Viscoplastic Part Research Articles

Modeling of nonlinear viscoelastic–viscoplastic frictional contact problems

This paper presents a nonlinear time-dependent computational model for analyzing the quasistatic response of viscoelastic–viscoplastic frictional contact problems. Both material and geometrical nonlinearities are considered in the framework of the Lagrangian description. The material nonlinearity is due to the time–stress-dependency of the constitutive equation, while the geometrical nonlinearity is due to large displacements and rotations, but small strains. The model is derived based on an implicit time-integration method within a general displacement-based finite element analysis. The nonlinear Schapery’s single integral model is used to model the viscoelastic part, while the Perzyna model is adopted model the viscoplastic part. The exponential form of the Prony series is used to represent the transient component of the viscoelastic creep compliance, since it permits hereditary effects to be computed recursively. In addition, an incremental form of the viscoplastic strain component is derived in the framework of associative viscoplasticity based on implicit integration scheme. Throughout the contact interface, friction is simulated using a local-nonlinear friction law, while the Lagrange multiplier method is adopted to model the inequality contact constraints. In a presented case study, the complete contact configuration and internal stresses are analyzed. Results show the significant effects of material nonlinearity, viscoplastic flow, and friction on the response of contact problems.

Contribution of Primary Creep in Modeling the Mechanical Behavior of Polycrystalline Ice

In most of the models proposed for deformation of ice, in addition to instantaneous elastic and viscoelastic parts, a viscoplastic part is also considered. An expression used for material secondary creep is usually employed to describe the viscoplastic component. In this study, however, another viscoplastic deformation of primary creep type is also considered in addition to the secondary creep. Therefore the permanent contribution of deformation is suggested to consist of primary and secondary creep parts. The existence of the primary creep contribution is investigated and characterized by using experimental results reported in the literature. The identified primary creep contribution is then validated by other available experimental results. Finally, the significance of primary creep in the inelastic behavior will be discussed.

A fully coupled elastoviscoplastic damage model at finite strains for mineral filled semi-crystalline polymer

A phenomenological finite strain non-associated elastoviscoplastic model coupled with damage is proposed in order to simulate the behaviour of a 20% mineral filled semi-crystalline polymer for a large strain rate range under several loading conditions. In the proposed model, the direct relation between the logarithmic rate of the Eulerian Hencky strain tensor (work-conjugate of the Cauchy stress) with the additively decomposition of the stretching tensor in an elastic and a viscoplastic part is used. The viscoplastic formulation is developed with the association of a pressure dependent yield surface coupled with damage to take the rate and the pressure dependency into account. In order to capture the non-isochoric deformation involving expansion under tensile loading and compaction under compression loading, a viscoplastic potential is developed. This model is implemented in a user-material subroutine in an implicit finite element code with a fully implicit viscoplastic return scheme for solid and shell elements. The particular case of shell elements is dealt with by using a second iteration loop to ensure the zero-normal-stress condition. All the material parameters are identified from experimental tests carried out at several kinds and speed loadings. Numerical responses of the proposed model are close to the experiments for several kinds and speed loadings.

Elastoplastic–viscoplastic modelling of metal powder compaction: application to hot isostatic pressing

A constitutive model is proposed for the densification of metal powder during hot isostatic pressing. The model considers an inelastic deformation resulting from time dependent (viscoplastic) and time independent (plastic) mechanisms during loading. With employing the Abouaf’s formalism for viscoplastic part, the paper is focused mainly on the plasticity contribution including the hardening effects of both relative density and the equivalent plastic strain. The proposed plastic–viscoplastic model is summarised to an equation expressing the densification rate under hydrostatic loading. The gas atomised 316LN stainless steel powder was used in this study. Model parameters necessary for simulation of HIP process during a given pressure ramp at constant temperature are identified using previously published data. The identified model is then verified using experimental data obtained from HIP trials performed at another fixed temperature but with varied pressure ramp rates. A good agreement was found between the model prediction and the experimental results.

Viscoelastic-Viscoplastic Modelling of the Scratch Response of PMMA

This paper aims at understanding how to model the time-dependent behavior of PMMA during a scratch loading at a constant speed and at middle strain levels. A brief experimental study is first presented, consisting of the analysis of microscratches carried out at various scratching velocities and normal loads. The loading conditions have been chosen in such a way that neither (visco)elasticity nor (visco)plasticity of the PMMA may be neglected a priori. The main analyzed parameter is the tip penetration depth measured during the steady state. Then, a finite element model is used to investigate the potential of classical elastic-viscoplastic constitutive models to reproduce these experimental results. It is mainly shown that these models lead to unsatisfying results. More specifically, it is pointed out here that the time-independent Young modulus used in such models is not suitable. To take into account this feature, a viscoelastic-viscoplastic model based on the connection in series of a viscoelastic part with a viscoplastic part is proposed. It is shown that it leads to more acceptable results, which points out the importance of viscoelasticity in the scratch behavior of solid polymers.

Modeling nonlinear creep and recovery behaviors of synthetic fiber ropes for deepwater moorings

The synthetic fiber ropes such as the aramid and polyester ones applied to deepwater mooring systems always exhibit obvious time-dependent like creep and recovery behaviors due to the viscoelasticity and viscoplasticity of the materials, which affect not only the modulus evolution of mooring ropes but also the dynamic response and fatigue performance of the taut-wire mooring system. In the present work, the Schapery's theory combined with Owen's one-dimensional rheological model is proposed to describe both viscoelastic and viscoplastic behaviors of the aramid and polyester fiber ropes. In the viscoelastic part, the Prony series is chosen to describe the transient compliance, which is more accurate than other functions especially under complex loadings; in the viscoplastic part, the adopted viscoplastic function is more suitable for the strain hardening behaviors and the stable state of the materials under variable stress levels. Detailed methods for identifying the model parameters are proposed, which can be applied to any component of the fiber rope such as the fiber, yarn, sub-rope and rope. The present model is capable of quantitatively capturing the change-in-length properties of fiber ropes reported by Flory et al., and can be easily incorporated in the commercial software for mooring analysis. In order to examine the feasibility and precision of the model, the viscoelastic and viscoplastic strains are calculated and compared with experimental and other numerical simulation results. It is observed that there is a good agreement between the predicted and experimental data, and the physically irrational results caused by the key parameter DP previously noticed by Chailleux and Davies can be well eliminated. The present model provides a better tool to further understand the nonlinear behaviors of synthetic fiber ropes for deepwater moorings.

Effect of the molecular network on high-strain compression of cross-linked polyethylene

A series of polyethylene samples cross-linked by electron irradiation with various doses was prepared and deformed by plane-strain compression. The stress observed in the neat and cross-linked samples was resolved into an elastic and visco-plastic part. The elastic part manifested as a quasi-stationary residual stress in relaxation experiments while the visco-plastic component was related to the relaxed part of the stress. The elastic stress component, found independent of the deformation rate, consists of elastic response of the crystalline skeleton and the response of the molecular network within amorphous phase. On contrary, the relaxing, visco-plastic component of the stress depends strongly on the deformation rate. It can be approximated by a sum of two relaxation processes: short around 200s and long at the range of 104s.The rubber-like network stress, being a sub-component of the observed elastic part of the stress, increases substantially above the true strain of e=1.0. This stress component was found to be the main source of the strong strain hardening observed in all compressed samples at high strains.

Simulation of curing-induced viscoplastic deformation: a new approach considering chemo-thermomechanical coupling

A modelling and simulation approach for plastic deformation effects in curing resins is presented. For this purpose known rheological models of viscoelasticity and viscoplasticity are combined and a thermochemical element is added to account for chemical shrinkage and thermal expansion. The degree of cure of the resin has a major influence on the behaviour of the curing material, and therefore, the material model is formulated depending on the degree of cure. It affects the viscoelastic behaviour as well as the chemical shrinkage and the yield function of the viscoplastic part of the model. For the yield function a von Mises approach with isotropic hardening is chosen, where the initial yield stress as well as the yield surface depends on the degree of cure and the temperature.

Three-dimensional chemo-thermomechanically coupled simulation of curing adhesives including viscoplasticity and chemical shrinkage

Based on the one-dimensional material model developed by Liebl et al. (Arch Appl Mech, 2011) a three-dimensional viscoelastic-viscoplastic material model for small deformations of curing adhesives on the basis of continuum mechanics is proposed in this contribution. The model describes the most relevant phenomena which occur during curing processes in the automotive industry and includes the effects of temperature and degree of cure on the mechanical properties of the material. Thermal expansion as well as chemical shrinkage are also contained. The yield stress for the viscoplastic part of the model goes back to the work of Schlimmer and Mahnken (Int J Numer Meth Eng 63:1461---1477, 2005), but is formulated in reference to the degree of cure and the temperature. Therefore this model considers chemo-thermomechanical coupling and extends the plasticity approach of Schlimmer and Mahnken, which is devised for cured adhesives, to the whole curing range, from the uncured to the fully cured adhesive. A peculiar focus is hereby laid on epoxy resins used in the automotive industry as structural adhesives.

Use of a Continuum Damage Model Based on Energy Equivalence to Predict the Response of a Single-Crystal Superalloy

Anisotropic viscoplasticity coupled with anisotropic damage has been modeled in previous works by using the energy equivalence principle appropriately adjusted. Isotropic and kinematic hardenings are present in the viscoplastic part of the model and the evolution equations for the hardening variables incorporate both static and dynamic recovery terms. The main difference to other approaches consists in the formulation of the energy equivalence principle for the plastic stress power and the rate of hardening energy stored in the material. As a practical consequence, a yield function has been established, which depends, besides effective stress variables, on specific functions of damage. The present paper addresses the capabilities of the model in predicting responses of deformation processes with complex specimen geometry. In particular, multiple notched circular specimens and plates with multiple holes under cyclic loading conditions are considered. Comparison of predicted responses with experimental results confirms the convenience of the proposed theory for describing anisotropic damage effects.

A dynamic elastic-visco-plastic unilateral contact problem with normal damped response and Coulomb friction

We consider a dynamic frictional contact problem between an elastic-visco-plastic body and a foundation. The contact is modelled with a normal damped response condition of such a type that the normal velocity is restricted with unilateral constraint, associated with the Coulomb law in which the coefficient of friction may depend on the velocity. We derive a variational formulation of the problem which has the form of a system coupling an integro–differential equation for the stress field with an evolutionary variational inequality for the displacement field. This inequality is approximated by a variational equation using a smoothing of the friction and the penalty approximation of the unilateral condition. The existence of a weak solution to the variational equation is proved by the Galerkin method for an auxiliary problem with given viscoplastic part of the stress and a fixed point argument. The solvability of the original problem is proved by passing to the limit of the penalty parameter and the smoothing parameter. This convergence is based on a certain regularity of solutions which is verified with the use of a local rectification of the boundary and a translation method.

Viscoplastic response and modeling of asphalt-aggregate mixes

The strain response of asphalt-aggregate mixes to applied stresses is decomposed additively into a viscoelastic part and a viscoplastic part. The paper focuses on the response and modeling of the viscoplastic component; it includes the development of a multiaxial constitutive formulation that is capable of generating: (i) strain hardening when the loading is applied in one direction; (ii) strain softening immediately after stress reversals; (iii) volumetric changes under uniaxial conditions or isotropic conditions, or both; and (iv) directional non-symmetry. In order to investigate the model’s capabilities, four tests were performed sequentially on one asphalt sample. The tests were limited to pre-peak conditions and one temperature and consisted of creep and recovery sequences in uniaxial tension, uniaxial compression, isotropic compression and uniaxial tension-compression. Analysis of the results showed that the new theory, once calibrated, was able to adequately reproduce the viscoplastic strain component; its forecastability, however, was found limited.

Tensile creep of a long‐fibre glass mat thermoplastic (GMT) composite. II. Viscoelastic‐viscoplastic constitutive modeling

AbstractIn Part I of this article, the short‐term tensile creep of a 3‐mm‐thick continuous long‐fibre glass mat thermoplastic composite was characterized and found to be linear viscoelastic up to 20 MPa. Subsequently, a nonlinear viscoelastic model has been developed for stresses up to 60 MPa for relatively short creep durations. The creep response was also compared with the same composite material having twice the thickness for a lower stress range. Here in Part II, the work has been extended to characterize and model longer term creep and recovery in the 3‐mm composite for stresses up to near failure. Long‐term creep tests consisting of 1‐day loading followed by recovery were carried out in the nonlinear viscoelastic stress range of the material, i.e., 20–80 MPa in increments of 10 MPa. The material exhibited tertiary creep at 80 MPa and hence data up‐to 70 MPa has been used for model development. It was found that viscoplastic strains of about 10% of the instantaneous strains were developed under load. Hence, a non‐linear viscoelastic–viscoplastic constitutive model has been developed to represent the considerable plastic strains for the long‐term tests. Findley's model which is the reduced form of the Schapery non‐linear viscoelastic model was found to be sufficient to model the viscoelastic behavior. The viscoplastic strains were modeled using the Zapas and Crissman viscoplastic model. A parameter estimation method which isolates the viscoelastic component from the viscoplastic part of the nonlinear model has been developed. The model predictions were found to be in good agreement with the average experimental curves. POLYM. COMPOS., 2009. © 2008 Society of Plastics Engineers

Embedded polycrystal plasticity and adaptive sampling

A simulation capability for multi-scale embedded polycrystal plasticity is demonstrated with over two orders of magnitude wall-clock speedup compared to direct embedding. In the coarse-scale material model, the visco-plastic part of the material response is based on parameters determined from polycrystal level fine-scale calculations. Polycrystal plasticity parameters are approximated from fine-scale calculations using adaptive sampling to substantially reduce the total number of expensive fine-scale calculations which must be performed. The adaptive sampling method uses Kriging models for local interpolation of the fine-scale plasticity parameters and a metric-tree database for storage and retrieval of the fine-scale response models. Efficacy of the method is demonstrated through a variety of example problems involving both quasi-static and dynamic loading scenarios.

Micro–macro modelling of the effects of the grain size distribution on the plastic flow stress of heterogeneous materials

When the mean grain size of polycrystalline materials is larger than ∼100nm, it is commonly accepted for metals, intermetallics or ceramics that the plastic flow stress scales linearly with the inverse square root of the mean grain size (the so-called Hall–Petch behaviour). However, in this classic formalism, only the mean grain size is considered in a semi phenomenological way, and, the fact that the grains form a population of stochastic nature with different sizes and shapes is not stated. Here, a new self-consistent model making use of the “translated fields” technique for elastic–viscoplastic materials is developed as micro–macro scale transition scheme, and, the aggregate is composed of spherical randomly distributed grains with a grain size distribution following a log-normal statistical function. The constitutive behaviour of each grain is described by a partitioned strain rate into an elastic part and a viscoplastic part. The viscoplastic strain rate is described by an isotropic power law including the grain diameter through the reference stress. Numerical results firstly display that the plastic flow stress of the material depends on both the mean grain size and the grain size dispersion of the distribution. Besides, the role of the dispersion is more important when the mean grain size is on the order of the μm and the trend is a decrease of the flow stress with an increase of the dispersion. Secondly, predictions of second order internal stresses within the material indicate an increase in the internal stresses when grain size dispersion is increased, so that the plastic flow stress of the material depends on a competition between the respective distributions of internal stresses and individual flow stresses of the grains.

Viscoplastic deformation of an epoxy resin at elevated temperatures

AbstractThe tensile behavior under monotonic loading and stress‐relaxation testing of an epoxy resin has been studied. Experimental data at various strain rates and three temperatures from ambient up to just below Tg were performed, to study the transition from the brittle behavior to a ductile and therefore viscoplastic one. Dynamic mechanical analysis was applied to study the glass transition region of the material. Furthermore, a three‐dimensional viscoplastic model was used to simulate the experimental results. This model incorporates all features of yield, strain softening, strain hardening, and rate/temperature dependence. The multiplicative decomposition of the deformation tensor into an elastic and viscoplastic part has also been applied, following the element arrangement in the mechanical model. A stress‐dependent viscosity was controlling the stress–strain material behavior, involving model parameters, calculated from the Eyring plots. A new equation for the evolution of the activation volume with deformation was proposed, based on a probability density function. The model capability was further verified by applying the same set of parameters to predict with a good accuracy the stress‐relaxation data as well. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 101: 2027–2033, 2006

Tensile creep behavior of unidirectional glass-fiber polymer composites

The tensile creep behavior of unidirectional glass-fiber polymer composites was studied at three different temperatures, namely 298, 333, and 353 K. Testing was performed on the pure epoxy matrix, the 0° specimens as well as off-axis at 15, 30, and 60 degrees in respect to the axis of tension. The creep strain rate was negligible at room temperature, while it was considerable at the higher temperatures examined. The materials exhibit nonlinear viscoelastic behavior, and the creep response of the composites was treated as a thermally activated rate process. The creep strain was considered to include an elastic, a viscoelastic and a viscoplastic part. The viscoplastic part was calculated through a functional form, developed in a previous work, assuming that viscoplastic response of polymer composites arises mainly from the matrix viscoplasticity. The model predictions in terms of creep compliances were found to be satisfactory, compared with the experimental results. POLYM. COMPOS. 26:287–292, 2005. © 2005 Society of Plastics Engineers.

On fast fracture in an elastic-(plastic)-viscoplastic solid Part II – The motion of crack

The motion of a crack in an elastic-(plastic )-viscoplastic medium is studied in terms of an energetic analysis. Combined with the stress and velocity fields obtained in Part 1, Kishimoto's energy integral, Ĵ, is used as a crack driving force to determine its motion. The major results obtained are: (1) dependence of crack speed on a modified near-field parameter, KI tip, (or equivalently, a modified dynamic energy release, GI tip), which is different from the usual stress intensity factor KI of an elastic crack-tip field but is related to it; (2) influence of inelastic effect, such as the viscoplastic exponent n, on the motion of the crack; and (3) stability condition of crack motion. In particular, for the last point, it has been found that, for a given loading and material coefficients, there exist two possible motions of the crack: one is stable crack growth and the other is unstable fracture. The lower and upper bounds of crack motion are also discussed. It is finally shown that the maximum crack velocity is lower than the Rayleigh wave speed, and is dependent on the viscoplastic exponent of the material.

Experimental and Three-Dimensional Finite Element Study of Scratch Test of Polymers at Large Deformations

An experimental and numerical study of the scratch test on polymers near their surface is presented. The elastoplastic response of three polymers is compared during scratch tests at large deformations: polycarbonate, a thermosetting polymer and a sol-gel hard coating composed of a hybrid matrix (thermosetting polymer-mineral) reinforced with oxide nanoparticles. The experiments were performed using a nanoindenter with a conical diamond tip having an included angle of 30 deg and a spherical radius of 600 nm. The observations obtained revealed that thermosetting polymers have a larger elastic recovery and a higher hardness than polycarbonate. The origin of this difference in scratch resistance was investigated with numerical modelling of the scratch test in three dimensions. Starting from results obtained by Bucaille (J. Mat. Sci., 37, pp. 3999–4011, 2002) using an inverse analysis of the indentation test, the mechanical behavior of polymers is modeled with Young’s modulus for the elastic part and with the G’sell-Jonas’ law with an exponential strain hardening for the viscoplastic part. The strain hardening coefficient is the main characteristic parameter differentiating the three studied polymers. Its value is equal to 0.5, 4.5, and 35, for polycarbonate, the thermosetting polymer and the reinforced thermosetting polymer, respectively. Firstly, simulations reveals that plastic strains are higher in scratch tests than in indentation tests, and that the magnitude of the plastic strains decreases as the strain hardening increases. For scratching on polycarbonate and for a penetration depth of 0.5 μm of the indenter mentioned above, the representative strain is equal to 124%. Secondly, in agreement with experimental results, numerical modeling shows that an increase in the strain hardening coefficient reduces the penetration depth of the indenter into the material and decreases the depth of the residual groove, which means an improvement in the scratch resistance.

A phenomenological constitutive model for the sintering of alumina powder

This paper develops a phenomenological model for the densification and viscous behavior of alumina powder during sintering. The free sintering strain rate part and the viscoplastic strain rate part of the constitutive model were determined from experimental data by using the experimental method proposed for WC–Co mixture by Bouvard and co-workers. From densification data under free sintering by using reference and stairway heating cycles with samples of three different green densities, a simple densification equation was identified. This equation expresses the densification rate as a function of temperature, relative density, and green density. The predictive capability of the identified densification equation was discussed by investigating the densification of alumina powder during various sintering cycles and with different green densities. For the viscoplastic strain rate part of the constitutive model, the axial viscosity and the viscous Poisson's ratio of an alumina powder compact were determined by using an intermittent loading method.