Finite element analysis of a class of stress-free martensitic microstructures

This work is concerned with the finite element approximation of a class of stress-free martensitic microstructures modeled by multi-well energy minimization. Finite element energy-minimizing sequences are first constructed to obtain bounds on the minimum energy over all admissible finite element deformations. A series of error estimates are then derived for finite element energy minimizers.

     

Nonconforming finite element approximation of crystalline microstructure

We consider a class of nonconforming finite element approximations of a simply laminated microstructure which minimizes the nonconvex variational problem for the deformation of martensitic crystals which can undergo either an orthorhombic to monoclinic (double well) or a cubic to tetragonal (triple well) transformation. We first establish a series of error bounds in terms of elastic energies for the approximation of derivatives of the deformation in the direction tangential to parallel layers of the laminate, for the approximation of the deformation, for the weak approximation of the deformation gradient, for the approximation of volume fractions of deformation gradients, and for the approximation of nonlinear integrals of the deformation gradient. We then use these bounds to give corresponding convergence rates for quasi-optimal finite element approximations.

     

The equations of elastostatics in a Riemannian manifold

To begin with, we identify the equations of elastostatics in a Riemannian manifold, which generalize those of classical elasticity in the three-dimensional Euclidean space. Our approach relies on the principle of least energy, which asserts that the deformation of the elastic body arising in response to given loads minimizes over a specific set of admissible deformations the total energy of the elastic body, defined as the difference between the strain energy and the potential of the loads. Assuming that the strain energy is a function of the metric tensor field induced by the deformation, we first derive the principle of virtual work and the associated nonlinear boundary value problem of nonlinear elasticity from the expression of the total energy of the elastic body. We then show that this boundary value problem possesses a solution if the loads are sufficiently small (in a sense we specify).     

A Regularized Sharp Interface Model for Martensitic Phase Transformations at Large Strains

The macroscopic mechanical behavior of many functional materials crucially depends on the formation and evolution of their microstructure. When considering martensitic shape memory alloys, this microstructure typically consists of laminates with coherent twin boundaries. We suggest a variational-based phase field model for the dissipative evolution of microstructure with coherence-dependent interface energy and construct a suitable gradient-extended incremental variational framework for the proposed dissipative material. We use our model to predict laminate microstructure in martensitic CuAlNi. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Un théorème d'existence pour une coque non linéairement élastique ≪ en flexion ≫

We establish an existence theorem for the two-dimensional equations of a nonlinearly elastic “flexural” shell, recently justified by V. Lods and B. Miara by the method of formal asymptotic expansions applied to the corresponding three-dimensional equations of nonlinear elasticity. To this end, we show that the associated energy has at least one minimizer over the corresponding set of admissible deformations. The strain energy is a quadratic expression in terms of the “exact” change of curvature tensor, between the deformed and undeformed middle surfaces; the set of admissible deformations is formed by the deformations of the undeformed middle surface that preserve its metric and satisfy boundary conditions of clamping or simple support.     

Injectivity and self-contact in second-gradient nonlinear elasticity

We prove the existence of globally injective weak solutions in mixed boundary-value problems of second-gradient nonlinear elastostatics via energy minimization. This entails the treatment of self-contact. In accordance with the classical (first-gradient) theory, the model incorporates the unbounded growth of the potential energy density as the local volume ratio approaches zero. We work in a class of admissible vector-valued deformations that are injective on the interior of the domain. We first establish a rigorous Euler–Lagrange variational inequality at a minimizer. We then define a self-contact coincidence set for an admissible deformation in a natural way, which we demonstrate to be confined to a closed subset of the boundary of the domain. We then prove the existence of a non-negative (Radon) measure, vanishing outside of the coincidence set, which represents the normal contact-reaction force distribution. With this in hand, we obtain the weak form of the equilibrium equations at a minimizer.     

Experimental and numerical investigation of metal foams undergoing large deformations

Open cell aluminum metal foams are a new kind of material that are used in composite structures to reduce their weight, to increase their sound or energy absorption capability or to decrease their thermal conductivity. The design and analysis of such structures requires a macroscopic constitutive model of the foam that has to be determined by various experiments under different loading conditions. We support this procedure by analyzing the microstructure of the metal foam numerically under large deformations. To this end, we employ the finite cell method that can deal with large deformations and allows for an automatic and efficient discretization of the CT-image of the foam. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Lamination Upper Bound to the Free Energy of Shape Memory Alloys

Modeling the energetic behavior of materials showing martensitic phase transformations usually leads to non-convex energy formulations. In a variety of models based on quasi-convex analysis, the Reuß lower bound, which neglects the compatibility constraint for the deformation fluctuations, is used as an estimate for the so-called energy of mixing. We present an upper bound that is on the one hand based on the lamination mixture formula, which gives an estimate of the free energy of two-variant materials and is extended to a specialized n-variant case in our work. On the other hand, we rely on experimentally well established assumptions about the type of microstructure that forms in such alloys. More precisely, we restrict the set of physically admissible microstructures to the subset of second order laminated microstructres consisting of austenite and twinned martensites. We further refine our upper bound by taking into account the notion of twin-compatibility. For the physically relevant examples of 13- and 7-variant Cu-Al-Ni shape memory alloys, striking congruence is obtained in the comparison of the Reuß lower and our upper bound for fixed volume fractions. Furthermore, we show results of global minimization of the energy obtained by each bound over the volume fractions of the variants. Similarities and differences in the energy-minimizing volume fractions are discussed. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling and Simulation of Rate-Dependent Magneto-Active Polymers

In the present work, the magneto-viscoelastic behavior of MAPs is studied by a thermodynamically consistent constitutive model. A finite deformation based framework of nonlinear magneto-viscoelastic coupling is introduced with a multiplicative decomposition of the deformation gradient. The viscosity is captured by evolution equations of the internal variables introduced. We propose energy functions for pure magnetic and magneto-mechanical coupling such that saturation behavior of the magnetostriction and magnetization is captured. After having established the general framework, the model is studied for homogeneous deformations for the purpose of a least-square-based parameter identification from experimental data. The model predictions of non-linear magneto-mechanical responses with strong rate and field dependency are presented. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An approach to the homogenization of nonlinear elastomers via the theory of unbounded functionals

The homogenization process for some energies of integral type arising in the modelling of rubber-like elastomers is carried out. The main feature of the variational problems taken into account is the presence of pointwise oscillating constraints on the gradients of the admissible deformations. The classical homogenization result is established also in this framework, both for Dirichlet with affine boundary data, Neumann, and mixed problems, by proving that the limit energy is again of integral type, gradient constrained. An explicit computation for the homogenized integrand relative to energy density in a particular relevant case is derived.     

Explicit energy-minimizers of incompressible elastic brittle bars under uniaxial extension

A rectangular bar made of a hyperelastic, but brittle, incompressible homogeneous and isotropic material is subject to uniaxial extension. We prove that the energy minimizers are, depending on the toughness coefficient of the material, either the homogeneous deformation, or the family of deformations for which a horizontal fracture breaks the material in two rectangular pieces, each of which is a rigid motion of the undeformed piece.     

Approximation of a laminated microstructure for a rotationally invariant,double well energy density

Summary. We give error estimates for the approximation of a laminated microstructure which minimizes the energy for a rotationally invariant, double well energy density . We present error estimates for the convergence of the deformation in the convergence of directional derivatives of the deformation in the “twin planes,” the weak convergence of the deformation gradient, the convergence of the microstructure (or Young measure) of the deformation gradients, and the convergence of nonlinear integrals of the deformation gradient. Received July 25, 1995 / Revised version received November 20, 1995     

On smeared and micromechanical approaches to modeling martensitic transformations in SMA

This paper explores a link between a recently proposed macroscopic “smeared” approach and a microscopic/mesoscopic approach of sequential laminate type to model martensitic transformations. In addition, a numerical simulation of the stress-induced martensitic transformation in a single crystal has been performed upon simplification to small deformations. One significant observation in the results of such a simulation is the counter-intuitive change of preferred martensitic plate even under proportional loading conditions. It remains to be seen if it is an artifact of the procedure adopted or the actual shift of active martensitic plate system. A further step toward modeling polycrystal behavior using homogenization with simple bounds has been attempted. Hysteresis results show that there is no clear demarcation of critical stress at which the transformation occurs. This may be critical to the functional fatigue behavior of shape memory materials.     

Linking macroscopic deformation processes to microstructure evolution using an FE-FFT-based micro-macro transition and non-conserved phase-fields

The purpose of this work is the multiscale FE-FFT-based prediction of macroscopic material behavior, micromechanical fields and bulk microstructure evolution in polycrystalline materials subjected to macroscopic mechanical loading. The macroscopic boundary value problem (BVP) is solved using implicit finite element (FE) methods. In each macroscopic integration point, the microscopic BVP is embedded, the solution of which is found employing fast Fourier transform (FFT), fixed-point and Green's function methods. The mean material response is determined by the stress-strain relation at the micro scale or rather the volume average of the micromechanical fields. The evolution of the microstructure is modeled by means of non-conserved phase-fields. As an example, the proposed methodology is applied to the modeling of stress-induced martensitic phase transformations in metal alloys. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of microstructures for some martensitic transformations

We analyze the stability of laminated microstructure for martensitic crystals that undergo cubic to trigonal, orthorhombic to triclinic, and trigonal to monoclinic transformations. We show that the microstructure is unique and stable for all laminates except when the lattice parameters satisfy certain identities.     

A constitutive model for shape memory alloy fibres and its applicability to prestressed composites

This contribution is concerned with a constitutive model for shape memory fibres. The 1D-constitutive model accounts for the pseudoplastic and shape memory effect (SME). The macroscopic answer of the material is determined by the evolution from a twinned martensitic lattice into a deformed and detwinned one. On the macroscopic scale these effects are responsible for the upper boundary of the hysteresis which is situated around the origin of the stress-strain-diagram. During the phase transition process inelastic strains arise. When the lattice is fully detwinned, a linear elastic branch at the end of the hysteresis is observed. The initial state of the material is recovered by unloading and heating the material subsequently. The constitutive model is derived from the Helmholtz' free energy and fulfils the 2nd law of thermodynamics. For the present model five internal state variables are employed. Two of them are used to describe the inelastic strain and a backstress. The others represent the martensitic volume fraction and are necessary to describe the SME. The latter variables are depending on the deformation state as well as on temperature. A change on temperature goes along with a reduction of the inelastic strain. The model is incorporated in a fibre matrix discretization to prestress the surrounding structure. The boundary value problem is solved for a truss element applying the finite element method. Examples will demonstrate the applicability in engineering structures. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Large deformations of bodies of revolution made of elastic homogeneous and fiber-reinforced materials 1. Torsion of toroidal bodies

Equations of a mathematical model for bodies of revolution made of elastic homogeneous and fiber-reinforced materials and subjected to large deformations are presented. The volume content of reinforcing fibers is assumed low, and their interaction through the matrix is neglected. The axial lines of the fibers can lie both on surfaces of revolution whose symmetry axes coincide with the axis of the body of revolution and along trajectories directed outside the surfaces. The equations are obtained for the macroscopically axisymmetric problem statement where the parameters of macroscopic deformation of the body vary in its meridional planes, but are constant in the circumferential directions orthogonal to them. The equations also describe the torsion of bodies of revolution and their deformation behavior under the action of inertia forces in rotation around the symmetry axis. The results of a numerical investigation into the large deformations of toroidal bodies made of elastic homogeneous and unidirectionally reinforced materials under torsion caused by a relative rotation of their butt-end sections around the symmetry axis are presented.     

Deformation energy function of large human blood vessels

A function of the specific energy of deformation, selected in the form of a number of exponents, is proposed. It describes well the stress-strain state of anisotropic human blood vessel at large deformations. The constants of the material included in the deformation energy function are determined by experiments for a monoaxial tensioning, along the main anisotropy axes. As an example, they were found for the human abdominal aorta, taken during an autopsy (male, age 29 years), by approximation of the experimental data on a computer by the method of least squares.     

A variational approach to optimal meshes

Summary. A variational approach for the optimization of triangular or tetrahedral meshes is presented. Starting from some very basic assumptions we will rigorously demonstrate that the functional controlling optimality is of a certain type related to energy functionals in non linear elasticity. It will be proved that these functionals attain their minima over admissible sets of mesh deformations which respect boundary conditions. In addition the injectivity of the deformed mesh is discussed. Thereby it is possible to construct suitable meshes for various numerical applications. Received March 14, 1994 / Revised version received August 8, 1994