Easy-to-Compute Tensors With Symmetric Inverse Approximating Hencky Finite Strain and Its Rate

1998 ◽  
Vol 120 (2) ◽  
pp. 131-136 ◽  
Author(s):  
Zdeneˇk P. Bazˇant
Keyword(s):  
Finite Strain ◽  
Strain Tensor ◽  
Taylor Series Expansion ◽  
Inverse Transformation ◽  
Quadratic Term ◽  
Original Strain ◽  
Volume Changes ◽  
Hencky Strain ◽  
Deformation Rate Tensor ◽  
Deviatoric Part

It is shown that there exist approximations of the Hencky (logarithmic) finite strain tensor of various degrees of accuracy, having the following characteristics: (1) The tensors are close enough to the Hencky strain tensor for most practical purposes and coincide with it up to the quadratic term of the Taylor series expansion; (2) are easy to compute (the spectral representation being unnecessary); and (3) exhibit tension-compression symmetry (i.e., the strain tensor of the inverse transformation is minus the original strain tensor). Furthermore, an additive decomposition of the proposed strain tensor into volumetric and deviatoric (isochoric) parts is given. The deviatoric part depends on the volume change, but this dependence is negligible for materials that are incapable of large volume changes. A general relationship between the rate of the approximate Hencky strain tensor and the deformation rate tensor can be easily established.

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.

A Corotational Elastoplastic Formulation With Subloading Surface Model for Hardening Materials

Volume 1 ◽  
2004 ◽  
Author(s):  
Ali Reza Saidi ◽  
Koichi Hashiguchi
Keyword(s):  
Constitutive Model ◽  
Strain Rate ◽  
Simple Shear ◽  
Deformation Theory ◽  
Elastoplastic Deformation ◽  
Strain Tensor ◽  
Surface Model ◽  
Strain Rate Tensor ◽  
Hencky Strain ◽  
Stress Components

In this paper a corotational constitutive model for the large elastoplastic deformation of hardening materials using subloading surface model is formulated. This formulation is obtained by refining the large deformation theory of Naghdabadi and Saidi (2002) adopting the corotational logarithmic (Hencky) strain rate tensor and incorporating it into the subloading surface model of Hashiguchi (1980, 2003) falling within the framework of the unconventional plasticity. As an application of the proposed constitutive model, the large Elastoplastic deformation of simple shear example has been solved and the results have been compared with classical elasto-plastic model using the Hencky strain tensor. Also the effect of the choice of corotational rates on stress components has been studied.

scholarly journals 2-D finite displacements and strain from particle imaging velocimetry (PIV) analysis of tectonic analogue models with TecPIV

Solid Earth ◽  
2019 ◽  
Vol 10 (4) ◽  
pp. 1123-1139 ◽  
Author(s):  
David Boutelier ◽  
Christoph Schrank ◽  
Klaus Regenauer-Lieb
Keyword(s):  
Software Package ◽  
Finite Strain ◽  
Deformation Gradient ◽  
Strain Tensor ◽  
Small Strain ◽  
Time Lapse ◽  
Image Correlation ◽  
Strain Rate Tensor ◽  
Analogue Models ◽  
Finite Displacements

Abstract. Image correlation techniques have provided new ways to analyse the distribution of deformation in analogue models of tectonics in space and time. Here, we demonstrate, using a new version of our software package (TecPIV), how the correlation of successive time-lapse images of a deforming model allows not only to evaluate the components of the strain-rate tensor at any time in the model but also to calculate the finite displacements and finite strain tensor. We illustrate with synthetic images how the algorithm produces maps of the velocity gradients, small-strain tensor components, incremental or instantaneous principal strains and maximum shear. The incremental displacements can then be summed up with Eulerian or Lagrangian summation, and the components of the 2-D finite strain tensor can be calculated together with the finite principal strain and maximum finite shear. We benchmark the measures of finite displacements using specific synthetic tests for each summation mode. The deformation gradient tensor is calculated from the deformed state and decomposed into the finite rigid-body rotation and left or right finite-stretch tensors, allowing the deformation ellipsoids to be drawn. The finite strain has long been the only quantified measure of strain in analogue models. The presented software package allows producing these finite strain measures while also accessing incremental measures of strain. The more complete characterisation of the deformation of tectonic analogue models will facilitate the comparison with numerical simulations and geological data and help produce conceptual mechanical models.

Spontaneous strain in synthetic titanite, CaTiOSiO4

2001 ◽  
Vol 65 (6) ◽  
pp. 709-715 ◽  
Author(s):  
T. Malcherek
Keyword(s):  
Phase Transition ◽  
Room Temperature ◽  
Finite Strain ◽  
Paraelectric Phase ◽  
Strain Tensor ◽  
Strain Component ◽  
Volume Strain ◽  
Single Crystal Diffraction ◽  
Paraelectric Phase Transition ◽  
Strain Coupling

AbstractLattice parameters of synthetic titanite powder, CaTiOSiO4, have been determined between room temperature and 1023 K. Only the e11 and e13 components contribute significantly to the strain tensor associated with the antiferroelectric-paraelectric phase transition at Tc = 487 K. A finite strain component e13 is observed in the paraelectric phase for 487 K < T < 825 K. The disappearance of this shear strain marks the isosymmetric transition near 825 K. The temperature evolution of the volume strain and of e11 is proportional to the squared order parameter observed in single-crystal diffraction experiments. The magnitude of the volume strain is sufficiently large to relate the observed near tricritical behaviour of the antiferroelectric-paraelectric phase transition to strain coupling.

scholarly journals An ellipticity domain for the distortional Hencky logarithmic strain energy

2015 ◽  
Vol 471 (2184) ◽  
pp. 20150510 ◽  
Author(s):  
Ionel-Dumitrel Ghiba ◽  
Patrizio Neff ◽  
Robert J. Martin
Keyword(s):  
Elastic Energy ◽  
Strain Energy ◽  
Strain Tensor ◽  
Energy Criterion ◽  
Logarithmic Strain ◽  
Hencky Energy ◽  
Von Mises ◽  
Deviatoric Part

We describe ellipticity domains for the isochoric elastic energy F ↦ ∥ dev n log ⁡ U ∥ 2 = ∥ log ⁡ F T F ( det F ) 1 / n ∥ 2 = 1 4 ∥ log ⁡ C ( det C ) 1 / n ∥ 2 for n =2,3, where C = F T F for F ∈ GL + ( n ). Here, dev n log U = log U − ( 1 / n )   tr ( log U ) ⋅ 1 is the deviatoric part of the logarithmic strain tensor log ⁡ U . For n =2, we identify the maximal ellipticity domain, whereas for n =3, we show that the energy is Legendre–Hadamard (LH) elliptic in the set E 3 ( W   H iso , LH , U , 2 3 ) := { U ∈ PSym ( 3 ) | ∥ dev 3 log ⁡ U ∥ 2 ≤ 2 3 } , which is similar to the von Mises–Huber–Hencky maximum distortion strain energy criterion. Our results complement the characterization of ellipticity domains for the quadratic Hencky energy W   H ( F ) = μ ∥ dev 3 log ⁡ U ∥ 2 + ( κ / 2 ) [ tr ( log ⁡ U ) ] 2 , U = F T F with μ >0 and κ > 2 3 μ , previously obtained by Bruhns et al.

scholarly journals 2-D finite displacements and finite strain from PIV analysis of plane-strain tectonic analogue models

2019 ◽  
Author(s):  
David Boutelier ◽  
Christoph Schrank ◽  
Klaus Regenauer-Lieb
Keyword(s):  
Finite Strain ◽  
Deformation Gradient ◽  
Strain Tensor ◽  
Small Strain ◽  
Time Lapse ◽  
Complete Characterization ◽  
Image Correlation ◽  
Strain Rate Tensor ◽  
Analogue Models ◽  
Finite Displacements

Abstract. Image correlation techniques have provided new ways to analyze the distribution in space and time of deformation in analogue models of tectonics. Here we demonstrate how the correlation of successive time-lapse images of a deforming model allows not only to evaluate the components of the strain-rate tensor at any time in the model but also calculate the finite displacements and finite strain tensor. We illustrate, using synthetic images, the ability of the algorithm to produce maps of the velocity gradients, small-strain tensor components, but also incremental or instantaneous principal strains and maximum shear. The incremental displacements can then summed up using a Eulerian or a Lagrangian summation, and the components of the 2-D finite strain tensor can be calculated together with the finite principal strain and maximum finite shear. We benchmark the measures of finite displacements using specific synthetic tests for each summation mode. The deformation gradient tensor is calculated from the deformed state, and decomposed into the finite rigid-body rotation and left or right finite stretch tensors, allowing the deformation ellipsoids to be drawn. The finite strain has long been the only quantified measure of strain in analogue models. The presented software package allows producing these finite strain measures while also accessing incremental measures of strain. The more complete characterization of the deformation of tectonic analogue models will facilitate the comparison with numerical simulations and geological data, and help produce conceptual mechanical models.

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 Rank-one convexity implies polyconvexity for isotropic, objective and isochoric elastic energies in the two-dimensional case

2017 ◽  
Vol 147 (3) ◽  
pp. 571-597 ◽  
Author(s):  
Robert J. Martin ◽  
Ionel-Dumitrel Ghiba ◽  
Patrizio Neff
Keyword(s):  
Energy Function ◽  
Strain Tensor ◽  
Dimensional Case ◽  
Logarithmic Strain ◽  
Two Dimensional ◽  
Rank One ◽  
Isotropic Functions ◽  
Deviatoric Part ◽  
Convexity Conditions ◽  
Quasi Convexity

We show that, in the two-dimensional case, every objective, isotropic and isochoric energy function that is rank-one convex on GL+(2) is already polyconvex on GL+(2). Thus, we answer in the negative Morrey's conjecture in the subclass of isochoric nonlinear energies, since polyconvexity implies quasi-convexity. Our methods are based on different representation formulae for objective and isotropic functions in general, as well as for isochoric functions in particular. We also state criteria for these convexity conditions in terms of the deviatoric part of the logarithmic strain tensor.

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