Energy-Conservation Error Due to Use of Green–Naghdi Objective Stress Rate in Commercial Finite-Element Codes and Its Compensation

2013 ◽  
Vol 81 (2) ◽  
Author(s):  
Zdeňek P. Bažant ◽  
Jan Vorel
Keyword(s):  
Finite Element ◽  
Energy Conservation ◽  
Strain Tensor ◽  
High Volume ◽  
Cauchy Stress ◽  
Commercial Software ◽  
Steel Cylinder ◽  
Kirchhoff Stress ◽  
Jaumann Rate ◽  
Objective Stress

The objective stress rates used in most commercial finite element programs are the Jaumann rate of Kirchhoff stress, Jaumann rates of Cauchy stress, or Green–Naghdi rate. The last two were long ago shown not to be associated by work with any finite strain tensor, and the first has often been combined with tangential moduli not associated by work. The error in energy conservation was thought to be negligible, but recently, several papers presented examples of structures with high volume compressibility or a high degree of orthotropy in which the use of commercial software with the Jaumann rate of Cauchy or Kirchhoff stress leads to major errors in energy conservation, on the order of 25–100%. The present paper focuses on the Green–Naghdi rate, which is used in the explicit nonlinear algorithms of commercial software, e.g., in subroutine VUMAT of ABAQUS. This rate can also lead to major violations of energy conservation (or work conjugacy)—not only because of high compressibility or pronounced orthotropy but also because of large material rotations. This fact is first demonstrated analytically. Then an example of a notched steel cylinder made of steel and undergoing compression with the formation of a plastic shear band is simulated numerically by subroutine VUMAT in ABAQUS. It is found that the energy conservation error of the Green–Naghdi rate exceeds 5% or 30% when the specimen shortens by 26% or 38%, respectively. Revisions in commercial software are needed but, even in their absence, correct results can be obtained with the existing software. To this end, the appropriate transformation of tangential moduli, to be implemented in the user's material subroutine, is derived.

Errors Caused by Non-Work-Conjugate Stress and Strain Measures and Necessary Corrections in Finite Element Programs

2010 ◽  
Vol 77 (4) ◽  
Author(s):  
Wooseok Ji ◽  
Anthony M. Waas ◽  
Zdeněk P. Bažant
Keyword(s):  
Finite Element ◽  
Fiber Composite ◽  
Strain Tensor ◽  
Stress And Strain ◽  
Hencky Strain ◽  
Tangential Stiffness ◽  
Jaumann Rate ◽  
Lagrangian Strain ◽  
Strain Formulation ◽  
Accurate Expression

Many finite element programs including standard commercial software such as ABAQUS use an incremental finite strain formulation that is not fully work-conjugate, i.e., the work of stress increments on the strain increments does not give a second-order accurate expression for work. In particular, the stress increments based on the Jaumann rate of Kirchhoff stress are work-conjugate with the increments of the Hencky (logarithmic) strain tensor but are paired in many finite element programs with the increments of Green’s Lagrangian strain tensor. Although this problem was pointed out as early 1971, a demonstration of its significance in realistic situations has been lacking. Here it is shown that, in buckling of compressed highly orthotropic columns or sandwich columns that are very “soft” in shear, the use of such nonconjugate stress and strain increments can cause large errors, as high as 100% of the critical load, even if the strains are small. A similar situation may arise when severe damage such as distributed cracking leads to a highly anisotropic tangential stiffness matrix, or when axial cracks between fibers severely weaken a uniaxial fiber composite or wood. A revision of these finite element programs is advisable, and will in fact be easy—it will suffice to replace the Jaumann rate with the Truesdell rate. Alternatively, the Green’s Lagrangian strain could be replaced with the Hencky strain.

scholarly journals Work conjugacy error in commercial finite-element codes: its magnitude and how to compensate for it

2012 ◽  
Vol 468 (2146) ◽  
pp. 3047-3058 ◽  
Author(s):  
Zdeněk P. Bažant ◽  
Mahendra Gattu ◽  
Jan Vorel
Keyword(s):  
Finite Element ◽  
Finite Strain ◽  
Strain Tensor ◽  
Biological Tissues ◽  
Organic Soils ◽  
Osteoporotic Bone ◽  
Cauchy Stress ◽  
Ceramic Foams ◽  
Commercial Program ◽  
Updated Lagrangian

Most commercial finite-element programs use the Jaumann (or co-rotational) rate of Cauchy stress in their incremental (Riks) updated Lagrangian loading procedure. This rate was shown long ago not to be work-conjugate with the Hencky (logarithmic) finite strain tensor used in these programs, nor with any other finite strain tensor. The lack of work-conjugacy has been either overlooked or believed to cause only negligible errors. Presented are examples of indentation of a naval-type sandwich plate with a polymeric foam core, in which the error can reach 28.8 per cent in the load and 15.3 per cent in the work of load (relative to uncorrected results). Generally, similar errors must be expected for all highly compressible materials, such as metallic and ceramic foams, honeycomb, loess, silt, organic soils, pumice, tuff, osteoporotic bone, light wood, carton and various biological tissues. It is shown that a previously derived equation relating the tangential moduli tensors associated with the Jaumann rates of Cauchy and Kirchhoff stresses can be used in the user’s material subroutine of a black-box commercial program to cancel the error due to the lack of work-conjugacy and make the program perform exactly as if the Jaumann rate of Kirchhoff stress, which is work-conjugate, were used.

scholarly journals Consistent numerical implementation of hypoelastic constitutive models

2020 ◽  
Vol 71 (5) ◽  
Author(s):  
Mehrdad Palizi ◽  
Salvatore Federico ◽  
Samer Adeeb
Keyword(s):  
Constitutive Models ◽  
Elasticity Tensor ◽  
Element Analysis ◽  
Kirchhoff Stress ◽  
User Subroutine ◽  
Jaumann Rate ◽  
The Finite Element Analysis ◽  
Kirchhoff Stress Tensor ◽  
Objective Stress Rate ◽  
Objective Stress

Abstract 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 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.

Finite Element Modeling of the Punch Stretching of Square Plates

1988 ◽  
Vol 55 (3) ◽  
pp. 667-671 ◽  
Author(s):  
E. Nakamachi ◽  
R. Sowerby
Keyword(s):  
Finite Element ◽  
Strain Distribution ◽  
Pure Aluminum ◽  
Strain Tensor ◽  
Deformation Mode ◽  
Cauchy Stress ◽  
Normal Anisotropy ◽  
Stick Model ◽  
Punch Geometry ◽  
Updated Lagrangian

The paper presents an updated Lagrangian-type finite element procedure, formulated with reference to a surface embedded coordinated system. Membrane shell theory is employed, and an attempt is made to calculate the strain distribution incurred by a peripherally clamped square plate, when impressed by a rigid punch. Three different punch geometries were considered. The material is treated as a rate insentive, elastic work-hardening solid, which obeys the J2 flow theory; both finite deformation and normal anisotropy can be considered. A linear relationship between the Jaumann rate of Cauchy stress and the Eulerian rate of Green’s strain tensor is derived. A slip-stick model was adopted for the interfacial frictional conditions. This was achieved by considering the equilibrium of a constant strain-element node in contact with the tools, and deciding whether such a node would stick or slip under Coulomb friction conditions. It is demonstrated that the punch geometry and frictional conditions exert a strong influence on the deformation mode, and hence, upon the overall strain distribution. The predictions were checked against experimental observations when stretch-forming square plates of pure aluminum, 0.5-mm thick. Contours of equal height on the deforming blanks were determined using a Moir´e fringe technique. The agreement between theory and experiment was favorable.

Elastic Soft-Core Sandwich Plates: Critical Loads and Energy Errors in Commercial Codes Due to Choice of Objective Stress Rate

2013 ◽  
Vol 80 (4) ◽  
Author(s):  
Jan Vorel ◽  
Zdeněk P. Bažant ◽  
Mahendra Gattu
Keyword(s):  
Finite Element ◽  
Differential Equations ◽  
Finite Strain ◽  
Strain Tensor ◽  
Stress Rate ◽  
Soft Core ◽  
Sandwich Plates ◽  
The Core ◽  
Objective Stress Rate ◽  
Objective Stress

Most commercial finite element codes, such as ABAQUS, LS-DYNA, ANSYS and NASTRAN, use as the objective stress rate the Jaumann rate of Cauchy (or true) stress, which has two flaws: It does not conserve energy since it is not work-conjugate to any finite strain tensor and, as previously shown for the case of sandwich columns, does not give a correct expression for the work of in-plane forces during buckling. This causes no appreciable errors when the skins and the core are subdivided by several layers of finite elements. However, in spite of a linear elastic behavior of the core and skins, the errors are found to be large when either the sandwich plate theory with the normals of the core remaining straight or the classical equivalent homogenization as an orthotropic plate with the normals remaining straight is used. Numerical analysis of a plate intended for the cladding of the hull of a light long ship shows errors up to 40%. It is shown that a previously derived stress-dependent transformation of the tangential moduli eliminates the energy error caused by Jaumann rate of Cauchy stress and yields the correct critical buckling load. This load corresponds to the Truesdell objective stress rate, which is work-conjugate to the Green–Lagrangian finite strain tensor. The commercial codes should switch to this rate. The classical differential equations for buckling of elastic soft-core sandwich plates with a constant shear modulus of the core are shown to have a form that corresponds to the Truesdell rate and Green–Lagrangian tensor. The critical in-plane load is solved analytically from these differential equations with typical boundary conditions, and is found to agree perfectly with the finite element solution based on the Truesdell rate. Comparisons of the errors of various approaches are tabulated.

Modeling the HSK Toolholder-Spindle Interface

2002 ◽  
Vol 124 (3) ◽  
pp. 734-744 ◽  
Author(s):  
Ihab M. Hanna ◽  
John S. Agapiou ◽  
David A. Stephenson
Keyword(s):  
Finite Element ◽  
Material Properties ◽  
System Reliability ◽  
High Volume ◽  
Finite Element Models ◽  
Operational Experience ◽  
Bending Load ◽  
Load Carrying ◽  
Detailed Evaluation ◽  
Contact Pressures

The HSK toolholder-spindle connection was developed to overcome shortcomings of the 7/24 steep-taper interface, especially at higher speeds. However, the HSK system was standardized quickly, without detailed evaluation based on operational experience. Several issues concerning the reliability, maintainability, and safety of the interface have been raised within the international engineering community. This study was undertaken to analytically investigate factors which influence the performance and limitations of the HSK toolholder system. Finite Element Models were created to analyze the effects of varying toolholder and spindle taper geometry, axial spindle taper length, drawbar/clamping load, spindle speed, applied bending load, and applied torsional load on HSK toolholders. Outputs considered include taper-to-taper contact pressures, taper-to-taper clearances, minimum drawbar forces, interface stiffnesses, and stresses in the toolholder. Static deflections at the end of the holder predicted by the models agreed well with measured values. The results showed that the interface stiffness and load-carrying capability are significantly affected by taper mismatch and dimensional variations, and that stresses in the toolholder near the drive slots can be quite high, leading to potential fatigue issues for smaller toolholders subjected to frequent clamping-unclamping cycles (e.g., in high volume applications). The results can be used to specify minimum toolholder material properties for critical applications, as well as drawbar design and spindle/toolholder gaging guidelines to increase system reliability and maintainability.

Modeling and Experimental Study of a Wire Clamp for Wire Bonding

2015 ◽  
Vol 137 (1) ◽  
Author(s):  
Fuliang Wang ◽  
Dengke Fan
Keyword(s):  
Experimental Study ◽  
Finite Element ◽  
Energy Conservation ◽  
Design Process ◽  
Gold Wire ◽  
Wire Bonding ◽  
Fe Model ◽  
Amplification Ratio ◽  
Model Studies ◽  
Thermosonic Wire Bonding

A wire clamp is used to grip a gold wire with in 1–2 ms during thermosonic wire bonding. Modern wire bonders require faster and larger opening wire clamps. In order to simplify the design process and find the key parameters affecting the opening of wire clamps, a model analysis based on energy conservation was developed. The relation between geometric parameters and the amplification ratio was obtained. A finite element (FE) model was also developed in order to calculate the amplification ratio and natural frequency. Experiments were carried out in order to confirm the results of these models. Model studies show that the arm length was the major factor affecting the opening of the wire clamp.

scholarly journals Evaluation of Pile’s Buckling Under Axial Load by B-Spline Method and Comparison With Finite Element Method and Exact Solution

2018 ◽  
Vol 8 (2) ◽  
pp. 29-34
Author(s):  
A. Moghaddam ◽  
A. Nayeri ◽  
S.M. Mirhosseini
Keyword(s):  
Finite Element Method ◽  
Exact Solution ◽  
Finite Element ◽  
Numerical Methods ◽  
High Volume ◽  
Spline Method ◽  
B Spline ◽  
Volume Calculations ◽  
Reaction Coefficient ◽  
Element Method

Abstract Although various analytical and numerical methods have been proposed by researchers to solve equations, but use of numerical tools with low volume calculations and high accuracy instead of other numerical methods with high volume calculations is inevitable in the analysis of engineering equations. In this paper, B-Spline spectral method was used to study buckling equations of the piles. Results were compared with the calculated amounts of the exact solution and finite element method. Uniform horizontal reaction coefficient has been used in most of proposed methods for analyzing buckling of the pile on elastic base. In reality, soil horizontal reaction coefficient is nonlinear along the pile. So, in this research by using B-Spline method, buckling equation of the pile with nonlinear horizontal reaction coefficient of the soil was investigated. It is worth mentioning that B-Spline method had not been used for buckling of the pile.

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