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.

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.

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.

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.

Nonlinear Viscoelastic Finite Element Analysis of Adhesive Joints

1988 ◽  
Vol 16 (3) ◽  
pp. 146-170 ◽  
Author(s):  
S. Roy ◽  
J. N. Reddy
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Viscoelastic Behavior ◽  
Bonded Joints ◽  
Adhesively Bonded ◽  
Element Analysis ◽  
Adhesively Bonded Joints ◽  
Nonlinear Viscoelastic ◽  
Structural Failures ◽  
Updated Lagrangian

Abstract A good understanding of the process of adhesion from the mechanics viewpoint and the predictive capability for structural failures associated with adhesively bonded joints require a realistic modeling (both constitutive and kinematic) of the constituent materials. The present investigation deals with the development of an Updated Lagrangian formulation and the associated finite element analysis of adhesively bonded joints. The formulation accounts for the geometric nonlinearity of the adherends and the nonlinear viscoelastic behavior of the adhesive. Sample numerical problems are presented to show the stress and strain distributions in bonded joints.

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.

Study and Prediction of Bone Strength of Osteoporosis Model Based on Synchrotron Radiation

2021 ◽  
Author(s):  
Wanyu Li ◽  
Jun Xu ◽  
Shunan Zhang ◽  
Han Guo ◽  
Jianqi Sun ◽  
...  
Keyword(s):  
Finite Element ◽  
Synchrotron Radiation ◽  
Bone Strength ◽  
Lumbar Vertebrae ◽  
Bone Microstructure ◽  
Ovariectomized Rats ◽  
Osteoporotic Bone ◽  
Mineral Density ◽  
Model Based ◽  
Osteoporosis Diagnosis

Abstract Background: As the gold standard for clinical osteoporosis diagnosis, bone mineral density has significant limitations in bone strength assessment and fracture risk prediction. The purpose of this study is to explore a new osteoporotic bone quality evaluation criteria from both diagnosis site selection and bone strength prediction. Methods: Ovariectomized rats with different intensity swimming therapy were investigated in this study. The lumbar vertebrae and femurs of all the rats were scanned by synchrotron radiation computed tomography. Bone microstructure analysis and finite element analysis were combined to obtain bone microstructure parameters and estimated bone strength. And the sensitivity of different skeletal sites to therapy was explored. An elastic network regression model was established to predict bone strength by integrating additional bone microstructure characteristics besides bone mass.Results: Histomorphometry analysis showed that swimming therapy could reduce the risk of osteoporosis of lumbar vertebrae and femur and suggested that the femur might be a more suitable site for osteoporosis diagnosis and efficacy evaluation than the lumbar vertebrae. The average coefficient of determination and average root mean squared error of our predictive model were 0.774 and 0.110. Bland-Altman analysis showed that our model could be a good alternative to the finite element method. Conclusions: The present study developed a machine learning model for prediction of bone strength of osteoporosis model based on synchrotron x-ray imaging and demonstrated that different skeletal sites had different sensitivity to therapy, which is of great significance for the early diagnosis of osteoporosis, the prevention of fractures and the monitoring of therapy.

FINITE ELEMENT APPROACH TO IMMERSED BOUNDARY METHOD WITH DIFFERENT FLUID AND SOLID DENSITIES

2011 ◽  
Vol 21 (12) ◽  
pp. 2523-2550 ◽  
Author(s):  
DANIELE BOFFI ◽  
NICOLA CAVALLINI ◽  
LUCIA GASTALDI
Keyword(s):  
Finite Element ◽  
Immersed Boundary Method ◽  
Stokes Equations ◽  
Numerical Procedure ◽  
Biological Tissues ◽  
Immersed Boundary ◽  
Fluid Structure ◽  
Finite Element Approach ◽  
Boundary Method ◽  
Element Approach

The Immersed Boundary Method (IBM) has been designed by Peskin for the modeling and the numerical approximation of fluid-structure interaction problems, where flexible structures are immersed in a fluid. In this approach, the Navier–Stokes equations are considered everywhere and the presence of the structure is taken into account by means of a source term which depends on the unknown position of the structure. These equations are coupled with the condition that the structure moves at the same velocity of the underlying fluid. Recently, a finite element version of the IBM has been developed, which offers interesting features for both the analysis of the problem under consideration and the robustness and flexibility of the numerical scheme. Initially, we considered structure and fluid with the same density, as it often happens when dealing with biological tissues. Here we study the case of a structure which can have a density higher than that of the fluid. The higher density of the structure is taken into account as an excess of Lagrangian mass located along the structure, and can be dealt with in a variational way in the finite element approach. The numerical procedure to compute the solution is based on a semi-implicit scheme. In fluid-structure simulations, nonimplicit schemes often produce instabilities when the density of the structure is close to that of the fluid. This is not the case for the IBM approach. In fact, we show that the scheme enjoys the same stability properties as in the case of equal densities.

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