Finite Deformation Regime Research Articles

Phase‐field model for ductile fracture in the stress resultant geometrically exact shell

SummaryWe present a phase‐field fracture model for a stress resultant geometrically exact shell in finite deformation regime where the configuration manifold evolves according to deformation and fracture. The Reissner–Mindlin shell problem is first solved via the finite element method, where the independent unknown fields are the displacement and director. The phase‐field ductile fracture model is then coupled with the verified geometrically exact shell by enriching the three‐field elasto‐plastic free energy with a regularized crack surface energy. To capture the geometrical nonlinearity due to the large plastic deformation exhibited in the processing zone, the corresponding finite‐strain stress resultant elasto‐plasticity model is coupled with the phase field model. This coupling between the stress resultant elasto‐plasticity model and the phase‐field fracture model enables us to predict the different amounts of energy dissipation attributed to plastic deformation and fracture and hence simulate the crack propagation in the ductile regime properly. We introduce a mixed finite element model with displacement, director, and phase‐field order parameters as the nodal degree of freedoms formulated on the mid‐surface. An energy‐based arc‐length method is generalized to track the equilibrium path and mitigate instabilities arising from material nonlinearity. Four numerical examples are presented to validate the implementation of the model and demonstrate its capability to simulate ductile fracture in shell structures. These examples include a plane‐stress tension/shear fracture model, the Muscat‐Fenech and Atkins plate, axial tension of a notched cylindrical shell, and ductile fracture of a simply supported plate under uniform pressure.

A Poroelastic Approach for Modelling Myocardial Oedema in Acute Myocarditis.

Myocarditis is a general set of mechanisms that manifest themselves into the inflammation of the heart muscle. In 2017, more than 3 million people were affected by this disease worldwide, causing about 47,000 deaths. Many aspects of the origin of this disease are well known, but several important questions regarding the disease remain open. One of them is why some patients develop a significantly localised inflammation while others develop a much more diffuse inflammation, reaching across large portions of the heart. Furthermore, the specific role of the pathogenic agent that causes inflammation as well as the interaction with the immune system in the progression of the disease are still under discussion. Providing answers to these crucial questions can have an important impact on patient treatment. In this scenario, computational methods can aid specialists to understand better the relationships between pathogens and the immune system and elucidate why some patients develop diffuse myocarditis. This paper alters a recently developed model to study the myocardial oedema formation in acute infectious myocarditis. The model describes the finite deformation regime using partial differential equations to represent tissue displacement, fluid pressure, fluid phase, and the concentrations of pathogens and leukocytes. A sensitivity analysis was performed to understand better the influence of the most relevant model parameters on the disease dynamics. The results showed that the poroelastic model could reproduce local and diffuse myocarditis dynamics in simplified and complex geometrical domains.

Total Lagrange implementation of a finite-deformation continuum dislocation dynamics model of mesoscale plasticity

We present a computational algorithm for solving the recently developed finite-deformation continuum dislocation dynamics theory of mesoscale plastic deformation of single crystals (Starkey et al., 2020). This CDD theory is based on a vector density representation of dislocations governed by curl-type transport-reaction equations subjected to the divergence-free constraint of the appropriate dislocation density. These density evolution equations are to be solved simultaneously with the finite-deformation crystal mechanics. Specifically, our algorithm aims to solve the referential form of the governing equations for a representative volume element (RVE) subject to remote uniform loading. The mechanical fields at the mesoscale are thus split into RVE-averages plus fluctuating components and treated using a strain-driven homogenization scheme. A virtual work-based total Lagrange formulation was used to discretize the governing mechanics equations. A first-order system least squares finite element formulation was used to solve the transport equations. The two schemes are coupled in a staggered fashion. As a part of the crystal mechanics discretization, we derive a consistent tangent modulus and show that the stress update for this model is both linear and global. This linear stress update comes at the cost of solving the dislocation transport equations at every time step to update the plastic distortion caused by dislocation motion. Several test problems are given, demonstrating the ability of the discretization scheme to solve the problem, including the expansion of dislocation loop-like bundles under constant velocity and driven by a mean deformation gradient, dynamic recovery of two oppositely oriented tilt boundaries in a single crystal, and a uniaxial tension test of a single crystal with one slip system activated. In most of these examples, the evolution behavior of the dislocations in the finite deformation regime is demonstrated.

A finite micro-rotation material point method for micropolar solid and fluid dynamics with three-dimensional evolving contacts and free surfaces

This paper introduces an explicit material point method designed specifically for simulating the micropolar continuum dynamics in the finite deformation and finite microrotation regime. The material point method enables us to simulate large deformation problems while circumventing the potential mesh distortion without remeshing. To eliminate rotational motion damping and loss of angular momentum during the projection, we introduce the mapping for microinertia and angular momentum between particles and grids through the affine particle-in-cell approach. The microrotation and the curvature at each particle are updated through zero-order forward integration of the microgyration and its spatial gradient. We show that the microinertia and the angular momentum are conserved during the projections between particles and grids in our formulation. We verify the formulation and implementation by comparing with the analytical dispersion relation of micropolar waves under the small strain and small microrotation, as well as the analytical soliton solution for solids undergoing large deformation and large microrotation. We also demonstrate the capacity of the proposed computational framework to handle a wide spectrum of simulations that exhibit size effects in the geometrical nonlinear regime through three representative numerical examples, i.e., a cantilever beam torsion problem, a fragment-impact penetration problem, and a micropolar fluid discharging problem.

Material modeling for parametric, anisotropic finite strain hyperelasticity based on machine learning with application in optimization of metamaterials

AbstractMechanical metamaterials such as open‐ and closed‐cell lattice structures, foams, composites, and so forth can often be parametrized in terms of their microstructural properties, for example, relative densities, aspect ratios, material, shape, or topological parameters. To model the effective constitutive behavior and facilitate efficient multiscale simulation, design, and optimization of such parametric metamaterials in the finite deformation regime, a machine learning‐based constitutive model is presented in this work. The approach is demonstrated in application to elastic beam lattices with cubic anisotropy, which exhibit highly nonlinear effective behaviors due to microstructural instabilities and topology variations. Based on microstructure simulations, the relevant material and topology parameters of selected cubic lattice cells are determined and training data with homogenized stress‐deformation responses is generated for varying parameters. Then, a parametric, hyperelastic, anisotropic constitutive model is formulated as an artificial neural network, extending a recent work of the author extending a recent work of the author, Comput Mech., 2021;67(2):653‐677. The machine learning model is calibrated with the simulation data of the parametric unit cell. The authors offer public access to the simulation data through the GitHub repository https://github.com/CPShub/sim‐data. For the calibration of the model, a dedicated sample weighting strategy is developed to equally consider compliant and stiff cells and deformation scenarios in the objective function. It is demonstrated that this machine learning model is able to represent and predict the effective constitutive behavior of parametric lattices well across several orders of magnitude. Furthermore, the usability of the approach is showcased by two examples for material and topology optimization of the parametric lattice cell.

A localized mass-field damage model with energy decomposition: Formulation and FE implementation

In this article, we present a new computational damage approach based on localized mass loss concept and its detailed finite element implementation for brittle fracture under quasi-static loading condition. Formulation of this approach is derived in a general way by means of finite deformation regime and nonlinear isotropic materials. The underlying idea of the theory lies in the fact that cracks are created by massive breakage of atomic bonds, diffusing in a volume of characteristic size, resulting in highly localized mass loss area, and nor geometric description of cracks are required. In degraded domain, the traditional law of conservation of mass is violated locally and that is replaced by the local mass-balance equation, accounting for mass flow in the area of degraded material. A coupled system of equilibrium and local mass-balance equations is thus introduced to govern deformation of the body and evolution of the mass density. In this setting, the failure, in contrast to conventional damage models, is thus driven by localized mass loss, which has physical meaning, nor internal parameters such as phase-field or damage variables are defined. We also introduce new constitutive laws for mass source and mass flux by means of energy decomposition, where only the positive part of strain energy density (SED) function involves in the degradation of mass, accounting for distinction of fracture behavior in tension and compression. The discrete forms of governing equations are solved by a staggered nonlinear algorithm. Importantly, equally linear interpolations for the standard finite elements are used for both displacements and mass density, nor mismatch of approximation exits, convenient in the implementation. In addition, we present a special detection technique to find so-called updated nodes, which aims to prevent numerically unstable issues caused by random distortions of deleted elements. The staggered algorithm is thus modified so that the displacements and mass density are updated separately. This detection technique is then integrated into the staggered algorithm. We also present a procedure for calibration of the required quantities used for the localized mass loss analysis. Numerical experiments for brittle fracture are studied to show the accuracy and performance of the developed damage approach.

Finite deformation hyperplasticity theory for crushable, cemented granular materials

The work is focused on the formulation of a thermodynamically–based constitutive theory for granular, cemented geomaterials, often characterized by a open structure with high porosity and voids of large diameter. Upon mechanical degradation processes such as bond rupture and grain crushing, these material undergo large volumetric and shear strains, and in some cases the deformations are so large that the usual assumption of linearized kinematics may be not applicable. In the first part of this work, the theory of hyperplasticity is extended to the finite deformation regime by adopting a multiplicative split of the deformation gradient into an elastic and a plastic part, under the assumption of material isotropy. Grain breakage and bond damage processes are accounted for through two micromechanically–inspired internal variables. A specific constitutive model for carbonatic cemented sands and calcarenites is proposed as a relevant example of application. In the second part, an implicit stress–point algorithm has been developed which is amenable to closed form linearization, for the implementation of the model into standard FE platforms. A series of numerical tests have demonstrated the accuracy and efficiency of the proposed algorithm. The simulation of plane strain biaxial tests, modeled as boundary–value problems, has highlighted the role played by geometric non–linearity in determining the evolution of the specimen deformation upon reaching a bifurcation condition.

A computational model for dynamic strain localization in unsaturated elasto‐viscoplastic soils

SummaryStrain localization has been the subject of extensive theoretical and numerical studies over the last four decades. The static and dynamic strain localization in both dry and water‐saturated geomaterials has widely been investigated numerically. However, studies on unsaturated geomaterials have been insufficient. This paper presents a computational model for nonlinear dynamic analysis of strain localization in multiphase geomaterials. The discretization of the governing equations is achieved using the finite element method in the finite deformation regime with the updated Lagrangian description. An elasto‐viscoplastic model extended to include the effect of suction on the behavior of partially saturated clay is used in the formulation. The nonlinear kinematic hardening rule and softening due to the structural degradation of soil particles are taken into account in the model. The inception and progression of localization in fully saturated and unsaturated clay under dynamic compressive loading are investigated. In the unsaturated case, the strain localization becomes more pronounced and much clearer shear bands are formed. More brittleness is observed in the unsaturated case due to the collapse of suction force caused by shearing. This demonstrates that even though suction improves the mechanical properties of soil, ie, increasing the stiffness and shear strength, the hypothesis of fully saturation is not at all on the side of safety due to the drastic loss of strength associated with the collapse of suction which may cause a strong strain localization.

Swelling-Driven Crack Propagation in Large Deformation in Ionized Hydrogel

Stepwise crack propagation is evidently observed in experiments both in geomaterials and in hydrogels. Pizzocolo et al. (2012, “Mode I Crack Propagation in Hydrogels is Step Wise,” Eng. Fract. Mech., 97(1), pp. 72–79) show experimental evidence that mode I crack propagation in hydrogel is stepwise. The pattern of the intermittent crack growth is influenced by many factors, such as porosity of the material, the permeability of the fluid, the stiffness of the material, etc. The pause duration time is negatively correlated with the stiffness of the material, while the average propagation length per step is positively correlated. In this paper, we integrate extended finite element method (XFEM) and enhanced local pressure (ELP) method, and incorporate cohesive relation to reproduce the experiments of Pizzocolo et al. in the finite deformation regime. We investigate the stepwise phenomenon in air and in water, respectively, under mode I fracture. Our simulations show that despite the homogeneous material properties, the crack growth under mode I fracture is stepwise, and this pattern is influenced by the hydraulic permeability and the porosity of the material. Simulated pause duration is negatively correlated with stiffness, and the average propagating length is positively correlated with stiffness. In order to eliminate the numerical artifacts, we also take different time increments into consideration. The staccato propagation does not disappear with smaller time increments, and the pattern is approximately insensitive to the time increment. However, we do not observe stepwise crack growth scheme when we simulate fracture in homogeneous rocks.

A finite-deformation dislocation density-based crystal viscoplasticity constitutive model for calculating the stored deformation energy

A previously-developed infinitesimal strain and dislocation density based crystal viscoplasticity constitutive theory is reformulated in the finite deformation regime in a thermodynamically consistent manner. The constitutive equations are then numerically implemented into the Abaqus/Standard finite element package. The constitutive model is primarily verified and calibrated with respect to single crystal aluminum experiments and then the distribution of stored deformation energy in aluminum bicrystals and polycrystals is investigated to estimate the driving forces for grain boundary migration due to the stored energy difference across grain boundary. Our statistical analysis shows that the onset of strain induced grain boundary migration in a polycrystalline microstructure is basically dependent on the spatial distribution of the stored deformation energy rather than the overall stored deformation energy value; and it preferably occurs in a few predictable specific regions where certain grain orientations meet at a grain boundary across which high values of stored energy difference happens.

Asymptotic analysis of microscopic impenetrability constraints for atomistic systems

We analyze systems of atomistic interactions on a triangular lattice allowing for fracture under a geometric condition on the triangles corresponding to a microscopic impenetrability constraint. Such systems can be thought as a computational simulation of materials undergoing brittle fracture. We show that in the small-deformation regime such approximation can be validated analytically in the framework of variational models of fracture. Conversely, in a finite-deformation regime various pathologies show that the continuum approximation of such a system differs from the usual variational representations of fracture and either needs new types of formulations on the continuum, or a proper interpretation of the atomistic constraints limiting their range and adapting them to a dynamical framework.

Nonlinear electro-magneto-mechanical constitutive modelling of monolayer graphene

Using the classical theory of invariants for the specific class of graphene's symmetry, we constitutively characterize electro-magneto-mechanical interactions of graphene at continuum level. Graphene's energy depends on five arguments: the Finger strain tensor, the curvature tensor, the shift vector, the effective electric field intensity and the effective magnetic induction. The Finger strain tensor describes in- surface phenomena, the curvature tensor is responsible for the out-of-surface motions, while the shift vector is used due to the fact that graphene is a multilattice. The electric and the magnetic fields are described by the effective electric field intensity and the effective magnetic induction, respectively. An energy with the above arguments that also respects graphene's symmetries is found to have 42 invariants. Using these invariants, we evaluate all relevant measures by finding derivatives of the energy with respect to the five arguments of the energy. We also lay down the field equations that should be satisfied. These are the Maxwell equations, the momentum equation, the moment of momentum equation and the equation ruling the shift vector. Our framework is general enough to capture fully coupled processes in the finite deformation regime.

Computational Wear and Contact Modeling for Fretting Analysis with Isogeometric Dual Mortar Methods

A finite element framework based on dual mortar methods is presented for simulating fretting wear effects in the finite deformation regime. The mortar finite element discretization is realized with Lagrangean shape functions as well as isogeometric elements based on non-uniform rational B-splines (NURBS) in two and three dimensions. Fretting wear effects are modeled in an incremental scheme with the help of Archard’s law and the worn material is considered as additional contribution to the gap function. Numerical examples demonstrate the robustness and accuracy of the presented algorithm.

Bayesian uncertainty analysis of finite deformation viscoelasticity

The viscoelasticity of the dielectric elastomer, VHB 4910, is experimentally characterized, modeled, and analyzed using Bayesian uncertainty analysis. Whereas these materials are known for their large-field induced deformation and broad applications in smart structures, the rate-dependent viscoelastic effects are not well understood. To address this issue, we quantify both the hyperelastic and viscoelastic constitutive behavior and use Bayesian uncertainty analysis to assess several key modeling attributes. Specifically, we compare an Ogden-based phenomenological model to a nonaffine hyperelastic model and couple hyperelasticity to both linear and nonlinear viscoelasticity. The utilization of Bayesian statistics is shown to provide insight into quantifying nonlinear viscoelasticity behavior as a function of internal state variables. The results are validated experimentally in the finite deformation regime over a range of stretch rates spanning four orders of magnitude (6.7×10-5–0.67Hz). A unique set of hyperelastic parameters are identified, independent of the stretch rate. In addition, comparisons of the linear and nonlinear viscoelastic models demonstrate a reduction in modeling error by approximately a factor of three. Finally, the viscoelastic time constant is shown to produce an inverse stretch rate power law dependence regardless of which hyperelastic model is used.

A multiscale overlapped coupling formulation for large-deformation strain localization

We generalize the multiscale overlapped domain framework to couple multiple rate-independent standard dissipative material models in the finite deformation regime across different length scales. We show that a fully coupled multiscale incremental boundary-value problem can be recast as the stationary point that optimizes the partitioned incremental work of a three-field energy functional. We also establish inf-sup tests to examine the numerical stability issues that arise from enforcing weak compatibility in the three-field formulation. We also devise a new block solver for the domain coupling problem and demonstrate the performance of the formulation with one-dimensional numerical examples. These simulations indicate that it is sufficient to introduce a localization limiter in a confined region of interest to regularize the partial differential equation if loss of ellipticity occurs.

A finite deformation constitutive model for shape memory polymers based on Hencky strain

In many engineering applications, shape memory polymers (SMPs) usually undergo arbitrary thermomechanical loadings at finite deformation. Thus, development of 3D constitutive models for SMPs within the finite deformation regime has attracted a great deal of interest. In this paper, based on the classical framework of thermodynamics of irreversible processes, employing the logarithmic (or Hencky) strain as a more physical measure of strain, a 3D large-strain macromechanical model is presented. In the constitutive model development, we adopt a multiplicative decomposition of the deformation gradient into elastic and stored parts. In addition, employing the averaging scheme, the logarithmic elastic strain tensor is decomposed into the rubbery and glassy parts. The evolution equations for internal variables are introduced for both cooling and heating processes. The time-discrete form of the proposed model in the implicit form is also presented. Comparing the predicted results with experimental data reported in the literature, the model is validated. Finally, using the finite element method, two boundary value problems e.g., a 3D beam and a medical stent made of SMPs are numerically simulated.

Multiscale thermomechanical contact: Computational homogenization with isogeometric analysis

A computational homogenization framework is developed in the context of the thermomechanical contact of two boundary layers with microscopically rough surfaces. The major goal is to accurately capture the temperature jump across the macroscopic interface in the finite deformation regime with finite deviations from the equilibrium temperature. Motivated by the limit of scale separation, a two-phase thermomechanically decoupled methodology is introduced, wherein a purely mechanical contact problem is followed by a purely thermal one. In order to correctly take into account finite size effects that are inherent to the problem, this algorithmically consistent two-phase framework is cast within a self-consistent iterative scheme that acts as a first-order corrector. For a comparison with alternative coupled homogenization frameworks as well as for numerical validation, a mortar-based thermomechanical contact algorithm is introduced. This algorithm is uniformly applicable to all orders of isogeometric discretizations through non-uniform rational B-spline basis functions. Overall, the two-phase approach combined with the mortar contact algorithm delivers a computational framework of optimal efficiency that can accurately represent the geometry of smooth surface textures. Copyright © 2013 John Wiley & Sons, Ltd.

Realization of uniform deformation of soil specimen under undrained plane strain condition based on soil–water coupled finite deformation analysis considering inertia forces

Based on a soil–water coupled finite deformation analysis, theoretical considerations and numerical calculations were carried out under the undrained plane strain condition in order to reproduce a uniform deformation field. Rather than the “quasi-static” equation of motion, which does not include inertia forces, a dynamic equation of motion which includes inertia forces was used. At first, a theoretical consideration was carried out to realize uniform deformation for a saturated soil that satisfied the element-wise undrained/constant-volume condition. This presents an “infinitely slow loading” case without ignoring the inertia term based on the u–p formulation. In other words, it can be seen that under general slow loading that is not infinitely slow, a gradient in the pore water pressure will always be produced, resulting in the migration of pore water and loss/collapse of uniformity. This first conclusion is useful for verifying numerical analysis code made in the finite deformation regime. Next, the uniform deformation of a plane strain rectangular soil specimen was measured under constant cell pressure and undrained boundary conditions using a dynamic soil–water coupled analysis in which the SYS Cam-clay model was employed as the elasto-plastic constitutive model for the soil skeleton. In addition, the effects of the loading rates as well as loading applications, with/without inertia forces, on the loss of uniformity in deformation were shown to have a significant influence on the inertia term even though the loss itself was extremely small.

Thermodynamic modeling of fully coupled finite-deformation thermo-electro-magneto-mechanical behavior for multifunctional applications

In this paper, we model fully coupled thermo-electro-magneto-mechanical (TEMM) behavior in the finite-deformation regime by (i) developing for the first time a comprehensive catalogue of free energies, state variables, and state equations, and (ii) combining this catalogue with the first principles of nonlinear continuum electrodynamics. We develop our catalogue in a thermodynamically consistent manner, and circumvent the ambiguities and challenges inherent in nonlinear continuum electrodynamics, by connecting with classical equilibrium thermodynamics. We use its formalism as a blueprint for characterizing a fundamental energetic process, that is, one where internal energy is the characterizing potential, the independent variables are extensive, and the dependent variables are intensive. A key feature of identifying this fundamental energetic process is the resulting ability to transparently and rigorously introduce new free energies – many appearing in the finite-deformation TEMM literature for the first time – that employ any set of intensive or extensive quantities as independent variables. We also develop novel mathematical transformations that accommodate alternative electromagnetic work conjugates as independent variables.Each thermodynamic potential in our comprehensive catalogue characterizes a particular thermo-electro-magneto-mechanical process. Each process, in turn, correlates with a particular experiment, the independent variables being controlled and the dependent variables being the measured responses. Our framework will thus enable the development of constitutive models for multifunctional materials under different experimental conditions. Additionally, the research presented herein can be used to convert targeted performance properties that are inherently nonlinear, three dimensional, and anisotropic into a “recipe” for multifunctional material design.

Approximation of incompressible large deformation elastic problems: some unresolved issues

Several finite element methods for large deformation elastic problems in the nearly incompressible and purely incompressible regimes are considered. In particular, the method ability to accurately capture critical loads for the possible occurrence of bifurcation and limit points, is investigated. By means of a couple of 2D model problems involving a very simple neo-Hookean constitutive law, it is shown that within the framework of displacement/pressure mixed elements, even schemes that are inf-sup stable for linear elasticity may exhibit problems when used in the finite deformation regime. The roots of such troubles are identified, but a general strategy to cure them is still missing. Furthermore, a comparison with displacement-based elements, especially of high order, is presented.