Showing papers on "Constitutive equation published in 2019"

The hardening soil model: Formulation and verification

Abstract: A new constitutive model is introduced which is formulated in the framework of classical theory of plasticity. In the model the total strains are calculated using a stress-dependent stiffness, different for both virgin loading and un-/reloading. The plastic strains are calculated by introducing a multi-surface yield criterion. Hardening is assumed to be isotropic depending on both the plastic shear and volumetric strain. For the frictional hardening a non-associated and for the cap hardening an associated flow rule is assumed. First the model is written in its rate form. Therefor the essential equations for the stiffness modules, the yield-, failure- and plastic potential surfaces are given. In the next part some remarks are given on the models incremental implementation in the Plaxis computer code. The parameters used in the model are summarized, their physical interpretation and determination are explained in detail. The model is calibrated for a loose sand for which a lot of experimental data is available. With the so calibrated model undrained shear tests and pressuremeter tests are back-calculated. The paper ends with some remarks on the limitations of the model and an outlook on further developments.

Mechanical Behavior and Damage Constitutive Model of Rock Subjected to Water-Weakening Effect and Uniaxial Loading

Abstract: Water-weakening effect is one of the most important causes triggering large deformation and failure of soft-rock engineering; however, few studies have paid attention to damage evolution and constitutive relationship of rock in water-weakening process In this paper, laboratory tests are first carried out to estimate the evolution of mechanical properties along with changes of immersion time for shale samples Then with the aid of X-ray diffraction and scanning electron microscope, mechanism of parameter degradation for shale under immersion conditions is investigated from the microscopic perspective Based on the generalized strain equivalent principle and the theory of statistical microscopic damage mechanics, a damage constitutive model of rock subjected to water-weakening effect and uniaxial loading is established by considering the influence of void-compression stage, and the proposed model is verified to be in good agreement with the experiment results This paper provides an effective approach to analyze the constitutive relationship of rock subjected to water-weakening effect and uniaxial loading

Thermodynamically consistent data-driven computational mechanics

Abstract: In the paradigm of data-intensive science, automated, unsupervised discovering of governing equations for a given physical phenomenon has attracted a lot of attention in several branches of applied sciences. In this work, we propose a method able to avoid the identification of the constitutive equations of complex systems and rather work in a purely numerical manner by employing experimental data. In sharp contrast to most existing techniques, this method does not rely on the assumption on any particular form for the model (other than some fundamental restrictions placed by classical physics such as the second law of thermodynamics, for instance) nor forces the algorithm to find among a predefined set of operators those whose predictions fit best to the available data. Instead, the method is able to identify both the Hamiltonian (conservative) and dissipative parts of the dynamics while satisfying fundamental laws such as energy conservation or positive production of entropy, for instance. The proposed method is tested against some examples of discrete as well as continuum mechanics, whose accurate results demonstrate the validity of the proposed approach.

Finite Element Analysis of Additive Manufacturing Based on Fused Deposition Modeling: Distortions Prediction and Comparison With Experimental Data

Abstract: Additive manufacturing (or three-dimensional (3D) printing) is constantly growing as an innovative process for the production of complex-shape components. Among the seven recognized 3D printing technologies, fused deposition modeling (FDM) covers a very important role, not only for producing representative 3D models, but, mainly due to the development of innovative material like Peek and Ultem, also for realizing structurally functional components. However, being FDM a production process involving high thermal gradients, non-negligible deformations and residual stresses may affect the 3D printed component. In this work we focus on meso/macroscopic simulations of the FDM process using abaqus software. After describing in detail the methodological process, we investigate the impact of several parameters and modeling choices (e.g., mesh size, material model, time-step size) on simulation outcomes and we validate the obtained results with experimental measurements.

A general fluid–sediment mixture model and constitutive theory validated in many flow regimes

Abstract: We present a thermodynamically consistent constitutive model for fluid-saturated sediments, spanning dense to dilute regimes, developed from the basic balance laws for two-phase mixtures. The model can represent various limiting cases, such as pure fluid and dry grains. It is formulated to capture a number of key behaviours such as: (i) viscous inertial rheology of submerged wet grains under steady shearing flows, (ii) the critical state behaviour of grains, which causes granular Reynolds dilation/contraction due to shear, (iii) the change in the effective viscosity of the fluid due to the presence of suspended grains and (iv) the Darcy-like drag interaction observed in both dense and dilute mixtures, which gives rise to complex fluid–grain interactions under dilation and flow. The full constitutive model is combined with the basic equations of motion for each mixture phase and implemented in the material point method (MPM) to accurately model the coupled dynamics of the mixed system. Qualitative results show the breadth of problems which this model can address. Quantitative results demonstrate the accuracy of this model as compared with analytical limits and experimental observations of fluid and grain behaviours in inhomogeneous geometries.

Derivation of heterogeneous material laws via data-driven principal component expansions

Abstract: A new data-driven method that generalizes experimentally measured and/or computational generated data sets under different loading paths to build three dimensional nonlinear elastic material law with objectivity under arbitrary loadings using neural networks is proposed. The proposed approach is first demonstrated by exploiting the concept of representative volume element (RVE) in the principal strain and stress spaces to numerically generate the data. A computational data-training algorithm on the generalization of these principal space data to three dimensional objective isotropic material laws subjected to arbitrary deformation is given. To validate these data-driven derived material laws, large deformation and buckling analysis of nonlinear elastic solids with reference material models and engineering structure with microstructure are performed. Numerical experiments show that only seven sets of data under different stress loading paths on RVEs are required to reach reasonable accuracy. The requirements for constitutive law such as objectivity are preserved approximately. The consistent tangent modulus is also derived. The proposed approach also shows a great potential to obtain the material law between different scales in the multiscale analysis by pure data.

From Yielding to Shear Jamming in a Cohesive Frictional Suspension.

Abstract: Simulations are used to study the steady shear rheology of dense suspensions of frictional particles exhibiting discontinuous shear thickening and shear jamming, in which finite-range cohesive interactions result in a yield stress. We develop a constitutive model that combines yielding behavior and shear thinning at low stress with the frictional shear thickening at high stresses, in good agreement with the simulation results. This work shows that there is a distinct difference between solids below the yield stress and in the shear-jammed state, as the two occur at widely separated stress levels, with an intermediate region of stress in which the material is flowable.

A physically based visco-hyperelastic constitutive model for soft materials

Abstract: Viscoelasticity is an essential mechanical property of soft materials. Many constitutive models have been proposed to capture this mechanical behavior, but few physically based models were developed due to the challenge to incorporate the micro-mechanisms of viscoelasticity into the formulation. In a previous paper, we proposed a constitutive model to characterize the hyperelastic response under different deformation states with only three parameters (Xiang et al., 2018). Based on the above work and the tube theory of polymer dynamics, we develop herein a physically based viscoelastic constitutive model in this paper. We start from a new microscopic picture at the molecular chain scale, and decompose the stress into a hyperelastic part which comes from the elastic ground network (crosslinked network and entanglement network), and a viscous part which is originated from free chains. Utilizing the same scheme from the previous work (Xiang et al., 2018), we are able to decompose the free chains into two parts, the untangled crosslinked network and the entanglement network. The contour length relaxation and disentanglement from the free chain's networks give the viscous behavior. We test VHB 4910 to validate our model. The model is applied to the mechanical behavior of the soft digital materials (DMs). In addition, the model can explain the Shore A index independent relaxation time of DMs and the asymmetry of viscoelasticity under loading and unloading. This viscoelastic model can be used to predict the mechanical response of soft materials.

Predictive Modeling of Machining Temperatures with Force⁻Temperature Correlation Using Cutting Mechanics and Constitutive Relation.

Abstract: Elevated temperature in the machining process is detrimental to cutting tools-a result of the effect of thermal softening and material diffusion. Material diffusion also deteriorates the quality of the machined part. Measuring or predicting machining temperatures is important for the optimization of the machining process, but experimental temperature measurement is difficult and inconvenient because of the complex contact phenomena between tools and workpieces, and because of restricted accessibility during the machining process. This paper presents an original analytical model for fast prediction of machining temperatures at two deformation zones in orthogonal cutting, namely the primary shear zone and the tool⁻chip interface. Temperatures were predicted based on a correlation between force and temperature using the mechanics of the cutting process and material constitutive relation. Minimization of the differences between calculated material flow stresses using a mechanics model and a constitutive model yielded an estimate of machining temperatures. Experimental forces, cutting condition parameters, and constitutive model constants were inputs, while machining forces were easily measurable by a piezoelectric dynamometer. Machining temperatures of AISI 1045 steel were predicted under various cutting conditions to demonstrate the predictive capability of each presented model. Close agreements were observed by verifying them against documented values in the literature. The influence of model inputs and computational efficiency were further investigated. The presented model has high computational efficiency that allows real-time prediction and low experimental complexity, considering the easily measurable input variables.

A constitutive relation of AZ80 magnesium alloy during hot deformation based on Arrhenius and Johnson–Cook model

Abstract: In order to understand the constitutive behavior of as-cast AZ80 with large grain size, the uniaxial hot compression tests were carried out over a series of isothermal upsetting experiments. The maximum deformation degree was 65%. The experimental temperatures were 523 K, 573 K, 623 K and 673 K and the strain rate was 0.001 s(-1), 0.01 s(-1), 0.1 s(-1), and 1 s(-1). The stress-strain curves can be divided into three stages which are work hardening stage, softening stage, and steady-state stage at low strain rate and high temperature, while the steady-state stage cannot be observed at low forming temperature and high strain rate because of incomplete dynamic recrystallization. The Arrhenius type relation predicts the peak stress with high accuracy but cannot satisfy the strain relevant requirement. The Johnson-Cook model shows an inappropriate ability to describe the constitutive behavior in this case. Therefore, a new mathematic model (a segmented model) with high prediction accuracy based on the modified Arrhenius type relation (including strain rate) and Johnson-Cook model is proposed. The modified Arrhenius type relation is used to reflect the constitutive behavior before the peak strain and the modified Johnson-Cook model is aimed at showing the stages after peak strain.

Anisotropic fractional viscoelastic constitutive models for human descending thoracic aortas.

Abstract: The generalized fractional Maxwell model, formulated for hyperelastic material within the framework of the nonlinear viscoelasticity with internal variables, is applied to identify viscoelastic constitutive equations from layer-specific experimental data obtained by uniaxial harmonic loading of ex-vivo human descending thoracic aortas. The constitutive parameters are identified by using a genetic algorithm for the optimal fitting of the experimental data. The accuracy of the fitted fractional model is compared to the fitted integer order model with the same number of Maxwell elements. The formulation of an original strain energy density function for anisotropic nonlinear viscoelasticity is introduced and constitutive parameters are obtained from the experiments.

Modeling the rheology of thixotropic elasto-visco-plastic materials

Abstract: To describe the macroscopic rheological behavior of thixotropic elasto-visco-plastic (TEVP) materials, phenomena that take place in their microstructure must be accounted for. To this end, we couple the tensorial constitutive model by Saramito for EVP materials with thixotropy, extending the ideas of isotropic hardening, and with kinematic hardening (KH), to account for back stresses. We use a scalar variable that describes the level of structure at any instance and a modified Armstrong–Frederick KH equation, thus providing rules governing the dynamics of the apparent yield stress. The material viscosity, yield stress, and back stress modulus feature a nonlinear dependence on the structural parameter, enabling the model to make accurate predictions with a single structural parameter. To avoid unphysical stress evolution in both shear and extensional flows, we propose a modified back stress constitutive equation that keeps the components of the stress tensor bounded. The predictions of the new model are compared to experimental data and predictions of previously proposed TEVP models in simple rheometric flows, including steady and step-shear tests, flow reversal, intermittent step tests, small amplitude oscillatory shear (SAOS) and large amplitude oscillatory shear. In most cases, the proposed model reproduces more accurately these experimental data than the other models, highlighting its predictive capabilities. Moreover, SAOS illustrates that introducing viscoplasticity via the Saramito model necessarily reduces G″ to zero in the linear strain regime. This calls for model adjustments in the solid state. Finally, we examined the proposed model in uniaxial elongation and concluded that it is important to include this flow in the rheological characterization and modeling of such systems.To describe the macroscopic rheological behavior of thixotropic elasto-visco-plastic (TEVP) materials, phenomena that take place in their microstructure must be accounted for. To this end, we couple the tensorial constitutive model by Saramito for EVP materials with thixotropy, extending the ideas of isotropic hardening, and with kinematic hardening (KH), to account for back stresses. We use a scalar variable that describes the level of structure at any instance and a modified Armstrong–Frederick KH equation, thus providing rules governing the dynamics of the apparent yield stress. The material viscosity, yield stress, and back stress modulus feature a nonlinear dependence on the structural parameter, enabling the model to make accurate predictions with a single structural parameter. To avoid unphysical stress evolution in both shear and extensional flows, we propose a modified back stress constitutive equation that keeps the components of the stress tensor bounded. The predictions of the new model are co...

A fabric-based sand plasticity model with reversal surfaces within anisotropic critical state theory

Abstract: The paper describes the formulation and capabilities of a new constitutive model that accounts for the effects of fabric anisotropy on the response of granular materials under monotonic loading. It is developed within the framework of bounding surface plasticity in conjunction with the concept of (stress) reversal surfaces, i.e., the use of the last stress reversal point as projection center for defining the image stress on the bounding surface. A key constitutive ingredient is the fabric anisotropy variable A, relating the fabric tensor to the plastic strain rate direction, that acquires the value A = 1 as the third requirement for critical state according to the anisotropic critical state theory. This A is used in the definition of dilatancy, the plastic modulus and the evolution equation of the fabric tensor, thus simulating experimental results that show more dilative and stiff response when the loading is applied along the direction of the fabric. Model performance is verified against a large database of monotonic shearing tests on samples of Toyoura sand prepared with three different methods, as well as similar tests on samples whose (initially horizontal) deposition plane was rotated by up to 90 degrees. All simulations are performed with a single set of constants, thus validating the efficiency of the model to account for density, stress level and, most importantly, fabric anisotropy effects on the monotonic shearing response. The paper shows that considering dependence of strength and dilatancy on Lode angle θ and state parameter ψ does not suffice for simulating fabric anisotropy effects on sand response. It ends with a discussion of how fabric effects on sand response are more pronounced under undrained, versus drained conditions.

Nonlinear dynamic responses of rotating pretwisted cylindrical shells

Abstract: A rotating pretwisted cylindrical shell model with a presetting angle is established to investigate nonlinear dynamic responses of the aero-engine compressor blade. The centrifugal force and the Coriolis force are considered in the model. The aerodynamic pressure is obtained by the first-order piston theory. The strain–displacement relationship is derived by the Green strain tensor. Based on the first-order shear deformation theory and the isotropic constitutive law, nonlinear partial differential governing equations are derived by using the Hamilton principle. Discarding the Coriolis force effect, Galerkin approach is utilized to reduce nonlinear partial differential governing equations into a two-degree-of-freedom nonlinear system. According to nonlinear ordinary differential equations, numerical simulations are performed to explore nonlinear transient dynamic responses of the system under the effect of the single point excitation and nonlinear steady-state dynamic responses of the system under the effect of the uniform distribution excitation. The effects of the excitation parameter, damping coefficient, rotating speed, presetting angle and pretwist angle on nonlinear dynamic responses of the system are fully discussed.

A variable order fractional constitutive model of the viscoelastic behavior of polymers

Abstract: The multiple timescale evolution of polymers’ microstructure due to an applied load is a well-known challenge in building models that accurately predict its mechanical behavior during deformation. Here, a constitutive model involving a variable order fractional derivative with piecewise definition is presented to describe the viscoelasticity of polymers under the condition of uniaxial loading at constant strain rates. It is shown that our model requires three parameters for small strains while five parameters are defined for large deformations. By comparing the predictions made by the proposed model with published experimental data and an existing model for polymers, we demonstrate that our model has higher accuracy while it benefits from its simple form of linearly decreasing order function to predict large deformations. An illustration based on the mechanism of molecular chain resistance indicates that the hardening process and the rate dependence of polymers are captured by the variation of fractional order. We conclude that the evolution of microstructure and mechanical properties of polymers during deformation is well represented by the variable order fractional constitutive model.

The yield normal stress

Abstract: Normal stresses in complex fluids lead to new flow phenomena because they can be comparable to, or even larger than, the shear stress. In addition, they are of paramount importance for formulating and testing constitutive equations for predicting nonviscometric flow behavior. Very little attention has thus far been paid to the normal stresses of yield stress fluids, which are difficult to measure. We report the first systematic study of the first and second normal stress differences in both continuous and slow oscillatory shear of three model nonthixotropic yield stress fluids, with N1 > 0 and N2 < 0. We show that both normal stress differences are quadratic functions of the shear stress both above and below the shear yield stress, leading to the existence of a yield normal stress. However, the contribution of the normal stresses to the von Mises yield criterion for these materials is small.

Constitutive equation including acoustic stress work and plastic strain for modeling ultrasonic vibration assisted friction stir welding process

Abstract: To reveal the underlying mechanism of the synchronous interaction between the ultrasonic vibration exerted onto the welding tool along welding direction and the thermo-mechanical behavior induced by the tool’ rotation and traverse, two available constitutive equations are combined together by including various influencing factors such as strain, strain rate and temperature. A modified constitutive equation considering the acoustic stress work and a method determining the flow stress are proposed and experimentally calibrated. Both are employed in modeling ultrasonic vibration assisted friction stir welding process (UaFSW) to characterize the effect of acoustic softening on the plastic deformation of aluminum alloy in UaFSW process. Numerical simulation of flow stress, strain/strain rate and plastic material flow as well as ultrasonic field is carried out to explain the reason why ultrasonic vibration exerted onto the tool in this way could reduce the flow stress and promote the plastic material flow in UaFSW. The model is validated by comparing the predicted and experimentally measured profile of thermo-mechanically affected zone.