Showing papers on "Constitutive equation published in 2013"

Finite Element Analysis of Composite Materials using Abaqus

Abstract: Mechanics of Orthotropic Materials Lamina Coordinate System Displacements Strain Stress Contracted Notation Equilibrium and Virtual Work Boundary Conditions Continuity Conditions Compatibility Coordinate Transformations Transformation of Constitutive Equations 3D Constitutive Equations Engineering Constants From 3D to Plane Stress Equations Apparent Laminate Properties Introduction to Finite Element Analysis Basic FEM Procedure General Finite Element Procedure Solid Modeling, Analysis, and Visualization Elasticity and Strength of Laminates Kinematic of Shells Finite Element Analysis of Laminates Failure Criteria Predefined Fields Buckling Eigenvalue Buckling Analysis Continuation Methods Free Edge Stresses Poisson's Mismatch Coefficient of Mutual Influence Computational Micromechanics Analytical Homogenization Numerical Homogenization Local-Global Analysis Laminated RVE Viscoelasticity Viscoelastic Models Boltzmann Superposition Correspondence Principle Frequency Domain Spectrum Representation Micromechanics of Viscoelastic Composites Macromechanics of Viscoelastic Composites FEA of Viscoelastic Composites Continuum Damage Mechanics One-Dimensional Damage Mechanics Multidimensional Damage and Effective Spaces Thermodynamics Formulation Kinetic Law in Three-Dimensional Space Damage and Plasticity Discrete Damage Mechanics Overview Approximations Lamina Constitutive Equation Displacement Field Degraded Laminate Stiffness and CTE Degraded Lamina Stiffness Fracture Energy Solution Algorithm Delaminations Cohesive Zone Method Virtual Crack Closure Technique Appendix A: Tensor Algebra Appendix B: Second-Order Diagonal Damage Models Appendix C: Software Used Index Problems appear at the end of each chapter.

Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations

Abstract: United States. National Aeronautics and Space Administration (Microgravity Fluid Sciences (Code UG) for support of this research under grant no. NNX09AV99G)

Describing and prescribing the constitutive response of yield stress fluids using large amplitude oscillatory shear stress (LAOStress)

Abstract: Large amplitude oscillatory shear (LAOS) is used as a tool to probe the nonlinear rheological response of a model elasto-viscoplastic material (a Carbopol microgel). In contrast to most recent studies, these large amplitude measurements are carried out in a stress-controlled manner. We outline a descriptive framework of characterization measures for nonlinear rheology under stress-controlled LAOS, and this is contrasted experimentally to the strain-controlled framework that is more commonly used. We show that this stress-controlled methodology allows for a physically intuitive interpretation of the yielding behavior of elasto-viscoplastic materials. The insight gained into the material behavior through these nonlinear measures is then used to develop two constitutive models that prescribe the rheological response of the Carbopol microgel. We show that these two successively more sophisticated constitutive models, which are based on the idea of strain decomposition, capture in a compact manner the important features of the nonlinear rheology of the microgel. The second constitutive model, which incorporates the concept of kinematic hardening, embodies all of the essential behaviors exhibited by Carbopol. These include elasto-viscoplastic creep and time-dependent viscosity plateaus below a critical stress, a viscosity bifurcation at the critical stress, and Herschel–Bulkley flow behavior at large stresses.

Finite Element Analysis of Composite Materials Using ANSYS

Abstract: Mechanics of Orthotropic Materials Material Coordinate System Displacements Strain Stress Contracted Notation Equilibrium and Virtual Work Boundary Conditions Continuity Conditions Compatibility Coordinate Transformations Transformation of Constitutive Equations 3D Constitutive Equations Engineering Constants From 3D to Plane Stress Equations Apparent Laminate Properties Suggested Problems References Introduction to Finite Element Analysis Basic FEM Procedure General FEM Procedure FE Analysis with CAE Systems Suggested Problems References Elasticity and Strength of Laminates Kinematics of Shells FE Analysis of Laminates Failure Criteria Suggested Problems References Buckling Bifurcation Methods Continuation Methods Suggested Problems References Free Edge Stresses Poisson's Mismatch Coefficient of Mutual Influence Suggested Problems References Computational Micromechanics Analytical Homogenization Numerical Homogenization Local-Global Analysis Laminated RVE Suggested Problems References Viscoelasticity Viscoelastic Models Boltzmann Superposition Correspondence Principle Frequency Domain Spectrum Representation Micromechanics of Viscoelastic Composites Macro-Mechanics of Viscoelastic Composites FEA of Viscoelastic Composites Suggested Problems References Continuum Damage Mechanics One-Dimensional Damage Mechanics Multi-Dimensional Damage and Effective Spaces Thermodynamics Formulation Kinetic Law in Three-Dimensional Space Damage and Plasticity Suggested Problems References Discrete Damage Mechanics Theoretical Formulation Numerical Implementation Model Identification Laminate Damage References Bibliography Delaminations Two-Dimensional Delamination Delamination in Composite Plates Suggested Problems References Appendices ANSYS BMI3 References Index

Nonlocal rheology of granular flows across yield conditions.

Abstract: The rheology of dense granular flows is studied numerically in a shear cell controlled at constant pressure and shear stress, confined between two granular shear flows. We show that a liquid state can be achieved even far below the yield stress, whose flow can be described with the same rheology as above the yield stress. A nonlocal constitutive relation is derived from dimensional analysis through a gradient expansion and calibrated using the spatial relaxation of velocity profiles observed under homogeneous stresses. Both for frictional and frictionless grains, the relaxation length is found to diverge as the inverse square root of the distance to the yield point, on both sides of that point.

On the existence of a simple yield stress fluid behavior

Abstract: Materials such as foams, concentrated emulsions, dense suspensions or colloidal gels, are yield stress fluids Their steady flow behavior, characterized by standard rheometric techniques, is usually modeled by a Herschel–Bulkley law The emergence of techniques that allow the measurement of their local flow properties (velocity and volume fraction fields) has led to observe new complex behaviors It was shown that many of these materials exhibit shear banding in a homogeneous shear stress field, which cannot be accounted for by the standard steady-state constitutive laws of simple yield stress fluids In some cases, it was also observed that the velocity fields under various conditions cannot be modeled with a single constitutive law and that nonlocal models are needed to describe the flows Doubt may then be cast on any macroscopic characterization of such systems, and one may wonder if any material behaves in some conditions as a Herschel–Bulkley material In this paper, we address the question of the existence of a simple yield stress fluid behavior We first review experimental results from the literature and we point out the main factors (physical properties, experimental procedure) at the origin of flow inhomogeneities and nonlocal effects It leads us to propose a well-defined procedure to ensure that steady-state bulk properties of the materials are studied We use this procedure to investigate yield stress fluid flows with MRI techniques We focus on nonthixotropic dense suspensions of soft particles (foams, concentrated emulsions, Carbopol gels) We show that, as long as they are studied in a wide (as compared to the size of the material mesoscopic elements) gap geometry, these materials behave as ‘simple yield stress fluids’: they are homogeneous, they do not exhibit steady-state shear banding, and their steady flow behavior in simple shear can be modeled by a local continuous monotonic constitutive equation which accounts for flows in various conditions and matches the macroscopic response

Fibre‐reinforced concrete in fib Model Code 2010: principles, models and test validation

Abstract: In the fib Model Code for Concrete Structures 2010, fibre-reinforced concrete (FRC) is recognized as a new material for structures. This introduction will favour forthcoming structural applications because the need of adopting new design concepts and the lack of international building codes have significantly limited its use up to now. In the code, considerable effort has been devoted to introducing a material classification to standardize performance-based production and stimulate an open market for every kind of fibre, favouring the rise of a new technological player: the composite producer. Starting from standard classification, the simple constitutive models introduced allow the designer to identify effective constitutive laws for design, trying to take into account the major contribution in terms of performance and providing good orientation for structural uses. Basic new concepts such as structural characteristic length and new factors related to fibre distribution and structural redistribution benefits are taken into account. A few examples of structural design starting from the constitutive laws identified are briefly shown. FRC can be regarded as a special concrete characterized by a certain toughness after cracking. For this reason, the most important constitutive law introduced is the stress-crack opening response in uniaxial tension. A wide discussion of the constitutive models introduced to describe this behaviour, which controls all the main contributions of fibres for a prevailing mode I crack propagation, is proposed. The validity of the models is discussed with reference to ordinary cross-sections as well as thin-walled elements by adopting plane section or finite element models.

Coarse-grained local and objective continuum description of three-dimensional granular flows down an inclined surface

Abstract: Dry, frictional, steady-state granular flows down an inclined, rough surface are studied with discrete particle simulations. From this exemplary flow situation, macroscopic fields, consistent with the conservation laws of continuum theory, are obtained from microscopic data by time-averaging and spatial smoothing (coarse-graining). Two distinct coarse-graining length scale ranges are identified, where the fields are almost independent of the smoothing length w. The smaller, sub-particle length scale, w ≪ d, resolves layers in the flow near the base boundary that cause oscillations in the macroscopic fields. The larger, particle length scale, w ≈ d, leads to smooth stress and density fields, but the kinetic stress becomes scale-dependent; however, this scale-dependence can be quantified and removed. The macroscopic fields involve density, velocity, granular temperature, as well as strain-rate, stress, and fabric (structure) tensors. Due to the plane strain flow, each tensor can be expressed in an inherently anisotropic form with only four objective, coordinate frame invariant variables. For example, the stress is decomposed as: (i) the isotropic pressure, (ii) the “anisotropy” of the deviatoric stress, i.e., the ratio of deviatoric stress (norm) and pressure, (iii) the anisotropic stress distribution between the principal directions, and (iv) the orientation of its eigensystem. The strain rate tensor sets the reference system, and each objective stress (and fabric) variable can then be related, via discrete particle simulations, to the inertial number, I. This represents the plane strain special case of a general, local, and objective constitutive model. The resulting model is compared to existing theories and clearly displays small, but significant deviations from more simplified theories in all variables – on both the different length scales.

A hyperelastic constitutive model for rubber-like materials

Abstract: Hyperelastic behavior of isotropic incompressible rubbers is studied to develop a strain energy function which satisfies all the necessary characteristic properties of an efficient hyperelastic model. The proposed strain energy function includes only three material parameters which are somehow related to the physical quantities of the material molecular network. Moreover, the model benefits from mathematical simplicity, well suitting in all ranges of stretch and possessing the property of deformation-mode-independency. This reduces the required number of experimental tests for parameter calibration of the model. Results of the proposed model are compared with results of some available models as well as experimental data. Moreover, complete analysis of the Mooney plot over a wide range of stretch in extension–compression is carried out. It is found that the proposed model gives reasonable predictions in comparison with those of experiments.

Microplane model M7 for plain concrete. I: Formulation

Abstract: Mathematical modeling of the nonlinear triaxial behavior and damage of such a complex material as concrete has been a long-standing challenge in which progress has been made only in gradual increments. The goal of this study is a realistic and robust material model for explicit finite-element programs for concrete structures that computes the stress tensor from the given strain tensor and some history variables. The microplane models, which use a constitutive equation in a vectorial rather than tensorial form and are semimultiscale by virtue of capturing interactions among phenomena of different orientation, can serve this goal effectively. This paper presents a new concrete microplane model, M7, which achieves this goal much better than the previous versions M1–M6 developed at Northwestern University since 1985. The basic mathematical structure of M7 is logically correlated to thermodynamic potentials for the elastic regime, the tensile and compressive damage regimes, and the frictional slip regi...

Flow of an eyring-powell non-newtonian fluid over a stretching sheet

Abstract: This article is devoted to the study of the boundary layer flow of a non-Newtonian fluid over a stretching sheet. The non-Newtonian behavior of the fluid is characterized by the constitutive equation due to Powell and Eyring (1944). A second-order approximation of the Eyring-Powell model is used to obtain the flow equations. A local similarity solution of the governing problem is obtained numerically using an implicit finite difference scheme known as the Keller box method. The influence of pertinent non-Newtonian fluid parameters M and λ on the velocity and skin-friction coefficient is analyzed through graphical and tabular results.

A unified approach to model elasto-viscoplastic thixotropic yield-stress materials and apparent yield-stress fluids

Abstract: A constitutive model for elasto-viscoplastic thixotropic materials is proposed. It consists of two differential equations, one for the stress and the other for the structure parameter, a scalar quantity that indicates the structuring level of the microstructure. In contrast to previous models of this kind, the structure parameter varies from zero to a positive and typically large number. The lower limit corresponds to a fully unstructured material, whereas the upper limit corresponds to a fully structured material. When the upper limit is finite, the model represents a highly shear-thinning, thixotropic, and viscoelastic liquid that possesses an apparent yield stress. When it tends to infinity, the behavior of a true yield-stress material is achieved. Predictions for rheometric flows such as constant shear rate tests, creep tests, SAOS, and large-amplitude oscillatory shear (LAOS) are presented, and it is shown that, in all cases, the trends observed experimentally are faithfully reproduced by the model. Within the framework of the model, simple explanations are given for the avalanche effect and the shear banding phenomenon. The LAOS results obtained are of particular importance because they provide a piece of information that so far is absent in the literature, namely a quantitative link between the Lissajous–Bowditch curve shapes and rheological effects such as elasticity, thixotropy, and yielding.

A simple anisotropic hyperelastic constitutive model for textile fabrics with application to forming simulation

Abstract: A simple hyperelastic constitutive model is developed to characterize the anisotropic and large deformation behavior of textile fabrics. In the model, the strain energy function is decomposed into two parts representing fiber stretches and fiber–fiber interaction (cross-over shearing) between weft and warp yarns. The proposed constitutive model is demonstrated on a balanced plain woven fabric. The actual forms of the strain energy functions are determined by fitting uni-axial tensile and picture-frame shear tests of the woven fabrics. The developed model is validated by comparing numerical results with experimental bias extension data, and then applied to simulation of a benchmark double dome forming, demonstrating that the proposed anisotropic hyperelastic constitutive model is highly suitable to predict the large deformation behavior of textile fabrics.

Criteria for Shear Banding in Time-Dependent Flows of Complex Fluids

Abstract: We study theoretically the onset of shear banding in the three most common time-dependent rheological protocols: step stress, finite strain ramp (a limit of which gives a step strain), and shear startup. By means of a linear stability analysis we provide a fluid-universal criterion for the onset of banding for each protocol, which depends only on the shape of the experimentally measured time-dependent rheological response function, independent of the constitutive law and internal state variables of the particular fluid in question. Our predictions thus have the same highly general status, in these time-dependent flows, as the widely known criterion for banding in steady state (of negatively sloping shear stress vs shear rate). We illustrate them with simulations of the Rolie-Poly model of polymer flows, and the soft glassy rheology model of disordered soft solids.

Coupled hydromechanical and electromagnetic disturbances in unsaturated porous materials.

Abstract: A theory of cross-coupled flow equations in unsaturated soils is necessary to predict (1) electroosmotic flow with application to electroremediation and agriculture, (2) the electroseismic and the seismoelectric effects to develop new geophysical methods to characterize the vadose zone, and (3) the streaming current, which can be used to investigate remotely ground water flow in unsaturated conditions in the capillary water regime. To develop such a theory, the cross-coupled generalized Darcy and Ohm constitutive equations of transport are extended to unsaturated conditions. This model accounts for inertial effects and for the polarization of porous materials. Rather than using the zeta potential, like in conventional theories for the saturated case, the key parameter used here is the quasi-static volumetric charge density of the pore space, which can be directly computed from the quasi-static permeability. The apparent permeability entering Darcy's law is also frequency dependent with a critical relaxation time that is, in turn, dependent on saturation. A decrease of saturation increases the associated relaxation frequency. The final form of the equations couples the Maxwell equations and a simplified form of two-fluid phases Biot theory accounting for water saturation. A generalized expression of the Richard equation is derived, accounting for the effect of the vibration of the skeleton during the passage of seismic waves and the electrical field. A new expression is obtained for the effective stress tensor. The model is tested against experimental data regarding the saturation and frequency dependence of the streaming potential coupling coefficient. The model is also adapted for two-phase flow conditions and a numerical application is shown for water flooding of a nonaqueous phase liquid (NAPL, oil) contaminated aquifer. Seismoelectric conversions are mostly taking place at the NAPL (oil)/water encroachment front and can be therefore used to remotely track the position of this front. This is not the case for other geophysical methods.

Steady bubble rise in Herschel–Bulkley fluids and comparison of predictions via the Augmented Lagrangian Method with those via the Papanastasiou model

Abstract: The steady, buoyancy-driven rise of a bubble in a Herschel–Bulkley fluid is examined assuming axial symmetry. The variation of the rate-of-strain tensor around a rising bubble necessitates the coexistence of fluid and solid regions in this fluid. In general, a viscoplastic fluid will not be deforming beyond a finite region around the bubble and, under certain conditions, it will not be deforming either just behind it or around its equatorial plane. The accurate determination of these regions is achieved by introducing a Lagrange multiplier and a quadratic term in the corresponding variational inequality, resulting in the so-called Augmented Lagrangian Method (ALM). Additionally here, the augmentation parameters are determined following a non-linear conjugate gradient procedure. The new predictions are compared against those obtained by the much simpler Papanastasiou model, which uses a continuous constitutive equation throughout the material, irrespective of its state, but does not determine the boundary between solid and liquid along with the flow field. The flow equations are solved numerically using the mixed finite-element/Galerkin method on a mesh generated by solving a set of quasi-elliptic differential equations. The accuracy of solutions is ascertained by mesh refinement and comparison with our earlier and new predictions for a bubble rising in a Newtonian and a Bingham fluid. We determine the bubble shape and velocity and the shape of the yield surfaces for a wide range of material properties, expressed in terms of the Bingham, Bn, Bond, and Archimedes numbers. As Bn increases, the bubble decelerates, the yield surfaces at its equatorial plane and away from it approach each other and eventually merge immobilizing the bubble. For small and moderate Bingham numbers, the predictions using the Papanastasiou model satisfactorily approximate those of the discontinuous Herschel–Bulkley model for sufficiently large values of the normalization exponent (⩾104). On the contrary, as Bn increases and the rate-of-strain approaches zero almost throughout the fluid-like region, much larger values of the exponent are required to accurately compute the yield surfaces. Bubble entrapment does not depend on the power law index, i.e. a bubble in a Herschel–Bulkley fluid is entrapped under the same conditions as in a Bingham fluid.

Constitutive relationships of hot stamping boron steel B1500HS based on the modified Arrhenius and Johnson–Cook model

Abstract: Constitutive relationship of boron steel is one of the most necessary mathematical models in the numerical simulation of hot stamping; it describes the relationship of the flow stress with strain, strain rate and temperature. In order to attain the constitutive relationship of boron steel B1500HS, four types of samples with microstructure of austenite, ferrite+pearlite, bainite or martensite are prepared by the Gleeble 1500D thermo-mechanical simulator. Isothermal uniaxial tension testings for these specimens are performed at 20–900 °C at the strain rates of 0.01 s –1 , 0.1 s –1 , 1.0 s –1 and 10 s –1 by Gleeble 1500D, and the true stress–strain curves at the relative conditions are gained. The experimental results show that, the flow stress of samples with relative microstructure rises with the decrease of the deformation temperature, and with the increase of the strain rate. The modified Arrhenius model is used to describe the hot deformation of samples with austenite microstructure, and the modified Johnson–Cook model is used to describe the deformation process of samples with ferrite+pearlite, bainite or martensite microstructure. The constitutive equations depending on the strain, strain rate and temperature are attained by the regression analysis for the experimental data of flow stress, strain, strain rate, temperature, etc. The comparison of the computational data and the experimental results shows that, the computational data using the constitutive relationships are well consistent with the experimental data.

A finite-strain constitutive model for magnetorheological elastomers: Magnetic torques and fiber rotations

Abstract: This paper is concerned with the development of constitutive models for a class of magnetoelastic composites consisting of stiff, aligned cylindrical fibers of a magnetizable material that are embedded firmly in a soft elastomeric matrix. The fibers have elliptical cross section and their (transverse) in-plane axes are also aligned, but their distribution is random and characterized by “elliptical” two-point correlations. Estimates are obtained for the macroscopic response and stability of this new type of magnetorheological elastomer (MRE) under combined in-plane mechanical and magnetic loading by means of the finite-strain homogenization framework and “partial decoupling approximation” of Ponte Castaneda and Galipeau (2011) . The resulting macroscopic magnetoelastic constitutive model accounts for the microstructure of the composite and its evolution under finite strains and rotations, as well as for the nonlinear magnetic behavior of the fibers, including the effect of magnetic saturation. When the loading directions are not aligned with the fiber axes, the model predicts magnetic and mechanical torques on the fibers, leading to their in-plane rotation, which is found to have significant effects on the coupled magnetoelastic response of the composite, including the possible development of macroscopic torques on a given finite-size sample of the composite. To eliminate these macroscopic torques, while maintaining the advantageous effects of the fiber rotations, we also investigate the response of a laminated composite consisting of plus/minus orientations of the fibers relative to the layering direction, and subjected to magnetic and mechanical loadings along the layering direction. The results for the actuation tractions, magnetostrictive strain and magnetoelastic moduli demonstrate that the microstructure of these laminated MRE samples can be designed optimally for significantly enhanced magnetoelastic effects. In particular, the actuation tractions and magnetostrictive strains can be made several times larger than the corresponding tractions and strains for isotropic MREs with spherical (circular) particles.

MHD three-dimensional flow of couple stress fluid with Newtonian heating

Abstract: Effects of Newtonian heating on the magnetohydrodynamic (MHD) three-dimensional flow past a stretching surface are analyzed. Mathematical formulation is completed using constitutive equations of couple stress fluid. A constant magnetic field normal to the surface is applied. Viscous dissipation and Joule heating effects are present. The transformation procedure reduces the involved partial differential equations into the ordinary differential equations. Series solutions of the resulting systems are constructed. The convergence of the obtained series solutions is seen through graphical results and tabular values. Numerical values of skin friction and the Nusselt number for different parameters are also tabulated and analyzed.

On the constitutive equations for fluid particle breakage

Abstract: Abstract The mechanisms and modeling concepts proposed in the literature for the fluid particle (i.e., bubble and drops) breakage phenomena in laminar and turbulent dispersions are examined. Four categories are considered for the breakage mechanisms: (i) turbulent motions and inertial stresses, (ii) viscous shear stresses, (iii) shearing-off processes, and (iv) interfacial instability. In the first category, six breakage criteria have been proposed. The constitutive equations for the breakage frequency and daughter size probability density functions for these mechanisms are examined. A brief survey of experimental analyses performed to understand the behavior of these functions is provided. It is concluded that extensive, well-planned, model-based experiments are required to elucidate the underlying mechanisms for fluid particle breakage. In particular, single particle experiments in known flow regimes are needed to determine the functionality of the breakage frequency, number of daughters, and the daughter size distribution functions and for parameter fitting. Size distribution measurements in the same flow regimes are useful for model validation studies.