Finite Deformation, Finite Strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids with Rheology

This paper presents a mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and ordered rate constitutive theories in Lagrangian description derived using entropy inequality and the representation theorem for thermoviscoelastic solids (TVES) with rheology. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics and are based on contravariant deviatoric second Piola-Kirchhoff stress tensor and its work conjugate covariant Green’s strain tensor and their material derivatives of up to order m and n respectively. All published works on nonlinear dynamics of TVES with rheology are mostly based on phenomenological mathematical models. In rare instances, some aspects of CBL are used but are incorrectly altered to obtain mass, stiffness and damping matrices using space-time decoupled approaches. In the work presented in this paper, we show that this is not possible using CBL of CCM for TVES with rheology. Thus, the mathematical models used currently in the published works are not the correct description of the physics of nonlinear dynamics of TVES with rheology. The mathematical model used in the present work is strictly based on the CBL of CCM and is thermodynamically and mathematically consistent and the space-time coupled finite element methodology used in this work is unconditionally stable and provides solutions with desired accuracy and is ideally suited for nonlinear dynamics of TVES with memory. The work in this paper is the first presentation of a mathematical model strictly based on CBL of CCM and the solution of the mathematical model is obtained using unconditionally stable space-time coupled computational methodology that provides control over the errors in the evolution. Both space-time coupled and space-time decoupled finite element formulations are considered for obtaining solutions of the IVPs described by the mathematical model and are presented in the paper. Factors or the physics influencing dynamic response and dynamic bifurcation for TVES with rheology are identified and are also demonstrated through model problem studies. A simple model problem consisting of a rod (1D) of TVES material with memory fixed at one end and subjected to harmonic excitation at the other end is considered to study nonlinear dynamics of TVES with rheology, frequency response as well as dynamic bifurcation phenomenon.

Finite Deformation, Finite Strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids

This paper presents the mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and the constitutive theories derived using entropy inequality and representation theorem for thermoviscoelastic solids (TVES) matter without memory. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics. This mathematical model is thermodynamically and mathematically consistent and is ideally suited to study nonlinear dynamics of TVES and dynamic bifurcation and is used in the work presented in this paper. The finite element formulations are constructed for obtaining the solution of the initial value problems (IVPs) described by the mathematical models. Both space-time coupled as well as space-time decoupled finite element methods are considered for obtaining solutions of the IVPs. Space-time coupled finite element formulations based on space-time residual functional (STRF) that yield space-time variationally consistent space-time integral forms are considered. This approach ensures unconditional stability of the computations during the entire evolution. In the space-time decoupled finite element method based on Galerkin method with weak form for spatial discretization, the solutions of nonlinear ODEs in time resulting from the decoupling of space and time are obtained using Newmark linear acceleration method. Newton’s linear method is used to obtain converged solution for the nonlinear system of algebraic equations at each time step in the Newmark method. The different aspects of the deformation physics leading to the factors that influence nonlinear dynamic response and dynamic bifurcation are established using the proposed mathematical model, the solution method and their validity is demonstrated through model problem studies presented in this paper. Energy methods and superposition techniques in any form including those used in obtaining solutions are neither advocated nor used in the present work as these are not supported by calculus of variations and mathematical classification of differential operators appearing in nonlinear dynamics. The primary focus of the paper is to address various aspects of the deformation physics in nonlinear dynamics and their influence on dynamic bifurcation phenomenon using mathematical models strictly based on CBL of CCM using reliable unconditionally stable space-time coupled solution methods, which ensure solution accuracy or errors in the calculated solution are always identified. Many model problem studies are presented to further substantiate the concepts presented and discussed in the paper. Investigations presented in this paper are also compared with published works when appropriate.

Algorithmic tangent modulus at finite strains based on multiplicative decomposition

The algorithmic tangent modulus at finite strains in current configuration plays an important role in the nonlinear finite element method. In this work, the exact tensorial forms of the algorithmic tangent modulus at finite strains are derived in the principal space and their corresponding matrix expressions are also presented. The algorithmic tangent modulus consists of two terms. The first term depends on a specific yield surface, while the second term is independent of the specific yield surface. The elastoplastic matrix in the principal space associated with the specific yield surface is derived by the logarithmic strains in terms of the local multiplicative decomposition. The Drucker-Prager yield function of elastoplastic material is used as a numerical example to verify the present algorithmic tangent modulus at finite strains.

Virtual Element Formulation for Finite Strain Elastodynamics Dedicated to Professor Karl Stark Pister for his 95th birthday

The virtual element method(VEM)can be seen as an extension of the classical finite element method(FEM)based on Galerkin projection.It allows meshes with highly irregular shaped elements,including concave shapes.So far the virtual element method has been applied to various engineering problems such as elasto-plasticity,multiphysics,damage and fracture mechanics.This work focuses on the extension of the virtual element method to efficient modeling of nonlinear elasto-dynamics undergoing large deformations.Within this framework,we employ low-order ansatz functions in two and three dimensions for elements that can have arbitrary polygonal shape.The formulations considered in this contribution are based on minimization of potential function for both the static and the dynamic behavior.Generally the construction of a virtual element is based on a projection part and a stabilization part.While the stiffness matrix needs a suitable stabilization,the mass matrix can be calculated using only the projection part.For the implicit time integration scheme,Newmark-Method is used.To show the performance of the method,various two-and three-dimensional numerical examples in are presented.

An Extended Chaboche's Viscoplastic Law at Finite Strains: Theoretical and Numerical Aspects

This paper presents a newly extended Chaboche’s viscoplastic law at finite strains, so that the classical Chaboche’s theories can be applied to the physical and numerical simulation of metals processing and behavior description of spatial metal structures. The extension is based on a new dissipation inequality at finite strains. The evolution equations are formulated in terms of the corotational rates of the logarithmic elastic strain and the strain-like internal variable conjugate to the back stress as well as the material time derivative of the accumulated plastic strain. The stress equation is expressed on the hyperelastic theory. Therefore, the possible inconsistency with elasticity, caused by the hypoelastic equations, is completely removed. A set of numerical examples with finite deformations are presented to prove the effectivities of the new model and numerical algorithms.

On Numerical Modelling of Industrial Powder Compaction Processes for Large Deformation of Endochronic Plasticity at Finite Strains

Compaction processes are one the most important par ts of powder forming technology. The main applications are focused on pieces for a utomotive, aeronautic, electric and electronic industries. The main goals of the compaction processes are to obtain a compact with the geometrical requirements, without cracks, and with a uniform distribution of density. Design of such proc esses consist, essentially, in determine the sequence and relative displacements of die and punches in order to achieve such goals. A.B. Khoei presented a gener al framework for the finite element simulation of powder forming processes based on the following aspects; a large displacement formulation, centred on a total and updated Lagrangian formulation; an adaptive finite element strategy based on error estimates and automatic remeshing techniques; a cap model based on a hard ening rule in modelling of the highly non-linear behaviour of material; and the use of an efficient contact algorithm in the context of an interface element fo rmulation. In these references, the non-linear behaviour of powder was adequately desc ribed by the cap plasticity model. However, it suffers from a serious deficiency when the stress-point reaches a yield surface. In the flow theory of plasticit y, the transition from an elastic state to an elasto-plastic state appears more or less abruptly. For powder material it is very difficult to define the locati on of yield surface, because there is no distinct transition from elastic to ela stic-plastic behaviour. Results of experimental test on some hard met al powder show that the plastic effects were begun immediately upon loading. In such mater ials the domain of the yield surface would collapse to a point, so making the di rection of plastic increment indeterminate, because all directions are normal to a point. Thus, the classical plasticity theory cannot deal with such materials and an advanced constitutive theory is necessary. In the present paper, the constitutive equations of powder materials will be discussed via an endochronic theory of plasticity. This theory provides a unifi ed point of view to describe the elastic-plastic behaviour of material since it places no requirement for a yield surface and a ’loading function’ to disting uish between loading an unloading. Endochronic theory of plasticity has been app lied to a number of metallic materials, concrete and sand, but to the knowledge of authors, no numerical scheme of the model has been applied to powder material . In the present paper, a new approach is developed based on an endochronic rate independent, density-dependent plasticity model for describing the isothermal deformation behavior of metal powder at low homologous temperature. Although the concept of yield surface has not been explicitly assumed in endochronic theory, it is shown that the cone-cap plasticity yield surface (Fig.1), which is the m ost commonly used plasticity models for describing the behavior of powder materi al can be easily derived as a special case of the proposed endochronic theory. Fig.1 Trace of cone-cap yield function on the meridian pl ane for different relative density As large deformation is observed in powder compaction process, a hypoelastic-pl astic formulation is developed in the context of finite deformation plasticity. Constitutive equations are stated in unrotated frame of reference that greatly s implifies endochronic constitutive relation in finite plasticity. Constitutive e quations of the endochronic theory and their numerical integration are establish ed and procedures for determining material parameters of the model are demonstra ted. Finally, the numerical schemes are examined for efficiency in the model ling of a tip shaped component, as shown in Fig.2. Fig.2 A shaped tip component. a) Geometry, boundary conditio n and finite element mesh; b) density distribution at final stage of

Nonlinear Free Vibration of Reinforced Skew Plates with SWCNs Due to Finite Strain

This paper is an attempt to investigate the nonlinear free vibration of skew plates reinforced by carbon nanotubes(CNTs)due to finite strain tensor.The material properties of the nano-composite are estimated using the molecular dynamic results and the rule of mixture.Also,the differential equations governing the motions are derived on the basis of Classical Plate Theory(CPT)regarding the nonlinear Green-Lagrange strain tensor.In order to solve the nonlinear equations,Galerkin’s method,Frechet derivative and differential quadrature method are used.The effects of volume fraction of functionally graded materials(FGM),skew angle,distribution of CNTs and geometrical features of the plate on the nonlinear vibration of system have been studied.The results of this study have been compared with other researches and a good agreement has been achieved.

An investigation on multilayer shape memory polymers under finite bending through nonlinear thermo-visco-hyperelasticity

This study presents a semi-analytical solution to describe the behavior of shape memory polymers(SMPs) based on the nonlinear thermo-visco-hyperelasticity which originates from the concepts of internal state variables and rational thermodynamics.This method is developed for the finite bending of multilayers in a dual-shape memory effect(SME) cycle.The layer number and layering order are investigated for two different SMPs and a hyperelastic material.In addition to the semi-analytical solution,the finite element simulation is performed to verify the predicted results,where the outcomes demonstrate the excellent accuracy of the proposed solution for predicting the behavior of the multilayer SMPs.Since this method has a much lower computational cost than the finite element method(FEM),it can be used as an effective tool to analyze the SMP behavior under different conditions,including different materials,different geometries,different layer numbers,and different layer arrangements.

FINITE ELEMENT ANALYSIS ABOUT STRESS AND STRAIN OF SURFACE PEELING IN Cu-Fe-P SHEET

The microstructure of surface peeling in finish rolled Cu-0.1Fe-0.03P sheetis analyzed by scanning electron microscope and energy dispersive spectroscope. Fe-rich areas ofdifferent contents are observed in the matrix. The stress distributions and strain characteristicsat the interface between Cu matrix and Fe particle are studied by elastic-plastic finite elementplane strain model. Larger Fe particles and higher deforming extent of finish rolling are attributedto the intense stress gradient and significant non-homogeneity equivalent strain at the interfaceand accelerate surface peeling of Cu-0.1Fe-0.03P lead frame sheet.

Deformation characteristics of granitic rocks in Erguna ductile shear zone,NE China

The Erguna ductile shear zone is situated in the Erguna Massif,which has been exposed along the eastern bank of the Erguna River in northeastern China.The authors present comprehensive study results on the macro-and micro-structures,finite strain and kinematic vorticity,quartz electron backscatter diffraction(EBSD)fabrics,and geochronology of granitic rocks in the Erguna ductile shear zone.The deformed granitic rocks have experienced significant SE-trending dextral strike-slip shearing.Finite strain and kinematic vorticity in all deformed granitic rocks indicate that the deformation is characterized by simple sheardominated general shearing with S-L tectonites.Mineral deformation behaviors and quartz C-axis textures demonstrate that the deformed granitic rocks developed under greenschist to amphibolite facies conditions at deformation temperatures ranging from 450 to 550℃.New LA-ICP-MS zircon U-Pb ages indicate that these granitic rocks were formed in Early Triassic(~248.6 Ma)and Early Cretaceous(~136.7 Ma).All the evidence indicates that this deformation may have occurred in Early Cretaceous and was related to the compression resulting from the final closure of the Mongol-Okhotsk Ocean.

Power-law Distribution of Normal Fault Displacement and Length and Estimation of Extensional Strain due to Normal Faults:A Case Study of the Sierra de San Miguelito, Mexico

The Sierra de San Miguelito is a relatively uplifted area and is constituted by a large amount of silicic volcanic rocks with ages from middle to late Cenozoic. The normal faults of the Sierra de San Miguelito are Domino-style and nearly parallel. The cumulative length and displacement of the faults obey power-law distribution. The fractal dimension of the fault traces is -1.49. Using the multi-line one-dimensional sampling, the calculated exponent of cumulative fault displacements is -0.66. A cumulative curve combining measurements of all four sections yielded a slope of -0.63. The displacement-length plot shows a non-linear relationship and large dispersion of data. The large dispersion in the plot is mainly due to the fault linkage during faulting. An estimation of extensional strain due to the normal faults is ca. 0.1830. The bed extension strain is always less than or equal to the horizontal extension strain. The deformation in the Sierra de San Miguelito occurred near the surface, producing pervasive faults and many faults are too small to appear in maps and sections at common scales. The stretching produced by small faults reach ca. 33% of the total horizontal elongation.

SOME MISUNDERSTANDINGS ON ROTATION OF CRYSTALS AND REASONABLE PLASTIC STRAIN RATE

It is pointed out that crystals are discrete but not continuous materials. Hence the rotation R in decomposition F = RU and spin TY in (F) over dot F-1 ore not correct. Errors will arise in plastic deformation rare if it is directly expressed with amounts of velocity of slips in glide systems such as (gamma) over dot upsilon circle times n. The geometrical figure of crystal lattices does nor change after slips and based on this idea a simple way in mechanics of continuous media to get the plastic deformations rare induced by slips is proposed. Constitutive equations are recommended.

Necking of anisotropic micro-films with strain-gradient effects

Necking of stubby micro-films of aluminum is investigated numerically by considering tension of a specimen with an initial imperfection used to onset localisation. Plastic anisotropy is represented by two different yield criteria and strain-gradient effects are accounted for using the visco-plastic finite strain model. Furthermore, the model is extended to isotropic anisotropic hardening （evolving anisotropy）. For isotropic hardening plastic anisotropy affects the predicted overall nominal stress level, while the peak stress remains at an overall logarithmic strain corresponding to the hardening exponent. This holds true for both local and nonlocal materials. Anisotropic hardening delays the point of maximum overall nominal stress.

Fifty years of finite elements——a solid mechanics perspective

The finite element method has been considered as one of the most significant engineering advances of the twentieth century. This computational methodology has made substantial impact on many fields in science and also has profoundly changed engineering design procedures and practice. This paper, mainly froln a solid mechanics perspective, and the Swansea viewpoint in particular, describes very briefly the origin of the methodology, then summaries selected milestones of the technical developments that have taken place over the last fifty years and illustrates their application to some practical engineering problems.

PREDICTION FOR FORMING LIMIT OF AL2024T3 SHEET BASED ON DAMAGE THEORY USING FINITE ELEMENT METHOD

This paper presents the application of anisotropic damage theory to the study of forming limit diagram of A12024T3 aluminum alloy sheet. In the prediction of limiting strains of the aluminum sheet structure, a finite element cell model has been constructed. The cell model consists of two phases, the aluminum alloy matrix and the intermetallic cluster. The material behavior of the aluminum alloy matrix is described with a fully coupled elasto-plastic damage constitutive equation. The intermetallic cluster is assumed to be elastic and brittle. By varying the stretching ratio, the limiting strains of the sheet under biaxial stretching have been predicted by using the necking criterion proposed. The prediction is in good agreement with the experimental findings. Moreover, the finite element cell model can provide information for understanding the microscopic damage mechanism of the aluminum alloy. Over-estimation of the limit strains may result if the effect of material damage is ignored in the sheet metal forming study.

Comparative Study on Constitutive Modeling of Tantalum and Tantalum Tungsten Alloy

As a model bee metal, tantalum and its alloys have wide applications in defense-related fields. The KHL （Khan, Huang, Liang, 1999） model and the constitutive model proposed by Nemat-Nasser et al （Nemat-Nasser and Kapoor, 2001） for tantalum and its alloys were analyzed and compared with each other. A set of published data recorded during elastic-plastic deformations of tantalum, tantalum alloy containing tungsten of 2.5% （Ta-2.5W）, over a wide range of strains, strain rates, and temperatures were used to correlate the two models. Overall, it can be concluded that KHL model correlates much better with the data than the model used by Nemat-Nasser et al.

Strength differential effect and influence of strength criterion on burst pressure of thin-walled pipelines

In the framework of the finite deformation theory, the plastic collapse analysis of thin-walled pipes subjected to the internal pressure is conducted on the basis of the unified strength criterion （USC）. An analytical solution of the burst pressure for pipes with capped ends is derived, which includes the strength differential effect and takes the influence of strength criterion on the burst pressure into account. In addition, a USC- based analytical solution of the burst pressure for end-opened pipes under the internal pressure is obtained. By discussion, it is found that for the end-capped pipes, the influence of different yield criteria and the strength differential effect on the burst pressure are significant, while for the end-opened pipes, the burst pressure is independent of the specific form of the strength criterion and strength difference in tension and compression.

The efficiency of direct integration methods in elastic contact-impact problems

A comparison of direct integration methods is madeand their efficiency is investigated for impact problems.New-mark,Wilson-θ,Central Difference and Houbolt Methodsare used as direct integration methods.Impact analysisincludes that of elastic and large deformation based uponupdated Lagrangian including buckling check.The resultsshow that the direct integration methods give differentresults in different contact-impact cases.

Theoretical Assessment of Burst Failure of Internally Pressurized Defect-Free Pipelines with Plastic Anisotropy

In the framework of finite deformation theory, the burst failure analysis of end-opened defect-free pipes with plastic anisotropy under internal pressure is carried out. The analytical solutions of burst pressure and the corresponding equivalent stress and strain are obtained for thin-walled pipes, which can take into account the effects of material plastic anisotropy and strain hardening exponent. The influences of plastic anisotropy on the burst pressure and the corresponding equivalent stress and strain are discussed. It is shown that the burst pressure and the corresponding equivalent stress and strain are dependent upon the plastic anisotropy of material, and the degree of dependence is related to the strain hardening exponent of material. In addition, the effects of the strain hardening exponent on burst failure are investigated.

Mathematical and numerical modelling of large creep deformations for annular rotating disks

A simulation model is presented for the creep process of the rotating disks under the radial pressure in the presence of body forces. The finite strain theory is applied. The material is described by the Norton-Bailey law generalized for true stresses and logarithmic strains. A mathematical model is formulated in the form of a set of four partial differential equations with respect to the radial coordinate and time. Necessary initial and boundary conditions are also given. To make the model complete, a numerical procedure is proposed. The given example shows the effectiveness of this procedure. The results show that the classical finite element method cannot be used here because both the geometry and the loading （body forces） change with the time in the creep process, and the finite elements need to be redefined at each time step.