Volumetric-deviatoric Split Research Articles

Phase field approach to predict mixed-mode delamination and delamination migration in composites

In this work, we present a mixed-mode phase field approach to model delamination progression and its migration. The proposed model includes the volumetric and deviatoric effects that occur during the mixed-mode delamination and migration. The volumetric–deviatoric split is combined with a power law criterion of delamination to capture the mixed-mode effects. An anisotropic tensor is used in the crack surface density function for capturing the anisotropic fracture. Two history parameters are introduced to ensure the irreversibility of the damage field throughout the evolution process of delamination and to consider the maximum value of the strain energy. A staggered approach is implemented for solving the equilibrium and evolution equations. The algorithm enables to obtain stable numerical solutions with faster convergence. Several examples of mode I and mode II as per standard tests have been incorporated to demonstrate the working of the proposed method. The numerical examples are validated by comparison with experimental results from the literature. It is observed that the proposed model can predict crack paths and angles closer to the experimental results. Examples of multiple delamination and delamination migration are also studied to understand the kinking in the delamination scenario. It is observed that the delamination migration is governed by the ratio of notch length a to the distance of the load from the edge L, and by the stress state in front of the crack tip. In the case of laminates with multiple interphases, the event of delamination impinging and kinking at interphase depends on the fracture toughness ratio of the bulk and the interphase. The example of the single-edge notched composite lamina and open-hole tension test with different fiber orientations has been validated with the literature and experiments.

A modified RI-IGABEM with only weakly singular integral for viscoelastic analysis

In this paper, an improved isogeometric boundary element method based on radial integration method (RI-IGABEM) is proposed to solve the viscoelastic problems. Prony series are utilized to discretize the stress with the two most recent time-step strains. The boundary integral equation (BIE) for the viscoelastic problems is derived. The domain integral related to stress is decomposed into the linear combination of two sub-integrals by volumetric–deviatoric split. The radial integration method (RIM) is applied to transform all the domain integrals into equivalent boundary integrals. The present method has several advantages. Firstly, the present method eliminates the need for stress calculation at each time step, thus avoiding the error accumulation associated with the traction recovery method (TRM). Secondly, the BIE for the viscoelastic problems contains only weakly singular integrals. In addition, we provide an analytical solution for the radial component of the equivalent boundary integrals, which is transformed using RIM. To sum up, the present method improves the accuracy and efficiency of RI-IGABEM by reducing the required computational steps, eliminating error accumulation, and providing analytical solutions for certain integrals. Numerical examples demonstrate that the proposed RI-IGABEM is an efficient and accurate method suitable for viscoelastic problems.

Impact of mathematical requirements on the invariant-based anisotropic constitutive models for non-linear biomaterials

Finite element simulations are widely used to study the non-linear mechanical behavior of various biomaterials. Constructing an anisotropic strain energy function within the framework of hyperelasticity is an effective approach to describe the material constitutive behavior. The constructed strain energy function needs to satisfy several mathematical requirements to ensure the framework indifference, the material (near) incompressibility, and the material stability. While the framework indifference, or objectivity, can be naturally satisfied by the invariant-based constitutive formulation, how to enforce the material incompressibility and stability requires detailed discussions and careful treatments. The scope of this paper is to examine in detail the impacts of the mathematical requirements on the constitutive formulation for non-linear anisotropic biomaterials. Particularly, theoretical analyses and numerical simulations are combined to investigate the influences of the material incompressibility as a mathematical constraint and various convexity conditions on the invariant-based constitutive modeling. Through a constructed boundary value problem, analytical solutions are derived via the energy minimization and further used to quantify the influences of the material anisotropic and isotropic components on the material responses. The impact of the volumetric–deviatoric split is demonstrated separately for the strictly incompressible and nearly incompressible materials. In order to ensure the material stability, two commonly used convexity conditions, including the strong ellipticity condition and the polyconvexity condition, are discussed in detail. Several numerical examples are provided to demonstrate their impacts on the material stability under different loading conditions. These discussions are particularly relevant to model biomaterials that exhibit non-linear and anisotropic behaviors under complex loading conditions.

A nonlocal softening plasticity based on microplane theory for concrete at finite strains

Parts of the irreversible response of materials can be described by a plasticity approach. For modelling quasi-brittle materials such as concrete, the microplane approach is a powerful method. In order to predict the load–displacement as well as the stress–strain relation, numerous constitutive models developed for small strains have been used successfully. Nevertheless, for example, at high hydrostatic pressure, extremely large deformations occur even in concrete materials with no damage or voids. As consequence, the microplane-plasticity model at small strains needs to be extended to the finite strain framework for largely deformed structures. The elastoplastic microplane approach based on the kinematic constraint of the volumetric-deviatoric split and the Drucker-Prager yield criterion at finite strains are implemented in the present work. Furthermore, due to strain localisation, an implicit gradient enhanced approach is applied in here to obtain stable and mesh insensitive solutions. Numerical examples including the comparison of simulated results to existing experimental data are provided to validate the proposed formulation.

A note on the volumetric-deviatoric split on the anisotropic constitutive model for fiber-reinforced materials

In the literature, it has been suggested that for a class of anisotropic constitutive laws for fiberreinforced materials, the volumetric-deviatoric split should only be performed on the isotropic (matrix) term, but not on the anisotropic (fiber) term. In this research note, we follow up on the theoretical and numerical analyses adopted in these early publications with an intuitive example that allows us to directly analyze the effect of this split. We demonstrate that performing such split on the anisotropic term leads to non-physical volume growth of the material sample. Therefore, we consolidate the observation that the volumetric-deviatoric split should not be applied to the anisotropic (fiber) term of the total strain energy.

Hexahedral-Based Smoothed Finite Element Method Using Volumetric-Deviatoric Split for Contact Problem

A first-order hexahedral (H8)-element-based smoothed finite element method (S-FEM) with a volumetric-deviatoric split for nearly incompressible materials was developed for highly accurate deformation analysis of large-strain problems. In the proposed method, the isovolumetric part of the deformation gradient at the integration point is derived from F based on the beta finite element method (i.e., an S-FEM), whereas the volumetric part of the deformation gradient is derived from F on the basis of the standard FEM with reduced integration elements. This method targets H8 elements that are automatically generated from tetrahedral elements, which makes it quite practical. This is because the FE mesh can be created automatically even if the targeted object has a complex shape. This method eliminates the phenomena of volumetric and shear locking, and reduces pressure oscillations. The proposed method was implemented in the commercial FE software Abaqus and applied to the large-deformation contact problem to verify its effectiveness.

An Anisotropic Multi-plane Elastic-Damage Model with Axial and Shear Damage and Its Application to Concrete Behavior

The prediction of material response is necessary for the analysis of structure under static or dynamic loading. Mathematical modeling of the nonlinear tri-dimensional mechanical behavior of quasi-brittle materials like concrete caused by damaging and plasticity effects are one of the most serious classical challenges we face in the engineering science. Among phenomena of different orientation, the micro-plane models, like multi-planes models which use a constitutive equation in a vectorial form rather than tensorial form by means of capturing interactions, can serve this goal adequately. This paper presents a simple realistic and robust damage based model in the multi-planes framework accomplished with a few parameters for calibration and suitable for engineering purposes without volumetric-deviatoric split of strain tensor and its problems. This damage formulation has been built on the basis of two types of fundamental damage, axial damage and shear damage, that essentially can be occurred on each micro-plane and based on this concept two new axial and shear damage functions are presented. By comparing between the results of the proposed model and experimental data, model verification has been examined under different loading/unloading/reloading stress/strain paths.

An implicit gradient formulation for microplane Drucker-Prager plasticity

A microplane plasticity model is regularized by an implicit gradient enhancement. The plasticity model is defined by a Drucker-Prager yield criterion and within the thermodynamically consistent framework of the microplane theory. Thus, the advantages of both the microplane approach and the pressure sensitive Drucker-Prager yield criterion are combined within a nonlocal implicit gradient type method. The microplane approach allows for the description of induced anisotropy which is observed in the failure of quasi-brittle materials. Based on the volumetric–deviatoric split, a microplane version of the Drucker-Prager yield function is defined. The implicit gradient enhancement is employed to remedy the pathological mesh sensitivity. It yields the advantage of being strongly nonlocal and equivalent to the integral-type models, while it preserves the mathematical locality of the equations and keeps the implementation straightforward within the classical finite element method. The proposed formulation is implemented in a 3D finite element code. The stress return mapping algorithm and the consistent tangent are derived. Numerical examples are simulated to demonstrate the capability of the proposed method to regularize the model and eliminate the pathological mesh dependency.

An anisotropic gradient damage model based on microplane theory

In this paper, a thermodynamically consistent formulation and numerical implementation of a gradient-enhanced anisotropic microplane damage model are proposed. The microplane model is derived based on the volumetric–deviatoric split and the kinematic constraint assumption. The mixed finite element formulation of displacement and nonlocal strains field is developed to simulate anisotropic quasi-brittle fracture. The proposed model is used to describe the mechanical behavior of anisotropic quasi-brittle materials by numerical simulations of uniaxial tension, simple shear, tension of a bar with localized deformation, and a rectangular specimen with a material imperfection. The results show the ability of the proposed approach to predict mesh-independent results for quasi-brittle damage behavior accompanied by the localization of deformation. Comparison between numerical and experimental results shows that the relatively simple model based on microplane theory together with the standard finite elements implementation is capable to realistically simulate complex behaviors related to fracture of quasi-brittle material such as concrete.

Regularization of microplane damage models using an implicit gradient enhancement

The microplane model allows for the description of damage induced anisotropy in a natural manner by introducing constitutive laws for quantities on individual microplanes at each material point. However, if damage or other strain softening constitutive laws are used within the microplane approach, the well-known problem of localization arises leading to spurious results and mesh dependency. This problem demands some regularization method to stabilize the solution. The paper focuses on the efficient implementation of implicit gradient enhancement for microplane damage models. Previous works enhanced the strain tensor, thus resulting in large number of extra degrees of freedom, which limits the use of this method for large scale 3D simulations. A new method which enhances the equivalent strain driving the damage on each microplane is introduced in this work. The new method limits the number of additional degrees of freedom to one, while preserving the regularizing effect. The two methods are implemented within a 3D finite element code to compare their performance. The microplane model used is based on a thermodynamically consistent formulation and on a volumetric–deviatoric split of strains on each microplane. Furthermore, an exponential damage law is used and an equivalent strain expression which distinguishes between compression and tension is applied to simulate the behavior of concrete. The capabilities of the proposed formulation are demonstrated by comparison to published experiments on plain concrete.

A thermodynamically-consistent microplane model for shape memory alloys

In microplane theory, it is assumed that a macroscopic stress tensor is projected to the microplane stresses. It is also assumed that 1D constitutive laws are defined for associated stress and strain components on all microplanes passing through a material point. The macroscopic strain tensor is obtained by strain integration on microplanes of all orientations at a point by using a homogenization process. Traditionally, microplane formulation has been based on the Volumetric–Deviatoric–Tangential split and macroscopic strain tensor was derived using the principle of complementary virtual work. It has been shown that this formulation could violate the second law of thermodynamics in some loading conditions. The present paper focuses on modeling of shape memory alloys using microplane formulation in a thermodynamically-consistent framework. To this end, a free energy potential is defined at the microplane level. Integrating this potential over all orientations provides the macroscopic free energy. Based on this free energy, a new formulation based on Volumetric–Deviatoric split is proposed. This formulation in a thermodynamic-consistent framework captures the behavior of shape memory alloys. Using experimental results for various loading conditions, the validity of the model has been verified.

An enhanced concept of rheological models to represent nonlinear thermoviscoplasticity and its energy storage behavior, part 2: spatial generalization for small strains

The paper provides the spatial formulation of the enhanced concept of rheological models (Brocker and Matzenmiller in Cont Mech Thermodyn 25(6):749–778, 2013) for small deformations and the graphical representation of the rheological network for three-dimensionalmaterial behavior. The rheological components are redefined as tensor-valued elements, and afterward, they are assembled into a network of thermoviscoplasticity with volumetric–deviatoric split. The related constitutive equations are derived analogously as in the uniaxial model in Brocker and Matzenmiller (Cont Mech Thermodyn 25(6):749–778, 2013), providing the von-Mises yield function of metal plasticity and the associated flow rule in a natural way.

Microplane model M7f for fiber reinforced concrete

Model M7f is a new model for fiber reinforced concretes under static and dynamic loads, which features two kinds of improvement over the earlier versions: (1) It is built on M7, a new, greatly improved, microplane model for plain concrete; and (2) it includes a more realistic description of the fiber pullout and breakage. The former include: (a) the absence of volumetric deviatoric split of elastic strains, which eliminates excessive lateral expansions or contractions and stress locking in far post-peak extensions; (b) simulation of the differences between hydrostatic compression and uniaxial compression under rigid lateral confinement; and (c) high shear dilatancy of low strength concretes; and realistic description of unloading, reloading and load cycles, even if they cross between tension and compression. The latter includes an improved continuous dependence of the effect of fibers on the fiber volume fraction. The fiber resistance is a function of the strain representing the average opening of cracks of given spacing and, as in model M5f, a horizontal plateau as a function of the type of fiber and fiber volume fraction has been employed and used systematically for all fits. In this study, this horizontal plateau is justified using uniformly distributed crack bridging fibers. The model behavior is calibrated and verified by fitting of the main test data from the literature. The match of experimental observations and the computational results is closer than in the previous models.

Towards variational constitutive updates for non-associative plasticity models at finite strain: Models based on a volumetric-deviatoric split

In this paper, an enhanced variational constitutive update suitable for a class of non-associative plasticity theories at finite strain is proposed. In line with classical numerical formulations for plasticity models, such as the by now established return-mapping algorithm, variational constitutive updates represent a numerical method for computing the unknown state variables. However, in contrast to conventional algorithms, variational constitutive updates are fully variational, i.e., all unknown variables follow jointly from minimizing a certain potential. In addition to the physical and mathematical elegance of these variational schemes, they show several practical advantages as well. For instance, numerically efficient and robust optimization schemes can be directly employed for solving the resulting minimization problem. Since mathematically, plasticity is a non-smooth problem and often, it leads to highly singular systems of equations as known from single crystal plasticity, a robust implementation is of utmost importance. So far, variational constitutive updates have been developed for different classes of standard dissipative solids, i.e., solids characterized by associative evolution equations and flow rules. In the present paper, this framework is extended to a certain class of non-associative plasticity models at finite strain. All models falling into this class show a volumetric-deviatoric split of the Helmholtz energy and the yield function. Typical prototypes are Drucker-Prager or Mohr-Coulomb models playing an important role in soil mechanics. The efficiency and robustness of the resulting algorithmic formulation is demonstrated by means of selected numerical examples.

Microplane modelling of shape memory alloys

A three-dimensional (3D) constitutive model based on a statically constrained microplane theory with volumetric–deviatoric split is proposed for polycrystalline shape memory alloys (SMAs) under multiaxial loading paths. Microplane governing equations are 1D stress–strain relations for normal and shear stresses on each microplane, in which suitable relationships between the microscopic and macroscopic quantities are considered so that switching between elastic and inelastic local responses automatically occurs according to the macroscopic response of SMA without additional constraint. Shear stress on each microplane is expressed by the resultant shear component within the plane to overcome directional bias and to prevent the appearance of shear strain in a pure axial loading or axial strain in a pure shear loading while microplane formulations based on two shear directions may predict such impractical results. The behaviour of SMA under simple and complicated loadings has been studied. In nonproportional loading paths, the model shows interaction between stress components, as well as deviation from normality. Predicted results from the model are in good agreement with those of the existing theoretical and experimental investigations.

Modeling of Shape Memory Alloys Based on Microplane Theory

A three-dimensional microplane constitutive model utilizing statically constrained formulation with volumetric—deviatoric split is presented for shape memory alloys (SMAs). Shear stress within each microplane is described by resultant shear component on the plane. One-dimensional stress—strain laws are used for normal and shear stresses on microplanes by considering suitable adjustments between the macroscopic and the microscopic quantities. The behavior of SMAs under simple and complicated loadings is studied. The model represents interaction between the stress components and the deviation from normality in the case of nonproportional loadings. The results are in good agreement with the existing theoretical and experimental findings.

Identification and Interpretation of Microplane Material Laws

The present paper addresses the so-called microplane formulation which became recently more and more popular for the description of quasi-brittle materials. The essential feature of this material formulation is a split of the local microplane strains and stresses allowing one to resort to simplified or in certain cases even unidirectional constitutive laws. The main attraction of the microplane concept is that an initial or evolving anisotropic material behavior can be described in a natural and simple way. Motivated from a macroscopic viewpoint, it is advocated to restrict the microplane concept to the pure volumetric-deviatoric split, as a constraint subset of the most often applied volumetric-deviatoric-tangential split. This variant has the particular advantage that typical macroscopic responses are directly reflected on the mesoscale. It will be shown that in certain cases the present version of a microplane formulation is closely related to well-known macroscopic models although being much more general than those macroscopic formulations. This close relation is exploited to derive physically sound microplane constitutive laws. Therefore the characteristic damage mechanisms of materials at two levels of observation, (1) at the macroscale in the sense of classical continuum damage mechanics, and (2) at the mesoscale utilizing the so-called microplane concept, are examined. The comparison of the microplane formulation to a well-known macroscopic one-parameter damage model enables the identification and interpretation of the microplane constitutive laws. The constitutive formulations are embedded in a thermodynamically consistent framework. Finally, the performance of the attained microplane formulation is analyzed in a mixed-mode fracture simulation of concrete.

A comparison of damage models formulated on different material scales

This contribution aims at characteristic damage mechanisms of materials at two levels of observation, (a) at the macro-scale in the sense of classical continuum damage mechanics, and (b) at the meso-scale utilizing the so-called microplane concept. The constitutive formulations are embedded in a thermodynamically consistent framework. The microplane formulation is based on a volumetric–deviatoric split (V–D), which is motivated from a macroscopic viewpoint. The main advantage of this formulation is that the material behavior at a material point is characterized by constitutive laws formulated on the individual microplanes, allowing to describe anisotropic material behavior in a very natural and simple way. One particular advantage of the present version of a microplane formulation is also its close relation to macroscopic models which enables the interpretation and identification of the microplane constitutive laws in terms of well-known macroscopic constitutive laws. The appropriate choice of the microplane loading functions is illustrated by a comparison of the microplane damage formulation with a macroscopic two-parameter damage model.

Finite strain generalization of smallstrain constitutive relations for any finite strain tensor and additive volumetric-deviatoric split

The paper deals with finite strain generalization of small-strain constitutive equations for isotropic materials for which the strain is split into a volumetric part and a deviatoric part (the latter characterizing the isochoric strain, i.e. a strain at constant volume). The volumetric-deviatoric split has so far been handled by a multiplicative decomposition of the transformation tensor; but the existing sophisticated complex constitutive models for small strains of cohesive pressure-sensitive dilatant materials, such as concrete and geomaterials, involve an additive decomposition and would be difficult to convert a multiplicative decomposition. It is shown that an additive decomposition of any finite strain tensor, and of the Green-Lagrange strain tensor in particular, is possible, provided that the higher-order terms of the deviatoric strain tensor are allowed to depend on the volumetric strain. This dependence is negligible for concrete and geomaterials because the volumetric strains are normally small, whether or not the deviatoric strains are large. Furthermore, the related question of the choice of the finite-strain measure to be used for the finite-strain generalization is analysed. A transformation of the Green-Lagrange finite strain tensor whose parameters approximately reflect the degrees of freedom equivalent to replacing the small strain tensor by any other possible finite strain measure is proposed. Finally a method by which the stress tensor that is workconjugate to any finite strain tensor can be converted to the Green-Lagrange strain tensor is presented.