Logarithmic strain tensor. logarithmic strain rate, Hencky, corotational.

Logarithmic strain tensor For infinitesimal deformations of a continuum body, in which the displacement gradient tensor (2nd order tensor) is small compared to unity, i. In other words, the strain rate tensor, d, is the corotational rate of the Hencky strain tensor On the interpretation of the logarithmic strain tensor in an arbitrary system of representation Marcos Latorre, Francisco Javier Mont´ans Escuela T ecnica Superior de Ingenieros Aeron auticos, Universidad Polit ecnica de Madrid Pza. NORRIS Eulerian is made explicit by the polar decomposition F=RU=VR; quantities associated with or defined by U and This strain measure is convenient computationally for problems involving large motions but only small strains, because, as we will show later, its generalization to a strain tensor in any three-dimensional motion can be computed directly from the deformation gradient without requiring solution for the principal stretch ratios and their directions. (2) as the correct expression for the equivalent strain. Similarly principal values of the Lagrangian strain tensor: Ch2-Kinematics Page 8 . , Montáns F. Section 4. Acta Mech 124:89–105 Furthermore, logarithmic strain measures - also known as Hencky strain - are considered for the kinematics, while the decomposition of the total deformation into elastic and plastic parts is based We investigate a family of isotropic volumetric-isochoric decoupled strain energies (Formula Presented. We establish the simple formula Recall that the deformation gradient tensor, Logarithmic strain, logarithmic spin and logarithmic rate. 243. N. The resulting loga-rithmic strain space formulation has proven as most accurate, stable, and efficient by its implementation into Therefore, by transforming tensors from the Lagrangean to the logarithmic Lagrangean strain space it is possible to map the large-strain scope in the Lagrangean space to the small-strain scope in the logarithmic Lagrangean space. Subsequently, logarithmic strain tensor [3] is used and based on the logarithmic flow rule proposed by Naghdabadi et al. REINHARDT and R. ‖ ‖, it is possible to perform a geometric linearization of any one of the finite strain tensors used in finite strain theory, e. 这里的 dl 和 dL 分别代表现时和初始构形线元的长度，是一个标量； dx 和 dX 代表现时和初始线元，是一个矢量， F 是变形梯度， E 是Green应变张量（Green-Lagrange Strain Tensor），也称为物质应变张量（Material Strain Tensor）。 Recently these authors have proved [46, 47] that a smooth spin tensor Ω log 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. Save. Generally it can be computed from the three eigenvalues λ a and eigenvectors N a of C ( Miehe and Lambrecht, 2001 ). Guansuo Dui. logUis the referential (Lagrangian) logarithmic strain tensor, k. However, this is only being used to transform the deformation gradient from the LS-Dyna For q = r = 0, the tensors of logarithmic strain and stress emerge from the {q,r}-generalized strains and stresses together with the corresponding fourth-order logarithmic transformation tensors. : On the interpretation of the logarithmic strain tensor in an arbitrary system of representation. O. Eulerian description) is defined as Logarithmic strain, Natural strain, True strain, or Hencky strain () = Strain tensors and strain measures in nonlinear elasticity Patrizio Neff, Bernhard Eidel and Robert J. 302] and Murphy [142]. A strain measure of significant interest can be found in the logarithmic strain measure, known as the total Hencky strain H (also called the logarithmic strain) in the reference configuration defined through (2) H = 1 2 log C, where C is the right Cauchy–Green deformation tensor defined by (3) C = F T F. and Bruhns, Strongly related to the logarithmic strains are issues concerning the conjugate strain of the Cauchy stress tensor itself. Ψ = Ψ eq A + Ψ neq A e where A and A e are the Green–Lagrange strain tensors obtained from the total deformation gradient X and the internal elastic gradient X (3) was invalid since the logarithmic spin tensor [39], [48] has been derived where the rate of deformation tensor, D, is the objective co-rotational rate of the logarithmic strain. . An additive decomposition of logarithmic strain has been used . Becker, who was 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. 적용 시기: 고무, 플라스틱, 생체 조직과 같은 Geometry of logarithmic strain measures in solid mechanics Patrizio Ne 1, Bernhard Eidel 2 and Robert J. An extensive overview of the properties of the logarithmic strain tensor and its applications can be found in [209] and [154]. The plots of The correct interpretation of the components of the logarithmic strain tensor in any system of representation is a key for obtaining a correct and accuratedescription for such models. On the dual variable of the Cauchy stress tensor in isotropic finite hyperelasticity. 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. Representations of Strain Rate and Spin Tensors in Lagrangian and Eulerian Triads Decomposition of Deformation Gradient Tensor into Isochoric and Volumetric Parts Introduction to Finite Strain Theory for Continuum Elasto-Plasticity Some Basis-Free Formulae for the Time Rate and Conjugate Stress of Logarithmic Strain Tensor. Volumetric Strain The volumetric strain is the change of volume relative to the undeformed volume. k is the Frobenius tensor norm, and devn X= X− 1 n tr(X) ·11 is the deviatoric part of a second order tensor X∈ Rn×n (see Section 2 for other notations). 1007/BF01213020 [2] Xiao, H. C. For the further analysis, five scale functions tensors and RE is the elastic rotation tensor. Examples of MD simulation. 63. CILAMCE-202 3 . • “True” strain ( ): • Logarithmic strain ( ): • Lagrange strain ( ): First of all, there is nothing innately better about any of these strain measures — they all legitimate quantify the deformation. Martin3 In memory of Giuseppe Grioli (*10. In fact, the sum is the trace of the true strain tensor. 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 The new logarithmic strain measure shows monotonic behavior under simple shear as opposed to the non-monotonic behavior of Hencky strain Latorre M. based on the Hencky-logarithmic (true, natural) strain tensor logU, where μ>0 is the infinitesimal shear modulus, \(\kappa=\frac{2\mu+3\lambda}{3}>0\) is the infinitesimal bulk modulus with λ the first Lamé constant, \(k,\widehat{k}\) are additional dimensionless material parameters, F=∇φ is the gradient of deformation, \(U=\sqrt{F^{T} F}\) is the right stretch tensor In this article we derive explicit formulas for the time rate of change of the logarithmic strains ln U and ln V, where U and V are the right and left stretch tensors, respectively. Int J Solids Struct 40:1455–1463. Subjects: Differential Geometry (math. Further, new rate-form constitutive models based on this objective tensor-rate are established. Structural Mechanics 2. Furthermore, as we show below, if a good understanding of the strain tensor is achieved, some useful expressions involving functions of such ten- The general tensor representation of the logarithmic strain is often called Hencky strain. This work presents a hyper-viscoelastic model based on the Hencky-logarithmic strain tensor to model the response of a Tire Derived Material (TDM) undergoing moderately large deformations. The Eulerian finite strain tensor, or Eulerian-Almansi finite strain tensor, referenced to the deformed configuration (i. , Cauchy strain tensor) for characterizing the behavior of real materials at large deformations. Both quantities are non-symmetric tensors, aranged as a 9-dimensional vector • Almansi-Eulerian strain • Logarithmic strain Conventional notions of strain in 1D Consider a uniform bar of some material before and after motion/deformation. J. This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor \(\varepsilon ={{\mathrm{sym}}}\nabla u\), which 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. Whatever approach is used, it is widely admitted that one of the In this paper a flow rule for rigid plastic hardening materials based on von Mises yield criterion is introduced. Solids Structures22, 1019–1032 The logarithmic Hencky strain H is based on the right stretch tensor U or the related right Cauchy–Green tensor C in Eq. 2001; 25. 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. Remark on the objectivity of logarithmic strain and Hencky strain. 1). Then, some compact basis-free representations for the time rate and conjugate stress of 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. e. The evolution equation for the kinematic The expressions obtained by proving this theorem for convector tensors show that each tensor from the Doyle–Ericksen family is associated with a single strain-consistent convective tensor rate 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. p. Brünig [14] proved that the logarithmic strain tensor can be used for nonlinear finite elements analyses considering large deformations and elastoplasticity in 2d solids. 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. We establish the simple formula 1, which holds for arbitrary three-dimensional motions. 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. A sim-ple Cartesian description for Heneky strain-rate in the Lagrangian state is obtained. g. In this chapter we have discussed three strain tensors— Lagrangian strain, Eulerian strain and logarithmic strain. The logarithmic or Hencky strain measure is a favored measure of strain due to its remarkable properties in large deformation problems. Here σ is the Cauchy stress and D the stretching tensor. With the proper constitutive model, any of these strains can be related to a choice of stress tensor (Chapter 4). Variants of the considered model, using the Cauchy stress tensor rather than the Kirchhoff stress tensor, have been developed in literature as well We introduce a new family of strain tensors—a family of symmetrically physical (SP) strain tensors—which is also a subfamily of the well-known Hill family of strain tensors. 51, 1507–1515 (2014) Article Google Scholar Additional motivations of the logarithmic strain tensor were also given by Vall ee [205, 206], Roug ee [182, p. The spin tensor log is Logarithmic strain tensor in the positional f ormulation of FEM . Examples include constitutive laws The formulation uses logarithmic strain measures in order to be teamed with spline-based hyperelasticity. It is a direct generalization of the classical Hooke's law for isotropic This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor ε = sym ∇u, which is the symmetric part of the 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 The logarithmic strain framework discussed in the previous paragraph consists merely as a pre-processing and a post-processing stages of the behavior integration. 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 The present article describes a solution to take into account the finit strains based on the Lagrangian logarithmic strain framework introduced by Miehe et al. A very useful interpretation of the deformation gradient is that it causes In this paper, two kinds of tensor equations are studied and their solutions are derived in general cases. T. This work-conjugate relation is independent of any notion of a reference configuration, although it logarithmic strain rate, Hencky, corotational. This strain measure is appropriate for the small-strain, large-rotation approximation used in these elements. The latter generates and is strongly connected to the rotated 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. Ref. We investigate a family of isotropic volumetric-isochoric decoupled strain energies (Formula Presented. 1 Introduction Logarithmic strain is the preferred measure of strain used by materials scientists, who typically refer to it as the 'true strain'. Compared with other strain measures, e. pdf | Find, read and cite all the research you need The consistent tangent moduli are defined as the sensitivity of the stress tensor to the conjugate logarithmic strain tensor (43) E algo = d ε σ = E e-2 μ ∂ ε ε p, where E e: = ∂ ε σ = κ 1 ⊗ 1 + 2 μ P, P: = I-(1 / 3) (1 ⊗ 1) and I denotes the symmetric fourth-order identity tensor. Since k:kis orthogonally invariant, i. Int. -This logarithmic strain measure has been shown to have certain advantages in the formulation of the deformation theory of 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 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 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 $\begingroup$ You seem to imply that the true (logarithmic) strain is fundamentally better than other large strain measures (e. Here, μ > 0 is the infinitesimal shear modulus, κ = (2 μ + 3 λ) / 3 > 0 is the infinitesimal bulk modulus with λ the first Lamé constant, k, k ^ are additional dimensionless parameters, F = ∇ φ is the gradient of deformation, U = F T F is the right stretch tensor and dev 2 log U = log U − 1 2 tr (log U) · 1 is the deviatoric part of the strain tensor log U. Furthermore, the use of a quadratic hyperelastic energy function of the logarithmic strains and an exponential integration allows for simple, yet accurate stress integration algorithms in large Xiao H, Chen LS (2003) Henckys logarithmic strain and dual stress-strain and strain-stress relations in isotropic finite hyperelasticity. Part I: Constitutive issues and rank-one convexity For simplicity, the effects of the plasticity were not considered. All strain tensors, by the de nition employed here, can be seen as equivalent: since The Green-Lagrange strain tensor defines the strain in the undeformed configuration as: 图 5. Solids Struct. In this appendix, a MATLAB® Code is presented to show that Hencky strain, logarithmic strain and asymmetric logarithmic strain are objective tensor quantities with respect to Galilean transformation [Citation 31] and they are not equal to each other in general. [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. This framework, whose very appealing features compared to other finite strain framework are described in Section 2, is available in general purpose finite element solvers used by engineers working in the In response to a recent paper by Latorre and Montáns (2014), which employs fictitious paths between the second-order identity tensor and the total logarithmic strain tensor to analyse its The logarithmic strain tensor h = ln V, referred sometimes to as the left Hencky strain tensor, has the unique property among spatial strain tensors that it allows uncoupled additive separation of the volumetric and isochoric parts of the deformation (Flory, 1961; Criscione, Humphrey, Douglas, & Hunter, 2000; Xiao, Bruhns, & Meyers, 2004), according to 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. The logarithmic strain, treated as a scalar, is widely used to describe the one-dimensional extension of a rod. Nominal strain (Biot’s strain) its generalization to a strain tensor in any three-dimensional motion can be computed directly from the deformation gradient without requiring solution for the principal stretch ratios and their directions. In the Logarithmic strain, logarithmic spin and logarithmic rate H. Google Scholar Xiao H, Bruhns OT, Meyers A (1997) Logarithmic strain, logarithmic spin and logarithmic rate. , [7,20,36,37,60]) which are responsible for the In Equation (26), only the symmetric part of the logarithmic strain tensor contributed to the integration. the Lagrangian finite strain tensor, and the Eulerian finite strain tensor. 3843: The exponentiated Hencky-logarithmic strain energy. is the right Cauchy-Green deformation tensor, the Almansi strain tensor [2] E 1(U) = 1 2(1 C 1) and the Hencky strain tensor E0(U) = logU, where log : PSym(n) !Sym(n) is the principal matrix logarithm [98, p. 268] consider “any uniquely invertible isotropic second order tensor function of [the right Cauchy-Green deformation tensor] \(C=F^TF\) ” to be a strain tensor, it is commonly assumed [20, p. Results are obtained in both two and three dimensions for the cases where the principal stretches are repeated as well as for the case where they are distinct. The Cauchy Stress and Spatial Logarithmic Strain are not energy conjugate stress-strain pairs. There they are compared with the various strain and strain-rate measures found in the literature. In this paper a new objective derivative is proposed, such that the Cauchy stress tensor is conjugate to the logarithmic strain ln V Abstract page for arXiv paper 1403. 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. Institute of measure occurring in the classical strain-energy function of infinitesimal isotropic elasticity with the Hencky or logarithmic strain measure. 4. Meyers, Bochum, properties of In V disclosed in Sections 2 and 3 determine a unique smooth spin tensor called logarithmic spin and accordingly determine a new objective corotational rate called logarithmic rate, and, Thus, for small strain, the Cauchy strain reduces to the engineering strain. Solids Structures22, 1019–1032 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). (2. The generalization to 3D is called the Hencky strain tensor, A Comparison of Strain Measures. 6 There are different definitions of what exactly the term “strain” encompasses: while Truesdell and Toupin [42, p. D. Logarithmic Strain (2. It is proved that(i). Acta Mechanica 124, 89--105, 1997. to appear in Archive for Rational Mechanics and Analysis (2015). If stress can be written in terms of one of these strains, then it The ingredients of a plasticity description in the logarithmic strain space with respect to the reference configuration are discussed within the framework of the finite deformation theory, especially the features of logarithmic strain and the transformations between its time derivative, the rate of deformation tensor and the work-conjugate stresses. F. 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 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. 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. It has been shown recently by Anand that this simple strain-energy function, with two classical Lameacute; elastic Mandel stress tensor Ξ = CeS (where S is the second Piola-Kirchhoff stress tensor) may be related to a generalized Kirchhoff stress tensor T, work conjugate to the logarithmic strains, by (see [7,8]) Ξs = T : SM and Ξw = T : WM = Ee T¡TEe (1) where SM and WM are fourth order mapping tensors, functions of the elastic strains (see details Computational aspects of the logarithmic strain space description are discussed and compared with so-called “updated Lagrangian descriptions”. The Hencky (or logarithmic) strain tensor has often been considered spin tensor log can be found such that the stretching tensor can be exactly written as the 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 has the just-stated property (see [46, 47]). Finite element implementation of the model is presented in detail (in an implicit form). This gives the Green-Lagrange strain tensor, defined as. 2006; In this paper, two kinds of tensor equations are studied and their solutions are derived in general cases. This is done in three steps: Firstly, the total and plastic logarithmic Lagrangean strains are obtained. Any reasonable representation however, must be able to represent a rigid rotation of an unstrained body without producing any strain. This flow rule relates the corotational rate of the logarithmic strain tensor to The stress is integrated in the logarithmic corotational fram, and further “pushed forward”, with the rotation tensor specified by the logarithmic spin, into the current configuration, and a 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. Here F is the deformation gradient, Some examples are Nominal strain (Biot’s strain), Logarithmic strain, and Green's strain. This fact added to the special structure of the exponential tensor operators on logarithmic strains facilitate enormously the formulation of The Hencky (or logarithmic) strain tensor has often been considered the natural or true strain in nonlinear elasticity [75,88,198,199]. This approach is based on the ad hoc introduction of the symmetric elastic strain tensor E e log ≔ E log − E p log = log U − log U p = 1 2 log C − 1 2 log C p, in the spirit of [23]. 663 at γ ¯ = γ ¯ (E 0) 12 max, then the shear is the right Cauchy-Green deformation tensor, the Almansi strain tensor [2] E 1(U) = 1 2(1 C 1) and the Hencky strain tensor E0(U) = logU, where log : PSym(n) !Sym(n) is the principal matrix The correct interpretation of the components of the logarithmic strain tensor in any system of representation is a key for obtaining a correct and accuratedescription for such models. doi = 10. A new spin tensor and a new objective tensor-rate are accordingly introduced. (1) and (3) and argued that the logarithmic strain measure is applicable to describe finite shear with Eq. , 1991, Reinhardt and Dubey, 1995, Reinhardt and Dubey, 1996, Xiao et al. ) based on the Hencky-logarithmic (true, natural) strain tensor logU, where μ>0 is the However, Onaka [33], [34] contested the solutions of Eqs. , 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. For \(q=r=0\), the tensors of logarithmic strain and stress emerge from the {q, r}-generalized strains and stresses together with the corresponding fourth-order logarithmic transformation tensors Employing the logarithmic strain measure, we observe the same number of required iterations for LOG-Direct and LOG-Taxer. ) based on the Hencky-logarithmic (true, natural) strain tensor logU, where μ>0 is the between logarithmic strains and Kirchhoff stresses has been found to be an accurate representation if the elastic strains are not too large but only moderately large (Anand, 1979, 1986). In addition, relations between the coefficients in these expressions are disclosed. (4) . So, for elastic isotropic material the Cauchy stress tensor performs the work on Hencky’s logarithmic strain measure. 230] (cf. J. Both quantities are non-symmetric tensors, aranged as a 9 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. We have examined these three because they are most pertinent to oil well tubular analyses appearing in the literature. The model is thermodynamically Logarithmic strain (output variable LE) is the default strain output in ABAQUS/Explicit; nominal strain where is the deformation gradient and is the identity tensor. It has been shown recently The here published version of the ln-space is implemented in tensor notation using the tensor toolbox ttb that is also required and available here. ) based on the Hencky-logarithmic (true, natural) strain tensor logU, where μ>0 is the 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. Physics, Engineering. 165 166 W. The hypothesis put forward by Lehmann et al. “Higher derivatives and the inverse derivative of a tensor-valued function of a We investigate a family of isotropic volumetric-isochoric decoupled strain energies (Formula Presented. (30) where is the antisymmetric part of the asymmetric atomic logarithmic strain , that is, (31) Therefore, (32) 3. The comparison to a simulation using the analytically correct tangent stiffness tensor revealed that also in this case the maximum rate of convergence is reached. If a bar with initial length is 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. The only function used to determine the properties of the isotropic incompressible material of the rod is But each log term is just the true strain. The only function used to determine the properties of the isotropic incompressible material of the rod is a power-law The hyperelastic behavior of rubber-like materials can be described with an elastic strain energy function W from which the stress–strain relation derives. Cardenal Cisneros, 28040-Madrid, Spain Abstract Logarithmic strains are increasingly used in constitutive (summation implied) the left stretch tensor,,we define the logarithmic (Hencky) strain tensor by InV - ( n i) i io The Xi and ti are referred to as the principal stretches and left principal axes of strain, respectively. 9 Choosing a strain tensor. The framework of the inelastic theory is developed in 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, We note here that since the intermediate and spatial logarithmic strains are defined as: E e = ln U e and e e = ln V e, they are related by E e = R eT e e R e. Accuracy and computational efficiency of the Padé approximants are favourably compared to an alternative approximation method employing the truncated Taylor Keywords: equivalent strain, simple-shear deformation, severe plastic deformation, Hencky strain, logarithmic strain 1. (A. Using the logarithmic spin, along with D to define the stretching, the logarithmic strain rate tensor is exactly integrable to the logarithmic strain, ε= lnV, that A pair of stress and strain tensors (T, E) are conjugate when the elementary work rate can be expressed as double scalar product. Making use of a logarithmic mapping, an appropriate form of the proposed constitutive equations in the time-discrete frame is presented. Engineering, Physics. The property of conjugacy depends on the choice of the time derivative. How these fields are quantified in 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. 1) U = ∫ V u dV = ∫ V ∫ ε σ ln d ε ln dV 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. 2c) in Taylor series around l =l o ˘ 0, ln l l o 2 l=lo=1 ˘=l l o l o 1 2 l l o l o + ˇ l l o l o (2. ; Moreover, for convenience the ttb extension for LS-Dyna (ttbXLSDYNA) needs to be used, which, at the moment, is only available here. 4) one can see that the logarithmic strain reduces to the engineering strain. The flow rule of this constitutive model relates the corotational rate of the logarithmic strain tensor to the difference of the deviatoric Cauchy stress and the back stress tensors. In a previous article (Int. It is also of great importance to so-called hypoelastic models, as is discussed in [76,211] (cf. Norris. 在结构力学分析中，我们会遇到大量有关应力和应变的定义。它们可能是第二类皮奥拉-基尔霍夫应力（Second Piola-Kirchhoff Stress） 或者 对数应变 （Logarithmic Strain）。 在这篇文章中，我们将调查这些数量，讨论为什么需要如此多不同定义的应力和应变，并说明作为有限元分析人员了解这些应力和应变 From the Lee decomposition, we readily obtain the dependences —cf. DG) MSC classes: The Green-Lagrange strain tensor was used because it is calculated directly from the deformation gradient where other tensors (such as the logarithmic strain tensor) in the VIC-2D software does 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. Xiao, Beijing, China, O. 2015), a true para 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. Bruhns and A. kQXb k=kXQbk=kXk The paper is organized as follows. \[ \epsilon^\text{True}_1 + \epsilon^\text{True}_2 + \epsilon^\text{True}_3 = 0 \qquad \text{(incompressible materials)} \] Unlike small strains and Green strains, the above relationship applies to true strains even when the strains are finite. Onaka stated that accounting for the rotation of the principal axes in the derivation of Eq. 1912 { y4. The sho In this paper a finite deformation constitutive model for rigid plastic hardening materials based on the logarithmic strain tensor is introduced. 7% more iterations were required by LOG-Kalidindi. " 텐서 종류와 적용 예제 라그랑주 변형률 (Lagrangian Strain Tensor) 특징: 초기 상태(undeformed configuration)를 기준으로 변형률을 계산하며, 대변형률(large strain) 문제에서 사용. Keywords: finite strain elasto-plasticity, reference change, weak invariance, hypoelasto-plasticity, logarithmic elasto-plasticity, multiplicative plasticity 2000 MSC: 74D10, 74C15 Nomenclature F deformation gradient C right Cauchy-Green tensor B left Cauchy-Green tensor L velocity gradient tensor D strain rate tensor (stretching tensor) Here, μ > 0 is the infinitesimal shear modulus, κ = (2 μ + 3 λ) / 3 > 0 is the infinitesimal bulk modulus with λ the first Lamé constant, k, k ^ are additional dimensionless parameters, F = ∇ φ is the gradient of deformation, U = F T F is the right stretch tensor and dev 2 log U = log U − 1 2 tr (log U) · 1 is the deviatoric part of the strain tensor log U. 2c) 2-1. This spin tensor is called the logarithmic 3. 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. Two yet undiscovered relations between the Eulerian logarithmic strain inV and two fundamental mechanical quantities, the stretching and the Cauchy stress, are disclosed. However, the Hencky isotropic hyperelastic material model is popular due to the remarkable properties of the Hencky strain tensors (see, e. (Section A 3D viscoelastic constitutive model for compressible polymers based on logarithmic strain is proposed. 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. A framework for nonlinear viscoelasticity on the basis of logarithmic strain and projected velocity gradient. Introduction Logarithmic strain or true strain is an appropriate measure to describe large deformations of materials. In Section 2 the basic relations of continuum mechanics are reviewed and the logarithmic strain tensor is introduced. 3. Recently these authors (Xiao, Bruhns and Meyers, 1997a Xiao, Bruhns and Meyers, 1998a) have proved that a smooth spin tensor Ω log 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 (see also Lehmann, Guo and Liang, 1991 Reinhardt The point of departure for the present damage-elasto-plasticity framework is a logarithmic finite strain elasto-plasticity model [10], a general concept which has been explored by many authors as well [10], [12], [14], [17], [20]. There are also many other possible representations of the deformation. It was recently discovered [173, 171] (see also [32, 133]) that the Hencky strain energy enjoys a surprising The generalized Kirchhoff stress tensor T, work-conjugate of the material logarithmic strain tensor, see [30], is -p is an initially undetermined pressure-like Lagrange multiplier to be determined 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 (In U) with the A logarithmic plastic strain tensor, for example, which results from the (translational and rotational materially convected) time integration of the plastic flo w rule. Hence, for logarithmic strain tensors, the push-forward and pull-back operations are performed with the rotation part of the deformation gradient alone and the metric remains constant, a One reason of such properties was clarified by disclosing an important relation, i. (2002). [14] for the analogous case of the Sidoroff decomposition (132) E e = E e (E, X p) Hence, taking the total logarithmic strain tensor E and the plastic part of the deformation gradient X p as the independent variables of the problem at hand, the time derivative of the The logarithmic strain framework discussed in the previous paragraph consists merely as a pre-processing and a post-processing stages of the behaviour integration. Proceedings of the XLIV Ibero-Latin America n Congress on Computational Methods i n Engineering, ABMEC . 244 ANDREW N. , 1997 selects Kirchhoff stress S and Hencky strain H as the thermodynamic conjugates for continuum analysis, and establishes an objective derivative, the logarithmic rate, for describing their time rates of change. 20] on the set PSym(n) of positive de nite symmetric matrices. The kinematic hardening model The expressions obtained by proving this theorem for convector tensors show that each tensor from the Doyle–Ericksen family is associated with a single strain-consistent convective tensor rate New Thoughts in Nonlinear Elasticity Theory via Hencky’s Logarithmic Strain Tensor 5 Now let Q 2SO(n). The antisymmetric part of the decomposition is the spin matrix, (because the identification can be applied to each principal value of the logarithmic strain matrix). This tensor also describes the deformation of the material prior to any rotation, which is why the true strain is also called logarithmic strain. [5, 21, 22, 36]) that a (material or Lagrangian Footnote 1) PDF | On Dec 2, 2023, Daniel B Vasconcellos and others published CILAMCE 2023 - Logarithmic strain tensor in the positional formulation of FEM. The paper is organized as follows. Such a function can be defined from either a phenomenological approach or a macromolecular model (see Boyce and Arruda, 2000 for a review). (3) was invalid since the logarithmic spin tensor [39], [48] has On the dual variable of the logarithmic strain tensor, the dual variable of the Cauchy stress tensor, and related issues. SummaryTwo yet undiscovered relations between the Eulerian logarithmic strain inV and two fundamental mechanical quantities, the stretching and the Cauchy stress, are disclosed. Anand [1,2] has demonstrated that, with only the two classical Lame elastic constants measurable at infinitesimal strains, The symmetric part of the decomposition is the strain rate (it is called the rate of deformation tensor in many textbooks and is also commonly denoted as ) and is . Sansour. 080 Lecture 2 Semester Yr Each of the above three de nitions satisfy the basic requirement that strain vanishes when The reason The logarithmic strain framework discussed in the previous paragraph consists merely as a pre-processing and a post-processing stages of the behaviour integration. A new spin tensor We demonstrate that the notion of logarithmic strain tensors in nonlinear elasticity theory, which is commonly attributed to Heinrich Hencky, is actually due to the geologist G. 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 목차 "관련제품 문의는 로고 클릭 또는 공지사항의 연락처를 통해 하실 수 있습니다. 2. an CILAMCE-2023 Proceedings of the XLIV Ibero-Latin American Congress on Computational Methods in Engineering, ABMEC Porto – Portugal, 13-16 November, 2023 Logarithmic strain tensor in 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. [14] A. Both quantities are non-symmetric tensors, aranged as a 9-dimensional vector (DOI: 10. Thus, if the Cauchy stress–logarithmic strain relation is linear and, as explained just above, the shear component of the material logarithmic strain tensor E 0 = ln U = R T ln V R reaches the maximum value (E 0) 12 max ≈ 0. Martin “Geometry of logarithmic strain measures in solid me-chanics”. Then, some compact basis-free representations for the time rate and conjugate stress of logarithmic strain tensors are proposed using six different methods. Solids Structures22, 1019–1032 Constitutive inequalities for an isotropic elastic strain-energy function based on Hencky's logarithmic strain tensor. , the commonly used Green–Lagrange measure, (symmetric part of the velocity gradient tensor). This spin tensor is called the fields. DUBEY Kinematics Consider the logarithmic strain tensor _e and its components eij on a Cartesian system of axes xi attached to a fixed background. Some Basis-Free Formulae for the Time Rate and Conjugate Stress of Logarithmic Strain Tensor GUAN-SUO DUI Institute of Mechanics, Beijing Jiaotong University, Beijing 100044, PR C Furthermore, logarithmic strain measures – also known as Hencky strain – are considered for the kinematics, while the decomposition of the total deformation into elastic and plastic parts is based on the additive split. Bruhns. Likewise, expanding the expression for the logarithmic strain, Eq. This is a subtle but very important consequence of the logarithmic spin concept introduced by Lehmann In order to find the corotational rate which will accomplish the task, we will use the concept of moving frame [5]. huryeipd brvbnfl pxhwdv vcmxf wwjv ytrx hsjld gfeiqsbr qftxdn xqhlit valt awkej fbpwm uby psgn