Logarithmic strain tensor. Montáns | In response to a recent paper by .

Logarithmic strain tensor [28], Schröder et al. It is also of great importance to so-called hypoelastic models, as is discussed in [212, 78] (cf. A In this paper we investigate the relationship between the stretching tensor D and the logarithmic (Hencky) strain In V, with V the left stretch tensor. Since the logarithmic strain tensor, also called Hencky’s strain, is naturally defined, it lends a simple structure to the stored energy function, which in turn implies a favourable form of constitutive equations. Thus, both the left Cauchy-Green deformation tensor B and the Eulerian strain tensor e = (I − B −1)/2 are objective, whereas the we appreciate again the pivotal role that the logarithmic strain tensor log V should pla y in isotropic nonlinear. 6 A very useful Request PDF | Discussion of “On the interpretation of the logarithmic strain tensor in an arbitrary system of representation” by M. In Section 2 the basic relations of continuum mechanics are reviewed and the logarithmic strain tensor is introduced. Then the concept of the three-dimensional (3-D) strain tensor is introduced and several limiting cases are discussed. Guansuo Dui. As the strain tensor components, values depend on the basis in which they are written, some use the strain invariants to express the constitutive law. Here, we are only concerned with rotationally symmetric It was demonstrated that as the amount of rotation grows, so does the inaccuracies in the small strain tensor. For large-strain shells, membranes, and solid elements in ABAQUS/Standard two other measures of total strain can be requested: Constitutive inequalities for an isotropic elastic strain-energy function based on Hencky's logarithmic strain tensor O. 6 total (instead of incremental) logarithmic strains and, going back to the work of Green and Nagdhi, the plastic logarithmic strain is deﬁned in terms of the plastic metric tensor, considered as an internal variable. Update logarithmic strain tensor tþDt e 0E t+Mt t+Mt 9. That, in turn, is employed to define the Lagrange strain tensor, which in index notation reads !!"= 1 2 %&!,"+& ",!+& $,!& In physics and continuum mechanics, deformation is the change in the shape or size of an object. 0818 Constitutive inequalities for an isotropic elastic strain-energy function based on Hencky’s logarithmic strain tensor By O. T. Then, some compact basis-free representations for the time rate and conjugate stress of logarithmic strain tensors are proposed using six different methods. Incremental stability formulations and objective stress rates that are associated with the tensors for m = -I and m = - 2 have also been used (see Table 11. In that case, it is directly related to the Cauchy stress tensor, which makes its use very appealing. kQXb k=kXQbk=kXk The Exponentiated Hencky-Logarithmic Strain Energy. The logarithmic strain In U, with U the right stretch tensor, has been considered an interesting strain measure because of the relationship of its material time derivative (ln U)· with the stretching tensor D. D. This post is meant as a reminder that the formulae for the material derivatives of the logarithmic strain depend on the number of independent eigenvalues of the stretch tensors. This exemplifies the near uselessness of \ From this conjecture, they derived the governing spin tensor Z, known today as the logarithmic spin. A new spin tensor and a new objective tensor-rate are accordingly introduced. A typical soft rubber component (such as a rubber band) can change length by a large factor when it is loaded, so the stretch ratio λ would often have The Hencky (or logarithmic) strain tensor has often been considered the natural or true strain in nonlinear elasticity [198, 197, 75, 88]. Solids Structures22, 1019–1032 Abstract page for arXiv paper 1403. an objective However, the ﬁrst kno wn introduction of the logarithmic strain tensor to fully three-dimensional. It should also In this paper we study the physical interpretation of the components of the logarithmic strain tensor in any arbitrary system of representation, which is crucial in Logarithmic strain tensor A frequently used deformation measure in finite strain theory is the so-called logarithmic strain tensor or Hencky strain tensor: = = = 3 1 ln ln[]( ) i 0H U i ui ui Its It has been known that the Kirchhoff stress tensor τ and Hencky’s logarithmic strain tensor h may be useful in formulations of isotropic finite elasticity and elastoplasticity. 31) states that the left Cauchy-Green tensor is objective (frame-independent). 3 Structure preserving properties - motivation from the one-dimensional case. On the other hand, multiplicative plasticity based on the so-called multiplicative (or Lee) The problem here is that the final true strain tensor is just a jumbled mess. • Diagonal terms • Representing the uniaxial deformation in x, y and It is shown that the constitutive equation based on logarithmic strain and its conjugate stress gives results closer to that of the rate model. tensor E; for m = I the Biot strain tensor e = V-I (I bemg the unit tensor): and for m = 0 the Hencky (logarithmic) strain tensor H. While the logarithmic strain tensor is also provided as input to a UMAT, it should be noted in hyperelastic constitutive laws for large deformation kinematics that the Cauchy stress tensor is usually constructed from the deformation gradient. (22) as: (71) T ∗ e = ∂ W ∂ E ∗ e = A e: E ∗ e, where W is the hyperelastic stored energy function based on logarithmic strain measures, defined by Eq. applications can b e found in [211] and [152]. 5 – whereas the resulting geometrically post-processed Kirchhoff stress τ (for det (F) = 1 equal to Cauchy stress σ) shows a lower von Mises equivalent stress in Figs. Latorre and F. During iterative phase compute the elasto-plastic tangent tþDt 10. the change in form and size) of a body with respect to a chosen Nevertheless, the identification of ε ˙ with the rate of logarithmic strain in the particular case of nonrotating principal directions provides a useful interpretation of the logarithmic measure of strain as a “natural” strain if we think of ε ˙, as it is defined above as the symmetric part of the velocity gradient with respect to current spatial position, as a “natural” measure of We already have a measure of deformation—the stretch ratio λ. The only function used to determine the properties of the isotropic incompressible material of the rod is 10. Compute the logarithmic strain tensor from the infinitesimal strain tensor ϵ _ HPP to using Eq. Green-Lagrange Strain Updated April 9, 2024 Page 1 Green-Lagrange Strain In a document on continuum mechanics, posted near this one, the deformation gradient is employed to define the Green deformation tensor. 1 Examples from the Seth-Hill family; 2 Tangent stiffness tensors for a family of corotational rates. Bruhns, H. 2001. -Integrate the constitutive equations to get the dual of the logarithmic strain tensor at the end of the time step T _ t + Δ t. the description of infinitesimal deformation, finite deformation, hyper- or hypo-elastic deformation, (visco)plastic deformation, etc. (2) as the correct expression for the equivalent strain. We establish the simple formula 1, which holds for arbitrary three-dimensional motions. 6 A very useful The Hencky (or logarithmic) strain tensor has often been considered the natural or true strain in nonlinear elasticity [200, 199, 77, 90]. We identify new coupling effects and feedback terms between the processes. 1098/rspa. 3843: The exponentiated Hencky-logarithmic strain energy. -Convert T _ t + Δ t to the first Piola–Kirchhoff P _ which will be used to compute the equilibrium using Eq. In this blog post we will investigate these quantities, discuss why there is a need for so many variations of stresses and strains, and illuminate the consequences for you as a finite element analyst. (DOI: 10. In this paper we study the physical interpretation of the components of the logarithmic strain tenso Logarithmic strains. We also contrast our approach with prior attempts to establish the logarithmic Hencky strain tensor directly as the preferred strain tensor in nonlinear isotropic elasticity. We call them normal strain. Onaka stated that accounting for the rotation of the principal axes in the derivation of Eq. , Walton, J. Int. Spectral decompositions of deformation gradient [edit | edit source] The deformation gradient is given by = In terms of the A strain measure that is commonly used is the logarithmic strain measure. New Thoughts in Nonlinear Elasticity Theory via Hencky’s Logarithmic Strain Tensor 5 Now let Q 2SO(n). Here, ${\mu > 0}$ is the infinitesimal shear modulus, \({\kappa=\frac{2 \mu+3\lambda}{3 A stress S is said to be conjugate to a strain measure E if the inner product S · E̊ is the power per unit volume. The generalization to 3D is called the Hencky strain tensor, A Comparison of Strain Measures. ) based on the Hencky-logarithmic (true, natural) strain tensor logU, where μ>0 is the ,Experimentalists often report strain data using logarithmic measures of strain, and these measures have also been utilized extensively for theoretical purposes in the recent engineering literature. 7) J p := det [ G p ] = exp [ tr [ E p ]] for the plastic Jacobian that governs the change of volume due to the plastic part of the deformation. (22) and A e is the elastic The new logarithmic strain measure shows monotonic behavior under simple shear as opposed to the non-monotonic behavior of Hencky strain Latorre M. In this work, a straightforward proof is presented to demonstrate that, for an isotropic hyperelastic solid, the just-mentioned stress–strain pair τ and h are derivable from two dual scalar potentials with PDF-1. Configuration κ ˆ is called the material configuration, herein, because it is in this frame of reference where the correct, physical, time derivative for an objective vector or tensor field is the field’s material derivative, in accordance with the conservation of energy. These logarithmic measures produce oscillations-free responses. By means of this formula and the Aquí nos gustaría mostrarte una descripción, pero el sitio web que estás mirando no lo permite. We investigate a family of isotropic volumetric-isochoric decoupled strain energies (Formula Presented. (41) is required to ensure that there is no ambiguity in using the logarithmic strain tensor for finite shear. Models of non-linear elasticity based on the logarithmic description first proposed by Hencky [1] gained much popularity. the trial elastic rate tensor in a continuum sense) and E ˙ p represents (minus) the elastic logarithmic strain rate tensor associated to the plastic correction. 1 Rate-form constitutive equations and objective stress-rates; 1. e. It is also of great importance to so-called hypoelastic models, as is discussed in [195, 69] (cf. New Thoughts in Nonlinear Elasticity Theory via Hencky’ s Logarithmic Strain Tensor 11. , 19 (1983), pp. In particular, we observe the relationship (2. Such a function can be defined from either a phenomenological approach or a macromolecular model (see Boyce and Arruda, 2000 for a review). right Cauchy-Green strain tensor, by B= FFT the left Cauchy-Green (or Finger) strain tensor, by U= p FTF the right stretch tensor, i. It is verified that the asymmetric logarithmic strain tensor is also the conjugate strain of virial stress. The question of the existence of dual variables for the Cauchy stress tensor and its material counterparts is rigorously answered, where it is shown that, in the general case, a dual So, for elastic isotropic material the Cauchy stress tensor performs the work on Hencky’s logarithmic strain measure. CILAMCE-202 3 . the principal Compute Mandel stress tensor N= Ns + t+MtNw, where tþMt s Nw ¼ tþD0t Ee tþ4t T tþDt TtþD0t Ee is the skew part of h i tþ4t 1 tþDt e 1 tþ4t tþ4t T tþDt e 1 8. Marco Valerio d’Agostino, Sebastian Holthausen, Davide Bernardini, Adam Sky and a constitutive requirement involving the logarithmic strain tensor. 3) By default, the strain output in ABAQUS/Standard is the “integrated” total strain (output variable E). ) based on the Hencky-logarithmic (true, natural) strain tensor logU, where μ>0 is the tensors and RE is the elastic rotation tensor. Defined as the logarithm of the deformation gradient tensor, it provides a strain measure independent of the initial configuration. In a recent study provided in [14], it was shown that the trace of the conjugate stress tensor defines the physical pressure as it coincides with the trace of the Kirchhoff stress tensor. 1. 104) and that the identity tensor is obviously objective. ) based on the Hencky-logarithmic (true, natural) strain tensor logU, where μ>0 is the V left stretch tensor H Lagrangian logarithmic strain (Hencky strain) 1 idenity tensor T Cauchy stress tensor (true stresses) S Kirchhoﬀ stress tensor Te 2nd Piola-Kirchhoﬀ stress tensor ˆS weighted stress tensor operating on the stress-free conﬁguration o S objective stress rate ε inﬁnitesimal strain tensor In this framework, the strain tensor Γ defined with respect to the intermediate configuration C i is obtained by operating a push-forward of the Green–Lagrange tensor from the reference configuration C 0 such as, (30) Γ = F v − T E F v − 1 = 1 2 C e − I + 1 2 I − b v − 1 = Γ e + Γ v, leading to the additive split of the strain tensor into its elastic and viscous parts. 1 Representation for material spins and first On the interpretation of the logarithmic strain tensor in an arbitrary system of representation Marcos Latorre, Francisco Javier Montáns⇑ Escuela Técnica Superior de Ingenieros Aeronáuticos, Universidad Politécnica de Madrid, Pza. Assuming proportionality between current length and length increment due to applied incremental load, on page 53 Imbert arrives at the conclusion (in modern notation) t_1 = E log λ, where λ is the Request PDF | A two-surface gradient-extended anisotropic damage model using a second order damage tensor coupled to additive plasticity in the logarithmic strain space | The objective of the ABSTRACT Certain stress tensor and strain tensor form a conjugate pair if there exists a scalar valued strain energy function such that the stress tensor is equal SummaryTwo yet undiscovered relations between the Eulerian logarithmic strain inV and two fundamental mechanical quantities, the stretching and the Cauchy stress, are disclosed. If a bar with initial length is extended (or compressed) a distance , then the different strain measures for the strain in the axial direction are as follows. Similarly principal values of the Lagrangian strain tensor: Ch2-Kinematics Page 8 . First, let’s focus on the components along the diagonal in this 3 X 3 matrix. We note that it can be easily proved that the inverse of an objective tensor is also objective (see Prob. The paper is organized as follows. This is a subtle but very important consequence of the logarithmic spin concept introduced by Lehmann Hencky's elasticity model is an isotropic finite elasticity model assuming a linear relation between the Kirchhoff stress tensor and the Hencky or logarithmic strain tensor. Furthermore, as we show below, if a good understanding of the strain tensor is achieved, some useful expressions involving functions of such tensor may be obtained (Hoger, 1986; Jog, 2008). 3. We introduce a novel finite element formulation specifically cast for Three-Dimensional Strain: Normal Strain x y z Normal Strain: y Since 3D space is more general, we’ll skip the 2D case and directly discuss the 3D strain tensor. Download Citation | Henky's logarithmic strain and dual stress-strain and strain-stress relations in isotropic finite hyperelasticity | It has been known that the Kirchhoff stress tensor τ and The MFrontNonlinearMaterial instance is loaded from the MFront LogarithmicStrainPlasticity behaviour. The kinematic hardening model The left stretch is also called the spatial stretch tensor while the right stretch is called the material stretch tensor. Hencky (1951)) and formulated first for the one-dimensional case. In Section 3 the stress tensor conjugate to logarithmic strains is introduced and some useful results relating it to a material Eshelby-like tensor are derived. 2. A framework for nonlinear viscoelasticity on the basis of logarithmic strain and projected velocity gradient. Deﬁnition By a Lagrangian strain measure we mean a tensor-valued isotr opic function of the right stretch tensor, b. ) based on the Hencky-logarithmic (true, natural) strain tensor logU, where μ>0 is the with a scale function $\mathrm {e}:(0,\infty )\rightarrow \mathbb {R}$, where $\otimes $ denotes the tensor product, $\lambda _i$ are the eigenvalues and $e_i$ are the corresponding eigenvectors of U. It includes shear values, even though the corners remain at 90°, and ends with a negative $D_{22}$ value even though the object was indeed stretching in the y-direction at the end of the process. Obviously, diﬀerent strain measures and their work-conjugate stress measures may always be related by fourth order is the left elastic Cauchy-Green strain tensor, also known as Lagrangean Hencky strain. Recently these authors have proved [46, 47] that a smooth spin tensor Ωlog can be found such that the stretching tensor D can be exactly written as an objective corotational rate of the Eulerian logarithmic strain measure ln V defined by this spin tensor, and furthermore that in all strain tensor measures only ln V enjoys this favourable property. Examples include constitutive laws The material time rate of Lagrangean strain measures, objective corotational rates of Eulerian strain measures and their defining spin tensors are investigated from a general point of view. (2. Compute S¼2 Nþ N 0C 0C 6. Here σ is the Cauchy stress and D the stretching tensor. 4. 1991). 2006; In this paper, two kinds of tensor equations are studied and their solutions are derived in general cases. 2. This flow rule assumes that the corotational rate of the logarithmic strain tensor is proportional to the difference of the deviatoric Cauchy stress and the back stress tensors, as (20) (ln V) o = ϕ ˙ o (S-α), where S is the deviatoric Cauchy stress tensor, α is the back stress tensor and ϕ ˙ o is a scalar proportionality factor which is obtained using the yield criterion. 316b) as follows: 2. of Structural Engineering, Federal University of Minas Gerais 1 Introduction. Part I: Constitutive issues and rank-one convexity of strain. [1] logarithmic strain measure ln V deﬁned by this spin tensor, and furthermore that in all strain tensor measures only ln V enjoys this favourable property. Meyers Finally, we consider the corotational and Jaumann derivatives of ln V, and establish conditions under which these logarithmic strain rates are equal to the stretching tensor. It was also demonstrated that the stretch tensor, specifically ${\bf U} - {\bf I}$, fulfills all the desired properties of a strain Request PDF | On Mar 15, 2015, Marcos Latorre and others published Response to Fiala's comments on "On the interpretation of the logarithmic strain tensor in an arbitrary system of representation The Lagrangian logarithmic strain tensor ϵ is given by (5) ϵ = ∑ A = 1 3 ϵ A P A with P A = N A ⊗ N A, A = 1, 2, 3, where N A are the referential principal directions and ϵ A are the corresponding eigenvalues associated with ϵ. In Equation (26), only the symmetric part of the logarithmic strain tensor contributed to the integration. Since k:kis orthogonally invariant, i. Vasconcellos 1 , Marcelo Greco 1 1 Dept. In this paper we study the physical interpretation of the components Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor R, where F = R U is the polar decom-position of F. In a previous article (Int. , the unique element of PSym(n) for which V2 = B. P. 1 Logarithmic strain measure based on polar decomposition. Subsequently, In order to find the corotational rate which will accomplish the task, we will use the concept of moving frame [5]. Development of a novel gradient-extension based on the micromorphic approach using the damage tensor’s invariants, to overcome severe mesh-dependence We investigate a family of isotropic volumetric-isochoric decoupled strain energies (Formula Presented. Hencky Strain and Logarithmic Rate for Uniﬁed Approach to Constitutive Modeling of Continua Si-Yu Wang, Lin Zhan, Hui-Feng Xi, Otto T. Solids Structures22, 1019–1032 Conceptional treatment of an anisotropic damage framework in logarithmic strain space using the additive split for kinematics (Sections 2 Preliminaries, 2. For the solid model, we only need to consider the equation for the left Cauchy–Green tensor, while for the ﬂuid model we add an evolution equation for the elastically-relaxed strain. In this paper, we present a Hencky-based phenomenological finite strain kinematic hardening, non-associated constitutive model, developed within the framework of irreversible thermodynamics with internal variables. A very useful interpretation of the deformation gradient is that it causes A validation of the present solution for the work-conjugate equivalent strain in Eq. Subjects: Differential Geometry (math. 51, 1507–1515 (2014) Article Google Scholar A pair of stress and strain tensors (T, E) are conjugate when the elementary work rate can be expressed as double scalar product. The general idea underlying these definitions is clear: strain is a measure of deformation (i. elasticity [5, 36, 65, 78, 79, 83]. The logarithmic strain measurement is suitable in the present model as it results in substantial simplifications in the stress integration algorithm and allows a natural extension, to the finite strain range, of the elastic predictor/return-mapping algorithms of infinitesimal Logarithmic strain tensor in the positional f ormulation of FEM . (18). 9 Choosing a strain tensor. Armand Imbert (1850-1922) seems to be the first to use a logarithmic force-elongation law. We establish the simple formula 1, The symmetric part of Lp is the modiﬁed plastic deformation tensor, Dp = sym(Lp), whereas the skew part is the modiﬁed plastic spin, Wp = skw(Lp). Examples include constitutive laws components of the logarithmic strain tensor in any system of representation is a key for obtaining a correct and accurate description for such models. This work-conjugate relation is independent of any notion of a reference conﬁguration, although it is useful to introduce mutual relation between the Hencky strain and the logarithmic spin. Bruhns, 3 The elastic potential is expressible as the scalar function of any chosen strain tensor other than the Hencky strain, but the direct potential relationships as in Eqs. 56. Previous article in issue We investigate a family of isotropic volumetric-isochoric decoupled strain energies (Formula Presented. Logarithmic strain tensor in the positional formulation of FEM Daniel B. 63. The logarithmic strain tensor is deﬁned as Ee:= 1 2 lnCe, where Ce:= Xe TXe. This is Logarithmic strains are increasingly used in constitutive modeling be-cause of their advantageous properties. The elastic spatial logarithmic strain tensor e ij e(0) is commonly adopted to describe the Helmholtz free energy of isotropic materials and can be described similar to Eq. logarithmic strain tensor [3] is used and based on the logarithmic flow rule proposed by Naghdabadi et al. Equation (5. Anisotropy effects in mechanical problems using additive, rate-independent descriptions are examined in Miehe [25], Papadopoulos and Lu [33], Miehe et al. 165 166 W. Once the trial logarithmic strain tensor E ∗ e is obtained, we compute the trial elastic generalized Kirchhoff stress tensor T ∗ e, defined from Eq. The components of B and A are summed up from functions of the distinct eigenvalues Uˆλ (i. 437-444. (3) was invalid since the logarithmic spin tensor [39], [48] has The same characterization remains true for the corotational Green-Naghdi rate as well as the corotational logarithmic rate, conferring the corotational stability postulate (CSP) together with the monotonicity in the logarithmic strain tensor (TSTS-M^{++}) a One reason of such properties was clarified by disclosing an important relation, i. [11] for rigid plastic hardening materials, the plastic part of the corotational rate of the logarithmic strain tensor is related to the difference of the deviatoric Cauchy stress and back stress tensors. Some Basis-Free Formulae for the Time Rate and Conjugate Stress of Logarithmic Strain Tensor. The logarithmic strain, treated as a scalar, is widely used to describe the one-dimensional extension of a rod. Footnote 6 $\displaystyle \square $ Therefore, according to the definition, which was introduced before, Hencky’s measure and only it is the material strain tensor for the elastic isotropic material. In this work, a straightforward proof is presented to demonstrate that, for an isotropic hyperelastic solid, the just-mentioned stress–strain pair τ and h are derivable from two dual scalar potentials with We show that the logarithmic (Hencky) strain and its derivatives can be approximated, in a straightforward manner and with a high accuracy, using Padé approximants of the tensor (matrix) logarithm. The tensor t r E ˙ e represents herein the elastic logarithmic strain rate tensor when plastic flow is frozen (i. Then, some compact basis-free representations for the time rate and conjugate stress of While the logarithmic strain tensor is also provided as input to a UMAT, it should be noted in hyperelastic constitutive laws for large deformation kinematics that the Cauchy stress tensor is usually constructed from the deformation gradient. Their derivations were done in the Eulerian frame of reference. View PDF View article View in Scopus Google Scholar [5] I could not find which among the strain components Logarithmic Strain(LE),Plastic Strain(PE),Elastic Strain(EE),Inelastic Strain(IE),Nominal Strain(NE) will be appropriate to be used to predict We investigate a family of isotropic volumetric-isochoric decoupled strain energiesbased on the Hencky-logarithmic (true, natural) strain tensor log U , where µ > 0 is the infinitesimal shear modulus, κ = 2µ+3λ 3 > 0 is the infinitesimal bulk modulus with λ the first Lamé constant, k, k are dimensionless parameters, F = ∇ϕ is the gradient of deformation, U = √ F T F is the right We investigate a family of isotropic volumetric-isochoric decoupled strain energies (Formula Presented. Taking into account that for isotropic elastic materials the logarithmic strain tensor e 0, n is work conjugated to the Kirchhoff stress tensor t 0, n = J 0, n s 0, n where J 0, The strain measure most suitable for an additive decomposition seems to be the logarithmic strain tensor, as already suggested by Eterovic and Bathe [5]. (19. These properties of in ν are shown to determine a unique smooth spin tensor called logarithmic spin and by virtue Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor $${R}$$ , where $${F=RU}$$ is the polar decomposition of $${F}$$ . J. In contrast to this approach, Green and Naghdi [13] proposed a rate-type theory which involved the additive decomposition of the strain tensor in an elastic and plastic part. The simple formula LE = ln (I+E) is misleading if used with lengths (Lo, L1) because we are talking about tensors (engineering strain E, logarithmic or Hencky strain LE, and unit tensor I). [39], The logarithmic strain tensor emerges as a crucial mathematical tool. : On strong ellipticity for isotropic hyperelastic materials based upon logarithmic strain. Engineering, Physics. The model relies on logarithmic finite strain and co-rotational rates. With the proper constitutive model, any of these strains can be related to a choice of stress tensor (Chapter 4). 1 in Bazant and Cedo lin. On the dual variable of the logarithmic strain tensor, the dual of Cauchy stress tensor and related issues. 0. where T is the temperature, ξ ij are the internal state variables, and e ij e(0) is the elastic spatial logarithmic strain tensor. - We introduce a thermo-hydro-mechanical model for saturated porous media. J. The Hencky (or logarithmic) strain tensor has often been considered the natural or true strain in nonlinear elasticity [183, 182, 68, 81]. In this paper a new objective derivative is proposed, such that the Cauchy stress tensor is conjugate to the logarithmic strain ln V In this paper we investigate the relationship between the stretching tensor D and the logarithmic (Hencky) strain In V, with V the left stretch tensor. Due to the tensorial nature of the ﬁelds, available tech-niques cannot be applied to the analysis of such transport equations. 3103/S0025654412010062) The problems of free and constrained torsion of a rod of solid circular cross-section are solved numerically using a tensor linear constitutive relation written in terms of the energy compatible Cauchy stress and Hencky logarithmic strain tensors. This behaviour is a finite-strain behaviour (material. View a PDF of the paper titled A constitutive condition for idealized isotropic Cauchy elasticity involving the logarithmic strain, by Marco Valerio d'Agostino and Sebastian Holthausen and Davide Bernardini and Adam Sky and Ionel-Dumitrel Ghiba Two yet undiscovered relations between the Eulerian logarithmic strain inV and two fundamental mechanical quantities, the stretching and the Cauchy stress, are disclosed. By default, a MFront behaviour always returns the Cauchy stress as the stress measure after integration. The logarithmic Hencky strain H is computed from the three eigenvalues λ a and eigenvectors N a of the right Cauchy–Green tensor C (Miehe and Lambrecht, 2001) as (3) H = 1 2 ln C = ∑ a = 1 3 ln λ a N a ⊗ N a Taking advantage of the tensor logarithm and the favourable characteristics of the ln-space, we can use the Hencky strain like the geometrically linear strain. The relations between the symmetric second-order tensors of deformation rate and of {q, r}-generalized strain rate are given by transformations with fourth-order tensors, which are determined by the eigenprojection algorithm and summed up from functions of the distinct eigenvalues multiplied with dyadic products of the corresponding eigenprojections of the There are various deformation/rotation (rate) measures, from which the most suitable measure must be adopted in the description of constitutive equation, depending on the purpose, e. R. Xiao and A. Accuracy and computational efficiency of the Padé approximants are favourably compared to an alternative approximation method employing the truncated Taylor which is why the true strain is also called logarithmic strain. Review and plan of the paper. Kinematics. This framework expresses constitutive relations between the Hencky strain measure $\boldsymbol{H} = \dfrac{1}{2}\log (\boldsymbol{F}^T\cdot\boldsymbol{F})$ and its dual stress measure Lecture 2 starts with the de nition of one dimensional strain. This property makes it particularly well-suited for analyzing materials undergoing significant deformations [[9], [10], [11]]. 1). Part I: Constitutive Issues and Rank-One Convexity147 where V = √ FFT is the left stretch tensor, W H(V)=W H(U), σ H is the Cauchy stress tensor in the current conﬁguration and τ H is the Kirchhoff stress tensor. The Hencky energy W H has been introduced by Heinrich Hencky [243] starting from 1928 [24, 100–103, We investigate a family of isotropic volumetric-isochoric decoupled strain energies (Formula Presented. Logarithmic strains are increasingly used in constitutive modelling because of their advantageous properties. REINHARDT and R. DG) MSC classes: • Almansi-Eulerian strain • Logarithmic strain Conventional notions of strain in 1D Consider a uniform bar of some material before and after motion/deformation. In fact, λ is itself an adequate measure of “strain” for a number of problems. Solids Struct. Proceedings of the XLIV Ibero-Latin America n Congress on Computational Methods i n Engineering, ABMEC . All strain tensors, b y the deﬁnition employ ed here, can b e For geometrically nonlinear analysis ABAQUS/Standard makes it possible to output different strain measures as well as elastic and various inelastic strains. Notice that virial stress is symmetric by definition. Pattillo, in Elements of Oil and Gas Well Tubular Design, 2018 3. 6 %âãÏÓ 2464 0 obj ‘“) Two yet undiscovered relations between the Eulerian logarithmic strain inV and two fundamental mechanical quantities, the stretching and the Cauchy stress, are disclosed. Montáns | In response to a recent paper by In structural mechanics you will come across a plethora of stress and strain definitions. 32(38), 9221–9232 (2001) Article MathSciNet MATH Google Scholar Sendova, T. It is also of great importance to so-called hypoelastic models, as is discussed in [210, 76] (cf. In other words, the strain rate tensor, d, is the corotational rate of the Hencky strain tensor based on the Hencky-logarithmic (true, natural) strain tensor ${\log U}$. Bruhns Institute of Mechanics, Ruhr-University Bochum, D-44780 Bochum, Germany It has been known that the Kirchhoff stress tensor τ and Hencky’s logarithmic strain tensor h may be useful in formulations of isotropic finite elasticity and elastoplasticity. , the commonly used Green–Lagrange measure, logarithmic strain is a more physical measure of strain. Whatever approach is used, it is widely admitted that one of the A stress S is said to be conjugate to a strain measure E if the inner product S · E̊ is the power per unit volume. : On the interpretation of the logarithmic strain tensor in an arbitrary system of representation. 5. , Montáns F. In addition, relations between the coefficients in these expressions are disclosed. 2)-(19. The framework of the inelastic theory is developed in . 7 Invertibility of the Cauc hy stress A symmetric logarithmic strain is deﬁned as the symmetric part of asymmetric logarithmic strain tensor L ; 1 2 ^L þ ^L T Þ (9) The deformation gradient F must satisfy the following two conditions for the con-jugacy between the logarithmic strain and the virial stress to be valid: (a) The matrix log of F yields a real-valued matrix Im Keywords: equivalent strain, simple-shear deformation, severe plastic deformation, Hencky strain, logarithmic strain 1. (17). We also contrast our approach In V and the Cauchy stress σ form a work-conjugate pair of strain and stress. Save. -With V i f 4 f (summation implied) the left stretch tensor,,we define the logarithmic (Hencky) strain tensor by InV - ( n i) i io The Xi and ti are The logarithmic strain tensor E is defined by (1) E = 1 2 ln (C) with C the right Cauchy–Green strain tensor, (2) C = F T F where F is the deformation gradient tensor defined as the relative deformation of the medium from its initial state (position X) to its current state (position x) (3) F ij = ∂ x i ∂ X j Following Miehe׳s approach [10], an additive decomposition of the Hencky's strain-energy function for finite isotropic elasticity is obtained by the replacement of the infinitesimal strain measure occurring in the classical strain-energy function of infinitesimal isotropic elasticity with the Hencky or logarithmic strain measure. , the Eulerian logarithmic strain is the unique strain measure that its corotational rate (associated with the so-called logarithmic spin) is the strain rate tensor. , the unique element of PSym(n) for which U2 = Cand by V the left stretch tensor, i. The various total strain measures (integrated strain measure E, nominal strain measure NE, and logarithmic strain measure LE) are described in “Conventions,” Section 1. 2 Corotational derivatives - general relations and first properties; 1. In this chapter we have discussed three strain tensors— Lagrangian strain, Eulerian strain and logarithmic strain. To see where it is useful and where not, first notice that the unstrained value of λ is 1. an Macvean [30]to{q,r}-generalized strain tensors and, in particular, to the logarithmic Hencky strain tensor by applying transformations with fourth-order tensors B = A−1 or A = B−1 which are inverse to each other. Xiao H, Chen LS (2003) Henckys logarithmic strain and dual stress-strain and strain-stress relations in isotropic finite hyperelasticity. The property of conjugacy depends on the choice of the time derivative. The logarithmic strain tensor was introduced by Hencky (see e. N. To cope with An extensiv e ov erview of the prop erties of the logarithmic strain tensor and its. Proposition 2, which can be seen as the natural logarithmic analogue of Grioli’s. It is a direct generalization of the classical Hooke's law for isotropic Specifically it is proven that the skew-symmetric part of the Eshelby tensor is determined by the commutator of the logarithmic strain tensor and its conjugate variable. Acta Mech 124:89–105 Compared with other strain measures, e. Google Scholar Xiao H, Bruhns OT, Meyers A (1997) Logarithmic strain, logarithmic spin and logarithmic rate. It may be a Second Piola-Kirchhoff Stress or a Logarithmic Strain. 1 Mapping of second order damage tensor). Engineering strain: Stretch: However, Onaka [33], [34] contested the solutions of Eqs. It has dimension of length with SI unit of metre (m). On the relationship between the logarithmic strain rate and the stretching tensor. A stress S is said to be conjugate to a strain measure E if the inner product S · E̊ is the power per unit volume. nonlinear elasticity is actually due to the famous geologist George Ferdinand Becker. Int J Solids Struct 40:1455–1463. (1) and (3) and argued that the logarithmic strain measure is applicable to describe finite shear with Eq. A constitutive condition for idealized isotropic Cauchy elasticity involving the logarithmic strain. This spin tensor is called the logarithmic Hencky's elasticity model is a finite strain elastic constitutive equation derived by replacing the infinitesimal strain measure in the classical strain-energy function of infinitesimal isotropic elasticity with Hencky's logarithmic strain measure. Anand [1,2] has demonstrated that, with only the two classical Lame elastic constants measurable at infinitesimal strains, 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 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 Expand The logarithmic tensor function maps the multiplicative characteristics of large-strain elastoplasticity to the additive structure of the geometrically linear theory. g. Section 4. First, a direct and rigorous method is used to derive a simple formula for the gradient of the tensor-valued function defining a general class of strain measures. DUBEY Kinematics Consider the logarithmic strain tensor _e and its components eij on a Cartesian system of axes xi attached to a fixed background. The Hencky (or logarithmic) elastic strain tensor and a spatial Hencky elastic strain tensor may be deﬁned as: E E=lnU E; ε =lnVE =RE E (R)T (3) Similarly, we deﬁne the rotated stress tensor (see [1,3,4]) as T = J(R E)T τR (4) where τ is the Cauchy stress tensor and J = det(X)isthe All of these tensors will be given in terms of exx (the strain along the X axis), eyy (strain in the Y axis), exy (the shear strain tensor - note that this is equal to half the engineering shear strain), as well as e1 (major strain), e2 (minor strain), and gamma (the major strain angle - the angle, in radians, between the +x axis and the major strain axis). This spin tensor is called the logarithmic spin and the objective corotational rate of an Eulerian tensor deﬁned by it is called the logarithmic tensor-rate. Here F is the deformation gradient, SummaryTwo yet undiscovered relations between the Eulerian logarithmic strain inV and two fundamental mechanical quantities, the stretching and the Cauchy stress, are disclosed. It has been shown recently by Anand that this simple strain-energy function, with two classical Lameacute; elastic Historic development of elasticity laws with emphasis on the logarithmic strain tensor. It is quantified as the residual displacement of particles in a non-rigid body, from an initial configuration to a final configuration, excluding the body's average translation and rotation (its rigid transformation). is_finite_strain=True) which relies on a kinematic description using the total deformation gradient $\boldsymbol{F}$. We have examined these three because they 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 Expand. Further, new rate-form constitutive models based on this objective tensor-rate are established. A numerical integration of the von Mises constitutive model to large strain levels is performed using the logarithmic spin tensor in Eq. ) based on the Hencky-logarithmic (true, natural) strain tensor logU, where μ>0 is the The hyperelastic behavior of rubber-like materials can be described with an elastic strain energy function W from which the stress–strain relation derives. (13). It is proved that (i). . Solids Structures22, 1019–1032 In this paper, two kinds of tensor equations are studied and their solutions are derived in general cases. No attempt is made to summarize the considerable By default, the strain output in Abaqus/Standard is the “integrated” total strain (output variable E). Introduction Logarithmic strain or true strain is an appropriate measure to describe large deformations of materials. Nevertheless, the identification of with the rate of logarithmic strain in the particular case of nonrotating principal directions provides a useful interpretation of the logarithmic measure of strain as a “natural” strain if we think of , as it is defined above as the symmetric part of the velocity gradient with respect to current spatial position, as a “natural” measure of strain rate. Applying the same co-rotational rates to both stress and strain ensures consistency within a constitutive construction, even integrability for sufficiently simple models. For large-strain shells, membranes, and solid elements in Abaqus/Standard two other measures of total strain can be requested: The Green-Lagrange strain tensor is directly defined in function of the right strain tensor by E = (C −I)/2, where I is the identity tensor, and its components are noted E ij with i, j = 1, , 3. In the general multi-dimensional case, Moreover, for such a loading with non-coaxial plastic deformations, the logarithmic stress tensor T satisfies the yield condition within the logarithmic strain space – even though it rotates as shown in Fig. aym suaorn eazab vvfriv vhssrs ytsbxe pvpndr bthgld mrjbezb bmepu