Microscopic and macroscopic instabilities in finitely strained fiber-reinforced elastomers

The present work is a detailed study of the connections between microstructural instabilities and their macroscopic manifestations — as captured through the effective properties — in finitely strained fiber-reinforced elastomers, subjected to finite, plane-strain deformations normal to the fiber direction. The work, which is a complement to a previous and analogous investigation by the same authors on porous elastomers, (Michel et al., 2007), uses the linear comparison, second-order homogenization (S.O.H.) technique, initially developed for random media, to study the onset of failure in periodic fiber-reinforced elastomers and to compare the results to more accurate finite element method (F.E.M.) calculations. The influence of different fiber distributions (random and periodic), initial fiber volume fraction, matrix constitutive law and fiber cross-section on the microscopic buckling (for periodic microgeometries) and macroscopic loss of ellipticity (for all microgeometries) is investigated in detail. In addition, constraints to the principal solution due to fiber/matrix interface decohesion, matrix cavitation and fiber contact are also addressed. It is found that both microscopic and macroscopic instabilities can occur for periodic microstructures, due to a symmetry breaking in the periodic arrangement of the fibers. On the other hand, no instabilities are found for the case of random microstructures with circular section fibers, while only macroscopic instabilities are found for the case of elliptical section fibers, due to a symmetry breaking in their orientation.     

Microscopic and macroscopic instabilities in finitely strained porous elastomers

The present work is an in-depth study of the connections between microstructural instabilities and their macroscopic manifestations—as captured through the effective properties—in finitely strained porous elastomers. The powerful second-order homogenization (SOH) technique initially developed for random media, is used for the first time here to study the onset of failure in periodic porous elastomers and the results are compared to more accurate finite element method (FEM) calculations. The influence of different microgeometries (random and periodic), initial porosity, matrix constitutive law and macroscopic load orientation on the microscopic buckling (for periodic microgeometries) and macroscopic loss of ellipticity (for all microgeometries) is investigated in detail. In addition to the above-described stability-based onset-of-failure mechanisms, constraints on the principal solution are also addressed, thus giving a complete picture of the different possible failure mechanisms present in finitely strained porous elastomers.     

FE-analysis of sheet metal forming processes using continuous contact treatment

Discrete meshes cause stepwise propagation of the contact nodes of a sheet despite the fact that the contact region in the actual forming process is altered very smoothly. This can cause problems of convergence and accuracy in contact-sensitive processes, such as a bending process. In this study, a scheme for a continuous contact treatment is proposed in order to consider the more realistic behavior of the contact phenomena during the forming process. For verification of the proposed method, the contact pressures and forming load are evaluated during the compression forming of a tube. The analysis of a hemispherical dome formed without a blank holder is also presented in order to investigate the effects of the proposed algorithm. The results show that the precise deformation mode is predicted by the utilization of the proposed method.     

On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: I—Theory

This work presents an analytical framework for determining the overall constitutive response of elastomers that are reinforced by rigid or compliant fibers, and are subjected to finite deformations. The framework accounts for the evolution of the underlying microstructure, including particle rotation, which results from the finite changes in geometry that are induced by the applied loading. In turn, the evolution of the microstructure can have a significant geometric softening (or hardening) effect on the overall response, leading to the possible development of macroscopic instabilities through loss of strong ellipticity of the homogenized incremental moduli. The theory is based on a recently developed “second-order” homogenization method, which makes use of information on both the first and second moments of the fields in a suitably chosen “linear comparison composite,” and generates fairly explicit estimates—linearizing properly—for the large-deformation effective response of the reinforced elastomers. More specific applications of the results developed in this paper will be presented in Part II.     

A generic approach towards finite growth with examples of athlete's heart, cardiac dilation, and cardiac wall thickening

The objective of this work is to establish a generic continuum-based computational concept for finite growth of living biological tissues. The underlying idea is the introduction of an incompatible growth configuration which naturally introduces a multiplicative decomposition of the deformation gradient into an elastic and a growth part. The two major challenges of finite growth are the kinematic characterization of the growth tensor and the identification of mechanical driving forces for its evolution. Motivated by morphological changes in cell geometry, we illustrate a micromechanically motivated ansatz for the growth tensor for cardiac tissue that can capture both strain-driven ventricular dilation and stress-driven wall thickening. Guided by clinical observations, we explore three distinct pathophysiological cases: athlete's heart, cardiac dilation, and cardiac wall thickening. We demonstrate the computational solution of finite growth within a fully implicit incremental iterative Newton-Raphson based finite element solution scheme. The features of the proposed approach are illustrated and compared for the three different growth pathologies in terms of a generic bi-ventricular heart model.     

Onset of failure in finitely strained layered composites subjected to combined normal and shear loading

A limiting factor in the design of fiber-reinforced composites is their failure under axial compression along the fiber direction. These critical axial stresses are significantly reduced in the presence of shear stresses. This investigation is motivated by the desire to study the onset of failure in fiber-reinforced composites under arbitrary multi-axial loading and in the absence of the experimentally inevitable imperfections and finite boundaries.By using a finite strain continuum mechanics formulation for the bifurcation (buckling) problem of a rate-independent, perfectly periodic (layered) solid of infinite extent, we are able to study the influence of load orientation, material properties and fiber volume fraction on the onset of instability in fiber-reinforced composites. Two applications of the general theory are presented in detail, one for a finitely strained elastic rubber composite and another for a graphite-epoxy composite, whose constitutive properties have been determined experimentally. For the latter case, extensive comparisons are made between the predictions of our general theory and the available experimental results as well as to the existing approximate structural theories. It is found that the load orientation, material properties and fiber volume fraction have substantial effects on the onset of failure stresses as well as on the type of the corresponding mode (local or global).     

Thermomechanical formulation of ductile damage coupled to nonlinear isotropic hardening and multiplicative viscoplasticity

In this paper, we present a thermomechanical framework which makes use of the internal variable theory of thermodynamics for damage-coupled finite viscoplasticity with nonlinear isotropic hardening. Damage evolution, being an irreversible process, generates heat. In addition to its direct effect on material's strength and stiffness, it causes deterioration of the heat conduction. The formulation, following the footsteps of Simó and Miehe (1992), introduces inelastic entropy as an additional state variable. Given a temperature dependent damage dissipation potential, we show that the evolution of inelastic entropy assumes a split form relating to plastic and damage parts, respectively. The solution of the thermomechanical problem is based on the so-called isothermal split. This allows the use of the model in 2D and 3D example problems involving geometrical imperfection triggered necking in an axisymmetric bar and thermally triggered necking of a 3D rectangular bar.     

A variational model for fracture mechanics: Numerical experiments

In the variational model for brittle fracture proposed in Francfort and Marigo [1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319-1342], the minimum problem is formulated as a free discontinuity problem for the energy functional of a linear elastic body. A family of approximating regularized problems is then defined, each of which can be solved numerically by a finite element procedure. Here we re-formulate the minimum problem within the context of finite elasticity. The main change is the introduction of the dependence of the strain energy density on the determinant of the deformation gradient. This change requires new, more general existence and Γ-convergence results. The results of some two-dimensional numerical simulations are presented, and compared with corresponding simulations made in Bourdin et al. [2000. Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48, 797-826] for the linear elastic model.     

Magnetoelastic buckling of a rectangular block in plane strain

Of interest here is the stability of a rectangular block subjected to a uniform magnetic field perpendicular to its longitudinal axis. The two ends of the block are frictionless and kept parallel to each other. This boundary value problem is motivated by the classical problem of magnetoelastic buckling in which a cantilever beam subjected to a transverse magnetic field buckles when the applied field reaches a critical value.This work presents a finite strain continuum mechanics formulation of the stability problem of a homogeneous, compressible, magnetoelastic rectangular block in plane strain subjected to a uniform transverse magnetic field. The applied variational approach employs an unconstrained energy minimization recently proposed by the authors.The analytical solution for the critical buckling fields for both the antisymmetric and symmetric modes are obtained for three different constitutive laws. The corresponding result for thin beams is extracted asymptotically for a special material and the solution is compared to previously published results. The critical magnetic field is shown to increase monotonically with the block's aspect ratio for each material and mode type. Antisymmetric modes are always the critical buckling modes for stress saturated and neo-Hookean materials, except for a narrow range of moderate aspect ratios (about 0.25) where symmetric modes become critical. For strain-saturated solids no buckling is possible above a maximum aspect ratio.     

Analysis of material instabilities in inelastic solids by incremental energy minimization and relaxation methods: evolving deformation microstructures in finite plasticity

We propose an approach to the definition and analysis of material instabilities in rate-independent standard dissipative solids at finite strains based on finite-step-sized incremental energy minimization principles. The point of departure is a recently developed constitutive minimization principle for standard dissipative materials that optimizes a generalized incremental work function with respect to the internal variables. In an incremental setting at finite time steps this variational problem defines a quasi-hyperelastic stress potential. The existence of this potential allows to be recast a typical incremental boundary-value problem of quasi-static inelasticity into a principle of minimum incremental energy for standard dissipative solids. Mathematical existence theorems for sufficiently regular minimizers then induce a definition of the material stability of the inelastic material response in terms of the sequentially weakly lower semicontinuity of the incremental variational functional. As a consequence, the incremental material stability of standard dissipative solids may be defined in terms of the quasi-convexity or the rank-one convexity of the incremental stress potential. This global definition includes the classical local Hadamard condition but is more general. Furthermore, the variational setting opens up the possibility to analyze the post-critical development of deformation microstructures in non-stable inelastic materials based on energy relaxation methods. We outline minimization principles of quasi- and rank-one convexifications of incremental non-convex stress potentials for standard dissipative solids. The general concepts are applied to the analysis of evolving deformation microstructures in single-slip plasticity. For this canonical model problem, we outline details of the constitutive variational formulation and develop numerical and semi-analytical solution methods for a first-level rank-one convexification. A set of representative numerical investigations analyze the development of deformation microstructures in the form of rank-one laminates in single slip plasticity for homogeneous macro-deformation modes as well as inhomogeneous macroscopic boundary-value problems. The well-posedness of the relaxed variational formulation is indicated by an independence of typical finite element solutions on the mesh-size.     

Coupled thermo-mechanical modelling of bulk-metallic glasses: Theory, finite-element simulations and experimental verification

A three-dimensional, finite-deformation-based constitutive model to describe the behavior of metallic glasses in the supercooled liquid region has been developed. By formulating the theory using the principles of thermodynamics and the concept of micro-force balance [Gurtin, M., 2000. On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. J. Mech. Phys. Solids 48, 989-1036], a kinetic equation for the free volume concentration is derived by augmenting the Helmholtz free energy used for a conventional metallic alloy with a flow-defect free energy which depends on the free volume concentration and its spatial gradient. The developed constitutive model has also been implemented in the commercially available finite-element program ABAQUS/Explicit (2005) by writing a user-material subroutine. The constitutive parameters/functions in the model were calibrated by fitting the constitutive model to the experimental simple compression stress-strain curves conducted under a variety of strain-rates at a temperature in the supercooled liquid region [Lu, J., Ravichandran, G., Johnson, W., 2003. Deformation behavior of the Zr-Ti-Cu-Ni-Be bulk metallic glass over a wide range of strain-rates and temperatures. Acta Mater. 51, 3429-3443].With the model calibrated, the constitutive model was able to reproduce the simple compression stress-strain curves for jump-in-strain-rate experiments to good accuracy. Furthermore stress-strain responses for simple compression experiments conducted at different ambient temperatures within the supercooled liquid region were also accurately reproduced by the constitutive model. Finally, shear localization studies also show that the constitutive model can reasonably well predict the orientation of shear bands for compression experiments conducted at temperatures within the supercooled liquid region [Wang, G., Shen, J., Sun, J., Lu, Z., Stachurski, Z., Zhou, B., 2005. Compressive fracture characteristics of a Zr-based bulk metallic glass at high test temperatures. Mater. Sci. Eng. A 398, 82-87].     

Finite deformation pseudo-elasticity of shape memory alloys – Constitutive modelling and finite element implementation

In this paper we suggest a new phenomenological material model for shape memory alloys. In contrast to many earlier concepts of this kind the present approach includes arbitrarily large deformations. The work is motivated by the requirement, also expressed by regulatory agencies, to carry out finite element simulations of NiTi stents. Depending on the quality of the numerical results it is possible to circumvent, at least partially, expensive experimental investigations. Stent structures are usually designed to significantly reduce their diameter during the insertion into a catheter. Thereby large rotations combined with moderate and large strains occur. In this process an agreement of numerical and experimental results is often hard to achieve. One of the reasons for this discrepancy is the use of unrealistic material models which mostly rely on the assumption of small strains. In the present paper we derive a new constitutive model which is no longer limited in this way. Further its efficient implementation into a finite element formulation is shown. One of the key issues in this regard is to fulfil “inelastic” incompressibility in each time increment. Here we suggest a new kind of exponential map where the exponential function is suitably computed by means of the spectral decomposition. A series expansion is completely avoided. Finite element simulations of stent structures show that the new concept is well appropriate to demanding finite element analyses as they occur in practically relevant problems.     

A return-map algorithm for general associative isotropic elasto-plastic materials in large deformation regimes

The present paper addresses a flexible solution algorithm for associative isotropic elasto-plastic materials, i.e. for materials whose elastic and plastic behaviors are described through an isotropic free-energy function, an isotropic yield function and an associative flow rule. The discussion is relative to a large deformation regime, while no hardening mechanisms are included. The algorithm is based on a combination of the operator split method and a return map scheme. Both the algorithm linearization and the requirements for the yield criterion convexity are discussed in detail. Finally, to show the algorithm flexibility and performance, the discussion is specialized to three yield criteria and some test problems are studied.     

Simulation and interpretation of diffuse mode bifurcation of elastoplastic solids

The diffuse mode bifurcation of elastoplastic solids at finite strain is investigated. The multiplicative decomposition of deformation gradient and the hyperelasto-plastic constitutive relationship are adapted to the numerical bifurcation analysis of the elastoplastic solids. First, bifurcation analyses of rectangular plane strain specimens subjected to uniaxial compression are conducted. The onset of the diffuse mode bifurcations from a homogeneous state is detected; moreover, the post-bifurcation states for these modes are traced to arrive at localization to narrow band zones, which look like shear bands. The occurrence of diffuse mode bifurcation, followed by localization, is advanced as a possible mechanism to create complex deformation and localization patterns, such as shear bands. These computational diffuse modes and localization zones are shown to be in good agreement with the associated experimental ones observed for sand specimens to ensure the validity of this mechanism. Next, the degradation of horizontal sway stiffness of a rectangular specimen due to plane strain uniaxial compression is pointed out as a cause of the bifurcation of the first antisymmetric diffuse mode, which triggers the tilting of the specimen. Last, circular and punching failures of a footing on a foundation are simulated.     

Embedded polycrystal plasticity and adaptive sampling

A simulation capability for multi-scale embedded polycrystal plasticity is demonstrated with over two orders of magnitude wall-clock speedup compared to direct embedding. In the coarse-scale material model, the visco-plastic part of the material response is based on parameters determined from polycrystal level fine-scale calculations. Polycrystal plasticity parameters are approximated from fine-scale calculations using adaptive sampling to substantially reduce the total number of expensive fine-scale calculations which must be performed. The adaptive sampling method uses Kriging models for local interpolation of the fine-scale plasticity parameters and a metric-tree database for storage and retrieval of the fine-scale response models. Efficacy of the method is demonstrated through a variety of example problems involving both quasi-static and dynamic loading scenarios.     

Onset of necking in electro-magnetically formed rings

The electromagnetic forming (EMF) process is an attractive manufacturing technique, which uses electromagnetic (Lorentz) body forces to fabricate metallic parts. One of the many advantages of EMF is the considerable ductility increase observed in several metals, with aluminum featuring prominently among them. The quantitative explanation of this phenomenon is the primary motivation of this work.To study the ductility increase due to EMF, an aluminum ring is placed outside a fixed coil, which is connected to a capacitor. Upon the capacitor's discharge, the time varying current in the coil induces a larger current in the ring specimen and the resulting Lorentz forces make it expand. The coupled coil-ring electromagnetic and thermomechanical problem is solved, using an experimentally obtained constitutive model for a particular aluminum alloy. Our results show that for realistic imperfections, the EMF ring starts necking at strains about six times larger than its static counterpart, as observed experimentally. This study establishes the importance of solving the fully coupled electromagnetic and thermomechanical problem and provides insight on how different constitutive parameters influence ductility in an EMF process.     

Cavitation in elastomeric solids: I—A defect-growth theory

It is by now well established that loading conditions with sufficiently large triaxialities can induce the sudden appearance of internal cavities within elastomeric (and other soft) solids. The occurrence of such instabilities, commonly referred to as cavitation, can be attributed to the growth of pre-existing defects into finite sizes. This paper introduces a new theory to study the phenomenon of cavitation in soft solids that: (i) allows to consider general 3D loading conditions with arbitrary triaxiality, (ii) applies to large (including compressible and anisotropic) classes of nonlinear elastic solids, and (iii) incorporates direct information on the initial shape, spatial distribution, and mechanical properties of the underlying defects at which cavitation can initiate. The basic idea is to first cast cavitation in elastomeric solids as a homogenization problem of nonlinear elastic materials containing random distributions of zero-volume cavities, or defects. This problem is then addressed by means of a novel iterated homogenization procedure, which allows to construct solutions for a specific, yet fairly general, class of defects. These include solutions for the change in size of the defects as a function of the applied loading conditions, from which the onset of cavitation — corresponding to the event when the initially infinitesimal defects suddenly grown into finite sizes — can be readily determined. In spite of the generality of the proposed approach, the relevant calculations amount to solving tractable Hamilton-Jacobi equations, in which the initial size of the defects plays the role of “time” and the applied load plays the role of “space”. When specialized to the case of hydrostatic loading conditions, isotropic solids, and defects that are vacuous and isotropically distributed, the proposed theory recovers the classical result of Ball (1982) for radially symmetric cavitation. The nature and implications of this remarkable connection are discussed in detail.     

Cavitation in elastomeric solids: II—Onset-of-cavitation surfaces for Neo-Hookean materials

In Part I of this work we derived a fairly general theory of cavitation in elastomeric solids based on the sudden growth of pre-existing defects. In this article, the theory is used to determine onset-of-cavitation surfaces for Neo-Hookean solids where the defects are isotropically distributed and vacuous. These surfaces correspond to the set of all critical Cauchy stress states at which cavitation ensues; general three-dimensional loadings are considered. Their computation requires the numerical solution of a nonlinear first-order partial differential equation in two variables. The theoretical results indicate that cavitation occurs only for stress states where the three principal Cauchy stresses are tensile, and that the required hydrostatic tensile component increases with increasing shear components. These results are confronted to finite-element simulations for the growth of a small spherical cavity in a Neo-Hookean block under multi-axial loading. Good agreement is found for a wide range of loading conditions. Comparisons with earlier results available in the literature are also provided and discussed. We conclude this work by devising a closed-form approximation to the theoretical surface, which is of remarkable accuracy and mathematical simplicity.