A variational framework for fiber‐reinforced viscoelastic soft tissues including damage

Fibrous soft biological tissues such as skin, ligaments, tendons, and arteries are non‐homogeneous composite materials composed of fibers embedded in a ground substance. Cyclic tensile tests on these type of materials usually show a hysteretic stress–strain behavior in which strain rate dependence (viscoelasticity) and softening (Mullins' effect) play a coupled role. The main contribution of the present paper is to present unified variational approach to model both coupled phenomena: nonlinear viscoelasticity and Mullins‐like softening behavior. The approach is labeled as variational because viscous‐strain and damage internal variables are updated based on the minimization of a hyperelastic‐like potential that takes a renewed value at each time step. Numerical examples explores (a) the versatility of the proposed model to account for the two described phenomena according to the chosen functions for the free‐energy and dissipative potentials, (b) the ability of the time‐integration scheme embedded in the incremental potential definition to allow for large time increments, and (c) the capability of the model to mimic experimentally obtained stress–strain cyclic curves of soft tissues. The model implementation on standard finite elements is also tested in which symmetric analytic tangent matrices are used as a natural consequence of the variational nature of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimization of fibrous laminates undergoing progressive damage

Laminates made of fibrous composites are optimized with the objective of maximizing the remaining stiffness at the end of some given load history, during which damage and deterioration of the material have occurred. The material behaviour is thus both highly non‐linear and history dependent. The design variables are the fibre orientations of each of the plies. The progressive and anisotropic damage is modelled by a continuum damage model in which a set of internal damage variables describes the decrease of the elastic moduli and Poisson ratios and yet another set describes the reduction of the strength properties in each of the plies. The evolution of the deterioration is based on a principle of maximum energy dissipation. The numerical model and the formulation of a consistent tangent operator is thoroughly described. The involved design sensitivity analysis is performed by direct differentiation in a manner fully compatible with the numerical model. The importance of taking the damage into consideration in the design process is clearly demonstrated by some computational examples where also the advantages of the optimization process become evident. Copyright © 2000 John Wiley & Sons, Ltd.     

A Comparative Study of Several Material Models for Prediction of Hyperelastic Properties: Application to Silicone-Rubber and Soft Tissues

Abstract:  The correct modelling of constitutive laws is of critical importance for the analysis of mechanical behaviour of solids and structures. For example, the understanding of soft tissue mechanics, because of the nonlinear behaviour commonly displayed by the mechanical properties of such materials, makes common place the use of hyperelastic constitutive models. Hyperelastic models however, depend on sets of variables that must be obtained experimentally. In this study the authors use a computational/experimental scheme, for the study of the nonlinear mechanical behaviour of biological soft tissues under uniaxial tension. The material constants for seven different hyperelastic material models are obtained via inverse methods. The use of Martins's model to fit experimental data is presented in this paper for the first time. The search for an optimal value for each set of material parameters is performed by a Levenberg–Marquardt algorithm. As a control measure, the process is fully applied to silicone-rubber samples subjected to uniaxial tension tests. The fitting accuracy of the experimental stress–strain relation to the theoretical one, for both soft tissues and silicone-rubber (typically nonlinear) is evaluated. This study intents also to select which material models (or model types), the authors will employ in future works, for the analysis of human soft biological tissues.     

A variational framework for fiber‐reinforced viscoelastic soft tissues

The mechanical properties of soft biological tissues vary depending on how the internal structure is organized. Classical examples of tissues are ligaments, tendons, skin, arteries, and annulus fibrous. The main element of such tissues is the fibers which are responsible for the tissue resistance and the main mechanical characteristic is their viscoelastic anisotropic behavior. The objective of this paper is to extend an existing model for isotropic viscoelastic materials in order to include anisotropy provided by fiber reinforcement. The incorporation of the fiber allows the mechanical behavior of these tissues to be simulated. The model is based on a variational framework in which its mechanical behavior is described by a free energy incremental potential whose local minimization provides the constraints for the internal variable updates for each load increment. The main advantage of this variational approach is the ability to represent different material models depending on the choice of suitable potential functions. Finally, the model is implemented in a finite‐element code in order to perform numerical tests to show the ability of the proposed model to represent fiber‐reinforced materials. The material parameters used in the tests were obtained through parameter identification using experimental data available in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

THE INFLUENCE OF THE HARDENING STATE ON TIME DEPENDENT DAMAGE AND ITS CONSIDERATION IN A UNIFIED DAMAGE MODEL

Abstract— A model is presented for the prediction of the lifetime of metals in the high-temperature range under arbitrary variable multiaxial load. The definition of an internal variable for damage in continuum damage mechanics is adopted, which allows indirect measurement of damage via the deformation behaviour. To acquire some knowledge of damage evolution, damage is measured in two ways during uniaxial strain controlled cyclic tests: (a) a change of the modulus of elasticity and (b) a decrease of the peak stress. Surprisingly, both methods lead to results which are in good agreement. A new damage law is then developed (with reference to known models and lifetime rules) which is a modification of the creep damage law of Rabotnov that is extended by a dependence on the inelastic strain rate instead of the dependence on internal variables to take into account the hardening state. Uniaxial as well as multiaxial formulations of the new damage model (Inelastic Strain Rate Modified (ISRM) model) are presented.
The parameters of the ISRM model are determined with a view to applying them to AISI 316 L(N) austenitic steel. Some of the parameters are derived from standard creep experiments. To determine further parameters, the ISRM model is applied to uniaxial cyclic tests. Both failure behaviour and damage evolution are well described.     

A procedure for the computational investigation of stress-relaxation phenomena

The analysis of stress-relaxation tests in viscoelastic materials is discussed. Stress-relaxation tests are performed by means of a two-stage process, as a stretching stage up to an assumed strain level, according to a given strain rate, followed by a constant strain stage. Major attention is devoted to the influence of experimental variables, namely the strain level and the strain rate during stretching stage, on the subsequent relaxation phenomena. A constitutive analysis to interpret the behaviour shown by experimental tests is performed, accounting for the influence of the above mentioned experimental variables, to get an actual definition of time dependent behaviour of the material. This procedure is applied to investigate stress-relaxation phenomena in polymeric materials and soft biological tissues, as ligaments. The results confirm the necessity to analyse data from stress-relaxation tests taking into account the experimental variables that characterize the different stages, and evaluating the influence of the specific variables outlined.     

Modelling of steel phenomena and their interactions – an internal variable approach

Stress‐ and strain‐dependent phase transformations (PT), transformation‐induced plasticity (TRIP) and its interaction with classical plasticity (CP) are important phenomena in the material behaviour of steel, and they may cause distortion of work‐pieces. These phenomena are intensively studied in experiments and their modelling is a current field of investigations. A complex model of material behaviour for simulation and distortion controlling must include the models of relevant phenomena. It is the aim of the current paper to contribute to the formulation of such a model. We use internal variables for describing special effects in the material behaviour in the context of macroscopic continuum mechanics and include them in the thermodynamically consistent modelling.     

Hyper‐reduction of mechanical models involving internal variables

We propose to improve the efficiency of the computation of the reduced‐state variables related to a given reduced basis. This basis is supposed to be built by using the snapshot proper orthogonal decomposition (POD) model reduction method. In the framework of non‐linear mechanical problems involving internal variables, the local integration of the constitutive laws can dramatically limit the computational savings provided by the reduction of the order of the model. This drawback is due to the fact that, using a Galerkin formulation, the size of the reduced basis has no effect on the complexity of the constitutive equations. In this paper we show how a reduced‐basis approximation and a Petrov–Galerkin formulation enable to reduce the computational effort related to the internal variables. The key concept is a reduced integration domain where the integration of the constitutive equations is performed. The local computations being not made over the entire domain, we extrapolate the computed internal variable over the full domain using POD vectors related to the internal variables. This paper shows the improvement of the computational saving obtained by the hyper‐reduction of the elasto‐plastic model of a simple structure. Copyright © 2008 John Wiley & Sons, Ltd.     

A simultaneous solution procedure for strong interactions of generalized Newtonian fluids and viscoelastic solids at large strains

The paper presents a methodology for numerical analyses of coupled systems exhibiting strong interactions of viscoelastic solids and generalized Newtonian fluids. In the monolithic approach, velocity variables are used for both solid and fluid, and the entire set of model equations is discretized with stabilized space–time finite elements. A viscoelastic material model for finite deformations, which is based on the concept of internal variables, describes the stress‐deformation behaviour of the solid. In the generalized Newtonian approach for the fluid, the viscosity depends on the shear strain rate, leading to common non‐Newtonian fluid models like the power‐law. The consideration of non‐linear constitutive equations for solid and fluid documents the capability of the monolithic space–time finite element formulation to deal with complex material models. The methodology is applied to fluid‐conveying cantilevered pipes in order to determine the influence of material non‐linearities on stability characteristics of coupled systems. Copyright © 2005 John Wiley & Sons, Ltd.     

Evaluating a strain energy fatigue method using cyclic plasticity models

For the development of constitutive equations that describe the behaviour of materials under cyclic plastic strains, different kinds of formulations can be adopted. Recently, an energy‐based fatigue damage parameter has been developed to present energy‐fatigue life curves using a calculation of the total strain energy. In this study, the damage criterion is examined by calculation of the plastic strain energy from stress–strain hysteresis loops in the cyclic plasticity models under condition of multi‐axial fatigue. These cyclic plasticity models are the Garud multi‐surface model and the Chaboche nonlinear kinematic hardening model. The models are briefly explained and the general features of their computational procedure are presented. Then, the hysteresis loops of these models will be obtained and the fatigue lives are predicted and compared to experimental data by the ratio of predicted life to experimental life. Consequently, a weighting factor on shear plastic work is presented to decrease the life factors.     

Large strain constitutive modelling and computation for isotropic,creep elastoplastic damage solids

A three-dimensional fully coupled creep elastoplastic damage model at finite strain for isotropic non-linear material is developed. The model is based on the thermodynamics of an irreversible process and the internal state variable theory. A hyperelastic form of stress–strain constitutive relation in conjunction with the multiplicative decomposition of the deformation gradient into elastic and inelastic parts is employed. The pressure-dependent plasticity with strain hardening and the damage model with two damage internal variables are particularly considered. The rounding of stress–strain curves appearing in cycling loading is reproduced by introduction of the creep mechanism into the model. A numerical integration procedure for the coupled constitutive equations with three hierarchical phases is proposed. A consistent tangent matrix with consideration of the fully coupled effects at finite strain is derived. Numerical examples are tested to demonstrate the capability and performance of the present model at large strain.     

Meso-scale deformation and damage in thermally bonded nonwovens

Thermal bonding is the fastest and the cheapest technique for manufacturing nonwovens. Understanding mechanical behaviour of these materials, especially related to damage, can aid in design of products containing nonwoven parts. A finite element (FE) model incorporating mechanical properties related to damage such as maximum stress and strain at failure of fabric’s fibres would be a powerful design and optimisation tool. In this study, polypropylene-based thermally bonded nonwovens manufactured at optimal processing conditions were used as a model system. A damage behaviour of the nonwoven fabric is governed by its single-fibre properties, which are obtained by conducting tensile tests over a wide range of strain rates. The fibres for the tests were extracted from the nonwoven fabric in a way that a single bond point was attached at both ends of each fibre. Additionally, similar tests were performed on unprocessed fibres, which form the nonwoven. Those experiments not only provided insight into damage mechanisms of fibres in thermally bonded nonwovens but also demonstrated a significant drop in magnitudes of failure stress and respective strain in fibres due to the bonding process. A novel technique was introduced in this study to develop damage criteria based on the deformation and fracture behaviour of a single fibre in a thermally bonded nonwoven fabric. The damage behaviour of a fibrous network within the thermally bonded fabric was simulated with a FE model consisting of a number of fibres attached to two neighbouring bond points. Additionally, various arrangements of fibres’ orientation and material properties were implemented in the model to analyse the respective effects.     

A viscoelastic constitutive model with nonlinear evolutionary internal variables

Summary Based on the internal variable theory, a viscoelastic constitutive model of a highly deformable continuous medium is proposed. A set of second rank tensorial internal state variables corresponding to Biot's strain is introduced, and a nonlinear evolution law for these internal variables is suggested. The proposed model may be considered as an extension of the network theory of rubber elasticity to take the viscous effects into account. In order to verify the validity of the present model, uniaxial tension tests for HDPE are carried out at different strain rates. The prediction of the present model shows a good agreement with the experimental data. Finally, a discussion of the present constitutive model is given. It is found that the present constitutive model is more flexible to describe the strain rate sensitivity of polymeric materials in a wide range of strain rates.     

Numerical integration on the sphere and its effect on the material symmetry of constitutive equations—A comparative study

In the present paper, we consider a class of constitutive models based on numerical integration on the unit sphere. The directional behaviour of the quadrature schemes and its effect on the symmetry properties of these constitutive models are studied by subjecting the set of integration points on the sphere to arbitrary rigid rotations. We investigate a number of recently proposed integration schemes in application to a full network model of rubber elasticity and to an exponential model for soft tissues. In order to assess and compare these schemes, statistical methods are presented and applied. The analysis discloses a number of integration schemes that offer a good compromise between the numerical error and the number of integration points. However, as a general result it turns out that numerical integration is prone to introduce strong anisotropy into originally isotropic constitutive equations, in particular, for highly non‐linear integrand functions. The consequences for application of the investigated class of constitutive models in finite element calculations are highlighted in a benchmark‐like numerical example. Copyright © 2009 John Wiley & Sons, Ltd.     

Strain-life approach in thermo-mechanical fatigue evaluation of complex structures

This paper is a contribution to strain‐life approach evaluation of thermo‐mechanically loaded structures. It takes into consideration the uncoupling of stress and damage evaluation and has the option of importing non‐linear or linear stress results from finite element analysis (FEA). The multiaxiality is considered with the signed von Mises method. In the developed Damage Calculation Program (DCP) local temperature‐stress‐strain behaviour is modelled with an operator of the Prandtl type and damage is estimated by use of the strain‐life approach and Skelton's energy criterion. Material data were obtained from standard isothermal strain‐controlled low cycle fatigue (LCF) tests, with linear parameter interpolation or piecewise cubic Hermite interpolation being used to estimate values at unmeasured temperature points. The model is shown with examples of constant temperature loading and random force‐temperature history. Additional research was done regarding the temperature dependency of the K_p used in the Neuber approximate formula for stress‐strain estimation from linear FEA results. The proposed model enables computationally fast thermo‐mechanical fatigue (TMF) damage estimations for random load and temperature histories.     

CRACK TIP BEHAVIOUR AND CRACK PROPAGATION IN DUCTILE MATERIALS

Dilatational plastic equations, which can include the effects of ductile damage, are derived based on the equivalency in expressions for dissipated plastic work. Void damage developed internally at the large-strain stage is represented by an effective continuum being strain-softened and plastically dilated. Accumulation of this local damage leads to progressive failure in materials. With regard to this microstructural background, the constitutive parameters included for characterizing material behaviour have the sense of internal variables. They are not able to be determined explicitly by macroscopic testing but rather through computer simulation of experimental curves and data. Application of this constitutive model to mode-I cracking examples demonstrates that a huge strain concentration accompanied by a substantial drop of stress does occur near the crack tip. Eventually, crack propagation is simulated by using finite elements in computations. Two numerical examples show good accordance with experimental data. The whole procedure of study serves as a justification of the constitutive formulation proposed in the text.     

A regularized force‐based beam element with a damage–plastic section constitutive law

A new beam finite element is presented, with a generalized section constitutive law based on damage mechanics and plasticity, to analyse the cyclic structural response of plane frames. Both displacement‐based and force‐based (FB) approaches are used and compared, to demonstrate the significant advantages of the FB formulation in the presence of material non‐linearity. In order to overcome the analytical problems and the pathological mesh dependency of the numerical response in the presence of strain‐softening post‐peak behaviour, a classical non‐local regularization procedure is adopted first, based on the integral definition of the associated variable governing the damaging evolution process. Subsequently, for the FB element a new simple regularization technique is proposed based on a selected integration procedure along the element length, which predefines the location of the Gauss points in the beam region, where the localization phenomena take place. As for the other computational aspects, an iterative element state determination is adopted for the FB formulation and a local predictor–corrector algorithm is used to solve the incremental evolution problems of the damage and plastic internal variables. Finally, some examples are shown on simple beams and frames, subjected to monotonically increasing and cyclic loading conditions. Copyright © 2006 John Wiley & Sons, Ltd.