Hypoelastic Constitutive Model Research Articles

An efficient second-order cell-centered Lagrangian discontinuous Galerkin method for two-dimensional elastic-plastic flows

An efficient second-order cell-centered Lagrangian discontinuous Galerkin (DG) method for solving two-dimensional (2D) elastic-plastic flows with the hypo-elastic constitutive model and von Mises yield condition is presented. First, starting from the governing equations of conserved quantities in the Euler framework, the integral weak formulation of them in the Lagrangian framework is derived. Next, the DG method is used for spatial discretization of both the weak formulation of conserved quantities and the evolution equation of deviatoric stress tensor. The Taylor basis functions defined in the reference coordinates provide the piecewise polynomial expansion of the variables, including the conserved quantities and the deviatoric stress tensor. The vertex velocities and Cauchy stress tensor on the edges are computed using a nodal solver equipped with a variant of Li's new Harten-Lax-van Leer-contact approximate Riemann solver [Li et al., “An HLLC-type approximate Riemann solver for two-dimensional elastic-perfectly plastic model,” J. Comput. Phys. 448, 110675 (2022)], in which the longitudinal wave velocity in the plastic state is modified. Then the vertex velocities and Cauchy stress tensor on the edges are used to compute numerical fluxes. A second-order total variation diminishing Runge–Kutta scheme is used for time discretization of both the governing equations of conserved quantities and the evolution equation of deviatoric stress tensor. After solving the evolution equation of deviatoric stress tensor, a radial return algorithm is performed at the Gauss points of each element according to the von Mises yield condition. And then the coefficients of the DG expansion for the deviatoric stress tensor on each element are modified by a least squares procedure using the deviatoric stress tensors at these Gauss points. To achieve second-order accuracy, the least squares procedure is used for piecewise linear reconstruction of conserved quantities and the deviatoric stress tensor, and the Barth–Jespersen limiter is used to suppress the nonphysical numerical oscillation near the discontinuities. After that, the coefficients of the DG expansion are modified through L2 projection using the reconstructed polynomials. Finally, a second-order cell-centered Lagrangian DG scheme is established. Several tests demonstrate that the new scheme achieves second-order accuracy with good robustness, and that the DG method of updating the deviatoric stress tensor has comparable accuracy and much higher efficiency with mesh refinement compared with previous works.

Numerical analysis of the in-plane shear behavior of commingled fiber glass–polypropylene composite reinforcement using hypoelastic constitutive model

The purpose of this paper is to provide a detailed description of a hypoelastic constitutive model. An accurate and robust numerical algorithm is developed and implemented in the Abaqus/Explicit finite element code by means of a vectorized user material (VUMAT) subroutine written in FORTRAN. To validate this model, several elementary tests with different loading conditions were performed on two different types of commingled fiberglass–polypropylene woven composite. The numerical values obtained for the simple tensile test, simple shear test, and objectivity test, were in good agreement with the analytical results of the same tests. The second part of this study was dedicated to the extension of the validation process by performing more realistic simulations of the standard in-plane shear test (picture frame). The numerical responses obtained from the simulation of the picture frame and those from the benchmark curried out by Cao et al. are in good agreement.

Optimizaition for formability of plain woven carbon fiber fabrics

How to take full advantages of the strong designability of plain woven carbon fiber fabric is a key problem urgently needed to be addressed in the application of plain woven carbon fiber fabric reinforced thermoplastics (PW-CFRTPs). This study explored formability of plain woven carbon fiber fabric experimentally and numerically through stamping process, and attempted to develop efficient multiobjective optimization for further improving formability of plain woven carbon fiber fabric. First, the mechanical properties and formability of the carbon fiber fabric were characterized using the experimental tests, and then the finite element (FE) analysis involving the hypoelastic constitutive model was carried out to capture non-orthogonal behaviors of the fabric. Second, influences of blank holding force as well as blank holding area (BHA) on the formability of the fabric were investigated, and the forming characteristics including shear angle distribution, wrinkling strain and draw-in features were analyzed using the validated finite element model. Finally, a global multiobjective optimization algorithm based upon the Kriging Believer strategy was presented to optimize the formability of the fabric. The results indicated that the wrinkling strain along fiber increased with increasing shear angle; and the formability of carbon fiber fabric was largely improved through the optimization in comparison with the baseline design.

A Hypoelastic Dynamic Constitutive Model to Account for the Hysteretic Behaviour of Soil Subjected to Cyclic Loads

To reduce the difficulties associated with dynamic constitutive models, a model was established for soil in this study based on hypoelasticity. The stress‐strain relationship in soil under a cyclic load was divided into three stages: initial loading, unloading, and reloading. The stress‐strain relationship in each stage was ascertained using a hyperbolic equation. On this basis, the physical significance of the parameters in the model and their method of determination were described. The effects of the parameters on the stress‐strain relationship were investigated and the integration algorithm of the model was established. Finally, the rationality of the proposed model was verified by conducting triaxial tests under conventional and cyclic loads. The results show that the model is able to adequately demonstrate all the stress‐strain relations in the soil under both static and dynamic loads.

A hybrid lamination model for simulation of woven fabric reinforced thermoplastic composites solid-state thermo-stamping

Solid-state thermo-stamping of fabric reinforced composites has attracted extensive attention for its efficiency and potential energy saving in recent years. In this paper, a hybrid lamination model is proposed to describe the forming behavior of woven fabric reinforced thermoplastic composites (WFRTP) during solid-state thermo-stamping process. The woven carbon fiber (CF)/ Polyetheretherketone (PEEK) sheet/prepreg is modelled as laminate structure with woven reinforcement layers (shell elements) embedded in thermoplastic resin layer (solid elements). More specifically, a hypoelastic constitutive model is used to represent the anisotropic mechanical behavior of fabric reinforcement under large deformation, while the fabric model is calibrated and validated by uniaxial bias extension (UBE) test and tensile test of woven fabric. A phenomenological model is adopted to describe the viscoelastic-plastic deformation behavior of thermoplastic resin, while the resin model is calibrated by tensile tests of PEEK. The applicability of the hybrid model is demonstrated through comparing numerical results of UBE tests and Erichsen tests with their corresponding experiment results. The hybrid lamination model provides a theoretical foundation for numerical simulation of WFRTP solid-state thermo-stamping.

An exact Riemann solver for one‐dimensional multimaterial elastic‐plastic flows with Mie‐Grüneisen equation of state without vacuum

AbstractIn this article, we present exact Riemann solvers for the Riemann problem and the half Riemann problem, respectively, for one‐dimensional multimaterial elastic‐plastic flows with the Mie‐Grüneisen equation of state (EOS), hypoelastic constitutive model, and the von Mises' yielding condition. We first analyze the Jacobian matrices in the elastic and plastic states, and then build the relations of different variables across different type of waves. Based on these formulations, an exact Riemann solver is constructed with totally 36 possible cases of wave structures. A large number of tests prove the rightness of the new exact Riemann solver. Moreover, an exact Riemann solver is also deduced for the half Riemann problem and its validity is tested by two examples.

Consistent numerical implementation of hypoelastic constitutive models

In hypoelastic constitutive models, an objective stress rate is related to the rate of deformation through an elasticity tensor. The Truesdell, Jaumann, and Green–Naghdi rates of the Cauchy and Kirchhoff stress tensors are examples of the objective stress rates. The finite element analysis software ABAQUS uses a co-rotational frame which is based on the Jaumann rate for solid elements and on the Green–Naghdi rate for shell and membrane elements. The user subroutine UMAT is the platform to implement a general constitutive model into ABAQUS, but, in order to update the Jacobian matrix DDSDDE in UMAT, the model must be expressed in terms of the Jaumann rate of the Kirchhoff stress tensor. This study aims to formulate and implement various hypoelastic constitutive models into the ABAQUS UMAT subroutine. The developed UMAT subroutine codes are validated using available solutions, and the consequence of using wrong Jacobian matrices is elucidated. The UMAT subroutine codes are provided in the “Electronic Supplementary Material” repository for the user’s consideration.

A second-order cell-centered Lagrangian scheme with a HLLC Riemann solver of elastic and plastic waves for two-dimensional elastic-plastic flows

In this paper, we propose a fast and efficient second-order cell-centered Lagrangian scheme for 2D elastic-plastic flows with the hypo-elastic constitutive model and von Mises' yielding condition. First, we develop a novel HLLC-type Riemann solver with elastic and plastic waves (HLLCEP) for 2D elastic-plastic flows. Then, we present a two-directional momentum conservative method to determine the moving speed of grid vertexes. Moreover, we introduce a special symmetry-preserving second-order reconstruction method for scalars, vectors or tensors in order to keep the good symmetric property of the proposed second-order scheme. Finally, a second-order cell-centered Lagrangian scheme, based on the developed Riemann solver (HLLCEP) and the finite volume method framework, is constructed. A number of numerical tests have been carried out, and the numerical results show that the proposed scheme reaches the second-order accuracy for problems with smooth solutions, and is essentially non-oscillatory, and appears to be convergent and symmetric. Moreover, the current HLLCEP Riemann solver is more efficient than the FRRSE solver developed in [11].

A GPU Based Explicit Solid-Shell Finite Element Solver

In this work we present a co-rotational/updated Lagrangian, strain-rate based explicit finite element code which uses hexahedral solid-shell tri-linear elements, intended for simulation of the incremental sheet forming (ISF) process. This element is based heavily on the elements described in [1, 2, 3]: it is under-integrated with a single stack of stress integration points in the thickness direction passing through the elements center; it uses Assumed Natural Strain (ANS) interpolates for the thickness and transverse shear strains; it uses a single parameter Enhanced Assumed Strain (EAS) for the thickness strain; and it selectively scales the mass in the through thickness direction to increase the stable time-step. We combine these methods with a hypo-elastic constitutive model to simulate the ISF process. Initial results obtained with a GPU implementation of the element are presented.

3D numerical simulation of free-surface Bingham fluids interacting with structures using the PFEM

This paper presents a purely Lagrangian approach for the 3D simulation of Bingham free-surface fluids and their interaction with deformable solid structures. In the proposed numerical strategy, the fluid is handled using the Particle Finite Element Method (PFEM) to tackle the issues resulting from extreme changes of geometry, such as mesh distortion and free-surface evolution. Additionally, the Papanastasiou model is employed as a regularization technique to overcome the computational difficulties associated with the classical Bingham model. The solid structure, on the other hand, is represented by the hypoelastic constitutive model and simulated using the conventional Finite Element Method (FEM). The coupling between the fluid and the structure is achieved via a monolithic approach, called Unified formulation. Several numerical examples are presented to illustrate the correctness and the robustness of the proposed formulation, in 2D and in 3D. Special attention is devoted to the analysis of the convergence behavior of the proposed computational framework, the effect of the regularization on the numerical results and the 3D effects. Moreover, detailed comparisons between the simulated results and experimental data are performed so that the concerned problems and results can serve as benchmarks.

A Second-Order Cell-Centered Lagrangian Method for Two-Dimensional Elastic-Plastic Flows

AbstractFor 2D elastic-plastic flows with the hypo-elastic constitutive model and von Mises’ yielding condition, the non-conservative character of the hypo-elastic constitutive model and the von Mises’ yielding condition make the construction of the solution to the Riemann problem a challenging task. In this paper, we first analyze the wave structure of the Riemann problem and develop accordingly aFour-Rarefaction wave approximateRiemannSolver withElastic waves (FRRSE). In the construction of FRRSE one needs to use an iterative method. A direct iteration procedure for four variables is complex and computationally expensive. In order to simplify the solution procedure we develop an iteration based on two nested iterations upon two variables, and our iteration method is simple in implementation and efficient. Based on FRRSE as a building block, we propose a 2nd-order cell-centered Lagrangian numerical scheme. Numerical results with smooth solutions show that the scheme is of second-order accuracy. Moreover, a number of numerical experiments with shock and rarefaction waves demonstrate the scheme is essentially non-oscillatory and appears to be convergent. For shock waves the present scheme has comparable accuracy to that of the scheme developed by Maire et al., while it is more accurate in resolving rarefaction waves.

Data driven modeling of plastic deformation

In this paper the application of machine learning techniques for the development of constitutive material models is being investigated. A flow stress model, for strain rates ranging from 10−4 to 1012 (quasi-static to highly dynamic), and temperatures ranging from room temperature to over 1000 K, is obtained by beginning directly with experimental stress–strain data for Copper. An incrementally objective and fully implicit time integration scheme is employed to integrate the hypo-elastic constitutive model, which is then implemented into a finite element code for evaluation. Accuracy and performance of the flow stress models derived from symbolic regression are assessed by comparison to Taylor anvil impact data. The results obtained with the free-form constitutive material model are compared to well-established strength models such as the Preston–Tonks–Wallace (PTW) model and the Mechanical Threshold Stress (MTS) model. Preliminary results show candidate free-form models comparing well with data in regions of stress–strain space with sufficient experimental data, pointing to a potential means for both rapid prototyping in future model development, as well as the use of machine learning in capturing more data as a guide for more advanced model development.

Harten-Lax-van Leer-contact (HLLC) approximation Riemann solver with elastic waves for one-dimensional elastic-plastic problems

A Harten-Lax-van Leer-contact (HLLC) approximate Riemann solver is built with elastic waves (HLLCE) for one-dimensional elastic-plastic flows with a hypoelastic constitutive model and the von Mises’ yielding criterion. Based on the HLLCE, a third-order cell-centered Lagrangian scheme is built for one-dimensional elastic-plastic problems. A number of numerical experiments are carried out. The numerical results show that the proposed third-order scheme achieves the desired order of accuracy. The third-order scheme is used to the numerical solution of the problems with elastic shock waves and elastic rarefaction waves. The numerical results are compared with a reference solution and the results obtained by other authors. The comparison shows that the presented high-order scheme is convergent, stable, and essentially non-oscillatory. Moreover, the HLLCE is more efficient than the two-rarefaction Riemann solver with elastic waves (TRRSE).

Parametric Study and Centrifuge-Test Verification for Amplification and Bending Moment of Clay–Pile System Subject to Earthquakes

This paper examines seismic effects on fixed-head, end-bearing piles installed through soft clay. The numerical analyses were conducted using ABAQUS with a hypoelastic constitutive model for the clay. The dimensionless parameters involving the major parameters such as pile modulus, soil modulus, slenderness ratio, natural frequencies of clay layer and pile–raft, superstructure mass, density of the soil and peak ground acceleration were obtained from the parametric studies. The relationships for the amplification of ground motions and the maximum bending moment in the pile were developed based on regression of the numerical data. The computed results from the proposed relationships were compared with the results reported in the past studies.

A high-order cell-centered Lagrangian scheme for one-dimensional elastic–plastic problems

We construct the 2nd-order and 3rd-order cell-centered Lagrangian schemes for 1D elastic–plastic problems with the hypo-elastic constitutive model and the von Mises yield criterion. The basic procedure of the construction is the following: first, we carefully analyze the wave structure of the Riemann problem for elastic–plastic materials and develop a two-rarefaction Riemann solver with elastic waves (TRRSE). Then, based on the developed TRRSE, we propose the 2nd-order and 3rd-order cell-centered Lagrangian schemes for 1D elastic–plastic solid problems. Moreover, we show that our scheme is positivity-preserving, provided the time step is suitably small. A number of numerical experiments are carried out, and the numerical results show that our 3rd-order scheme achieves the desired order of accuracy. Finally, we apply our 2nd-order and 3rd-order schemes to the numerical solution of the problems with elastic shock waves and elastic rarefaction waves, and the numerical results are compared with the reference solution and with the results obtained by other authors. The comparison shows that the current high-order scheme appears to be convergent, stable and essentially non-oscillatory. Moreover, for shock waves the numerical results of our 2nd-order scheme agree very well with those computed by the 2nd-order scheme developed by Maire et al. (2013), while for rarefaction waves the current second-order scheme performs better than Maire et al.’s (2013) second-order scheme. Besides, our third-order scheme performs better than the 2nd-order scheme developed by Maire et al. (2013).

Influence of lateral initial pressure on axisymmetric wave propagation in hollow cylinder based on first power hypo-elastic model

Influence of identical applied initial pressures on the radial surfaces of a hollow cylinder which is compose of materials with first power hypo-elastic constitutive model was investigated. The basic equations of the problem were built up based on the framework of piecewise homogeneous body model with the use of three-dimensional linearized theory of elastic waves in initially stressed bodies (TLTEWISB). With the method proposed previously, this problem was then solved numerically. Moreover, the dispersion group velocity of the lowest order mode with different initial pressures was also studied. It can be concluded that the initial pressure and the geometry parameters will induce considerable changes of different degrees in dispersive relation between phase velocity and wave number in opposite trend (positive in initial pressure and negative in thickness).

Numerical Analysis of Piles Under Cyclic Lateral Load

The paper presents the results to study the response of piles in homogeneous and layered soils subjected to quasi-static and cyclic lateral loading using numerical analysis. The analyses are carried out using three-dimensional finite element software, ABAQUS v6.10. Hypoelastic constitutive model is used to simulate the small strain nonlinear behavior of soil observed during the application of cyclic lateral load on piles. The bending moment and deflection responses computed from ABAQUS analyses are compared with the results obtained from the reported case histories of field pile lateral load tests and centrifuge tests. The favorable comparisons established that the three-dimensional numerical analysis can be effective to model complex soil–pile systems. Numerical analysis was also able to simulate a cement injected layer as a reinforcement body to the soft foundation soil, which can affect the pile response. The study was further extended to understand, the effect of flow of surrounding soil around the laterally-loaded piles on the response of piles.

Nonlinear Cross-Anisotropic Model for Soils at Various Strain Levels

AbstractNonlinearity and anisotropy of stress-strain responses are important factors influencing the behavior of soil. To describe this complex behavior of soils, a hypoelastic constitutive model is developed based on cross-anisotropic elasticity, which involves four parameters: the bulk modulus, tangent Young’s modulus, coefficient of volume deformation, and Poisson’s ratio. To satisfactorily predict the stiffness of soils at various strain levels, especially in small strain regions, a new stress-strain fitting relationship with four parameters is proposed. For normally consolidated or weakly overconsolidated clay, the dilatancy equation of the Cam-clay model is used to determine the coefficient of volume deformation. Overall, the model is convenient to use. Ten material parameters are involved, which can be obtained from an isotropic compression test, a set of drained triaxial compression tests, and a uniaxial compression test. The computed results of the model are in good agreement with the data from t...

Modelling the Shear-Tension Coupling of Woven Engineering Fabrics

An approach to incorporate the coupling between the shear compliance and in-plane tension of woven engineering fabrics, in finite-element-based numerical simulations, is described. The method involves the use of multiple input curves that are selectively fed into a hypoelastic constitutive model that has been developed previously for engineering fabrics. The selection process is controlled by the current value of the in-plane strain along the two fibre directions using a simple algorithm. Model parameters are determined from actual experimental data, measured using the Biaxial Bias Extension test. An iterative process involving finite element simulations of the experimental test is used to normalise the test data for use in the code. Finally, the effectiveness of the method is evaluated and shown to provide qualitatively good predictions.

Simulation and tomography analysis of textile composite reinforcement deformation at the mesoscopic scale

The preforming stage of the RTM composite manufacturing process leads to fibrous reinforcement deformations which may be very large especially for double curvature shapes. The knowledge of the mesoscopic deformed geometry is important for reinforcement permeability computations and for damage prediction analyses of the composite. A simulation method for woven composite fabric deformation at mesoscopic scale is presented in this paper. A specific continuum hypo-elastic constitutive model is proposed for the yarn behavior. The associated objective derivative is based on the fiber rotation. Spherical and deviatoric parts of the transverse behavior are uncoupled. X-ray tomography is used to obtain experimental undeformed and deformed 3D geometries of the textile reinforcements. It provides, in particular, fiber distribution within the yarn. A comparison between deformed geometries obtained by mesoscopic simulation and by tomography is presented for biaxial tension and in-plane shear deformation.