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.     

Nonlinear constitutive models from nanoindentation tests using artificial neural networks

This paper presents a new approach using Artificial Neural Networks (ANNs) models to simulate the response during nanohardness tests of a variety of materials with nonlinear behavior. The ANNs continuous input and output variables usually include material parameters, indentation deflection, and resisting force. Different ANN models, including dimensionless input/output variables, are generated and trained with discrete finite-element (FE) simulations with different geometries and nonlinear material parameters. Only the monotonic loading part of the load–displacement indentation response is used to generate the trained ANN models. This is a departure from classical indentation simulations or tests where typically the unloading portion is used to determine the stiffness and hardness. The experimental part of this study includes nanoindentation tests performed on a silicon (Si) substrate with and without a nanocrystalline copper (Cu) film. The new ANN models are used to back-calculate (inverse problem) the in situ nonlinear material parameters for different copper material systems. The results are compared with available data in the literature. The proposed FE–ANN modeling approach is very effective and can be used in calibrating and predicting the in situ inelastic material properties using the monotonic part of the indentation response and for depths above 50 nm where the overall resisting force represents a continuum response.     

Gradient-enhanced softening material models

Classical constitutive models exhibit strong mesh dependency during softening and the numerical responses tend towards perfectly brittle behavior upon mesh refinements. Such sensitivity can be avoided by adopting the gradient-enhanced formulation. The implicit approach incorporates the gradient contributions indirectly via an additional Helmholtz equation and requires only C⁰ continuity. The explicit approach computes the gradient terms directly from the local field variables. Assuming a weak satisfaction of the yield function, C¹ continuity or C⁰ continuity with additional degrees of freedoms in the penalty approach is required. This makes the explicit method less attractive computationally. However, the explicit approach is able to fully regularize some material models where the standard implicit method fails to perform. Drawing analogy to the over-nonlocal integral formulation, the over-implicit-gradient framework is proposed. In addition, an alternative framework for the explicit gradient method requiring only C⁰ continuity is proposed. The regularizing effects of the abovementioned two gradient frameworks show promising applications to strain-softening materials.     

Indentation of a hard film on a soft substrate: Strain gradient hardening effects

The strain gradient work hardening is important in micro-indentation of bulk metals and thin metallic films, though the indentation of thin films may display very different behavior from that of bulk metals. We use the conventional theory of mechanism-based strain gradient plasticity (CMSG) to study the indentation of a hard tungsten film on soft aluminum substrate, and find good agreement with experiments. The effect of friction stress (intrinsic lattice resistance), which is important in body-center-cubic tungsten, is accounted for. We also extend CMSG to a finite deformation theory since the indentation depth in experiments can be as large as the film thickness. Contrary to indentation of bulk metals or soft metallic films on hard substrate, the micro-indentation hardness of a hard tungsten film on soft aluminum substrate decreases monotonically with the increasing depth of indentation, and it never approaches a constant (macroscopic hardness). It is also shown that the strain gradient effect in the soft aluminum substrate is insignificant, but that in the hard tungsten thin film is important in shallow indentation. The strain gradient effect in tungsten, however, disappears rapidly as the indentation depth increases because the intrinsic material length in tungsten is rather small.     

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.     

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.     

A large deformation theory for rate-dependent elastic–plastic materials with combined isotropic and kinematic hardening

We have developed a large deformation viscoplasticity theory with combined isotropic and kinematic hardening based on the dual decompositions F=F^eF^p [Kröner, E., 1960. Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Archive for Rational Mechanics and Analysis 4, 273–334] and [Lion, A., 2000. Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. International Journal of Plasticity 16, 469–494]. The elastic distortion F^e contributes to a standard elastic free-energy ψ^(e), while , the energetic part of F^p, contributes to a defect energy ψ^(p) – these two additive contributions to the total free energy in turn lead to the standard Cauchy stress and a back-stress. Since F^e=FF^p-1 and , the evolution of the Cauchy stress and the back-stress in a deformation-driven problem is governed by evolution equations for F^p and – the two flow rules of the theory.We have also developed a simple, stable, semi-implicit time-integration procedure for the constitutive theory for implementation in displacement-based finite element programs. The procedure that we develop is “simple” in the sense that it only involves the solution of one non-linear equation, rather than a system of non-linear equations. We show that our time-integration procedure is stable for relatively large time steps, is first-order accurate, and is objective.     

A Finite Element approach with patch projection for strain gradient plasticity formulations

Several strain gradient plasticity formulations have been suggested in the literature to account for inherent size effects on length scales of microns and submicrons. The necessity of strain gradient related terms render the simulation with strain gradient plasticity formulation computationally very expensive because quadratic shape functions or mixed approaches in displacements and strains are usually applied. Approaches using linear shape functions have also been suggested which are, however, limited to regular meshes with equidistanced Finite Element nodes. As a result the majority of the simulations in the literature deal with plane problems at small strains. For the solution of general three dimensional problems at large strains an approach has to be found which has to be computationally affordable and robust.     

Three dimensional combined fracture–plastic material model for concrete

This paper describes a combined fracture–plastic model for concrete. Tension is handled by a fracture model, based on the classical orthotropic smeared crack formulation and the crack band approach. It employs the Rankine failure criterion, exponential softening, and it can be used as a rotated or a fixed crack model. The plasticity model for concrete in compression is based on the Menétrey–Willam failure surface, the plastic volumetric strain as a hardening/softening parameter and a non-associated flow rule based on a nonlinear plastic potential function. Both models use a return-mapping algorithm for the integration of constitutive equations. Special attention is given to the development of an algorithm for the combination of the two models. The suggested combination algorithm is based on a recursive substitution, and it allows for the two models to be developed and formulated separately. The algorithm can handle cases when failure surfaces of both models are active, but also when physical changes such as crack closure occur. The model can be used to simulate concrete cracking, crushing under high confinement and crack closure due to crushing in other material directions. The model is integrated in a general finite element package ATENA and its performance is evaluated by comparisons with various experimental results from the literature.     

Deformation, stress state and thermodynamic force for a growing void in an elastic–plastic material

The growth of a spherical void in an elastic–plastic body, subjected to external pressure or tension and a gas pressure as well as a surface stress at the void surface, is investigated. The deformation, strain and stress state in the full body is presented. In addition, the local and global energy terms are calculated. Finally the total thermodynamic force on the void surface as well as the total dissipation are evaluated and compared allowing the calculation of the mechanical contribution to void growth due to diffusion of vacancies generated by plastification or irradiation.     

An atomistic-based finite deformation membrane for single layer crystalline films

A general methodology to develop hyper-elastic membrane models applicable to crystalline films one-atom thick is presented. In this method, an extension of the Born rule based on the exponential map is proposed. The exponential map accounts for the fact that the lattice vectors of the crystal lie along the chords of the curved membrane, and consequently a tangent map like the standard Born rule is inadequate. In order to obtain practical methods, the exponential map is locally approximated. The effectiveness of our approach is demonstrated by numerical studies of carbon nanotubes. Deformed configurations as well as equilibrium energies of atomistic simulations are compared with those provided by the continuum membrane resulting from this method discretized by finite elements.     

A two-dimensional finite element method for simulating the constitutive response and microstructure of polycrystals during high temperature plastic deformation

We describe a finite element method designed to model the mechanisms that cause superplastic deformation. Our computations account for grain boundary sliding, grain boundary diffusion, grain boundary migration, and surface diffusion, as well as thermally activated dislocation creep within the grains themselves. Front tracking and adaptive mesh generation are used to follow changes in the grain structure. The method is used to solve representative boundary value problems to illustrate its capabilities.     

Influence of reinforcement arrangement on the local reinforcement stresses in composite materials

The state of stress in and around reinforcements governs a number of physical processes in composite (multi-phase) materials, including the initiation of damage by either reinforcement cracking or interfacial decohesion. The stresses in the reinforcements have been observed to depend on the spatial distribution of the reinforcements, although the exact correlation is unclear. The present work determines the reinforcement stress for different reinforcement arrangements, ranging from a linear array of three uniformly spaced particles, to random and clustered microstructures. The stress calculations for elastic matrices were undertaken using a computationally efficient iterative technique. The technique was validated by comparing the results to finite element models, and the range of validity was determined. For the three-particle arrangements, the maximum reinforcement stress was observed when the particles were close to each other along the line of loading (a vertical arrangement). On the other hand, when the particle arrangement made a large angle with the loading direction, the reinforcement stress was low. Similar observations were recorded for the random and clustered arrangements where the location of the maximum reinforcement stress coincided with a vertical arrangement. The present work also develops a scheme for determining ‘representative volume elements’ for composite micromechanical models, based on the length scales of stress field interactions. These observations can be used to rationalize damage evolution mechanisms in commercial composites, and aid the development of physically based failure models for such materials.     

A multiscale method for dislocation nucleation and seamlessly passing scale boundaries

In the concurrent multiscale analysis, it is difficult to have truly seamless transition between the atomistic and continuum scale. This situation is even worse when defects pass through the boundary between different scales. For example, there is a lack of effective methods to handle the dislocation passing through scale boundaries which is important to investigate plasticity at the nanoscale. In this paper, the generalized particle (GP) method proposed by the first author is further developed so that a seamless transition and dislocation passing between different scales can be realized. Specifically, the linkage between different scales is through material neighbor-link cells (NLC) with scale duality. This indicates that material elements can be high-scale particles through a lumping process and can also be atoms via decomposition depending on the needs of the simulation. At the interface, the information transfer from bottom scale-up or from top scale-down is through the particles or atoms in the NLC. They are with the same material structure, all possess nonlocal constitutive behavior; thus, the smooth transition at the interface between different scales can be attained and validated to avoid non-physical responses. To save degrees of freedom, atoms are lumped together into a generalized particle in the domain in which the deformation gradient is near homogeneous. On the other hand, when defects such as dislocations in the atomistic domain are near the particle domain, the particles along dislocation propagation path and its surrounding region will be decomposed into atoms so dislocations can freely pass through the scale boundary and propagate inside the model just as it propagates in the deformed atomistic crystal structure. The method is verified first for seamless transition of variables at the scale boundary by a one-dimensional model and then verified for dislocation nucleation and propagation passing through scale boundaries in two cases, one is near the free surface and the other is inside of the copper nanowire. All the validations are through comparisons with fully atomistic analyses under same conditions. The comparison is satisfactory.     

A Taylor–Galerkin-based algorithm for viscous incompressible flow

In this paper the development and behaviour of a new finite element algorithm for viscous incompressible flow is presented. The stability and background theory are discussed and the numerical performance is considered for some benchmark problems. The Taylor–Galerkin approach naturally leads to a time-stepping algorithm which is shown to perform well for a wide range of Reynolds numbers (1 ? Re ? 400).

1 A conventional definition for Re is assumed.

Various modifications to the algorithm are investigated, particularly with respect to their effects on stability and accuracy.     

Debonding failure and size effects in micro-reinforced composites

Failure in micro-reinforced composites is investigated numerically using the strain-gradient plasticity theory of Gudmundson [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 52 (6) 1379–1406] in a plane strain visco-plastic formulation. Bi-axially loaded unit cells are used and failure is modeled using a cohesive zone at the reinforcement interface. During debonding a sudden stress drop in the overall average stress–strain response is observed. Adaptive higher-order boundary conditions are imposed at the reinforcement interface for realistically modeling the restrictions on moving dislocations as debonding occurs. It is found that the influence of the imposed higher-order boundary conditions at the interface is minor. If strain-gradient effects are accounted for a void with a smooth shape develops at the reinforcement interface while a smaller void having a sharp tip nucleates if strain-gradient effects are excluded. Using orthogonalization of the plastic strain gradient with three corresponding material length scales it is found that, the first length scale dominates the evaluated overall average stress–strain response, the second one only has a small effect and the third one has an intermediate effect. Finally, studies of reinforcement having elliptical cross-sections show rather significant gradients of stress which is not seen for the corresponding circular cross-sections. Also, an increased drop in the overall load carrying capacity is observed for cross-sections elongated perpendicular to the principal tensile direction compared to the corresponding circular cross-sections.