Inelastic Solids Research Articles

Homogenization of high-contrast media in finite-strain elastoplasticity

This work is devoted to the analysis of the interplay between internal variables and high-contrast microstructure in inelastic solids. As a concrete case-study, by means of variational techniques, we derive a macroscopic description for an elastoplastic medium. Specifically, we consider a composite obtained by filling the voids of a periodically perforated stiff matrix by soft inclusions. We study the Γ-convergence of the related energy functionals as the periodicity tends to zero, the main challenge being posed by the lack of coercivity brought about by the degeneracy of the material properties in the soft part. We prove that the Γ-limit, which we compute with respect to a suitable notion of convergence, is the sum of the contributions resulting from each of the two components separately. Eventually, convergence of the energy minimizing configurations is obtained.

A thermodynamically consistent modelling framework for strongly time-dependent bainitic phase transitions

In this work, a thermodynamically consistent constitutive framework is introduced that is capable of reproducing the significant time-dependent behaviour of austenite-to-bainite phase transformations. In particular, the aim is to incorporate the effect of these diffusion-controlled processes by plasticity-like evolution equations instead of incorporating related global diffusion equations. To this end, a variational principle for inelastic solids is adopted and enhanced by an additional term. This term essentially contributes to the evolution equations for the phase volume fractions of several crystallography-based bainite variants. Due to the specific modifications, special attention has to be paid with respect to the fulfilment of thermodynamical consistency, which can be shown to be unconditionally satisfied for the newly proposed modelling framework. The phase transformation model itself is based on the convexification of a multi-well energy density landscape in order to provide the effective material response for possible phase mixtures. Several material parameters are determined via parameter identification based on available experimental results for 51CrV4, which also allow the quantitative evaluation of the predicted results.

Variational updates for general thermo–chemo–mechanical processes of inelastic solids

The variational formulation of coupled mechanical problems has many advantages from the theoretical point of view and also guides the design of numerical methods that have attractive features such as symmetric tangents. In the current work we propose a novel variational principle for three-field, strongly coupled problems involving (inelastic, finite strain) mechanics, thermal transport, and mass diffusion. To obtain this result, it is key to redefine dissipative phenomena as driven by the free entropy thermodynamic potential. Such a reformulation provides a theoretical explanation of existing variational methods and paves the way for the formulation of variational updates applicable to nonlinear multi-field problems. Representative simulations are shown to illustrate the versatility and favorable features of the resulting methods in finite strain thermoplasticity, stress–diffusion, and thermo–chemo–mechanics.

Variational Methods for the Modelling of Inelastic Solids

This workshop brought together two communities working on the same topic from different perspectives. It strengthened the exchange of ideas between experts from both mathematics and mechanics working on a wide range of questions related to the understanding and the prediction of processes in solids. Common tools in the analysis include the development of models within the broad framework of continuum mechanics, calculus of variations, nonlinear partial differential equations, nonlinear functional analysis, Gamma convergence, dimension reduction, homogenization, discretization methods and numerical simulations. The applications of these theories include but are not limited to nonlinear models in plasticity, microscopic theories at different scales, the role of pattern forming processes, effective theories, and effects in singular structures like blisters or folding patterns in thin sheets, passage from atomistic or discrete models to continuum models, interaction of scales and passage from the consideration of one specific time step to the continuous evolution of the system, including the evolution of appropriate measures of the internal structure of the system.

Computational inelasticity for loading conditions on multiple time scales by adaptive step size control

AbstractThis contribution proposes an algorithm based on adaptive step size control for the simulation of inelastic solids and structures undergoing loading conditions at multiple time scales. Adaptivity in time integration of viscoelastic constitutive laws is directed by an refinement indicator which is constructed from integrators of different order, here a fourth‐order Runge‐Kutta (RK) method and linear Backward‐Euler. The key novel aspect is that by virtue of an recently established consistency condition the higher order methods, p ≥ 2, can achieve their full nominal order without fulfilling the weak form of balance of linear momentum in the RK stages, but only at the end of the time interval. A representative numerical example illustrates the performance of the present adaptive method and underpins the computational savings compared with uniform time step sizes. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Admitting Spontaneous Violations of the Second Law in Continuum Thermomechanics

We survey new extensions of continuum mechanics incorporating spontaneous violations of the Second Law (SL), which involve the viscous flow and heat conduction. First, following an account of the Fluctuation Theorem (FT) of statistical mechanics that generalizes the SL, the irreversible entropy is shown to evolve as a submartingale. Next, a stochastic thermomechanics is formulated consistent with the FT, which, according to a revision of classical axioms of continuum mechanics, must be set up on random fields. This development leads to a reformulation of thermoviscous fluids and inelastic solids. These two unconventional constitutive behaviors may jointly occur in nano-poromechanics.

An analytical framework to extend the general structural stability analysis by considering certain inelastic effects—theory and application to delaminated composites

An analytical framework which incorporates damage propagation/growth into the general structural stability analysis is presented. Therefore, the conventional total potential energy approach is extended by introducing an extended total potential energy-like functional capable of describing inelastic processes in which equilibrium holds between available and the required force for producing a change in structure. The work deals with systems which are described by I generalized coordinates and K damage parameters. The damage parameters are found to be functions of I generalized coordinates and M load parameters. The underlying variational principle for inelastic solids may be solved using discrete formulations or approximate methods such as a Rayleigh–Ritz formulation. This leads to a set of non-linear algebraic equations, comprising post-critical equilibrium paths and damage propagation. In order to verify the framework, it is applied to the well-known problem in which a delaminated composite strut/plate is subjected to an in-plane compressive load.

Nonuniform shear strains in torsional Kolsky bar tests on soft specimens

Abstract : We investigate inertial effects in torsional Kolsky bar tests on nearly incompressible, soft materials. The results are relevant for materials with instantaneous elastic shear modulus on the order of 1 1000 kPa and density on the order of water. Examples include brain tissue and many other soft tissues and tissue surrogates. We have conducted one- and three-dimensional analyses and simulations to understand the stress and strain states that exist in these materials in a torsional Kolsky bar test. We demonstrate that the short loading pulses typically used for high strain-rate (e.g., 700 = s) tests do not allow the softer specimens to ring-up to uniform stress and strain states and that consequently the shear stress versus shear strain data reported in the literature are erroneous. We also show that normal stress components, which are present even in quasistatic torsion of nonlinear elastic materials, can be amplified in dynamic torsion tests on soft materials. Kolsky bar tests are widely used to study strain-rate effects in inelastic solids [Gray 2000; Chen and Song 2011]. In a compression Kolsky bar test, a relatively thin specimen is sandwiched between two bars. Impact at one end of the incident bar generates a compressive wave that travels along the bar, through the specimen, and into a transmission bar. The goal of this test is to subject the specimen to uniform uniaxial stress and uniform (biaxial) strain at a prescribed axial strain-rate. These uniform conditions can often be achieved after an initial ring-up period involving multiple wave reflections from the specimentransmission bar and specimen-incident bar interfaces. Once uniform conditions have been achieved in the specimen, the axial stress, axial strain, and axial strain-rate histories in the specimen can be deduced from strain gage measurements on the elastic bars: the axial stress is proportional to the strain in the transmission bar, and the axial strain-rate is proportional to the strain in t

Stabilized numerical solution for transient dynamic contact of inelastic solids on rough surfaces

A simulation technique to deal with transient dynamic contact of tire rubber compounds on rough road surfaces is presented. The segment-to-surface approach is used for modeling the contact between tire tread rubber and road track. While the rubber components are deformable and described by a sophisticated viscoelastic damage constitutive model, the road surface is assumed to be rigid and characterized by an analytical function. A spectral approach based on an inverse computation of the 2D-Fast Fourier transform has been suggested for the reconstruction of rough surface profiles. The Newmark time-stepping method is used for the integration of transient dynamic equations. With the so-called contact-stabilized Newmark method the spurious oscillation at contact boundary has been removed. The detailed investigation on the dynamic contact of inelastic rubber block with rough road surfaces has been made. The robustness of the contact-stabilized Newmark method within a finite deformation framework is underlined by numerical studies, in which it is compared with several dissipation-based stabilization techniques selected from literature.

Mixed variational principles for the evolution problem of gradient–extended dissipative solids

AbstractThis work develops minimization and saddle point principles for a class of gradient–extended dissipative solids, incorporating micro–structural fields (micro–displacements, order parameters or generalized internal variables), whose gradients enter the energy storage and dissipation potential functions. In contrast to classical local continuum approaches to inelastic solids based on locally evolving internal variables, these global micro–structural fields are governed by additional balance–type partial differential equations including micro–structural boundary conditions. Typical examples are theories of phase field evolution, gradient damage or strain gradient plasticity. Such models incorporate non–local effects based on length scales, which reflect properties of the material micro–structure. We outline a unified variational framework for the evolution problem of first–order gradient–extended standard dissipative solids. Particular emphasis is put on mixed multi–field representations, where both the microstructural variable itself as well as the local driving forces are present (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Thermodynamic model formulation for viscoplastic solids as general equations for non-equilibrium reversible–irreversible coupling

Thermodynamic models for viscoplastic solids are often formulated in the context of continuum thermodynamics and the dissipation principle. The purpose of the current work is to show that models for such material behavior can also be formulated in the form of a General Equation for Non-Equilibrium Reversible–Irreversible Coupling (GENERIC), see, e.g., Grmela and Ottinger (Phys Rev E, 56:6620–6632, 1997), Ottinger and Grmela (Phys Rev E, 56:6633–6655, 1997), Grmela (J Non-Newtonian Fluid Mech, 165:980–986, 2010). A GENERIC combines Hamiltonian-dynamics-based modeling of time-reversible processes with Onsager–Casimir-based modeling of time-irreversible processes. The result is a model for the approach of non-equilibrium systems to thermodynamic equilibrium. Originally developed to model complex fluids, it has recently been applied to anisotropic inelastic solids in Eulerian (Hutter and Tervoort, in J Non-Newtonian Fluid Mech, 152:45–52, 2008; Hutter and Tervoort, in J Non-Newtonian Fluid Mech, 152:53–65, 2008; Hutter and Tervoort, in Adv Appl Mech, 42:254–317, 2008) and Lagrangian (Hutter and Svendsen, in J Elast 104:357–368, 2011) settings, as well as to damage mechanics. For simplicity, attention is focused in the current work on the case of thermoelastic viscoplasticity. Central to this formulation is a GENERIC-based form for the viscoplastic flow rule. A detailed comparison with the formulation based on continuum thermodynamics and the dissipation principle is given.

Finite element simulation of strain localization in inelastic solids by energy relaxation

A new approach for the treatment of strain localization in inelastic material is proposed. It is based on energy minimization principles associated with micro-structure developments. Shear bands are treated as micro-shearing of rank-one laminates. It is assumed that the thickness of the shear band represented by its volume fraction tends to zero, and the energy inside the shear band is a function of the norm of the strain field. The existence of shear bands in the structure leads to an ill-posed problem which can be solved by means of energy relaxation. The performance of the proposed concept is demonstrated through numerical simulation of tension test under plane strain conditions. Numerical results show that mesh sensitivity can be completely removed.

Damage evolution and thermal coupled effects in inelastic solids

Mechanical degradation and ductile failure in metal forming operations can be successfully modelled using Continuum Damage Mechanics. In addition to elastic–plastic deformations, heat transfer affects material behaviour, imposing further effects upon damage evolution. This paper addresses modelling aspects and presents a numerical discussion of the coupled effects between ductile damage and temperature evolution. The heat transfer problem is formulated based on a transient heat conduction approximation, in which heat transfer at the free surfaces and heating due to dissipation of the inelastic work are accounted for. Damage is modelled using an elastic–plastic fully coupled approximation, in which differences in tensile and compressive stress states are included.

On the Formulation of Continuum Thermodynamic Models for Solids as General Equations for Non-equilibrium Reversible-Irreversible Coupling

The purpose of this work is the comparison of some aspects of the formulation of material models in the context of continuum thermodynamics (e.g., Silhavý in The mechanics and thermodynamics of continuous media, Springer, Berlin, 1997) with their formulation in the form of a General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC: e.g., Grmela and Ottinger in Phys. Rev. E 56: 6620–6632, 1997; Ottinger and Grmela in Phys. Rev. E 56: 6633–6655, 1997; Ottinger in Beyond equilibrium thermodynamics, Wiley, New York, 2005; Grmela in J. Non-Newton. Fluid Mech. 165: 980–998, 2010). A GENERIC represents a generalization of the Ginzburg-Landau model for the approach of non-equilibrium systems to thermodynamic equilibrium. Originally developed to formulate non-equilibrium thermodynamic models for complex fluids, it has recently been applied to anisotropic inelastic solids in a Eulerian setting (Hutter and Tervoort in J. Non-Newton. Fluid Mech. 152: 45–52, 2008; 53–65, 2008; Adv. Appl. Mech. 42: 254–317, 2009) as well as to damage mechanics (Hutter and Tervoort in Acta Mech. 201: 297–312, 2008). In the current work, attention is focused for simplicity on the case of thermoelastic solids with heat conduction and viscosity in a Lagrangian setting (e.g., Silhavý in The mechanics and thermodynamics of continuous media, Springer, Berlin, 1997, Chaps. 9–12). In the process, the relation of the two formulations to each other is investigated in detail. A particular point in this regard is the concept of dissipation and its model representation in both contexts.

A multi-field incremental variational framework for gradient-extended standard dissipative solids

The paper presents a constitutive framework for solids with dissipative micro-structures based on compact variational statements. It develops incremental minimization and saddle point principles for a class of gradient-type dissipative materials which incorporate micro-structural fields (micro-displacements, order parameters, or generalized internal variables), whose gradients enter the energy storage and dissipation functions. In contrast to classical local continuum approaches to inelastic solids based on locally evolving internal variables, these global micro-structural fields are governed by additional balance equations including micro-structural boundary conditions. They describe changes of the substructure of the material which evolve relatively to the material as a whole. Typical examples are theories of phase field evolution, gradient damage, or strain gradient plasticity. Such models incorporate non-local effects based on length scales, which reflect properties of the material micro-structure. We outline a unified framework for the broad class of first-order gradient-type standard dissipative solids. Particular emphasis is put on alternative multi-field representations, where both the microstructural variable itself as well as its dual driving force are present. These three-field settings are suitable for models with threshold- or yield-functions formulated in the space of the driving forces. It is shown that the coupled macro- and micro-balances follow in a natural way as the Euler equations of minimization and saddle point principles, which are based on properly defined incremental potentials. These multi-field potential functionals are outlined in both a continuous rate formulation and a time–space-discrete incremental setting. The inherent symmetry of the proposed multi-field formulations is an attractive feature with regard to their numerical implementation. The unified character of the framework is demonstrated by a spectrum of model problems, which covers phase field models and formulations of gradient damage and plasticity.

Dynamic inelastic structural analysis by the BEM: A review

This review paper describes the most widely used techniques associated with the dynamic analysis of inelastic solids and structures by the boundary element method (BEM). Firstly, a historical overview is presented. Next, the various existing BEM formulations for dynamic analysis of two- and three-dimensional solids and structures as well as plates and shells are briefly described. Inelasticity refers to elastoplastic, damage or elastoplastic plus damage material behaviour. Then, five numerical examples from the literature and three new examples of the authors are presented to illustrate the applicability and accuracy of these boundary element methodologies. Finally, advantages and disadvantages of the various methods as well as future developments are presented in the conclusions.

Configurational–Force–Based Adaptive Methods in Nonlocal Gradient‐Type Inelastic Solids

AbstractWe propose a configurational‐force‐based framework for h‐adaptive finite element discretizations of solids with nonlocal, gradient‐type constitutive response. Typical applications are related to gradient‐type damage mechanics, strain gradient plasticity and regularized brittle fracture. On the theoretical side, we outline a general incremental variational framework for the multifield problem of gradient‐type dissipative solids, where generalized internal variable fields account for the current state of evolving microstructures. The Euler equations of the multifield variational principle define the macroscopic balance of momentum along with balance‐type evolution equations for the generalized internal variables in the physical space as well as the balance of configurational forces in the material space. We propose a staggered computational scheme for satisfying those balances in both the physical as well as the material space. The coupled micro‐ and macro‐structural balances of momentum and internal variables provide a solution in the physical space for a given finite element mesh. The balance in the material space is then used to provide an indicator for the quality of the finite element mesh and accounts for a subsequent h‐type mesh refinement. Such a configurational‐force‐based approach provides in a natural and unified format mesh refinement indicators for a broad class of complex nonlocal problems. This framework is applied to damage‐type regularized brittle fracture. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Homogenized elastic–viscoplastic behavior of plate-fin structures at high temperatures: Numerical analysis and macroscopic constitutive modeling

In this study, homogenized elastic–viscoplastic behavior of an ultra-fine plate-fin structure fabricated for compact heat exchangers is investigated. First, the homogenized behavior is numerically analyzed using a fully implicit mathematical homogenization scheme of periodic elastic–inelastic solids. A power-law creep relation is assumed to represent the viscoplasticity of base metals at high temperatures. The plate-fin structure is thus shown to exhibit significant anisotropy as well as noticeable compressibility in both the elastic and viscoplastic ranges of the homogenized behavior. Second, a non-linear rate-dependent macroscopic constitutive model is developed using the quadratic yield function proposed for anisotropic compressible plasticity. The resulting constitutive model is shown to be successful for simulating the anisotropy, compressibility, and rate dependency in the homogenized behavior in multi-axial stress states.

Computational model for predicting nonlinear viscoelastic damage evolution in materials subjected to dynamic loading

Many inelastic solids accumulate numerous cracks before failure due to impact loading, thus rendering any exact solution of the IBVP untenable. It is therefore useful to construct computational models that can accurately predict the evolution of damage during actual impact/dynamic events in order to develop design tools for assessing performance characteristics. This paper presents a computational model for predicting the evolution of cracking in structures subjected to dynamic loading. Fracture is modeled via a nonlinear viscoelastic cohesive zone model. Two example problems are shown: one for model validation through comparison with a one-dimensional analytical solution for dynamic viscoelastic debonding, and the other demonstrates the applicability of the approach to model dynamic fracture propagation in the double cantilever beam test with a viscoelastic cohesive zone.

A fully variational algorithmic formulation for wrinkling at finite strains

AbstractThis contribution is concerned with an efficient novel algorithmic formulation for wrinkling at finite strains. In contrast to previously published numerical implementations, the advocated method is fully variational. More precisely, the parameters describing wrinkles or slacks, together with the unknown deformation mapping, are computed jointly by minimizing the potential energy of the considered mechanical system. Furthermore, the wrinkling criteria are naturally included within the presented variational framework. The presented approach allows to employ three–dimensional constitutive models directly, i.e., plane stress conditions characterizing membranes are variationally enforced by minimizing the potential energy with respect to the transversal strains. Since the proposed formulation for wrinkling in membranes is fully variational, it can be conveniently combined with other variational methods (based on energy minimization). As an example, a variationally consistent framework for finite strain plasticity theory is considered. More precisely, the minimization principle characterizing wrinkling in elastic membranes and that describing plasticity in inelastic solids are coupled leading to a novel variational approach for inelastic membranes. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)