Logarithmic Strain Space 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.

Damage in a comprehensive model for shape memory alloys in logarithmic strain space

Shape memory alloys (SMAs) possess considerable complexity in response to thermomechanical load due to their strong dependence on strain, stress, temperature and history. This high complexity is a limiting factor in the fatigue analysis of SMA structures. With this in mind, a comprehensive phenomenological model is developed to address nearly all of the features in SMAs with a particular focus on the development of damage. Due to the fact that many SMA structures experience large deformations, the model is developed in finite strains using the logarithmic strain space approach. Besides the balance of linear momentum, the model includes the heat conduction equation and two micromorphic nonlocal balance laws. The latter are introduced to regularize the softening associated with the damage and transformation. The material model incorporates a damage evolution law that depends on the integration of a simplified dissipation term. The model is applied to a few examples to observe the interaction of the features and to illustrate the applicability of the model for rupture and fatigue analysis. The interaction of the features is shown to produce a reasonable representation of the cyclic loading of wires at different loading rates. Rupture is shown to occur due to a large amount of plastic strain and can be performed due to the nonlocal approach for the damage. The fatigue analysis shows some promising accuracy but only if the dissipated area is captured properly.

Dispersion‐type Anisotropic Viscoelasticity: Model Validation for Myocardium

AbstractThis contribution presents a novel constitutive model for rate‐dependent response of the passive myocardium. As a first step, we performed a comparative study on dispersion‐type anisotropic hyperelastic constitutive models [1–3] and assessed performance of various density distribution functions by fitting to experiments conducted on three distinct tissues [4]. Next, we proposed an angular integration type anisotropic viscoelastic constitutive model that uses bivariate von‐Mises distribution function to capture fiber dispersion in passive myocardium. The baseline hyperelasticity is described by a generalized structure tensor formulation proposed by GASSER ET AL. [1]. The non‐equilibrium part of the model utilizes a quadratic free energy function in the logarithmic strain space and a power‐type nonlinear evolution equation in orientation directions. The overstress response is then obtained by the numerical integration over the unit sphere by making use of 21 quadrature points. The proposed model parameters are obtained from cyclic triaxial shear and triaxial shear relaxation experiments on human passive myocardium [5].

Efficient gradient enhancements for plasticity with ductile damage in the logarithmic strain space

We contrast different gradient-enhancements for plasticity-damage material models in the logarithmic strain space and compare them to reference models based on multiplicative plasticity. The models being compared include plasticity - gradient-damage, where the gradient-enhancement is applied on the local damage variable, and gradient-plasticity - damage with a gradient-enhanced plastic hardening variable. Thereby, gradient-plasticity proved to be able to simultaneously regularise plastic and ductile (plasticity-driven) damage localisation as confirmed by numerical localisation analyses. This appears to be especially advantageous for logarithmic strain space plasticity-damage, because of the observed plastic localisation even for the case of plasticity with hardening. Even though gradient-plasticity appears to be numerically more demanding, two numerical examples indicate a good robustness and mesh objectivity in the softening regime. Moreover, the internal length for plasticity is able to adjust the damage zone width, similarly to gradient-damage, however ensuring a priori that damage takes place exclusively inside the plastic zone.

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.

Observations on additive plasticity in the logarithmic strain space at excessive strains

Additive plasticity in the logarithmic strain space is compared to multiplicative plasticity for various loading cases including coaxial and non-coaxial plastic deformations. Even though both finite plasticity approaches are based on total Lagrangian descriptions, the former is popular due to its inherent similarity to the infinitesimal theory and its easy extensibility. However, since its introduction several limitations of additive plasticity in the logarithmic strain space have been discovered. In this study, these problems such as stress rotation and softening are considered, revealing that fundamental differences compared to multiplicative plasticity occur for non-coaxial plastic deformations. We focus in particular on the observed softer response of the additive based approach, which is analysed in depth using three numerical examples including two well-known benchmarks for finite plasticity. By means of these finite element simulations the softer and possibly even localising response of additive plasticity in the logarithmic strain space is confirmed.

Anisotropic plasticity‐damage material model for sheet metal — Regularised single surface formulation

AbstractSheet metal forming as well as mechanical joining demand increasingly accurate and efficient material modelling to capture large deformations, the inherent sheet orthotropy and even process‐induced damage, which is expected to be influential. To account for large strains the additive logarithmic strain space is utilised that enables a straightforward incorporation of plastic anisotropy, herein modelled by a Hill48 yield function. A gradient‐enhancement is used to equip the ductile damage model with an internal length scale curing the damage‐induced localisation. An affine combination of the local and non‐local softening variable is derived enabling a more efficient single surface formulation for the regularised plasticity‐damage material model.

Parameter identification of strain rate dependent hardening for sheet metals

AbstractIn this contribution we examine the influence of strain rate on the isotropic hardening of two sheet metals. The nominal strain rates vary between 0.0005 s−1 and 20 s−1 and the full field deformation is captured with the help of digital image correlation (DIC) technique. For modelling the strain rate dependent behaviour, we introduce a rate dependent part in the definition of the hardening stress, distinguishing between approaches that incorporate the rate dependency in an additive or multiplicative manner. The hardening laws are thereby embedded in a large strain plasticity framework based on the logarithmic strain space. The material parameters are identified both in a “direct” fashion, where the gap between constitutive equation and measured stress‐strain curve is minimized and inverse, by using the finite element method (FEM) to generate force‐displacement curves and the virtual fields method, which compares internal and external virtual work.

Generalized strain space formulations and eigenprojection tensors

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 stretch tensors. These fourth-order transformation tensors also define the tensors of {q, r}-generalized stress which are work-conjugate to the corresponding {q, r}-generalized strain (rate). For finite deformations, every tensor pair of {q, r}-generalized stress and strain constitutes a distinct material model and forms a unique element of the generalized strain space formulations. 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. The resulting logarithmic strain space formulation has proven as most accurate, stable, and efficient by its implementation into the special-purpose finite element simulation tools AutoForm, Pafix, and Urmel.

Computational modeling of amorphous polymers: A Lagrangian logarithmic strain space formulation of a glass–rubber constitutive model

We present a reformulation of the finite strain, rate dependent inelastic glass–rubber material model suggested by Buckley and Jones (1995) and extended by Adams et al. (2000) for modeling the deformation of amorphous polymers in the Lagrangian logarithmic strain space. This not only warrants a hyperelastic characterization in the bonding part which remedies problems associated with hypoelastic approaches devising objective stress rates selected on ad hoc basis, see, e.g., Dooling et al. (2001) and Li and Buckley (2009), but also allows a transparent and naturally objective implementation analogous to the geometrically linear theory. A numerical implementation into Abaqus is pursued where algorithms for stress update and tangent moduli computations are reported. It is shown that significant reduction in nonlinear equation system size is possible in the computation of both bonding and conformational part. The characterization tests include constant-width tension, equi-biaxial tension, and simple shear. To demonstrate the robustness of the developed framework, two hypothetical problems of extreme deformation under tensile and combined tensile and torsion loading are considered. Finally, simulation of an injection stretch-blow molding process is presented as an application problem.

Micromorphic approach for gradient-extended thermo-elastic–plastic solids in the logarithmic strain space

The coupled thermo-mechanical strain gradient plasticity theory that accounts for microstructure-based size effects is outlined within this work. It extends the recent work of Miehe et al. (Comput Methods Appl Mech Eng 268:704–734, 2014) to account for thermal effects at finite strains. From the computational viewpoint, the finite element design of the coupled problem is not straightforward and requires additional strategies due to the difficulties near the elastic–plastic boundaries. To simplify the finite element formulation, we extend it toward the micromorphic approach to gradient thermo-plasticity model in the logarithmic strain space. The key point is the introduction of dual local–global field variables via a penalty method, where only the global fields are restricted by boundary conditions. Hence, the problem of restricting the gradient variable to the plastic domain is relaxed, which makes the formulation very attractive for finite element implementation as discussed in Forest (J Eng Mech 135:117–131, 2009) and Miehe et al. (Philos Trans R Soc A Math Phys Eng Sci 374:20150170, 2016).

Ductile failure with gradient plasticity coupled to the phase‐field fracture at finite strains

AbstractMost metals fail in a ductile fashion, i.e, fracture is preceded by significant plastic deformation. The modeling of failure in ductile metals must account for complex phenomena at micro‐scale, such as nucleation, growth and coalescence of micro‐voids. In this work, we start with von‐Mises plasticity model without considering void generation. The modeling of macroscopic cracks can be achieved in a convenient way by the continuum phase field approaches to fracture, which are based on the regularization of sharp crack discontinuities [1]. This avoids the use of complex discretization methods for crack discontinuities and can account for complex crack patterns. The key aspect of this work is the extension of the energetic and the stress‐based phase field driving force function in brittle fracture to account for a coupled elasto‐plastic response in line with our recent work [3]. We develop a new theoretical and computational framework for the phase field modeling of ductile fracture in elastic‐plastic solids. To account for large strains, the constitutive model is constructed in the logarithmic strain space, which simplify the model equations and results in a formulation similar to small strains. We demonstrate the modeling capabilities and algorithmic performance of the proposed formulation by representative simulations of ductile failure mechanisms in metals. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Analysis of some basic approaches to finite strain elasto-plasticity in view of reference change

There is a large variety of concepts used to generalize the classical Prandtl–Reuss relations of infinitesimal elasto-plasticity to finite strains. In this work, some basic approaches are compared in a qualitative way with respect to a certain invariance property. These basic approaches include the additive hypoelasto-plasticity with corotational stress rates, additive plasticity in the logarithmic strain space, and multiplicative hyperelasto-plasticity.The notion of weak invariance is introduced in this study. Roughly speaking, a material model is weakly invariant under a certain transformation of the local reference configuration if this reference change can be neutralized by a suitable transformation of initial conditions, leaving the remaining constitutive relations intact. We analyse the basic models in order to find out if they are weakly invariant under arbitrary volume-preserving transformations of the reference configuration.It is shown that the weak invariance property corresponds to a generalized symmetry which provides insights into underlying constitutive assumptions. This property can be used for a systematic study of different frameworks of finite strain elasto-plasticity. In particular, it can be used as a classification criterion.

Formulation and implementation of a constitutive model for semicrystalline polymers

The paper outlines a new constitutive model and its numerical implementation for the mechanical behaviour of semicrystalline polymers. Essential for the constitutive model is a homogenization approach that makes use of a representative mesostructure (RMS) and of the micromechanical description of the two characteristic phases of semicrystalline polymers, the amorphous and the crystalline phase. A homogenization approach is used to characterise the macroscopic element by extracting the macroscale characteristics of the material from the results of a microscale boundary value problem. For the microscopic amorphous constitutive model, a new evolution law is introduced for the athermal shear stress to the constitutive model proposed by Miehe et al. (2009), who developed a constitutive formulation for amorphous polymers in the logarithmic strain space. For the microscopic crystalline model, the approach proposed by Simo (1988a), who numerically implemented an algorithm for plasticity for crystalline materials, is transformed into a logarithmic kinematical framework. The modelling capability of the proposed formulation is verified by comparison of numerical simulations to experimental investigations carried out by the Leibniz-Institut für Polymerforschung (IPF) Dresden. The experiments were performed on isotactic polypropylene (iPP) and the mechanical tensile and compression tests were carried out at different temperatures. The model is also used for quasi-static (QS) and dynamic (DY) simulations of an iPP car bumper.

Modeling and simulation of viscous electro-active polymers

Electro-active materials are capable of undergoing large deformation when stimulated by an electric field. They can be divided into electronic and ionic electro-active polymers (EAPs) depending on their actuation mechanism based on their composition. We consider electronic EAPs, for which attractive Coulomb forces or local re-orientation of polar groups cause a bulk deformation. Many of these materials exhibit pronounced visco-elastic behavior. Here we show the development and implementation of a constitutive model, which captures the influence of the electric field on the visco-elastic response within a geometrically non-linear finite element framework. The electric field affects not only the equilibrium part of the strain energy function, but also the viscous part. To adopt the familiar additive split of the strain from the small strain setting, we formulate the governing equations in the logarithmic strain space and additively decompose the logarithmic strain into elastic and viscous parts. We show that the incorporation of the electric field in the viscous response significantly alters the relaxation and hysteresis behavior of the model. Our parametric study demonstrates that the model is sensitive to the choice of the electro-viscous coupling parameters. We simulate several actuator structures to illustrate the performance of the method in typical relaxation and creep scenarios. Our model could serve as a design tool for micro-electro-mechanical systems, microfluidic devices, and stimuli-responsive gels such as artificial skin, tactile displays, or artificial muscle.

An orthotropic viscoelastic material model for passive myocardium: theory and algorithmic treatment

This contribution presents a novel constitutive model in order to simulate an orthotropic rate-dependent behaviour of the passive myocardium at finite strains. The motivation for the consideration of orthotropic viscous effects in a constitutive level lies in the disagreement between theoretical predictions and experimentally observed results. In view of experimental observations, the material is deemed as nearly incompressible, hyperelastic, orthotropic and viscous. The viscoelastic response is formulated by means of a rheological model consisting of a spring coupled with a Maxwell element in parallel. In this context, the isochoric free energy function is decomposed into elastic equilibrium and viscous non-equilibrium parts. The baseline elastic response is modelled by the orthotropic model of Holzapfel and Ogden [Holzapfel GA, Ogden RW. 2009. Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Philos Trans Roy Soc A Math Phys Eng Sci. 367:3445–3475]. The essential aspect of the proposed model is the account of distinct relaxation mechanisms for each orientation direction. To this end, the non-equilibrium response of the free energy function is constructed in the logarithmic strain space and additively decomposed into three anisotropic parts, denoting fibre, sheet and normal directions each accompanied by a distinct dissipation potential governing the evolution of viscous strains associated with each orientation direction. The evolution equations governing the viscous flow have an energy-activated nonlinear form. The energy storage in the Maxwell branches has a quadratic form leading to a linear stress–strain response in the logarithmic strain space. On the numerical side, the algorithmic aspects suitable for the implicit finite element method are discussed in a Lagrangian setting. The model shows excellent agreement compared to experimental data obtained from the literature. Furthermore, the finite element simulations of a heart cycle carried out with the proposed model show significant deviations in the strain field relative to the elastic solution.

On a recursive formulation for solving inverse form finding problems in isotropic elastoplasticity

BackgroundInverse form finding methods allow conceiving the design of functional components in less time and at lower costs than with direct experiments. The deformed configuration of the functional component, the applied forces and boundary conditions are given and the undeformed configuration of this component is sought.MethodsIn this paper we present a new recursive formulation for solving inverse form finding problems for isotropic elastoplastic materials, based on an inverse mechanical formulation written in the logarithmic strain space. First, the inverse mechanical formulation is applied to the target deformed configuration of the workpiece with the set of internal variables set to zero. Subsequently a direct mechanical formulation is performed on the resulting undeformed configuration, which will capture the path-dependency in elastoplasticity. The so obtained deformed configuration is furthermore compared with the target deformed configuration of the component. If the difference is negligible, the wanted undeformed configuration of the functional component is obtained. Otherwise the computation of the inverse mechanical formulation is started again with the target deformed configuration and the current state of internal variables obtained at the end of the computed direct formulation. This process is continued until convergence is reached.ResultsIn our three numerical examples in isotropic elastoplasticity, the convergence was reached after five, six and nine iterations, respectively, when the set of internal variables is initialised to zero at the beginning of the computation. It was also found that when the initial set of internal variables is initialised to zero at the beginning of the computation the convergence was reached after less iterations and less computational time than with other values. Different starting values for the set of internal variables have no influence on the obtained undeformed configuration, if convergence can be achieved.ConclusionsWith the presented recursive formulation we are able to find an appropriate undeformed configuration for isotropic elastoplastic materials, when only the deformed configuration, the applied forces and boundary conditions are given. An initial homogeneous set of internal variables equal to zero should be considered for such problems.

Mixed Variational Principles and Robust Finite Element Design of Gradient Plasticity at Finite Strains

AbstractThe modeling of size effects in elastic‐plastic solids, such as the width of shear bands or the grain size dependence in polycrystals, must be based on non‐standard theories which incorporate length‐scales. This is achieved by models of strain gradient plasticity, incorporating spatial gradients of selected micro‐structural fields which describe the evolving dissipative mechanisms. The key aspect of this work is to provide a rigorous incremental variational formulation and mixed finite element design of additive finite gradient plasticity in the logarithmic strain space. We start from a mixed saddle point principle for metric‐type plasticity, which is specified for the important model problem of isochoric plasticity with gradient‐extended hardening/softening response. To this end, we propose a novel finite element design of the coupled problem incorporating a local‐global solution strategy of short‐ and long‐range fields. This includes several new aspects, such as extended Q1P0‐type and MINI‐type finite elements for gradient plasticity [4]. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Variational gradient plasticity at finite strains. Part II: Local–global updates and mixed finite elements for additive plasticity in the logarithmic strain space

The second part of our work on variational inelasticity with long-range effects outlines the formulation and finite element implementation of additive finite gradient plasticity in the logarithmic strain space. It is considered to be the most simple approach to finite plasticity, suitable for the purely phenomenological description of polycrystalline metals or amorphous materials, if structures of the geometrically linear theories are defined in the Lagrangian logarithmic strain space. We start from a mixed saddle point principle for metric-type additive plasticity, which is specified for the important model problem of isochoric von Mises plasticity with gradient-extended hardening/softening response. The mixed variational structure includes the hardening/softening variable itself as well as its dual driving force. The numerical implementation exploits the underlying variational structure, yielding a canonical symmetric structure of the monolithic problem. It results in a novel finite element design of the coupled problem incorporating a long-range hardening/softening parameter and its dual driving force. This allows a straightforward local definition of plastic loading–unloading driven by the long-range fields, providing very robust finite element implementations of gradient plasticity. It includes a rational method for the definition of elastic–plastic-boundaries (EPBs) in gradient plasticity along with a postprocessor that defines the plastic variables in the elastic range. We discuss alternative mixed finite element designs of the coupled problem, including a local–global solution strategy of short-and long-range fields. All methods are derived in a rigorous format from variational principles. Numerical benchmarks demonstrated the excellent performance of the proposed mixed variational approach to gradient plasticity in the logarithmic strain space.

On two different inverse form finding methods for hyperelastic and elastoplastic materials

AbstractThis contribution focuses on two different developments of mechanical‐computational methods for the optimal determination of the initial shape of formed functional components knowing the deformed configuration, the applied loads and the boundary conditions. The first method uses an inverse mechanical formulation and can be applied to materials with hyperelastic behaviors. For materials with elastoplastic properties this method is not advocated, without knowing the final plastic strains, due to the non uniqueness of the solution. The second method uses a shape optimization formulation in the sense of an inverse problem via successive iterations of the direct problem. For hyperelastic materials the inverse mechanical formulation is preferred for its velocity and the non exhibition of possible mesh distortions. In the shape optimization formulation mesh distortions can be avoided by an update of the reference configuration of the functional part. Both methods are using a formulation in the logarithmic strain space. A numerical example for materials with isotropic elastoplastic behaviors illustrates the shape optimization formulation. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)