A material model for finite elasto-plastic deformations considering a substructure

Developing further the substructure models proposed by Mandel and Dafalias a thermodynamically consistent system of differential and algebraic equations is derived to describe anisotropic elasto-plastic material behavior at finite deformations. Based on the multiplicative split of the deformation gradient an appropriate material law is formulated applying the principle of the maximum of plastic dissipation. Generalized basic relations of this material model containing a relation of hyperelasticity, evolutional equations for the internal variables describing different kinds of hardening, and the yield condition are presented. The capacity of the proposed material model is demonstrated on the example of a sheet with a hole. Presenting the evolution of yield surfaces the capability of the model to describe anisotropic hardening behavior is shown.     

Elasto-plastic postbuckling of damaged orthotropic plates

Based on the elasto-plastic mechanics and continuum damage theory, a yield criterion related to spherical tensor of stress is proposed to describe the mixed hardening of damaged orthotropic materials. Its dimensionless form is isomorphic with the Mises criterion for isotropic materials. Furthermore, the incremental elasto-plastic damage constitutive equations and damage evolution equations are established. Based on the classical nonlinear plate theory, the incremental nonlinear equilibrium equations of orthotropic thin plates considering damage effect are obtained, and solved with the finite difference and iteration methods. In the numerical examples, the effects of damage evolution and initial deflection on the elasto-plastic postbuckling of orthotropic plates are discussed in detail.     

Identification of Mechanical Material Behavior Through Inverse Modeling and DIC

Inverse methods offer a powerful tool for the identification of the elasto-plastic material parameters. One of the advantages with respect to classical material testing is the fact that those inverse methods are able to deal with heterogeneous deformation fields. The basic principle of the inverse method that is presented in this paper, is the comparison between experimentally measured strain fields and those computed by the finite element (FE) method. The unknown material parameters in the FE model are iteratively tuned so as to match the experimentally measured and the numerically computed strain fields as closely as possible. This paper describes the application of an inverse method for the identification of the hardening behavior and the yield locus of DC06 steel, based on a biaxial tensile test on a perforated cruciform specimen. The hardening behavior is described by a Swift type hardening law and the yield locus is modeled with a Hill 1948 yield surface.     

BUCKLING AND POSTBUCKLING ANALYSIS OF ELASTO-PLASTIC FIBER METAL LAMINATES

The elasto-plastic buckling and postbuckling of fiber metal laminates （FML） are studied in this research. Considering the geometric nonlinearity of the structure and the elasto- plastic deformation of the metal layers, the incremental Von Karman geometric relation of the FML with initial deflection is established. Moreover, an incremental elasto-plastic constitutive relation adopting the mixed hardening rule is introduced to depict the stress-strain relationship of the metal layers. Subsequently, the incremental nonlinear governing equations of the FML subjected to in-plane compressive loads are derived, and the whole problem is solved by the iterative method according to the finite difference method. In numerical examples, the effects of the initial deflection, the loading state, and the geometric parameters on the elasto-plastic buckling and postbuckling of FML are investigated, respectively.     

Damage analysis and dynamic response of elasto-plastic laminated composite shallow spherical shell under low velocity impact

Based on the elasto-plastic mechanics, the damage analysis and dynamic response of an elasto-plastic laminated composite shallow spherical shell under low velocity impact are carried out in this paper. Firstly, a yielding criterion related to spherical tensor of stress is proposed to model the mixed hardening orthotropic material, and accordingly an incremental elasto-plastic damage constitutive relation for the laminated shallow spherical shell is founded when a strain-based Hashin failure criterion is applied to assess the damage initiation and propagation. Secondly, using the presented constitutive relations and the classical nonlinear shell theory, a series of incremental nonlinear motion equations of orthotropic moderately thick laminated shallow spherical shell are obtained. The questions are solved by using the orthogonal collocation point method, Newmark method and iterative method synthetically. Finally, a modified elasto-plastic contact law is developed to determine the normal contact force and the effect of damage, geometrical parameters, elasto-plastic contact and boundary conditions on the contact force and the dynamic response of the structure under low velocity impact are investigated.     

Thermodynamic based model for the evolution equation of the backstress in cyclic plasticity

A nonlinear kinematic hardening rule is developed here within the framework of thermodynamic principles. The derived kinematic hardening evolution equation has three distinct terms: two strain hardening terms and a dynamic recovery term that operates at all times. The proposed hardening rule, which is referred in this paper as the FAPC (Fredrick and Armstrong–Phillips–Chaboche) kinematic hardening rule, shows a combined form of the Frederick and Armstrong backstress evolution equation, Phillips evolution equation, and Chaboche series rule. A new term is incorporated into the Frederick and Armstrong evolution equation that appears to have agreement with the experimental observations that show the motion of the center of the yield surface in the stress space is directed between the gradient to the surface at the stress point and the stress rate direction at that point. The model is further modified in order to simulate nonproportional cyclic hardening by proposing a measure representing the degree of nonproportionality of loading. This measure represents the topology of the incremental stress path. Numerically, it represents the angle between the current stress increment and the previous stress increment, which is interpreted through the material constants of the kinematic hardening evolution equation. This new kinematic hardening rule is incorporated in a material constitutive model based on the von Mises plasticity type and the Chaboche isotropic hardening type. Numerical integration of the incremental elasto-plastic constitutive equations is based on a simple semi-implicit return-mapping algorithm and the full Newton–Raphson iterative method is used to solve the resulting nonlinear equations. Experimental simulations are conducted for proportional and non-proportional cyclic loadings. The model shows good correlation with the experimental results.     

A quaternion-based formulation of Euler–Bernoulli beam without singularity

This paper proposes a singularity-free beam element with Euler–Bernoulli assumption, i.e., the cross section remains rigid and perpendicular to the tangent of the centerline during deformation. Each node of this two-nodal beam element has eight nodal coordinates, including three global positions and one normal strain to describe the rigid translation and flexible deformation of the centerline, respectively, four Euler parameters or quaternion to represent the attitude of cross section. Adopting quaternion instead of Eulerian angles as nodal variables avoids the traditionally encountered singularity problem. The rigid cross section assumption is automatically satisfied. To guarantee the perpendicularity of cross section to the deformed neutral axes, the position and orientation coordinates are coupled interpolated by a special method developed here. The proposed beam element allows arbitrary spatial rigid motion, and large bending, extension, and torsion deformation. The resulting governing equations include normalization constraint equations for each quaternion of the beam nodes, and can be directly solved by the available differential algebraic equation (DAE) solvers. Finally, several numerical examples are presented to verify the large deformation, natural frequencies and dynamic behavior of the proposed beam element.     

Design of microstructure-sensitive properties in elasto-viscoplastic polycrystals using multi-scale homogenization

Evolution of properties during processing of materials depends on the underlying material microstructure. A finite element homogenization approach is presented for calculating the evolution of macro-scale properties during processing of microstructures. A mathematically rigorous sensitivity analysis of homogenization is presented that is used to identify optimal forging rates in processes that would lead to a desired microstructure response. Macro-scale parameters such as forging rates are linked with microstructure deformation using boundary conditions drawn from the theory of multi-scale homogenization. Homogenized stresses at the macro-scale are obtained through volume-averaging laws. A constitutive framework for thermo-elastic–viscoplastic response of single crystals is utilized along with a fully-implicit Lagrangian finite element algorithm for modelling microstructure evolution. The continuum sensitivity method (CSM) used for designing processes involves differentiation of the governing field equations of homogenization with respect to the processing parameters and development of the weak forms for the corresponding sensitivity equations that are solved using finite element analysis. The sensitivity of the deformation field within the microstructure is exactly defined and an averaging principle is developed to compute the sensitivity of homogenized stresses at the macro-scale due to perturbations in the process parameters. Computed sensitivities are used within a gradient-based optimization framework for controlling the response of the microstructure. Development of texture and stress–strain response in 2D and 3D FCC aluminum polycrystalline aggregates using the homogenization algorithm is compared with both Taylor-based simulations and published experimental results. Processing parameters that would lead to a desired equivalent stress–strain curve in a sample poly-crystalline microstructure are identified for single and two-stage loading using the design algorithm.     

Deformation gradient based kinematic hardening model

A kinematic hardening model applicable to finite strains is presented. The kinematic hardening concept is based on the residual stresses that evolve due to different obstacles that are present in a polycrystalline material, such as grain boundaries, cross slips, etc. Since these residual stresses are a manifestation of the distortion of the crystal lattice a corresponding deformation gradient is introduced to represent this distortion. The residual stresses are interpreted in terms of the form of a back-stress tensor, i.e. the kinematic hardening model is based on a deformation gradient which determines the back-stress tensor. A set of evolution equations is used to describe the evolution of the deformation gradient. Non-dissipative quantities are allowed in the model and the implications of these are discussed. Von Mises plasticity for which the uniaxial stress–strain relation can be obtained in closed form serves as a model problem. For uniaxial loading, this model yields a kinematic hardening identical to the hardening produced by isotropic exponential hardening. The numerical implementation of the model is discussed. Finite element simulations showing the capabilities of the model are presented.     

Empirical plasticity models applied for paper sheets having different anisotropy and dry solids content levels

The rheological nature of paper or board is usually treated either as elasto-plastic or as viscoelastic depending on the studied paper making process or behavior in converting and end use. In this paper we study several stress–strain curve models and the determination of material parameters from an elasto-plastic point of view. Finally, a suitable approach for all stress–strain curves measured from 180 strips is constructed using a linear function for an elastic region and a nonlinear function for a strain hardening region. This model determines a proportional limit (elastic limit) and gives fairly elegant dependencies between material/fitting parameters and two important factors of mechanical properties of paper: dry solids content and anisotropy. In this paper the dependency of a plastic strain on dry solids content and anisotropy is estimated using the introduced stress–strain curve model. Correspondingly, the model can be used to estimate many other mechanical behaviors, for example, the tension differences arising from non-uniform moisture content of the paper web profile. However, the main target of this study is to produce competent parameters based on modeled stress–strain curves for further construction of a material model. This elasto-plastic material model will be utilized in out-of-plane deformation and fracture models.     

Contact laws between solid particles

This paper presents a comprehensive study for the contact laws between solid particles taking into account the effects of plasticity, strain hardening and very large deformation. The study takes advantage of the development of a so-called material point method (MPM) which requires neither remeshing for large deformation problems, nor iterative schemes to satisfy the contact boundary conditions. The numerical results show that the contact law is sensitive to impact velocity and material properties. The contact laws currently used in the discrete element simulations often ignore these factors and are therefore over-simplistic. For spherical particles made of elastic perfectly plastic material, the study shows that the contact law can be fully determined by knowing the relative impact velocity and the ratio between the effective elastic modulus and yield stress. For particles with strain hardening, the study shows that it is difficult to develop an analytical contact law. The same difficulty exists when dealing with particles of irregular shapes or made of heterogeneous materials. The problem can be overcome by using numerical contact laws which can be easily obtained using the material point method.     

On free energy-based formulations for kinematic hardening and the decomposition F = fpfe

Within the framework of linear plasticity, based on additive decomposition of the linear strain tensor, kinematical hardening can be described by means of extended potentials. The method is elegant and avoids the need for evolution equations. The extension of small strain formulations to the finite strain case, which is based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts, proved not straight forward. Specifically, the symmetry of the resulting back stress remained elusive. In this paper, a free energy-based formulation incorporating the effect of kinematic hardening is proposed. The formulation is able to reproduce symmetric expressions for the back stress while incorporating the multiplicative decomposition of the deformation gradient. Kinematic hardening is combined with isotropic hardening where an associative flow rule and von Mises yield criterion are applied. It is shown that the symmetry of the back stress is strongly related to its treatment as a truly spatial tensor, where contraction operations are to be conducted using the current metric. The latter depends naturally on the deformation gradient itself. Various numerical examples are presented.     

Anisotropic finite elastoplasticity with nonlinear kinematic and isotropic hardening and application to sheet metal forming

The paper discusses the derivation and the numerical implementation of a finite strain material model for plastic anisotropy and nonlinear kinematic and isotropic hardening. The model is derived from a thermodynamic framework and is based on the multiplicative split of the deformation gradient in the context of hyperelasticity. The kinematic hardening component represents a continuum extension of the classical rheological model of Armstrong–Frederick kinematic hardening. Introducing the so-called structure tensors as additional tensor-valued arguments, plastic anisotropy can be modelled by representing the yield surface and the plastic flow rule as functions of the structure tensors. The evolution equations are integrated by a new form of the exponential map that preserves plastic incompressibility and uses the spectral decomposition to evaluate the exponential tensor functions in closed form. Finally, the applicability of the model is demonstrated by means of simulations of several deep drawing processes and comparisons with experiments.     

A design of residual error estimates for a high order BDF‐DGFE method applied to compressible flows

We deal with the numerical solution of the non‐stationary compressible Navier–Stokes equations with the aid of the backward difference formula – discontinuous Galerkin finite element method. This scheme is sufficiently stable, efficient and accurate with respect to the space as well as time coordinates. The nonlinear algebraic systems arising from the backward difference formula – discontinuous Galerkin finite element discretization are solved by an iterative Newton‐like method. The main benefit of this paper are residual error estimates that are able to identify the computational errors following from the space and time discretizations and from the inexact solution of the nonlinear algebraic systems. Thus, we propose an efficient algorithm where the algebraic, spatial and temporal errors are balanced. The computational performance of the proposed method is demonstrated by a list of numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Relationship between multibody dynamics computation and nonlinear vibration theory

This paper presents the ground-work of implementing the multibody dynamics codes to analyzing nonlinear coupled oscillators. The recent developments of the multibody dynamics have resulted in several computer codes that can handle large systems of differential and algebraic equations (DAE). However, these codes cannot be used in their current format without appropriate modifications. According to multibody dynamics theory, the differential equations of motion are linear in the acceleration, and the constraints are appended into the equations of motion through Lagrange's multipliers. This formulation should be able to predict the nonlinear phenomena established by the nonlinear vibration theory. This can be achieved only if the constraint algebraic equations are modified to include all the system kinematic nonlinearities. This modification is accomplished by considering secondary nonlinear displacements which are ignored in all current codes. The resulting set of DAE are solved by the Gear stiff integrator. The study also introduced the concept of constrained flexibility and uses an instantaneous energy checking function to improve integration accuracy in the numerical scheme. The general energy balance is a single scalar equation containing all the energy component contributions. The DAE solution is then compared with the solution predicted by the nonlinear vibration theory. It also establishes new foundation for the use of multibody dynamics codes in nonlinear vibration problems. It is found that the simulation CPU time is much longer than the simulation of the original equations of the system.     

Contact dynamics of elasto-plastic thin beams simulated via absolute nodal coordinate formulation

Under the frame of multibody dynamics, the contact dynamics of elasto-plastic spatial thin beams is numerically studied by using the spatial thin beam elements of absolute nodal coordinate formulation (ANCF). The inter-nal force of the elasto-plastic spatial thin beam element is derived under the assumption that the plastic strain of the beam element depends only on its longitudinal deformation. A new body-fixed local coordinate system is introduced into the spatial thin beam element of ANCF for efficient con-tact detection in the contact dynamics simulation. The linear isotropic hardening constitutive law is used to describe the elasto-plastic deformation of beam material, and the classical return mapping algorithm is adopted to evaluate the plastic strains. A multi-zone contact approach of thin beams previ-ously proposed by the authors is also introduced to detect the multiple contact zones of beams accurately, and the penalty method is used to compute the normal contact force of thin beams in contact. Four numerical examples are given to demonstrate the applicability and effectiveness of the pro-posed elasto-plastic spatial thin beam element of ANCF for flexible multibody system dynamics.     

Nonlinear dynamic response and dynamic buckling of shallow spherical shells with circular hole

In this paper, the nonlinear equations of motion for shallow spherical shells with axisymmetric deformation including transverse shear are derived. The nonlinear static and dynamic response and dynamic buckling of shallow spherical shells with circular hole on elastically restrained edge are investigated. By using the orthogonal point collocation method for space and Newmark-β scheme for time, the displacement functions are separated and the nonlinear differential equations are replaced by linear algebraic equations to seek solutions. The numerical results are presented for different cases and compared with available data.     

Finite Elastic Deformations in Liquid-Saturated and Empty Porous Solids

Based on the Theory of Porous Media (TPM), a formulation of a fluid-saturated porous solid is presented where both constituents, the solid and the fluid, are assumed to be materially incompressible. Therefore, the so-called point of compaction exists. This deformation state is reached when all pores are closed and any further volume compression is impossible due to the incompressibility constraint of the solid skeleton material. To describe this effect, a new finite elasticity law is developed on the basis of a hyperelastic strain energy function, thus governing the constraint of material incompressibility for the solid material. Furthermore, a power function to describe deformation dependent permeability effects is introduced.After the spatial discretization of the governing field equations within the finite element method, a differential algebraic system in time arises due to the incompressibility constraint of both constituents. For the efficient numerical treatment of the strongly coupled nonlinear solid-fluid problem, a consistent linearization of the weak forms of the governing equations with respect to the unknowns must be used.     

Elasto-plastic buckling and post-buckling analysis of sandwich plates with functionally graded metal-metal face sheets and interfacial damage简

Based on the elasto-plastic theory, considering the effect of spherical stress tensor on the elasto-plastic deformation and using the slicing treatment to deal with the plasticity of functionally graded coatings, the elasto-plastic increment constitutive equations of the sandwich plates with functionally graded metal-metal face sheets can be derived. Applying the weak bonded theory to the interfacial constitutive relation and taking into account the geometric nonlinearity, the nonlinear increment differential equilibrium equations of the sandwich plates with functionally graded metal-metal face sheets are obtained by the minimum potential energy principle. The finite difference method and the iterative method are used to obtain the post-buckling path. When the effect of geometrical nonlinearity of the plate is ignored, the elasto-plastic critical buckling load of the sandwich plates with functionally graded metal-metal face sheets can be solved by the Galerkin method and the iterative method. In the numerical examples, the effects of the interface damages, the induced load ratio, the functionally graded index, and the geometry parameters on the elasto-plastic post-buckling path and the elasto-plastic critical buckling load are investigated.