Inelastic Constitutive Laws Research Articles

Data based constitutive modelling of rate independent inelastic effects in composite cables using Preisach hysteresis operators

AbstractThis contribution aims at introducing first steps to develop hysteresis operator type inelastic constitutive laws for Cosserat rods for the simulation of cables composed of complex interior components. Motivated by the basic elements of Cosserat rod theory, we develop a specific approach to constitutive modelling adapted for this application. Afterwards, we describe the hysteretical behaviour arising from cyclic bending experiments on cables by means of the Preisach operator. As shown in pure bending experiments, slender structures such as electric cables behave inelastically, and open hysteresis loops arise with noticeable difference between the first load cycle and the following ones. The Preisach operator plays an important role in describing the input-output relation in hysteresis behaviours, and it can be expressed as a superposition of relay operators. Hence, a mathematical formulation of the problem is introduced, and a first attempt is made to determine the hysteresis behaviour that describes the relation between curvature and bending moment. Therefore, a suitable kernel function is identified in a way that its integration over the Preisach plane results in the bending moment of the specimen, and a comparison between different kernel functions is performed.

Blast response of one-way arching masonry walls

The dynamic response of arching unreinforced masonry walls subjected to blast loads combines a spectrum of complex physical phenomena including opening and closing of cracks, unloading and reloading, accumulation of damage, rocking, coupling of in-plane and out-of-plane responses, and time variation of the arching force. These aspects contribute to the complexity of the dynamic behavior of the masonry wall and make the dynamic analysis challenging. This study develops a new model that integrates the aforementioned characteristics of the wall behavior and presents a quantitative tool for the assessment of the complex response. The model assumes one-way flexural action of an unreinforced masonry wall and takes the form of a multi-degree-of-freedom system, where the response is described by the displacements and rotations of the masonry units. The formulation employs variational principles and accounts for equilibrium conditions, cracking, geometrically nonlinear kinematics, and nonlinear and inelastic constitutive laws. The masonry units are assumed rigid bodies, whereas the flexural longitudinal and shear deformations of the wall are due to the mortar strains only. The mortar joints and the interfaces between adjacent masonry units are represented by arrays of nonlinear longitudinal and shear springs that introduce the normal, shear, and contact interactions between blocks. The model is examined and compared against experimental results taken from the literature. The investigated features include the cracking process, the inelastic behavior of the mortar material, and its effect on the wall response. The results demonstrate the capabilities of the analysis and throw light on the dynamic nonlinear response of the arching masonry wall under blast loads.

Time-Independent Plasticity Based on Thermodynamic Equilibrium and Its Stability

Within the thermodynamic framework with internal variables by Rice (1971, “Inelastic Constitutive Relations for Solids: An Internal Variable Theory and Its Application to Metal Plasticity,” J. Mech. Phys. Solids, 19(6), pp. 433–455), Yang et al. (2014, “Time-Independent Plasticity Related to Critical Point of Free Energy Function and Functional,” ASME J. Eng. Mater. Technol., 136(2), p. 021001) established a model of time-independent plasticity of three states. In this model, equilibrium states are the states with vanishing thermodynamic forces conjugate to the internal variables, and correspond to critical points of the free energy or its complementary energy functions. Then, the conjugate forces play a role of yield functions and further lead to the consistency conditions. The model is further elaborated in this paper and extended to nonisothermal processes. It is shown that the incremental stress–strain relations are fully determined by the local curvature of the free energy or its complementary energy functions at the critical points, described by the Hessian matrices. It is further shown that the extended model can be well reformulated based on the intrinsic time in the sense of Valanis (1971, “A Theory of Viscoplasticity Without a Yield Surface, Part I. General Theory,” Arch. Mech., 23(4), pp. 517–533; 1975, “On the Foundations of the Endochronic Theory of Viscoplasticity,” Arch. Mech., 27(5–6), pp. 857–868), by taking the intrinsic time as the accumulated length of the variation of the internal variables during inelastic processes. It is revealed within this framework that the stability condition of equilibrium directly leads to Drucker (1951, “A More Fundamental Approach to Stress–Strain Relations,” First U.S. National Congress of Applied Mechanics, pp. 487–497) and Il'yushin (1961, “On a Postulate of Plasticity,” J. Appl. Math. Mech., 25(2), pp. 746–750) inequalities, by introducing the consistency condition into the work of Hill and Rice (1973, “Elastic Potentials and the Structure of Inelastic Constitutive Laws,” SIAM J. Appl. Math., 25(3), pp. 448–461). Generalized inequalities of Drucker (1951, “A More Fundamental Approach to Stress–Strain Relations,” First U.S. National Congress of Applied Mechanics, pp. 487–497) and Il'yushin (1961, “On a Postulate of Plasticity,” J. Appl. Math. Mech., 25(2), pp. 746–750) for nonisothermal processes are established straightforwardly based on the connection.

Time-Independent Plasticity Related to Critical Point of Free Energy Function and Functional

In this paper, time-independent plasticity is addressed within the thermodynamic framework with internal variables 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). It is shown in this paper that the existence of a free energy function along with thermodynamic equilibrium conditions directly leads to associated flow rules. The time-independent inelastic behaviors can be fully determined by the Hessian matrix at the nondegenerate critical point of the free energy function. The normality rule of Hill and Rice (1973, “Elastic Potentials and the Structure of Inelastic Constitutive Laws,” SIAM J. Appl. Math., 25, pp. 448–461) or the Il'yushin (1961, “On a Postulate of Plasticity,” J. Appl. Math. Mech. 25, pp. 746–750) postulate is just a stability requirement of the thermodynamic equilibrium. The existence of a free energy functional which is not a direct function of the internal variables, along with thermodynamic equilibrium conditions also leads to associated flow rules. The time-independent inelastic behaviors with the free energy functional can be fully determined by the quasi Hessian matrix at the quasi critical point of the free energy functional. With the free energy functional, the thermodynamic forces conjugate to the internal variables are nonconservative and are constructed based on Darboux theorem. Based on the constructed nonconservative forces, it is shown that there may exist several possible thermodynamic equilibrium mechanisms for the thermodynamic system of the material sample. Therefore, the associated flow rules based on free energy functionals may degenerate into nonassociated flow rules. The symmetry of the conjugate forces plays a central role for the characteristics of time-independent plasticity.

Hamilton’s principle for Green-inelastic bodies

Abstract In this paper, Hamilton’s principle for Green-elastic bodies with conservative external forces (see Fung and Tong, 2001. Classical and Computational Solid Mechanics. World Scientific, Singapore, Chapter 11.1) is extended to Green-inelastic bodies. Inelastic constitutive laws are addressed within the thermodynamic framework with internal variables by Rice (1971. Inelastic constitutive relations for solids: an internal variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455). Generalized thermodynamic forces conjugate to internal variables are termed internal forces in this paper. The materials whose free energy functions are point functions of internal variables, are termed here Green-inelastic materials which embody Green-elastic materials as special cases and ensures that the internal forces are potential forces. It is shown that Hamilton’s principle for Green-elastic bodies can be extended to Green-inelastic bodies just by replacing the strain energy of a moving body with the corresponding free energy plus the potential energy of the internal forces.

Development Length of Straight FRP Composite Bars Embedded in Concrete

This article presents an analytical study performed on concrete girders reinforced with FRP bars with the goal of evaluating current design guidelines for FRP rebar development lengths. The analytical model is based on finite element formulation of embedded bar problems with inelastic constitutive laws for the constituent materials. The bond stress—slip behavior between FRP bars and concrete was calibrated from experimental data obtained from a large database set of 48 test specimens, most of which were collected from the open literature. The collected data was modified to emulate a specific bar size and embedment length for various scenarios. A large number of analytical simulations were conducted in order to investigate the effect of different parameters on the development length. These parameters include: bar type (GFRP, CFRP), bar strength, bar modulus of elasticity, bar diameter, bar surface coatings, concrete strength, and confinement. For each specimen, the analytical model was conducted for a set condition until an embedment length which satisfied a predetermined criterion was found. The analytical study resulted in the proposal of a refined design equation for development length of FRP bars for consideration by current building and bridge design codes.

On internal dissipation inequalities and finite strain inelastic constitutive laws: Theoretical and numerical comparisons

Internal dissipation always occurs in irreversible inelastic deformation processes of materials. The internal dissipation inequalities (specific mathematical forms of the second law of thermodynamics) determine the evolution direction of inelastic processes. Based on different internal dissipation inequalities several finite strain inelastic constitutive laws have been formulated for instance by Simo [Simo, J.C., 1992. Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Computer Methods in Applied Mechanics and Engineering 99, 61–112]; Simo and Miehe [Simo, J.C., Miehe, C., 1992. Associative coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation. Computer Methods in Applied Mechanics and Engineering 98, 41–104]; Lion [Lion, A., 1997. A physically based method to represent the thermo-mechanical behavior of elastomers. Acta Mechanica 123, 1–25]; Reese and Govindjee [Reese, S., Govindjee, S., 1998. A theory of finite viscoelasticity and numerical aspects. International Journal of Solids and Structures 35, 3455–3482]; Lin and Schomburg [Lin, R.C., Schomburg, U., 2003. A finite elastic–viscoelastic–elastoplastic material law with damage: theoretical and numerical aspects. Computer Methods in Applied Mechanics and Engineering 192, 1591–1627]; Lin and Brocks [Lin, R.C., Brocks, W., 2004. On a finite strain viscoplastic theory based on a new internal dissipation inequality. International Journal of Plasticity 20, 1281–1311]; and Lin and Brocks [Lin, R.C., Brocks, W., 2005. An extended Chaboche’s viscoplastic law at finite strains: theoretical and numerical aspects. Journal of Materials Science and Technology 21, 145–147]. These constitutive laws are consistent with the second law of thermodynamics. As the internal dissipation inequalities are described in different configurations or coordinate systems, the related constitutive laws are also formulated in the corresponding configurations or coordinate systems. Mathematically, these constitutive laws have very different formulations. Now, a question is whether the constitutive laws provide identical constitutive responses for the same inelastic constitutive problems. In the present work, four types of finite strain viscoelastic and endochronically plastic laws as well as three types of J 2-plasticity laws are formulated based on four types of dissipation inequalities. Then, they are numerically compared for several problems of homogeneous or complex finite deformations. It is demonstrated that for the same inelastic constitutive problem the stress responses are identical for deformation processes without rotations. In the simple shear deformation process with large rotation, the presented viscoelastic and endochronically plastic laws also show almost identical stress responses up to a shear strain of about 100%. The three laws of J 2-plasticity also produce the same shear stresses up to a shear strain of 100%, while different normal stresses are generated even at infinitesimal shear strains. The three J 2-plasticity laws are also compared at three complex finite deformation processes: billet upsetting, cylinder necking and channel forming. For the first two deformation processes similar constitutive responses are obtained, whereas for the third deformation process (with large global rotations) significant differences of constitutive responses can be observed.

Constitutive restrictions for a hyperelastic material, based upon strain invariants yielding orthogonal stress response terms

Internal dissipation always occurs in irreversible inelastic deformation processes of materials. The internal dissipation inequalities (specific mathematical forms of the second law of thermodynamics) determine the evolution direction of inelastic processes. Based on different internal dissipation inequalities several finite strain inelastic constitutive laws have been formulated for instance by Simo [Simo, J.C., 1992. Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Computer Methods in Applied Mechanics and Engineering 99, 61–112]; Simo and Miehe [Simo, J.C., Miehe, C., 1992. Associative coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation. Computer Methods in Applied Mechanics and Engineering 98, 41–104]; Lion [Lion, A., 1997. A physically based method to represent the thermo-mechanical behavior of elastomers. Acta Mechanica 123, 1–25]; Reese and Govindjee [Reese, S., Govindjee, S., 1998. A theory of finite viscoelasticity and numerical aspects. International Journal of Solids and Structures 35, 3455–3482]; Lin and Schomburg [Lin, R.C., Schomburg, U., 2003. A finite elastic–viscoelastic–elastoplastic material law with damage: theoretical and numerical aspects. Computer Methods in Applied Mechanics and Engineering 192, 1591–1627]; Lin and Brocks [Lin, R.C., Brocks, W., 2004. On a finite strain viscoplastic theory based on a new internal dissipation inequality. International Journal of Plasticity 20, 1281–1311]; and Lin and Brocks [Lin, R.C., Brocks, W., 2005. An extended Chaboche’s viscoplastic law at finite strains: theoretical and numerical aspects. Journal of Materials Science and Technology 21, 145–147]. These constitutive laws are consistent with the second law of thermodynamics. As the internal dissipation inequalities are described in different configurations or coordinate systems, the related constitutive laws are also formulated in the corresponding configurations or coordinate systems. Mathematically, these constitutive laws have very different formulations. Now, a question is whether the constitutive laws provide identical constitutive responses for the same inelastic constitutive problems. In the present work, four types of finite strain viscoelastic and endochronically plastic laws as well as three types of J2-plasticity laws are formulated based on four types of dissipation inequalities. Then, they are numerically compared for several problems of homogeneous or complex finite deformations. It is demonstrated that for the same inelastic constitutive problem the stress responses are identical for deformation processes without rotations. In the simple shear deformation process with large rotation, the presented viscoelastic and endochronically plastic laws also show almost identical stress responses up to a shear strain of about 100%. The three laws of J2-plasticity also produce the same shear stresses up to a shear strain of 100%, while different normal stresses are generated even at infinitesimal shear strains. The three J2-plasticity laws are also compared at three complex finite deformation processes: billet upsetting, cylinder necking and channel forming. For the first two deformation processes similar constitutive responses are obtained, whereas for the third deformation process (with large global rotations) significant differences of constitutive responses can be observed.

Thermodynamic conditions for stability in materials with rate-independent dissipation

A distinctive feature of the examined class of solids is that a part of the entropy production is due to rate-independent dissipation, as in models of plasticity, damage or martensitic transformations. The standard condition for thermodynamic stability is shown to be too restrictive for such solids and, therefore, an extended condition for stability of equilibrium is developed. The classical thermodynamic theory of irreversible processes is used along with the internal variable approach, with the emphasis on the macroscopic effects of micro-scale instabilities in the presence of two different scales of time. Specific conditions for material stability against internal structural rearrangements under deformation-sensitive loading are derived within the incremental constitutive framework of multi-mode inelasticity. Application to spontaneous formation of deformation bands in a continuum is presented. Conditions for stability or instability of a quasi-static process induced by varying loading are given under additional constitutive postulates of normality and symmetry. As illustration of the theory, the stability of equilibrium or a deformation path under uniaxial tension is analysed for a class of inelastic constitutive laws for a metal crystal deformed plastically by multi-slip or undergoing stress-induced martensitic transformation.

The Generalized Interpolation Material Point Method

The Material Point Method (MPM) discrete solution procedure for computational solid mechanics is generalized using a variational form and a Petrov- Galerkin discretization scheme, resulting in a family of methods named the Generalized Interpolation Material Point (GIMP) The generalization permits iden- tification with aspects of other point or node based dis- crete solution techniques which do not use a body-fixed grid, i.e. the methods. Similarities are noted and some practical advantages relative to some of these methods are identified. Examples are used to demon- strate and explain numerical artifact noise which can be expected in MPM calculations. This noise results in non- physical local variations at the material points, where constitutive response is evaluated. It is shown to destroy the explicit solution in one case, and seriously degrade it in another. History dependent, inelastic constitutive laws can be expected to evolve erroneously and report inac- curate stress states because of noisy input. The noise is due to the lack of smoothness of the interpolation func- tions, and occurs due to material points crossing compu- tational grid boundaries. The next degree of smoothness available in the GIMP methods is shown to be capable of eliminating cell crossing noise. keyword: MPM, PIC, meshless methods, Petrov- Galerkin discretization.

Springer handbook of enzymes, 2nd edn

Internal dissipation always occurs in irreversible inelastic deformation processes of materials. The internal dissipation inequalities (specific mathematical forms of the second law of thermodynamics) determine the evolution direction of inelastic processes. Based on different internal dissipation inequalities several finite strain inelastic constitutive laws have been formulated for instance by Simo [Simo, J.C., 1992. Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Computer Methods in Applied Mechanics and Engineering 99, 61–112]; Simo and Miehe [Simo, J.C., Miehe, C., 1992. Associative coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation. Computer Methods in Applied Mechanics and Engineering 98, 41–104]; Lion [Lion, A., 1997. A physically based method to represent the thermo-mechanical behavior of elastomers. Acta Mechanica 123, 1–25]; Reese and Govindjee [Reese, S., Govindjee, S., 1998. A theory of finite viscoelasticity and numerical aspects. International Journal of Solids and Structures 35, 3455–3482]; Lin and Schomburg [Lin, R.C., Schomburg, U., 2003. A finite elastic–viscoelastic–elastoplastic material law with damage: theoretical and numerical aspects. Computer Methods in Applied Mechanics and Engineering 192, 1591–1627]; Lin and Brocks [Lin, R.C., Brocks, W., 2004. On a finite strain viscoplastic theory based on a new internal dissipation inequality. International Journal of Plasticity 20, 1281–1311]; and Lin and Brocks [Lin, R.C., Brocks, W., 2005. An extended Chaboche’s viscoplastic law at finite strains: theoretical and numerical aspects. Journal of Materials Science and Technology 21, 145–147]. These constitutive laws are consistent with the second law of thermodynamics. As the internal dissipation inequalities are described in different configurations or coordinate systems, the related constitutive laws are also formulated in the corresponding configurations or coordinate systems. Mathematically, these constitutive laws have very different formulations. Now, a question is whether the constitutive laws provide identical constitutive responses for the same inelastic constitutive problems. In the present work, four types of finite strain viscoelastic and endochronically plastic laws as well as three types of J2-plasticity laws are formulated based on four types of dissipation inequalities. Then, they are numerically compared for several problems of homogeneous or complex finite deformations. It is demonstrated that for the same inelastic constitutive problem the stress responses are identical for deformation processes without rotations. In the simple shear deformation process with large rotation, the presented viscoelastic and endochronically plastic laws also show almost identical stress responses up to a shear strain of about 100%. The three laws of J2-plasticity also produce the same shear stresses up to a shear strain of 100%, while different normal stresses are generated even at infinitesimal shear strains. The three J2-plasticity laws are also compared at three complex finite deformation processes: billet upsetting, cylinder necking and channel forming. For the first two deformation processes similar constitutive responses are obtained, whereas for the third deformation process (with large global rotations) significant differences of constitutive responses can be observed.

Elastic and plastic analysis of thin-walled structures using improved hexahedral elements

In this paper a continuum based 3D-shell element for the nonlinear analysis of thin-walled structures is developed. Assumed natural strain method and enhanced assumed strain method are used to improve the relative poor element behaviour of a standard hexahedral displacement element. Different elastic and inelastic constitutive laws are considered. The anisotropic material behaviour of layered shells is modeled using a hyperelastic orthotropic material law in each layer. Furthermore, finite multiplicative J 2-plasticity is discussed. The models are characterized by an interface to the material setting of the boundary value problem. Several examples show the applicability and efficiency of the developed element formulation.

Polymorphic constitutive equations in finite element codes

This work presents an object-oriented design approach to the programming of finite element constitutive equations. A universal interface for arbitrary material laws is introduced from which easy addition and expansion of the material library can take place. Although such expansion using a generic interface is by definition polymorphic, the idiom is carried much further in several aspects of program design. Deriving from abstract examples of equation and algorithmic encapsulation, simple elastic damage and complex inelastic constitutive laws are developed. Examples of calculations show that adequate performance is given by C++ with respect to Fortran77 for simple to very complex material laws. The performance is seen to be a much stronger function of the compiler used than the actual numerical complexity.

Steady‐State Oscillation of Hysteretic Differential Model. II: Performance Analysis

The present study involves two aspects associated with the steady‐state performance of the versatile Bouc‐Wen hysteretic model. First, influence of the model parameters on steady‐state responses and hysteresis loops of the single‐degree‐of‐freedom (SDOF) hysteretic system is studied by the proposed Galerkin/ Levenberg‐Marquardt (GLM) method. The dynamic behavior corresponding to five kinds of hysteresis loops is identified. Second, the steady‐state characteristics of the Bouc‐Wen hysteretic model used for representing inelastic constitutive laws are recognized. As endochronic constitutive relationship, this smooth hysteretic model may locally violate the Drucker's (or Ilyushin's) stability postulate of plasticity when the steady‐state hysteresis loops are exhibited with intermediate unloading‐reloading. The study is emphasized on the energy dissipation properties in small unloading‐reloading cycles starting at an arbitrary deformation previously reached. The extent of local violation of the Drucker's postulate under steady‐state circumstances is observed through numerical analysis. The mitigation of this violation is addressed by parameter studies.

A physically motivated set of invariants and tensor generators in the case of transverse isotropy

A new set of invariants and symmetric and skew-symmetric second order tensor generators is introduced to represent the general form of a transversely isotropic constitutive law. In contrast to the standard system usually used in literature the new invariants and tensor generators have a well defined physical meaning which is strongly related to the deformation mechanisms of transversely isotropic materials. This substantially facilitates the decision which tensor generators should be considered in a constitutive law and on which invariants the scalar-valued functions should depend. Reducing the tensor function to a linear relation one gets a new frame independent representation of a fourth order transversely isotropic tensor. The relations between the different representations found in literature and that one introduced here are given. Their advantages are illustrated by the examples of elastic and inelastic constitutive laws of monocrystalline ice.

Elastic Potentials and the Structure of Inelastic Constitutive Laws

The existence of a work potential, governing material response when elastic behavior occurs, is shown to provide a concise framework for the structure of inelastic constitutive laws. The discussion includes the identification of differential invariants in conjugate stress and strain variables, an examination and reformulation of postulates leading to “normality rules”, and a study of the transmission of constitutive properties from subelements to overall behavior of heterogeneous systems. Also, internal variable representations of inelasticity are examined within this framework.