Logarithmic Strain Measure Research Articles

Finite strain modelling of Shape Memory Alloys in the logarithmic strain space: A comparative study with other finite strain approaches

Finite strain models for Shape Memory Alloys (SMAs) offer significant advantages over small strain models in that they excel in their ability to characterize SMA behaviour under large strain and finite rotations while eliminating the inherent assumptions associated with the small strain framework. The paper at hand presents a numerical model for SMAs at finite strain, employing the logarithmic strain measure to obtain an additive split of the total strain into its elastic and inelastic (transformation) parts. Notably, the model maintains the thermodynamic consistency validated using the Clausius–Duhem inequality. Additionally, we compare this model with two other finite strain SMA models employing logarithmic strain measures: a hypoelastic model, which splits the rate of deformation tensor additively, and a hyperelastic model, which decomposes the deformation gradient multiplicatively. The numerical formulation for the three finite strain models is specifically tailored to address superelasticity in SMAs. It is derived from the established small strain model and seamlessly extended into the domain of finite strain framework. Several numerical examples are carried out to showcase the logarithmic strain space approach applied to SMAs and to compare its response with that of the other aforementioned finite strain models in SMAs. The meticulous examination reveals that the logarithmic strain space approach aligns well with the existing finite strain models, demonstrating its effectiveness and compatibility.

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 present paper is to develop a thermodynamically consistent coupled damage-plasticity model at large deformations, which accounts for damage anisotropy. Moreover, a ‘two-surface’ approach allows modeling plasticity and damage independently. Thus, both phenomena are treated as separate dissipative mechanisms, making the model attractive for application to both brittle and ductile materials. The framework is based on Continuum Damage Mechanics. 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. Hence, the derivation of the model and its conjugated quantities takes place in the logarithmic strain space, but these are subsequently transformed to their Lagrangian counterparts to be applicable in standard finite element formulations. Consequently, the transformation of constitutively dependent quantities such as stresses, but also the various associated material sensitivities, are addressed here. Another main aspect of this work is the gradient-extension of the presented model in order to cure mesh sensitivity in case of material softening. To this end, a novel gradient extension is derived using the invariants of the second order damage tensor, which is based on the micromorphic approach. In addition to the theoretical framework, special attention is paid to the finite element implementation, the formulation of the local residuals, and additionally the computation of the material tangents to achieve quadratic convergence rate within the Newton–Raphson scheme. Single element studies as well as representative structural examples investigate the model’s response to various loading scenarios, the effect of damage anisotropy and further highlight its ability to provide mesh-independent results while undergoing large deformations.

Mapping and classifying large deformation from digital imagery: application to analogue models of lithosphere deformation

SUMMARY Particle image velocimetry (PIV), a method based on image cross-correlation, is widely used for obtaining velocity fields from time-series of images of deforming objects. Rather than instantaneous velocities, we are interested in reconstructing cumulative deformation, and use PIV-derived incremental displacements for this purpose. Our focus is on analogue models of tectonic processes, which can accumulate large deformation. Importantly, PIV provides incremental displacements during analogue model evolution in a spatial reference (Eulerian) frame, without the need for explicit markers in a model. We integrate the displacements in a material reference (Lagrangian) frame, such that displacements can be integrated to track the spatial accumulative deformation field as a function of time. To describe cumulative, finite deformation, various strain tensors have been developed, and we discuss what strain measure best describes large shape changes, as standard infinitesimal strain tensors no longer apply for large deformation. PIV or comparable techniques have become a common method to determine strain in analogue models. However, the qualitative interpretation of observed strain has remained problematic for complex settings. Hence, PIV-derived displacements have not been fully exploited before, as methods to qualitatively characterize cumulative, large strain have been lacking. Notably, in tectonic settings, different types of deformation—extension, shortening, strike-slip—can be superimposed. We demonstrate that when shape changes are described in terms of Hencky strains, a logarithmic strain measure, finite deformation can be qualitatively described based on the relative magnitude of the two principal Hencky strains. Thereby, our method introduces a physically meaningful classification of large 2-D strains. We show that our strain type classification method allows for accurate mapping of tectonic structures in analogue models of lithospheric deformation, and complements visual inspection of fault geometries. Our method can easily discern complex strike-slip shear zones, thrust faults and extensional structures and its evolution in time. Our newly developed software to compute deformation is freely available and can be used to post-process incremental displacements from PIV or similar autocorrelation methods.

Dynamic analysis of failure paths of truss structures: Benchmark examples including material degradation

We propose novel benchmark examples of dynamic failure paths of truss structures. Novelty arises from use of a logarithmic strain measure, within a total Lagrangian formulation, combined with a continuum material damage model. Previous benchmarks have considered the usual engineering strain measure, which is not always useful, as it can lead to material degeneration under finite levels of stress. Ductile material behavior has been considered in the literature, addressing plasticity and some material softening, but neglecting material degradation. Herein, damage accumulation is associated to the hydrostatic component of plastic strains, leading to a stable and explicit representation of material degradation. The original static formulation, presented elsewhere by the authors, is extended herein to the dynamic analysis of failure paths of truss structures. Several numerical examples from the literature are studied, and the differences in dynamic behavior are pointed out. In our implementation, as individual bars are damaged, elastic unloading and load redistribution are observed. When the critical damage hypothesized by Lemaitre is reached by individual bars, these are fully unloaded, with no material degeneration or numerical instability. Numerical results highlight the significant effects of material degradation in the dynamic behavior of truss structures under exceptional loads. We propose the set of examples addressed herein as the new benchmarks to which future developments in geometrical and material non-linear truss modelling will be compared.

A Modified Phase-Field Damage Model for Metal Plasticity at Finite Strains: Numerical Development and Experimental Validation

Steel structures are designed to operate in an elastic domain, but sometimes plastic strains induce damage and fracture. Besides experimental investigation, a phase-field damage model (PFDM) emerged as a cutting-edge simulation technique for predicting damage evolution. In this paper, a von Mises metal plasticity model is modified and a coupling with PFDM is improved to simulate ductile behavior of metallic materials with or without constant stress plateau after yielding occurs. The proposed improvements are: (1) new coupling variable activated after the critical equivalent plastic strain is reached; (2) two-stage yield function consisting of perfect plasticity and extended Simo-type hardening functions. The uniaxial tension tests are conducted for verification purposes and identifying the material parameters. The staggered iterative scheme, multiplicative decomposition of the deformation gradient, and logarithmic natural strain measure are employed for the implementation into finite element method (FEM) software. The coupling is verified by the ‘one element’ example. The excellent qualitative and quantitative overlapping of the force-displacement response of experimental and simulation results is recorded. The practical significances of the proposed PFDM are a better insight into the simulation of damage evolution in steel structures, and an easy extension of existing the von Mises plasticity model coupled to damage phase-field.

Closed-form solutions for the symmetric nonlinear free oscillations of pyramidal trusses

This work presents the development of analytical solutions regarding the symmetric free vibrations of pyramidal trusses, considering a general large strain measure. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. Illustrative examples considering the quadratic (Green–Lagrange) and logarithmic (Hencky) strain measures are used to exemplify the influence of the strain measure on the results. Pyramidal trusses are well-known bistable structures and in the unloaded configurations may present different analytical solutions depending on the initial conditions: small amplitude vibrations around each stable equilibrium configuration or large cross-well vibrations. These families of solution are separated by two homoclinic orbits, constituting the boundaries of the two potential wells, which are related to the saddle point initial conditions. In addition, closed-form time solutions for the undamped vibrations are derived and the nonlinear frequency–amplitude relations, which are a measure of the degree and type of nonlinearity (hardening or softening), are obtained. Finally, a semi-analytical procedure for the damped vibrations is also constructed.

Extending interior‐point methods to nonlinear second‐order cone programming: Application to finite‐strain elastoplasticity

AbstractInterior‐point methods (IPMs) are well suited for solving convex nonsmooth optimization problems which arise for instance in problems involving plasticity or contact conditions. This work attempts at extending their field of application to optimization problems involving either smooth but nonconvex or nonsmooth but convex objectives or constraints. A typical application for such kind of problems is finite‐strain elastoplasticity which we address using a total Lagrangian formulation based on logarithmic strain measures. The proposed interior‐point algorithm is implemented and tested on 3D examples involving plastic collapse and geometrical changes. Comparison with classical, Newton‐Raphson/return mapping methods show that the IPM exhibits good computational performance, especially in terms of convergence robustness. Similar to what is observed for convex small‐strain plasticity, the IPM is able to converge for much larger load steps than classical methods.

Nonlinear oscillations and dynamic stability of an elastoplastic pyramidal truss

Pyramidal space trusses are currently used in many present-day engineering applications, either as main parts or as a constitutive element. The present work studies, using a corotational finite element formulation considering large displacements and rotations and an elastoplastic material behavior with isotropic strain hardening, the nonlinear dynamic behavior of a pyramidal space truss. A bilinear hysteretic model with kinematic hardening is used to represent the material nonlinearity. An appropriate strain measure in such processes is the logarithmic strain. However, the engineering strain and the quadratic strain measures have been usually used to perform the elastic stability analysis of such structures. Thus, in the present work results using both the quadratic and the logarithmic strain measures are compared. First, the static nonlinear behavior is investigated in order to understand the influence of nonlinearities and static preload on the potential energy landscape, which controls the dynamics of the truss. These structures exhibit a two-well potential function that leads to coexisting in-well and cross-well solution branches, resulting in several coexisting periodic, quasi-periodic and chaotic attractors. The influence of the constitutive law, strain measure, truss geometry and support flexibility on the static stability is then investigated. Based on these results, a detailed parametric analysis of the nonlinear truss response under vertical load is conducted using bifurcations diagrams of the Poincare map to investigate the effect of the constitutive law, strain measure, truss geometry and load control parameters on the bifurcation scenario, coexisting attractors and dynamic buckling loads. The effect of the strain hardening parameter is particularly emphasized, allowing a more comprehensive picture of the dynamical behavior exhibited by the bistable system. Properties of the response are further illustrated by samples of time and phase plane responses and the related Poincare section plots. In addition to known behaviors, a rich class of solutions and bifurcations, including jump phenomena, symmetry-breaking, period-doubling cascades, fold and chaos is detected.

A brief history of logarithmic strain measures in nonlinear elasticity

AbstractWe 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. F. Becker, who was the first to consider a fully three‐dimensional logarithmic stress response in an 1893 article.

A non-ellipticity result, or the impossible taming of the logarithmic strain measure

Constitutive laws in terms of the logarithmic strain tensor logU, i.e. the principal matrix logarithm of the stretch tensor U=FTF corresponding to the deformation gradient F, have been a subject of interest in nonlinear elasticity theory for a long time. In particular, there have been multiple attempts to derive a viable constitutive law of nonlinear elasticity from an elastic energy potential which depends solely on the logarithmic strain measure ‖logU‖2, i.e. an energy function W:GL+(n)→R of the form (1)W(F)=Ψ(‖logU‖2)with a suitable function Ψ:[0,∞)→R, where ‖.‖ denotes the Frobenius matrix norm and GL+(n) is the group of invertible matrices with positive determinant.However, while such energy functions enjoy a number of favorable properties, we show that it is not possible to find a strictly monotone function Ψ such that W of the form (1) is Legendre–Hadamard elliptic.Similarly, we consider the related isochoric strain measure ‖devnlogU‖2, where devnlogU is the deviatoric part of logU. Although a polyconvex energy function in terms of this strain measure has recently been constructed in the planar case n=2, we show that for n≥3, no strictly monotone function Ψ:[0,∞)→R exists such that F↦Ψ(‖devnlogU‖2) is polyconvex or even rank-one convex. Moreover, a volumetric-isochorically decoupled energy of the form F↦Ψ(‖devnlogU‖2)+Wvol(detF) cannot be rank-one convex for any function Wvol:(0,∞)→R if Ψ is strictly monotone.

Evaluation of Strains and Thicknesses of Pipe Elbows on the Basis of Expressions Resulting from the Eudirective for the Case of Large and Small Deformations

The relations to calculate the maximum value of strains in processes of bending tubes on benders, in stretched layers of tubes, are presented in this work on the basis of the EU-Directive concerning production of pressure equipment. It has been shown that for large deformations that occur during bending of the pipes on knees, logarithmic strain measures (real) and relative strain measures give different values of strain but equal wall thicknesses in the bending zone. Logarithmic measures are frequently used in engineering practice and are valid for large and small deformations. Reverse expressions were also derived to calculate the required initial wall thickness of the tube to be bent, in order to obtain the desired wall thickness of the knee after bending.

A new class of plastic flow evolution equations for anisotropic multiplicative elastoplasticity based on the notion of a corrector elastic strain rate

We herein present a new continuum theory for both isotropic and anisotropic elastoplasticity at large strains. The new framework has the following properties: (1) It is valid for non-moderate large strains, (2) it is valid for both elastic and plastic anisotropy, (3) its description in rate form is parallel to that of the infinitesimal formulation, (4) it is compatible with the multiplicative decomposition, (5) results in a similar framework in any stress-strain work-conjugate pair, (6) it is consistent with the principle of maximum plastic dissipation and (7) does not impose any restriction on the plastic spin, which must be given as an independent constitutive equation. Furthermore, when formulated using logarithmic strain measures in the intermediate configuration: (8) it may be easily integrated using a classical backward-Euler rule resulting in an additive update. All these properties are obtained simply considering a plastic evolution in terms of a corrector rate of the proper elastic strain. This new continuum theory is a natural framework for elastoplasticity of both metals and soft materials and solves the (so-coined by Simo) rate issue.

Shear confusion: Identification of the appropriate equivalent strain in simple shear using the logarithmic strain measure

There is significant confusion surrounding the appropriateness of the logarithmic (Hencky) strain measure to describe simple shear deformation for finite strain. In simple shear loading of plastically deforming materials, the principal stress and strain directions do not remain coaxial which has led to conflicting derivations of the equivalent strain throughout the literature. The source of this confusion is attributed to a misapplication of the formulas for the von Mises equivalent strain and its increment that are only valid for proportional coaxial loading. In this work, a closed-form solution for the work-conjugate equivalent strain for an arbitrary yield function was derived for simple shear loading that is readily amenable to experimental characterization and is entirely consistent with the logarithmic strain measure. An analytical stress and strain integration of an elastic-plastic material was performed using the logarithmic objective rate to demonstrate that the stress, logarithmic strain, and principal directions are correctly determined within a hypo-elastic-plastic framework to finite strains. It was demonstrated that the integrated equivalent plastic strain is work-conjugate and the logarithmic strain measure is appropriate for finite simple shear. A review of the recent experimental literature for shear characterization has found that the misapplication of the von Mises equivalent strain formula for coaxial loading in simple shear loading is pervasive. The coaxial effective strain formula is the default measure in commercial digital image correlation (DIC) software and may significantly underestimate the equivalent strain in the simple shear loading condition. If the major principal strain in a simple shear test is lower than 50%, the error between the coaxial and work-conjugate equivalent strains is negligible. Otherwise, the error grows in a hyperbolic manner. Within the context of a finite element simulation of a simple shear test, if a hypo-elastic-plastic formulation is employed as in most commercial codes, the equivalent strain will be correctly computed from work conjugacy but the cumulative and principal logarithmic strain tensors will be incorrect at finite strains unless the logarithmic rate is employed.

Geometry of Logarithmic Strain Measures in Solid Mechanics

AbstractWe consider the two logarithmic strain measures$$\begin{array}{ll} {\omega_{\mathrm{iso}}} = ||{{\mathrm dev}_n {\mathrm log} U} || = ||{{\mathrm dev}_n {\mathrm log} \sqrt{F^TF}}|| \quad \text{ and } \quad \\ {\omega_{\mathrm{vol}}} = |{{\mathrm tr}({\mathrm log} U)} = |{{\mathrm tr}({\mathrm log}\sqrt{F^TF})}| = |{\mathrm log}({\mathrm det} U)|\,,\end{array}$$ωiso=||devnlogU||=||devnlogFTF||andωvol=|tr(logU)=|tr(logFTF)|=|log(detU)|,which are isotropic invariants of the Hencky strain tensor$${\log U}$$logU, and show that they can be uniquely characterized by purely geometric methods based on the geodesic distance on the general linear group$${{\rm GL}(n)}$$GL(n). Here,$${F}$$Fis the deformation gradient,$${U=\sqrt{F^TF}}$$U=FTFis the right Biot-stretch tensor,$${\log}$$logdenotes the principal matrix logarithm,$${\| \cdot \|}$$‖·‖is the Frobenius matrix norm,$${\rm tr}$$tris the trace operator and$${{\text dev}_n X = X- \frac{1}{n} \,{\text tr}(X)\cdot {\mathbb{1}}}$$devnX=X-1ntr(X)·1is the$${n}$$n-dimensional deviator of$${X\in{\mathbb {R}}^{n \times n}}$$X∈Rn×n. This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor$${\varepsilon={\text sym}\nabla u}$$ε=sym∇u, which is the symmetric part of the displacement gradient$${\nabla u}$$∇u, and reveals a close geometric relation between the classical quadratic isotropic energy potential$$\mu {\| {\text dev}_n {\text sym} \nabla u \|}^2 + \frac{\kappa}{2}{[{\text tr}({\text sym} \nabla u)]}^2 = \mu {\| {\text dev}_n \varepsilon \|}^2 + \frac{\kappa}{2} {[{\text tr} (\varepsilon)]}^2$$μ‖devnsym∇u‖2+κ2[tr(sym∇u)]2=μ‖devnε‖2+κ2[tr(ε)]2in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energy$$\mu {\| {\text dev}_n log U \|}^2 + \frac{\kappa}{2}{[{\text tr}(log U)]}^2 = \mu {\omega_{{\text iso}}^2} + \frac{\kappa}{2}{\omega_{{\text vol}}^2},$$μ‖devnlogU‖2+κ2[tr(logU)]2=μωiso2+κ2ωvol2,where$${\mu}$$μis the shear modulus and$${\kappa}$$κdenotes the bulk modulus. Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor$${R}$$R, where$${F=RU}$$F=RUis the polar decomposition of$${F}$$F. 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.

Determination of the tangent stiffness tensor in materials modeling in case of large deformations by calculation of a directed strain perturbation

The Finite Element Method in the field of materials modeling is often relying to the tangent stiffness tensor of the constitutive law. This so called Jacobian matrix is required at each time increment and describes the local material behavior. It assigns a stress increment to a strain increment and is of fundamental importance for the numerical determination of the equilibrium state. For increasingly sophisticated material models the tangent stiffness tensor can be derived analytically only with great effort, if at all. Numerical methods like the forward-difference, the central-difference, and the complex-step derivative approximation approach are widely used for its calculation. For each of these methods it is necessary to generate a specific strain perturbation. However, in large strain formulations it is not possible to perturb the strain directly but one can only modify the deformation gradient.We present our methods to generate a directed strain perturbation for the Green–Lagrange, Euler–Almansi and logarithmic strain measures as a function of the deformation gradient and compare them with other commonly used methods. An increase in accuracy and rate of convergence can be achieved with the proposed procedures.

Fully anisotropic finite strain viscoelasticity based on a reverse multiplicative decomposition and logarithmic strains

In this paper we present a novel formulation for phenomenological anisotropic finite visco-hyperelasticity. The formulation is based on a multiplicative decomposition of the equilibrated deformation gradient into nonequilibrated elastic and viscous contributions. The proposal in this paper is a decomposition reversed respect to that from Sidoroff allowing for anisotropic viscous contributions. Independent anisotropic stored energies are employed for equilibrated and non-equilibrated parts. The formulation uses logarithmic strain measures in order to be teamed with spline-based hyperelasticity. Some examples compare the results with formulations that use the Sidoroff decomposition and also show the enhanced capabilities of the present model.

Logarithmic strain and its material derivative for a QR decomposition of the deformation gradient

The well-known polar decomposition of the deformation gradient decomposes it into an orthogonal rotation and a symmetric stretch. We consider a Gram–Schmidt factorization of the deformation gradient, which decomposes it into a different orthogonal rotation with a right-triangular field that we call distortion. Properties of this distortion tensor are discussed, and a work-conjugate stress tensor is derived for this Lagrangian frame. The logarithm of distortion and its material derivative are then introduced, and their components are quantified, resulting in a new logarithmic measure for strain and its rate, distinct from Hencky strain (the logarithm of stretch) and its rate. An eigenprojection analysis and a first-order, differential, matrix equation solved using the technique of back substitution both produce the same matrix components describing the logarithm of distortion. Three homogeneous deformations illustrate similarities and differences between the logarithms of distortion and stretch. They are distinct measures of strain. The new logarithmic strain measure shows monotonic behavior under simple shear as opposed to the non-monotonic behavior of Hencky strain (see Fig. 2).

Determination of the thicknesses of pipe elbows on the basis of expressions resulting from EU-Directive

The relations to calculate the maximum value of relative strains, which occur in a process of bending of tubes on benders, in stretched layers of tubes, are presented in this work on the basis of EU-Directive concerning production of pressure equipment. It has been shown that for large deformations that occur during bending of the pipes on knees, logarithmic strain measures (real) and relative strain measures give different values of the relative distribution of strain and different wall thicknesses in the bending zone. Reverse expressions were also derived to calculate the initial wall thickness of the tube to be bent, in order to obtain the desired wall thickness of knee after bending.

Numerical calculation of the tangent stiffness matrix in materials modeling

AbstractThe Finite Element Method in the field of materials modeling is closely connected to the tangent stiffness matrix of the constitutive law. This so called Jacobian matrix is required at each time increment and describes the local material behavior. It assigns a stress increment to a strain increment and is of fundamental importance for the numerical determination of the equilibrium state. For increasingly sophisticated material models the tangent stiffness matrix can be derived analytically only with great effort, if at all. Numerical methods are therefore widely used for its calculation. We present our method to calculate the tangent stiffness matrix for the logarithmic strain measure. The approach is compared with other commonly used procedures. A significant increase in accuracy can be achieved with the proposed method. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Integrating a logarithmic-strain based hyperelastic formulation into a three-field mixed finite element formulation to deal with incompressibility in finite-strain elastoplasticity

This paper deals with the treatment of incompressibility in solid mechanics in finite-strain elastoplasticity. A finite-strain model proposed by Miehe, Apel and Lambrecht, which is based on a logarithmic strain measure and its work-conjugate stress tensor is chosen. Its main interest is that it allows for the adoption of standard constitutive models established in a small-strain framework. This model is extended to take into account the plastic incompressibility constraint intrinsically. In that purpose, an extension of this model to a three-field mixed finite element formulation is proposed, involving displacements, a strain variable and pressure as nodal variables with respect to standard finite element. Numerical examples of finite-strain problems are presented to assess the performance of the formulation. To conclude, an industrial case for which the classical under-integrated elements fail is considered.