Flow Potential Function Research Articles

A homogenized constitutive model for 2D woven composites under finite deformation: Considering fiber reorientation

Two-dimensional (2D) woven composites exhibit excellent mechanical properties along the fiber directions. The mechanical behaviors demonstrate nonlinearity in specific applications. Although plasticity methods can be applied to predict complex behaviors, however, fiber reorientation has been observed during finite deformation, indicating that the fiber directions are no longer along orthotropic material axes when the angle between fibers changes. The angular bisectors of two fiber directions can serve as the orthotropic material axes due to the rotational symmetries even in finite deformation scenarios. This study reports a homogenized nonlinear constitutive model based on the rotational symmetry axes, incorporating plasticity and fiber reorientation phenomena. The plasticity model contains a two-parameter flow potential and power function. Plastic deformations are computed using an explicit method. Fiber reorientation angles are computed both theoretically and numerically. The relationship between mechanical properties and fiber reorientation angles is studied using finite element method (FEM). Due to introduce of a novel approach to determining the strain and stress of 2D woven composites undergoing finite deformation, the proposed model should have potential in engineering predictions.

Boundary integral relations for submerged bodies and free-surface piercing ships and offshore structures

The linear flow model called rigid waterplane flow model, in which a body (ship, offshore structure) that pierces the free surface is treated as the smooth limit of a body that is submerged at a small depth, is considered. The flow within the thin layer between the horizontal rigid lid that closes the (open) body surface and the waterplane above the lid is analyzed. This analysis shows that the seemingly different boundary integral relations obtained by applying Green’s basic identity to the boundary value problems associated with the rigid waterplane flow model or a body that pierces the free surface are equivalent. Additionally, the rigid waterplane flow model yields three alternative weakly singular boundary integral relations, which only involve distributions of sources and weak dipoles that are continuous at the body surface. One of these three relations is notable because it does not involve a waterline integral, and only involves the flow potential (not its derivatives along directions tangent to the body surface) and smooth ordinary functions that are easily evaluated and integrated.

Numerical Integration Scheme for Coupled Elastoplastic–Viscoplastic Constitutive Law for Tunnels

The paper presents an efficient numerical integration scheme for coupled elastoplastic–viscoplastic constitutive behavior with internal-state variables standing for irreversible processes. In most quasi-static structural analyses, the solution to boundary value problems involving materials that exhibit time-dependent constitutive behavior proceeds from the equation integration, handled at two distinct levels. On the one hand, the first, or local, level refers to the numerical integration at each Gaussian point of the rate constitutive stress–strain relationships. For a given strain increment, the procedure of local integration is iterated for stresses and associated internal variables until convergence of the algorithm is achieved. On the other hand, the second, or global, level is related to structure equilibrium between internal and external forces achieved by the Newton–Raphson iterative scheme. A review of the elastoplastic and viscoplastic model is given, followed by the coupling between these models. Particular emphasis is given in this contribution to address the first level integration procedure, also referred to as the algorithm for stress and internal variable update, considering a general elastoplastic–viscoplastic constitutive behavior. The formulation is described for semi-implicit Euler schemes. The efficacy of the numerical formulation is assessed by comparison with analytical and numerical solutions derived for deep tunnels in coupled elastoplasticity–viscoplasticity. Finally, a parametric analysis is performed to show the importance that this model can have, in the long-term convergence profile, against other models. For the considered flow surfaces, potential functions, and properties, differences on the order of 23% to 52% are found in the long-term convergence profile.

Mechanical Characterization of the Elastoplastic Response of a C11000-H2 Copper Sheet.

This work presents an elastoplastic characterization of a rolled C11000-H2 99.90% pure copper sheet considering the orthotropic non-associated Hill-48 criterion together with a modified Voce hardening law. One of the main features of this material is the necking formation at small strains levels causing the early development of non-homogeneous stress and strain patterns in the tested samples. Due to this fact, a robust inverse calibration approach, based on an experimental–analytical–numerical iterative predictor–corrector methodology, is proposed to obtain the constitutive material parameters. This fitting procedure, which uses tensile test measurements where the strains are obtained via digital image correlation (DIC), consists of three steps aimed at, respectively, determining (a) the parameters of the hardening model, (b) a first prediction of the Hill-48 parameters based on the Lankford coefficients and, (c) corrected parameters of the yield and flow potential functions that minimize the experimental–numerical error of the material response. Finally, this study shows that the mechanical characterization carried out in this context is capable of adequately predicting the behavior of the material in the bulge test.

Internal Variable Theory Formulated by One Extended Potential Function

Abstract In the kinetic rate laws of internal variables, it is usually assumed that the rates of internal variables depend on the conjugate forces of the internal variables and the state variables. The dependence on the conjugate force has been fully addressed around flow potential functions. The kinetic rate laws can be formulated with two potential functions, the free energy function and the flow potential function. The dependence on the state variables has not been well addressed. Motivated by the previous study on the asymptotic stability of the internal variable theory by J. R. Rice, the thermodynamic significance of the dependence on the state variables is addressed in this paper. It is shown in this paper that the kinetic rate laws can be formulated by one extended potential function defined in an extended state space if the rates of internal variables do not depend explicitly on the internal variables. The extended state space is spanned by the state variables and the rate of internal variables. Furthermore, if the rates of internal variables do not depend explicitly on state variables, an extended Gibbs equation can be established based on the extended potential function, from which all constitutive equations can be recovered. This work may be considered as a certain Lagrangian formulation of the internal variable theory.

Can Moderately Rarefied Gas Transport Through Round and Flat Tight Channels of Fractured Porous Media be Described Accurately?

This paper provides critical insights into the rigorous formulation of moderately rarefied gas transport through narrow channels in naturally and induced fractured porous media, such as gas shale rocks, approximated as round (cylindrical) and flat (slit) types. This formulation considers critical improvements over the previous attempts by proper implementation of the effective equivalent mean hydraulic radius of tight flow channels, the wall-slip effect accommodation of Maxwell (Philos Trans R Soc Lond A 170:231–256, 1879), the variable cross-section hard sphere model of gas molecules and the modified bulk mean free-path of Bird (Phys Fluids 26(11):3222–3223, 1983. https://doi.org/10.1063/1.864095), the apparent viscosity and mean free path for the confined-state gas behavior modification, the flow through narrow capillary tubes represented by a Hagen–Poiseuille-type equation, the Knudsen diffusivity, and an improved relationship between the apparent permeability and the intrinsic permeability. The description of gas transport through extremely tight channels is accomplished by superposition of the Poiseuille bulk flow (convection) and the Knudsen transport (diffusion) mechanisms. This approach is applied to investigate the accuracy of several previous studies on the modeling of gas transport through extremely tight narrow channels of round and flat types under moderately rarefied conditions. Although the simulation results reported by the previous studies of Roy et al. (J Appl Phys 93(8):4870–4879, 2003), Javadpour (J Can Pet Technol 48(8):16–21, 2009), and Veltzke and Thoming (J Gas Mech 698:406–422, 2012) appear to follow the trends observed in the experimental studies of Roy et al. (2003) flowing argon gas through a round channel (tube) and Ewart et al. (J Gas Mech 584:337–356, 2007. https://doi.org/10.1017/S0022112007006374) flowing helium gas through a single straight flat narrow channel, it is concluded that these results are not actually accurate for several reasons. The accuracy of the basic model presented by Javadpour (2009) suffers from some formulation issues and the low-order accuracy of the numerical approximations. The complicated model presented by Veltzke and Thoming (2012) is impractical and difficult because of the two-dimensional solutions of the Navier–Stokes equations with no-slip boundary condition and produces inaccurate solutions because of the improper definition of the effective radius of the straight flat narrow channel. The improved pressure equation of the compressible rarefied gas flow in tight channels developed in the present paper is highly nonlinear. The possibility of numerical calculation errors associated with the solution of the differential pressure equation was eliminated completely by an application of an integral transformation by facilitating a pseudo-transfer or flow potential function, and the solution of this equation was obtained accurately and fully analytically. It is shown that the proper formulation and accurate analytical solution developed in this paper can indeed lead to significantly accurate matches of the same experimental data than those reported by the previous studies. Thus, the deviation of the previous simulation results from the experimental data cannot be attributed simply to possible experimental errors associated with the laboratory tests but also to the limitations in formulation and inaccuracies in numerical solution. The exercises presented in this paper reveal that the previous modeling efforts certainly involve various types of errors and the experimental data cannot be matched by the gas transport models simply by adjusting the values of the unknown model parameters unless the models and their parameters are theoretically meaningful.

Volume growth during uniaxial tension of particle-filled elastomers at various temperatures – Experiments and modelling

A common presumption for elastomeric material behaviour is incompressibility, however, the inclusion of filler particles might give rise to matrix-particle decohesion and subsequent volume growth. In this article, the volumetric deformation accompanying uniaxial tension of particle-filled elastomeric materials at low temperatures is studied. An experimental set-up enabling full-field deformation measurements is outlined and novel data are reported on the significant volume growth accompanying uniaxial tension of two HNBR and one FKM compounds at temperatures of −18, 0, and 23 °C. The volumetric deformation was found to increase with reduced temperature for all compounds. To explain the observed dilatation, in situ scanning electron microscopy was used to inspect matrix-particle debonding occurring at the surface of the materials. A new constitutive model, combining the Bergström–Boyce visco-hyperelastic formulation with a Gurson flow potential function is outlined to account for the observed debonding effects in a numerical framework. The proposed model is shown to provide a good correspondence to the experimental data, including the volumetric response, for the tested FKM compound at all temperature levels.

Stability Analysis of Elastic Plate in Axial Subsonic Flow

The stability and dynamics of the two-dimensional elastic plate with simply supported boundary conditions in uniform axial subsonic flow were studied. The governing equations of coupled elastic plate in axial flow were derived based on the potential theory. The finite difference method was employed to discrete the governing equation and the flow potential function. The governing equation can be expressed as the function of structural transverse vibration displacement by the matrix operations. The eigenvalue method was used to analyze the stability of the elastic plate, the results of which show that the models with simply supported boundary conditions undergo divergent instability when flow velocity reaches the critical value, the critical divergence velocity is in close agreement with theoretical result using other analytical approaches.

A creep model with damage based on internal variable theory and its fundamental properties

The creep damage is discussed within Rice irreversible internal state variable (ISV) thermodynamic theory. An ISV small-strain unified creep model with damage is derived by giving the complementary energy density function and kinetic equations of ISVs. The proposed model can describe viscoelasticity and, preferably, three phases of creep deformation. Creep strain results from internal structural adjustment, and different creep stages accompany different thermodynamic properties in terms of flow potential function and energy dissipation rate. During the viscoelastic process, the thermodynamic state of the material system tends to equilibrate spontaneously. The thermodynamic state of the material system without damage tends to equilibrate or achieve steady state after loading. Kinetic equations of ISVs can be derived by one single flow potential function, and the energy dissipation rate decreases monotonically over time. In the entire creep damage process, multiple potentials are needed to characterise evolution of ISVs, rotational fluxes are presented in affinity space, and the thermodynamic state of material system tends to depart from the steady or equilibrium state. The energy dissipation rate can be a measure of the distance between the current thermodynamic state and the equilibrium state. The time derivative of the rate can characterise the development trend of the material, and the integral value in the domain may be regarded as indices to evaluate the long-term stability of the structure.

A transformation of variables technique applicable to the boundary element method to simulate a special class of diffusive-advective potential problems

This paper describes a novel Boundary Element Technique developed for application to one-dimensional and a class of two-dimensional diffusive-advective potential problems. It is based on transformation of variable procedure to establish an integral equation inverse sentence, dealing only with boundary variables, using a fundamental solution associated with a diffusive problem. To apply the technique described here, the original differential equation is rewritten and flow potential functions are employed to contract terms which appear in the original equation, giving as a result an equivalent equation expressed in terms of the derivative of the product of two functions. This new form of the governing equation, together with the proposed transformation of variables is quite convenient for the application of the boundary element methodology: a very simple discretization procedure arises; the resulting algorithms require low CPU time and the numerical results are quite accurate.

Thermodynamic Significance and Basis of Damage Variables and Equivalences

In this article, thermodynamic significance and basis of damage variables and equivalences are addressed within the irreversible thermodynamic framework with internal variables by Rice [Rice, J.R. (1971). Inelastic Constitutive Relations for Solids: An Internal Variable Theory and its Application to Metal Plasticity. Journal of the Mechanics and Physics of Solids, 19:433—455]. It is revealed that, as soon as the multiscale kinematic relation is prescribed, the condition of free or complementary energy equivalence leads to the full equivalence of the microscale and macroscale thermodynamic formulations. In order to achieve the fully thermodynamic equivalence, the damage equivalence hypothesis should be based on free energy or (thermodynamic) complementary energy from a thermodynamic viewpoint. In this sense, the complementary energy equivalence hypothesis by Sidoroff [Sidoroff, F. (1981) Description of Anisotropic Damage Application to Elasticity, IUTAM Colloquium on Physical Nonlinearities in Structural Analysis, pp. 237—244, Springer, Berlin] is much better than the strain equivalence hypothesis by Lemaitre and Chaboche [Lemaitre, J. and Chaboche, J. L. (1978). Aspect Phenomnologique de la Rupture par Endommagement (in French). J. de Mecanique Appliquee, 2:317—365]. One single scalar damage variable plays a role of fictitious or intrinsic time. The introduction of the scalar damage variable implies that the microscale thermodynamic fluxes measured by the intrinsic time should be potential or irrotational and the corresponding flow potential function is just thermodynamic force conjugate to the damage variable.

Hamilton’s Principle of Entropy Production for Creep and Relaxation Processes

In this paper, two subspaces of the state space of constrained equilibrium states for solids are proposed and addressed. One subspace, constrained affinity space, is conjugate-force space with fixed temperature and internal variable. It is revealed in this paper that the remarkable properties of the kinetic rate laws of scalar internal variables, established by Rice (1971, “Inelastic Constitutive Relations for Solids: An Internal Variable Theory and Its Application to Metal Plasticity,” J. Mech. Phys. Solids, 19, pp. 433–455) and elaborated by Yang (2005, “Normality Structures With Homogeneous Kinetic Rate Laws,” ASME J. Appl. Mech., 72, pp. 322–329; 2007, “Normality Structures With Thermodynamic Equilibrium Points,” ASME J. Appl. Mech., 74, pp. 965–971), are all located in constrained affinity space. Furthermore, the flow potential function monotonically increases along any ray from the origin in constrained affinity space. Another subspace, constrained configuration space, is the state space with fixed external variables. It is shown that the specific free and complementary energies monotonically decrease and increase, respectively, along the path of motion of the thermodynamic system of the material sample in constrained configuration space. For conservative conjugate forces, Hamilton’s action principle is established in constrained configuration space, and the action is the entropy production of the thermodynamic system in a time interval. The thermodynamic processes in constrained configuration space are just creep or relaxation processes of materials. The Hamilton principle can be considered as a fundamental principle of rheology.

Thermodynamics of Infinitesimally Constrained Equilibrium States

This paper is concerned with infinitesimally constrained equilibrium states, which are nonequilibrium states and infinitesimally close to equilibrium states. The corresponding thermodynamics is established in this paper within the thermodynamic framework of Rice (1971, “Inelastic Constitutive Relations for Solids: An Internal Variable Theory and Its Application to Metal Plasticity,” J. Mech. Phys. Solids, 19, pp. 433–455). It is shown that the thermodynamics of infinitesimally constrained equilibrium states belongs to linear irreversible thermodynamics. The coefficient matrix is the Hessian matrix of the flow potential function at the equilibrium state. The process of a state change induced by an infinitesimal stress increment in time-independent plasticity can be viewed as a sequence of infinitesimally constrained equilibrium states. The thermodynamic counterpart of yield functions are flow potential functions, and their convexity is required by intrinsic dissipation inequality. Drucker and Il’yushin’s inequalities are not essential thermodynamic requirements.

A unified viscoplasticity constitutive model based on irreversible thermodynamics

A unified viscoplasticity constitutive model for metal materials is developed within the framework of irreversible thermodynamics, and an expression for the Helmholtz free energy function involving the parameters reflecting kinematic hardening and isotropic hardening is given. At the same time a non-associated flow potential function including the corresponding state variables is also given, from which the flow equation and the evolution equations of the internal state variables are derived. Thus, a general theoretical framework constructing a unified viscoplasticity constitutive model is given. Compared with the typical unified viscoplasticity constitutive models, the presented model evidently satisfies the irreversible thermodynamics laws. Moreover, this method not only provides a new theoretical foundation for further development of the unified viscoplasticity constitutive model, but also gives a new theoretical framework for the stress-strain analysis of more materials.

Coupled viscoplasticity damage constitutive model for concrete materials

A coupled viscoplasticity damage constitutive model for concrete materials is developed within the framework of irreversible thermodynamics. Simultaneously the Helmholtz free energy function and a non-associated flow potential function are given, which include the internal variables of kinematic hardening, isotropic hardening and damage. Results from the numerical simulation show that the model presented can describe the deformation properties of the concrete without the formal hypotheses of yield criterion and failure criteria, such as the volume dilatancy under the compression, strain-rate sensitivity, stiffness degradation and stress-softening behavior beyond the peak stress which are brought by damages and fractures. Moreover, we could benefit from the application of the finite element method based on this model under complex loading because of not having to choose different constitutive models based on the deformation level.

On Microscopic Thermodynamic Mechanisms of Damage Evolution Laws

Most existing phenomenological damage evolution laws can be covered by phenomenological equations or linear irreversible thermodynamics. In this paper, general microscopic thermodynamic mechanisms leading to nonlinear phenomenological equations are explored within the framework of ‘normality structures’ by Rice (Rice, J.R. (1971). Inelastic Constitutive Relations for Solids: An Internal Variable Theory and its Application to Metal Plasticity, Journal of the Mechanics and Physics of Solids, 19: 433-455, Rice, J.R. (1975). Continuum Mechanics and Thermodynamics of Plasticity in Relation to Microscale Deformation Mechanisms, In: Argon, A.S. (ed.), Constitutive Equations in Plasticity, MIT Press, Cambridge, MA, pp. 23-79.) at the level of microstructural rearrangements. Rice’s kinetic rate laws of local internal variables, with each rate being stress dependent only via its conjugate thermodynamic force, are cornerstones of the normality structure. It is revealed in this paper that nonlinear phenomenological equations and Onsager reciprocal relations emerge naturally from the normality structures if each rate is a homogeneous function of degree q in its conjugate force. Furthermore, the nonlinear phenomenological coefficient matrix is identical to the Hessian matrix of the flow potential function in conjugate forces scaled by q. Finally, as an application and demonstration, some fundamental issues on damage evolution laws for microcracked solids have been addressed based on the revealed remarkable properties. It is shown that the deduced flow potential functions of microcracked solids can be expressed in the forms of well-established Hill (Hill, R. (1950). The Mathematical Theory of Plasticity, Clarendon Press, Oxford.) anisotropic yield function and Karafillis and Boyce (Karafillis, A.P. and Boyce, D.B. (1993). A General Anisotropic Yield Criterion Using Bounds and a Transformation Weighting Tensor, Journal of the Mechanics and Physics of Solids, 46: 85-113.) isotropic yield surface.

Microscopic thermodynamic basis of normality structure of inelastic constitutive relations

The essential content of plasticity theory is the normality of plastic strain increments to the yield or flow potential surface at smooth points. The corresponding microscopic thermodynamic mechanism has been explored by Rice [Rice, J.R., 1971. Inelastic constitutive relations for solids: an integral variable theory and its application to metal plasticity. Journal of the Mechanics and Physics of Solids 19, 433; Rice, J.R., 1975. Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms, in: Argon, A.S. (Ed.), Constitutive Equations in Plasticity. MIT Press, Cambridge, MA, p. 23] in his thermodynamic framework with internal variables. Rice’s kinetic rate laws of local internal variables, with each rate being stress dependent only via its conjugate thermodynamic force, directly lead to a unifying normality structure and represent a wide class of inelastic behaviors. The aim of this paper is to reveal the underlying physical mechanism and basis of Rice’s kinetic rate laws. It is shown that Onsager fluxes which satisfy the so-called non-linear Onsager reciprocal relations Edelen [Edelen, D.G.B., 1972. A non-linear Onsager theory of irreversibility. International Journal of Engineering Science 10, 481; 1973. Asymptotic stability, Onsager fluxes and reaction kinetics, International Journal of Engineering Science 11, 819], can ensure the existence of the flow potential function and lead to the normality structure, and Rice’s kinetic rate laws of local internal variables are just certain specific Onsager fluxes. It is also shown that the convexity of the entropy production rate can ensure the convexity of the flow potential function based on homogeneous Onsager fluxes.

Normality Structures With Homogeneous Kinetic Rate Laws

Abstract In this paper, a homogeneous type of kinetic rate laws of local internal variables and its corresponding macroscopic behaviors, are explored within the framework of “normality structures” by Rice. Rice’s kinetic rate laws of local internal variables, with each rate being stress dependent only via its conjugate thermodynamic force, are corner stones of the normality structure. It is revealed in this paper that nonlinear phenomenological equations and Onsager reciprocal relations emerge naturally if each rate is a homogeneous function of degree q in its conjugate force. Furthermore, the nonlinear phenomenological coefficient matrix is identical to the Hessian matrix of the flow potential function in conjugate forces only scaled by q. It is further shown that the refined version of Griffith criterion proposed by Rice, (G−2γ)ȧ⩾0, can be derived from the normality structure with the homogeneous rate laws. Finally, some issues related to damage evolution laws have been discussed based on the remarkable properties.

A Viscoplastic Constitutive Theory for Monolithic Ceramics—I

This paper, which is the first of two in a series, provides an overview of a viscoplastic constitutive model that accounts for time-dependent material deformation (e.g., creep, stress relaxation, etc.) in monolithic ceramics. Using continuum principles of engineering mechanics, the complete theory is derived from a scalar dissipative potential function first proposed by Robinson (1978), and later utilized by Duffy (1988). Derivations based on a flow potential function provide an assurance that the inelastic boundary value problem is well posed, and solutions obtained are unique. The specific formulation used here for the threshold function (a component of the flow potential function) was originally proposed by Willam and Warnke (1975) in order to formulate constitutive equations for time-independent classical plasticity behavior observed in cement and unreinforced concrete. Here constitutive equations formulated for the flow law (strain rate) and evolutionary law employ stress invariants to define the functional dependence on the Cauchy stress and a tensorial state variable. This particular formulation of the viscoplastic model exhibits a sensitivity to hydrostatic stress, and allows different behavior in tension and compression.

An incremental stress-based constitutive modeling on anisotropic damaged materials

An ‘incremental form’ of anisotropic damage constitutive equation is proposed both for brittle and ductile materials. Based on the concept of irreversible thermodynamics that damage processes are history independent coupled with irreversible energy dissipation, two types of definition for damage representation are established, known as damage tensor D and damage strain tensor εd, to describe constitutive responses of damaged materials. A state variable coupled with damage and other observable state variables, i.e. εd, is formulated separately from other internal variables and defined as an equivalent damage variable. A constitutive relation due to damage is finally formulated by introducing ‘damage flow potential function’ employing the theory of irreducible integrity bases. A clear physical representation based on theoretical foundations and rigorous mathematical arguments of the conventional damage models defined in terms of ‘damage effect tensor M(D)’ is also elucidated. Validity of the proposed model is verified by comparing with the formulations of conventional damage effect tensor. A plastic potential function coupled with damage is also introduced by employing the anisotropic plastic flow theory, so that the proposed damage model can be applied to characterize a wide range of damage problems of practical engineering interest.