared with conventional artificial neural network.

Comparison of implicit and explicit numerical integration schemes for a bounding surface soil model without elastic range

Standard integration schemes for rate constitutive equations were designed within the classical theory of plasticity. Consequently, they rely on the assumption that a yield criterion defines a range of purely elastic material behaviour. Many constitutive models for non-cohesive soils discard that assumption. They account for inelastic deformations without employing a yield criterion. This simplifies the formulation of the models but raises questions concerning their numerical implementation. In particular, it is not fully clear which algorithmic method is most appropriate for the numerical integration of constitutive models without elastic range. To investigate this, two different stress point algorithms for a critical state bounding surface model for sands were developed and implemented. The explicit method employs substepping to automatically control the local error. The implicit method updates stresses and state variables through a local Newton iteration, the Jacobian of which is computed by numerical differentiation. The two algorithms were compared by means of calculations at integration point level as well as with respect to a boundary value problem. The results show that for a given level of accuracy, the explicit update procedure is significantly more efficient than the implicit one. This holds regardless of initial state parameters and input strain increment magnitudes.

A hypoelastic stress resultant shell approach for simulations of textile composite reinforcement forming

A stress resultant shell approach is proposed to simulate the draping of textile composite reinforcements with continuous fibers. This approach naturally makes it possible to take into account the specificity of the deformation of the fibrous reinforcements by considering independent bending and tension stiffness whereas they are linked in standard shell elements. The stress resultants and stress moments are related to membrane strains and curvatures by rate constitutive equations (hypoelastic laws). The experimental tests required to identify the material properties necessary for these constitutive laws in stress resultants are presented. This approach is implemented in the ABAQUS finite element code and can be used by all users of this software. A set of forming tests showed the effectiveness of the approach through comparisons between simulation and experimental results.

Hypo-viscoelastic modelling of in-plane shear in UD thermoset prepregs

In-plane shear of prepreg material is one of the key deformation modes that affects defect generation in composites manufacturing processes such as Automated Fibre Placement (AFP) and Thermoforming. Recent characterisation of the evolution of uncured prepreg in-plane shear behaviour during manufacturing has revealed a strong influence of deformation rate and temperature on the measured response. In the present contribution, a phenomenological viscoelastic model for the observed behaviour is formulated using rate constitutive equations. Each of the components is related to an assumed micro-mechanical phenomenon (e.g. friction between the fibres, tensile loading of the fibres, etc). The proposed model is implemented into a user material subroutine for the Finite Element (FE) package Abaqus/Explicit. Model validation is performed by modelling experiments with loading programmes distinctly different from those used for material parameter extraction and it is concluded that state-of-the-art numerical models for AFP need to better represent in-plane shear.

Unified thixotropic fluid model for soil liquefaction

The geotechnical earthquake engineering profession has struggled with the inherent complexity of the multiphase soil response to cyclic loading owing to the progressive nature of the generation of excess pore pressure (EPP) and degradation of soil stiffness and strength. One approach to improve understanding of the cyclic response and correlate the transition from a two-phase saturated soil to a single, fluid-phase liquefied soil is to treat the soil as a non-Newtonian viscous liquid. However, the work to date suggests that the viscous fluid model approach can only be implemented following the onset of sustained soil liquefaction. This paper presents a unified thixotropic fluid model and framework that effectively links the pre-shear soil fabric and its progressive cyclic response to the onset and maintained state of soil liquefaction. The framework treats the soil fabric as a fluid net-type structure proposed for use with thixotropic fluids, and presents the constitutive state and rate equations describing the deconstruction of the liquefiable soil fabric in response to cyclic loading. The unified framework uses physically meaningful soil parameters that can be obtained from common cyclic laboratory tests to seamlessly link the state-dependent and shear-strain-rate-dependent nature of soils to the generation of EPP, the latter of which is shown to increase in significance as EPP accumulates. The proposed thixotropic-induced excess pore pressure model should prove advantageous for use in forward modelling of the stress–strain rate response of liquefiable soil and generation of EPP.

Textile composite structural analysis taking into account the forming process

The manufacturing processes of composite materials modify the internal structure and therefore the mechanical properties of cured composite parts. In particular, draping of woven prepregs leads to angle changes between warp and weft yarns when the geometry of the part is double curved. We here propose to take the fiber reorientation caused by forming into account in the structural analysis of a cured composite. For this purpose, the simulation for the forming of prepregs was based on a rate constitutive equation. Then, a simple orthotropic elastic model of the cured composite was developed to take into account the angle change created by the forming. An experimental validation of this approach was performed in the case of traction on a cured test specimen after a bias extension test. Structural analyses of 3D composite parts obtained by draping also showed that it was necessary to take into account the reorientation of the fibers.

Numerical integration of rate constitutive equations in presence of large strains and rotations

The aim of the present work is to present two operational numerical methods allowing for integrating rate constitutive equations accounting for large strains and rotations. Both methods can be applied in the development of home-made computation codes or/and in the development of user material subroutines in commercial computation codes.

Simulation of strain localization with discrete element-Cosserat continuum finite element two scale method for granular materials

A multiscale method is proposed based on an individual particle provided with rotational freedom, which considers the kinematic connections and transformation consistency of physical parameters in micro-macro models, as well as the need for a regularization mechanism in the classical macroscopic continuum model to preserve the well-posedness of the localization problem. This method uses the discrete element method to incorporate both rolling resistances (rolling friction tangential force and rolling resistance moment) and the sliding friction tangential force between particles in the contact model on a microscopic scale, while the Cosserat continuum is used to describe the granular materials on a macroscopic scale. In addition, a consistent return mapping algorithm for the integration of the rate constitutive equation and the closed form of the consistent elastoplastic tangent modulus matrix for the generalized elastoplastic Cosserat continuum model are presented. The effectiveness of the developed multiscale method is demonstrated with two cases: one comparing discrete element computations with the Cosserat finite element analysis, and the other comparing a plane strain compression experiment using digital imaging measurements with the Cosserat finite element analysis. The rotational deformation and shear band failure modes are well reproduced in both cases. It also demonstrates that the present model has better performance in predicting the phenomena of shear bands than previous ones.

基于有限变形弹塑性模型模拟伪弹性合金全程变形行为

A new elastoplastic J2 flow model was proposed for shape memory alloys to comprehensively simulate the pseudo-elastic stage with perfect strain recovery, the plastic stage with partial strain recovery and the softening stage up to failure. To this end, a new explicit method based on any given uniaxial data was introduced to obtain the multi-axial expression for the constitutive quantities incorporated in this model. The advantage of this model lies in avoiding the usual complicated numerical procedures in treating nonlinear rate constitutive equations with a number of phase transition conditions and micro-macro averaging methods. Numerical examples give results in good accordance with experiment data.

Flow behavior and constitutive relationship for elevated temperature compressive deformation of a high Nb containing TiAl alloy with (α2 + γ) microstructure

The isothermal compression of a Ti-43.5Al-8Nb-0.2W-0.2B (at.%) alloy with (α2+γ) microstructure has been examined at temperature range of 950–1050°C under strain rate range of 10−2–10−5s−1. The results showed that the value of m (strain rate sensitivity factor) increased from 0.18 through 0.33 to 0.48 with decreasing diffusion compensated strain rates. This indicates a transition of deformation mechanism from dislocation creep to grain boundary sliding (GBS), which was observed in the microstructure characterization. The transition took place over about one and a half orders of magnitude in diffusion-compensated strain rate may also be predicted by employing unified rate constitutive equations for dislocation creep and GBS in an additive manner.

Efficient solution algorithms for multiaxial probabilistic elasto‐plastic constitutive simulations of soils

SummaryA Fokker‐Planck‐Kolmogorov (FPK) equation approach has recently been developed to probabilistically solve any elastic‐plastic constitutive equation with uncertain material parameters by transforming the nonlinear, stochastic constitutive rate equation into a linear, deterministic partial differential equation (PDE) and thereby simplifying the numerical solution process. For an uniaxial problem, conventional numerical techniques, such as the finite difference or finite element methods, may be used to solve the resulting univariate FPK PDE. However, for a multiaxial problem, an efficient algorithm is necessary for tractability of the numerical solution of the multivariate FPK PDE. In this paper, computationally efficient algorithms, based on a Fourier spectral approach, are presented for solving FPK PDEs in (stress) space and (pseudo) time, having space‐independent but time‐dependent coefficients and both space‐ and time‐dependent coefficients, that commonly arise in probabilistic elasto‐plasticity. The algorithms are illustrated by probabilistically simulating 2 common laboratory constitutive experiments in geotechnical engineering, namely, the unconfined compression test and the unconsolidated undrained triaxial compression test.

Nonlinear elastoplastic analysis of pressure sensitive materials

When structures undergo extreme loading conditions, the materials pass the elastic limits. Therefore, to preserve economy as well as safety, it is essential to perform a realistic elastoplastic analysis using the constitutive equations in plasticity. On the other hand, computing the stress alongside its associated variables on Gauss points is a delicate process and virtually the most important part of these analyses. In this study, an efficient stress-updating technique is presented for the constitutive rate equations of the pressure sensitive materials such as concrete, rock, soil and some kind of metals. Accordingly, the Drucker–Prager plasticity is utilized to consider the hydrostatic pressure in addition to the J 2-invariant of the deviatoric stress. Moreover, the isotropic and kinematic hardenings are used to take into account more realistic behavior of the materials. Finally, a wide range of numerical tests is carried out to show the performance of the presented method together with the application of the suggested formulations in elastoplastic analysis of a gravity dam.

A theory for non-Newtonian viscoelastic polymeric liquids

In many existing theories for incompressible polymeric liquids the Cauchy stress is decomposed as T=−p1+Sv+Se, where p is an arbitrary pressure, Sv=2μsD a deviatoric viscous stress with μs a viscosity and D the deviatoric stretching tensor, and Se is a deviatoric elastic stress which is introduced to account for stiffening arising from the alignment of long-chain polymer molecules during flow. A constitutive equation for Se needs to be prescribed and there are a large number of different proposals in the literature, with most proposals involving a hypoelastic rate constitutive equation for Se given in terms of a suitable frame-indifferent rate, which is usually taken as the Oldroyd or upper-convected rate. As is well-known, a hypoelastic equation for the stress is not thermodynamically consistent, in the sense that the constitutive equation for Se is not derived form a free energy function.The purpose of this paper is to present an alternative — thermodynamically-consistent and frame-indifferent — continuum theory for incompressible viscoelastic liquids. The theory is based on a Kröner-type multiplicative decomposition of the deformation gradient F of the form F = FeFp. In this theory the elastic stress Se is derived from a free-energy function which is prescribed in terms of a suitable measure based on the unimodular elastic distortion tensor Fe. This relation is supplemented by an evolution equation for the unimodular plastic distortion tensor Fp — the plastic flow rule.We study the response of the constitutive theory in steady simple shearing and steady extensional flows. We show: (i) that the theory qualitatively reproduces the experimentally-observed transient shear-thinning and normal stress effects during shearing flows of a polymer melt; and (ii) that it also reproduces the transient extensional response of a polymer melt.

Explicit, comprehensive modeling of multi-axial finite strain pseudo-elastic SMAs up to failure

A new elastoplastic J 2 − flow model with nonlinear combined hardening is proposed for the purpose of achieving an explicit, comprehensive simulation of pseudo-elastic SMAs over the whole range of strain up to failure. To this goal, a new, explicit methodology based on any given uniaxial data is introduced to obtain multi-axial expressions for the constitutive quantities incorporated in this model, in such a unified sense that all three successive deformation stages may be simultaneously represented, including a pseudo-elastic stage with perfect strain recovery, a hardening stage with partial strain recovery, and a softening stage up to failure. As such, any given shapes of uniaxial stress–strain curves from both loading and unloading in these three stages may be automatically reproduced from the proposed model, without involving usual complicated numerical procedures in treating nonlinear rate constitutive equations with a number of switching conditions. Numerical examples with test data are supplied to simulate a complete process including all three stages.

A new mixed finite-element approach for the elastoplastic analysis of Mindlin plates

The objective of this paper is to develop an accurate and efficient solution procedure for elastoplastic problems in structural mechanics in the framework of a two-field mixed variational principle. A novel solution algorithm is proposed and applied to the elastoplastic analysis of Mindlin plates. The Hellinger–Reissner principle is adopted to obtain the global finite-element equations of the problem. Instead of a static condensation, the stress-type field variables are preserved during the solution. According to the proposed approach, the strain increments within a nonlinear solution step are obtained directly at the nodal points from matrix operations instead of gradients of a displacement field. In the present implementation, the von Mises yield criterion with linear hardening is adopted. For the integration of the elastoplastic constitutive rate equations at the nodal points, a 3D fully implicit algorithm is employed. A layered approach is followed to enable the resolution of the plastic strains through the plate thickness. The mixed formulation of the Mindlin plate theory is shear-locking free by construction. The proposed solution strategy is verified by solving several benchmark problems that demonstrate the high accuracy and convergence rate of the presented layered mixed formulation for elastoplastic analyses.

An ALE method for penetration into sand utilizing optimization-based mesh motion

The numerical simulation of penetration into sand is one of the most challenging problems in computational geomechanics. The paper presents an arbitrary Lagrangian–Eulerian (ALE) finite element method for plane and axisymmetric quasi-static penetration into sand which overcomes the problems associated with the classical approaches. An operator-split is applied which breaks up solution of the governing equations over a time step into a Lagrangian step, a mesh motion step, and a transport step. A unique feature of the ALE method is an advanced hypoplastic rate constitutive equation to realistically predict stress and density changes within the material even at large deformations. In addition, an efficient optimization-based algorithm has been implemented to smooth out the non-convexly distorted mesh regions that occur below a penetrator. Applications to shallow penetration and pile penetration are given which make use of the developments.

Analysis of plug flow reactors with variable mass density

The design of plug flow reactors with variable mass density is examined. Equations which include a two‐term constitutive equation for the reaction rate are derived for the flow of liquids and for the flow of ideal gases in steady plug flow reactors. It is shown that the addition of the second term in the constitutive equation can have a significant effect on the calculation of the reactor volume needed to carry out a specific conversion of the reactant. Published experimental plug flow reactor data support the observation that a reaction rate constitutive equation with two terms can provide a good representation of the experimental data for variable mass density reactors. © 2014 American Institute of Chemical Engineers AIChE J 60: 4185–4189, 2014

Automatic consistency integrations for the stress update of J2 elastoplastic rate constitutive equations with combined hardening

SUMMARYA novel method is introduced to study numerical integrations of J2 elastoplastic rate constitutive equations with general combined hardening. The basic idea is to transform the usual time rate constitutive equations into those with reference to the equivalent plastic strain. By virtue of tensorial matrix operations, we show that these transformed equations may be converted to a linear differential system governing the shifted stress and the plastic multiplier. From this system, we derive explicit integrations for the shifted stress and then for the back stress and the Cauchy stress. We demonstrate that these results are accurate up to within a third order term of the equivalent plastic strain increment. In particular, for pure kinematic hardening, we show that the integrations obtained can achieve automatic enforcement of both the plastic consistency condition and the loading condition, thus bypassing the numerical treatment of the latter two. Furthermore, we explain that, with the new algorithm for the stress update, the continuum tangent moduli may be used to ensure a quadratic rate of convergency in Newton's iteration scheme for the balance equation. Numerical examples suggest that the new algorithm may be more accurate and efficient than the widely used return algorithm. Copyright © 2014 John Wiley & Sons, Ltd.

Energy absorption of three-dimensional angle-interlock woven composite under ballistic penetration based on a multi-scale finite element model

This paper reports the ballistic energy absorption of three-dimensional angle-interlock composite (3DAWC) based on a ballistic experiment and a theoretical model with high strain rate constitutive equation of fiber tows. The 3DAWC panels were penetrated under a hemispherical–cylindrical steel projectile, while time testers recorded the initial velocities and residual velocities of projectile in this process. The damage modes of 3DAWC are observed and analyzed from the view of penetrated damage morphologies and experimental velocity data. In order to demonstrate the energy absorption mechanism with more accuracy, a multi-scale finite element model of 3DAWC under ballistic penetration is specially designed and established to calculate this ballistic event. The constitutive relationship of the Twaron® filament yarn in microstructural model is derived from the springs and dashpots model and compiled into user-defined material subroutine in commercial-available finite element software package LS-DYNA, which can introduce strain rate sensitivity of fiber bundles to ballistic energy absorption process. A comparison of theoretical and experimental results shows a good agreement indicating an accurate validity of the multi-scale finite element model of 3DAWC. Moreover, this model can deeply reveal the ballistic energy absorption mechanism of 3DAWC which will help to evaluate the structural tolerance of ballistic protection composite material.

THREE-DIMENSIONAL PRESSURE-DEPENDENT ELASTOPLASTIC COSSERAT CONTINUUM MODEL AND FINITE ELEMENT SIMULATION OF STRAIN LOCALIZATION

A pressure-dependent elastoplastic Cosserat continuum model for three-dimensional problems is presented in this paper. The nonassociated Drucker–Prager yield criterion is particularly considered. Splitting the scalar product of the stress rate and the strain rate into the deviatoric and the spherical parts, the consistent algorithm of the pressure-dependent elastoplastic model is derived in the three-dimensional framework of Cosserat continuum theory, i.e., the return mapping algorithm for the integration of the rate constitutive equation and the closed form of the consistent elastoplastic tangent modulus matrix. The matrix inverse operation usually required in the calculation of elastoplastic tangent constitutive modulus matrix is avoided, that ensures the second order convergence rate and the computational efficiency of the model in numerical solution procedure. A comparison is performed between the classical and Cosserat continuum model through the numerical results of three-dimensional shear structure, tensile specimen, footing on a soil foundation, and soil slope stability. It illustrates that mesh dependency and numerical difficulties exist in classical model, while Cosserat model possesses the capability and performance in keeping the well-posedness of the boundary value problems with strain softening behavior incorporated. The relationship between the internal length scale and the width of shear band and the load-carrying capability of the structure has also been demonstrated.