Anisotropic Finite Strain Research Articles

A nonlinear finite viscoelastic formulation relative to the viscous intermediate configuration applied to plants

AbstractIn the contribution at hand, a new formulation for finite strain viscosity relative to the viscous intermediate configuration is presented. The evolution of the viscous deformations is based upon a new numerical approach, which allows for a consistent consideration of anisotropic finite strain viscoelasticity, according to the authors knowledge. A standard Maxwell model is used to describe viscous behaviour at finite deformations. Furthermore, the orthotropic Yeoh material model is extended to include a distinction between behaviour under tensile and compression loading. The proposed formulation is validated, and parameters of the model are identified by material tests on Sorghum bicolor plants. Subsequently, numerical examples are shown to demonstrate the capabilities of the model. In general, the proposed Yeoh material formulation is shown to accurately represent the inability of fibres to carry compression loading. Furthermore, the viscoelastic approach, developed relative to the viscous intermediate configuration, is demonstrated to be capable of producing plausible results. Additionally, the mechanical behaviour of Sorghum bicolor plants is simulated using the introduced formulation. The results show that the contribution at hand describes a novel methodology to simulate the viscoelastic behaviour of plant materials reliably.

An anisotropic finite strain elastoplastic model considering different plastic spin effects on the intermediate configuration

Large deformation anisotropic elastoplastic models have important application background and research value in engineering and materials fields. In this paper, a macroscopic phenomenological model of elastoplastic anisotropic deformation considering plastic spin is proposed. The model is based on the multiplication decomposition of deformation gradient. The free energy function is expressed as the isotropic function of the strain and the structural tensor on the intermediate configuration, which is a push-forward of initial configuration by using plastic deformation gradient. Advantageously, the non-equilibrium free energy function remains invariant under the superimposed rigid body rotation on the intermediate configuration, which is due to the non-uniqueness of the multiplicative decomposition of deformation gradient. However, the rate of superimposed rigid body rotation has effects on the model. The effects are discussed by considering three different spin assumptions. Numerical simulations show that the plastic spin has influence on the calculation results. Therefore, the plastic spin assumption should be carefully selected in the practical application of elastoplastic anisotropic model.

An anisotropic finite strain viscoelastic model considering different spin effects on the intermediate configuration

An anisotropic finite strain viscoelastic model considering different spin effects on the intermediate configuration

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.

Anisotropic finite strain viscoelasticity: Constitutive modeling and finite element implementation

A new anisotropic finite strain viscoelastic model is presented, which is based on the Holzapfel type anisotropic hyperelastic strain-energy function. The anisotropic viscous part is set to be independent from the isotropic viscous part. A corresponding multiplicative decomposition of the deformation gradient is presented, and a specific definition of the anisotropic viscous fiber term. A new method to develop the evolution equations of the viscous internal variables is also provided. The time derivatives of the internal variables for the isotropic and anisotropic viscous parts are obtained from the evolution equation of the second Piola–Kirchhoff stress for the viscous part. The corresponding analytical validation of non-negative dissipation using the second law of thermodynamics is provided. The incompressible plane stress case is used to achieve an analytical solution for the proposed constitutive model. A good agreement between the finite element results and the analytical solution is obtained. Finally, some numerical simulations are presented, including the viscous hysteresis response, experimental data fitting and a relaxation test.

Fully anisotropic finite strain viscoelasticity based on a reverse multiplicative decomposition and logarithmic strains

In this paper we present a novel formulation for phenomenological anisotropic finite visco-hyperelasticity. The formulation is based on a multiplicative decomposition of the equilibrated deformation gradient into nonequilibrated elastic and viscous contributions. The proposal in this paper is a decomposition reversed respect to that from Sidoroff allowing for anisotropic viscous contributions. Independent anisotropic stored energies are employed for equilibrated and non-equilibrated parts. The formulation uses logarithmic strain measures in order to be teamed with spline-based hyperelasticity. Some examples compare the results with formulations that use the Sidoroff decomposition and also show the enhanced capabilities of the present model.

Anisotropic finite strain viscoelasticity based on the Sidoroff multiplicative decomposition and logarithmic strains

In this paper a purely phenomenological formulation and finite element numerical implementation for quasi-incompressible transversely isotropic and orthotropic materials is presented. The stored en...

Modelling of tear propagation and arrest in fibre-reinforced soft tissue subject to internal pressure

The prediction of soft-tissue failure may yield a better understanding of the pathogenesis of arterial dissection and help to advance diagnostic and therapeutic strategies for the treatment of this and other diseases and injuries involving the tearing of soft tissue, such as aortic dissection. In this paper, we present computational models of tear propagation in fibre-reinforced soft tissue undergoing finite deformation, modelled by a hyperelastic anisotropic constitutive law. We adopt the appropriate energy argument for anisotropic finite strain materials to determine whether a tear can propagate when subject to internal pressure loading. The energy release rate is evaluated with an efficient numerical scheme that makes use of adaptive tear lengths. As an illustration, we present the calculation of the energy release rate for a two-dimensional strip of tissue with a pre-existing tear of length \(a\) under internal pressure \(p\) and show the effect of fibre orientation. This calculation allows us to locate the potential bifurcation to tear propagation in the \((a,p)\) plane. The numerical predictions are verified by analytical solutions for simpler cases. We have identified a scenario of tear arrest, which is observed clinically, when the surrounding connective tissues are accounted for. Finally, the limitations of the models and further directions for applications are discussed.

Anisotropic finite strain hyperelasticity based on the multiplicative decomposition of the deformation gradient

AbstractIn this contribution a new constitutive model for transversely isotropic materials is presented. The proposed model is based on the multiplicative decomposition of the deformation gradient into one part containing the deformation only in the direction of anisotropy and another part describing the remaining deformation. This clear assignment leads to a decoupling of the stress‐state. The model is investigated analytically in view of simple tension. Moreover, an inhomogenous deformation is solved using a finite elements simulation. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On a formulation for anisotropic elastoplasticity at finite strains invariant with respect to the intermediate configuration

The paper is concerned with a formulation of anisotropic finite strain inelasticity based on the multiplicative decomposition of the deformation gradient F = F e F p . A major feature of the theory is its invariance with respect to rotations superimposed on the inelastic part of the deformation gradient. The paper motivates and shows how such an invariance can be achieved. At the heart of the formulation is the mixed-variant transformation of the structural tensor, defined as the tensor product of the privileged directions of the material as given in a reference configuration, under the action of F p . Issues related to the plastic material spin are discussed in detail. It is shown that, in contrast to the isotropic case, any flow function formulated purely in terms of stress quantities, necessarily exhibits a non-vanishing plastic material spin. The possible construction of spin-free rates is discussed as well, where it is shown that the flow rule must then depend not only on the stress but on the strain as well.

Anisotropic elasto‐plastic damage formulation for finite strains using a damage structural tensor and the effective stress concept

AbstractUsing the strain equivalence principle and the effective stress concept an anisotropic finite strain damage model is proposed as a direct extension of the classical isotropic LEMAITRE damage model to the anisotropic finite strain case. The damage tensor is included as a structural tensor in the complementary energy potential. This approach allows to consider a wide range of anisotropic damage phenomena on the level of continuum mechanics. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An anisotropic fibre-matrix material model at finite elastic-plastic strains

In this paper a constitutive model for anisotropic finite strain plasticity, which considers the major effects of the macroscopic behaviour of matrix-fibre materials, is presented. As essential feature matrix and fibres are treated separately, which allows as many bundles of fibres as desired. The free energy function is additively split into a part related to the matrix and in parts corresponding to the fibres. Usually the free energy function is defined by the integrity basis of the main variable and structural tensors, which leads to complicated numerical schemes. Here, the continuum is considered as superimposed of the isotropic matrix and further one-dimensional continua each of them represents one bundle of fibres. The deformation gradient applies to all continua introducing a constraint, which links the different continua. One of the most striking features of the model is its suitability for a numerical treatment.

Numerical analyses of discontinuous material bifurcation: strong and weak discontinuities

In this paper an algorithmic formulation for numerical analyses of material bifurcation is presented. Conditions for the onset of both weak discontinuities (discontinuous strain rates) and strong discontinuities (discontinuous velocity fields) are summarized. Based on a recently proposed plasticity model formulated within the logarithmic strain space, the condition for the formation of strong discontinuities is extended to anisotropic finite strain plasticity theory. The resulting equations associated with the mode of bifurcation are solved numerically. For that purpose, an equivalent optimization problem is considered. The algorithmic formulation is based on Newton’s method using a consistent linearization. To enlarge the radius of convergence, a line search strategy is applied. The applicability of the proposed implementation as well as its performance and numerical robustness is investigated by means of three-dimensional numerical bifurcation analyses of a Drucker–Prager type plasticity model.

Nonlinear Viscoelastic Solids

1. Introduction. The classical theories of the mechanics of deformable materials are, for the most part, linear theories and ones in which the stress depends only on the kinematic situation existing at the time of measurement of the stress. For many materials, such as viscoelastic solids and liquids, these assumptions are not valid. Since the paper by Oldroyd [13], considerable advances have been made in formulating the continuum mechanics of such materials. In the present paper an account of this work will be given, particularly in relation to viscoelastic solids. Provided that any characteristic lengths which may exist iu the solid can be ignored, we may take as our starting point in the development of appropriate constitutive equations the assumption that the stress tensor: depends not only on the displacement gradients existing in the material considered at the instant at which the stress is measured, but also on those which existed in it at all previous times, i.e., the stress is a functional of the deformation gradient history up to the instant of measurement. If some reference state is adequately specified, the assumption that stress depends on the history of the deformation gradients and the assumption that stress depends on the history of the velocity or acceleration. gradients are quite equivalent, since either of the latter determine the deformation gradient history. The form of the functional dependence of the stress on the deformation gradient history cannot be arbitrarily chosen. It is subject to two major restrictions. The first of these, which is discussed in 2, arises from the fact that a simultaneous time-dependent rotation of the deformed body and reference system leaves the stress components unaltered. The second, which is discussed in 3 and 5, arises from any symmetry which the material may possess in its reference state. The important particular case when the material is isotropic in its reference state is discussed in 5. In certain circumstances, the history of the deformation gradients may be adequately described by a knowledge of the values of the deformation gradients and of the gradients of velocity, acceleration, second acceleration., etc., at the instant of measurement of the stress. The development of canonical forms for the constitutive equations in this case is discussed generally in 4 and for the particular circumstance when the material is isotropic in its reference state in 5. The constitutive equations which arise from these general studies are inevitably complicated. In 6 attention is drawn to the great simplifications whichmay arise