Continuum Constitutive Laws Research Articles

The effectiveness of published continuum constitutive laws to predict stress‐assisted densification of powder compacts

AbstractCommonly used constitutive laws for crystalline and viscous materials have been compared to predict the densification behavior under hot‐pressing and sinter‐forging. Experimental results, from literature for one loading condition, have been used to extract the constitutive laws for amorphous and crystalline materials and, these in‐turn, have been used to predict behavior under a different set of loading conditions. Ideally, the constitutive parameters obtained from one set of loading conditions and thermal history should apply to a different set of conditions. However, there is a lack of systematic experimental studies in which this can be checked. In this paper, we use constitutive parameters obtained from one set of conditions to predict the densification response under a different set of loading conditions. For both sintering of amorphous and crystalline materials, we use two different constitutive parameters and compare the predictions of these for the case where experimental results are not available. In addition, the effect of temperature on densification behavior for stress‐assisted sintering has been investigated. It is shown that the two commonly used constitutive models for viscous sintering (Scherer and Skorohod–Olevsky) predict similar behavior for amorphous materials. However, for crystalline materials, the predictions of the Riedel–Svoboda and the Kuhn–Sofronis–McMeeking (KSM) models are different. Finally, it is shown that the dependence of the normalized densification on temperature, under constant heating rate conditions, with parameters obtained from isothermal experiments, is a good test for the models.

A continuum consistent discrete particle method for continuum–discontinuum transitions and complex fracture problems

In numerical simulations where complex fracture behavior plays a prominent role in the material’s mechanical behavior, particle methods are an attractive computational tool since they adequately accommodate arbitrary discontinuities. However, existing particle methods are either limited in their constitutive flexibility, like the Discrete Element Method (DEM), or prone to instabilities, like Smoothed Particle Hydrodynamics (SPH) and Peridynamics. In this paper we present an alternative particle formulation, referred to as the Continuum Bond Method (CBM). The method has the same constitutive flexibility as conventional continuum methods like the Finite Element Method (FEM), while still being able to incorporate arbitrary discontinuities as in particle methods like DEM, SPH and Peridynamics. In CBM, the continuum body is divided into a series of material points where each material point carries a fraction of the body’s mass. A triangulation procedure establishes the bonds between the particles that interact with each other. The deformation gradient tensor is determined via a volume weighted averaging procedure over the volumes spanned by pairs of nearest neighboring particles. The obtained approximation of the continuum deformation field on the particles allows for a straightforward implementation of continuum constitutive laws. To assess this property in CBM, simulation outcomes for an elastic nonlinear plastic tensile bar are compared to FEM and SPH results. While the stress–strain curves obtained by FEM, CBM and SPH coincide quite accurately, it is found that the local plastic strains obtained by CBM are much closer to the FEM reference solution than the SPH results. The ability of CBM to account for arbitrary discontinuities is demonstrated via a series of dynamic fracture simulations. It is shown that, without the need of additional crack tracking routines, CBM can account for fracture instability phenomena like branches. In conclusion, CBM is suitable for the implementation of continuum constitutive behavior while maintaining the advantageous discontinuous fracture properties of particle methods.

On the use of continuum Finite Element and Equivalent Frame models for the seismic assessment of masonry walls

The assessment of the seismic vulnerability of masonry buildings requires reliable and computationally efficient numerical models. Different modelling strategies can be adopted when studying the global response of these structures, such as Continuum Constitutive Laws Models (CCLM) belonging to Finite Element (FE) models, which may be very accurate but whose use in practice presents several issues (high computational burden, requirement of many input data), and Equivalent Frame (EF) Models, that, even if based on strong simplifications, are now widespread in engineering practice, thanks to their computational efficiency and the need of few mechanical parameters for the structural analysis. The paper discusses the consistent use of these modelling techniques for the seismic analysis of masonry structures . To this aim, a comparison of two approaches (CCL and EF models) is presented focusing at first the attention on the calibration of the constitutive laws through analyses on single panels and then moving to the validation of some simplified assumptions made in the EFM through the analysis of a 2D regular URM wall. For the aim of validation, the CCL model is considered as the reference solution making an accurate comparison with the EF model in terms of generalized forces, drift and damage occurred at element scale and of pushover curve at global scale. • A procedure for cross-calibrating the mechanical parameters in Finite Element (FE) and Equivalent Frame (EF) models is discussed in the paper. • The procedure guarantees a consistent use of two modelling approaches for the seismic assessment of URM buildings. • The reliability of EF model in case of regular 2D URM walls is confirmed through a very detailed comparison with the FE model. • Different rules for the EF idealization of regular URM walls are tested and criteria to orient the choice on the most reliable ones are proposed.

Continuum constitutive laws to describe acoustic attenuation in glasses.

Nowadays metamaterials are at the focus of an intense research as promising for thermal and acoustic engineering. However, the computational cost associated to the large system size required for correctly simulating them imposes the use of finite-elements simulations, developing continuum models, able to grasp the physics at play without entering in the atomistic details. Still, a correct description should be able to reproduce not only the extrinsic scattering sources on waves propagation, as introduced by the metamaterial microstructure, but also the intrinsic wave attenuation of the material itself. This becomes dramatically important when the metamaterial is made out of a glass, which is intrinsically highly dissipative and with a wave attenuation strongly dependent on frequency. Here we propose a continuum mechanical model for a viscoelastic medium, able to bridge atomic and macroscopic scale in amorphous materials and describe phonon attenuation due to atomistic mechanisms, characterized by a defined frequency dependence. This represents a first decisive step for investigating the effect of a complex nano- or microstructure on acoustic attenuation, while including the atomistic contribution as well.

Effective Thermal Expansion Property of Consolidated Granular Materials.

Thermally-assisted compaction of granular materials is of keen interest in many engineering applications. A proper estimation of the material behavior of compacted granular materials is contingent upon the knowledge of microstructure formation, which is highly dependent on the bulk material properties and processing conditions, during the deformation stage. Originating from the pair interactions between particles, the macroscopic properties are obtained using various homogenization techniques and postulating continuum constitutive laws. While pioneers in this field have laid fundamental groundwork regarding effective medium descriptions, there exists a discrepancy between discrete and continuum level solutions. In our previous work, we elaborated a Particle Mechanics Approach (PMA) that integrates thermal contact and Hertzian deformation models to understand the thermo-mechanically-coupled consolidation problem. We also considered the analogous problem from the perspective of the conventional Continuum Mechanics Approach (CMA). In this study, following the multi-scale modeling framework, we propose an effective thermal expansion coefficient for the thermally-assisted compaction of granular materials.

A numerical-experimental approach to estimate cohesive laws based on continuum damage mechanics for ductile failure

Most cohesive zone models for the simulation of crack growth in ductile metals do not reflect continuum constitutive laws on traction–separation relationships across cohesive crack surfaces. In this study, a continuously-deforming material layer associated with a ductile failure process of void nucleation, growth and coalescence is converted into a separating pair of cohesive crack surfaces resisted by cohesive tractions. Representative volume elements of damageable and undamageable material layers are used to extract the porous portion of the deformation in the failure process zone, which is reduced to a cohesive law based on continuum damage mechanics. The failure process zone height is determined by matching the J-integral evaluated from the results of crack growth simulations using cohesive elements with that evaluated from the displacements measured in crack growth experiments. The present approach provides an efficient and effective way to estimate cohesive laws of ductile metals by combining experimental tests and numerical simulations of a cracked specimen.

Comparison between spring network models and continuum constitutive laws: Application to the large deformation of a capsule in shear flow

A capsule is a liquid drop enclosed by a solid, deformable membrane. To analyze the deformation of a capsule accurately, both the fluid mechanics of the internal and external fluids and the solid mechanics of the membrane must be solved precisely. Recently, many researchers have used discrete spring network models to express the membrane mechanics of capsules and biological cells. However, it is unclear whether such modeling is sufficiently accurate to solve for capsule deformation. This study examines the correlations between the mechanical properties of the discrete spring network model and continuum constitutive laws. We first compare uniaxial and isotropic deformations of a two-dimensional (2D) sheet, both analytically and numerically. The 2D sheet is discretized with four kinds of mesh to analyze the effect of the spring network configuration. We derive the relationships between the spring constant and continuum properties, such as the Young modulus, Poisson ratio, area dilation modulus, and shear modulus. It is found that the mechanical properties of spring networks are strongly dependent on the mesh configuration. We then calculate the deformation of a capsule under inflation and in a simple shear flow in the Stokes flow regime, using various membrane models. To achieve high accuracy in the flow calculation, a boundary-element method is used. Comparing the results between the different membrane models, we find that it is hard to express the area incompressibility observed in biological membranes using a simple spring network model.

The Relationship between Cell and Tissue Strain in Three-Dimensional Bio-Artificial Tissues

Continuum constitutive laws are needed to ensure that bio-artificial tissue constructs replicate the mechanical response of the tissues they replace, and to understand how the constituents of these constructs contribute to their overall mechanical response. One model designed to achieve both of these aims is the Zahalak model, which was modified by Marquez and co-workers to incorporate inhomogeneous strain fields within very thin tissues. When applied to reinterpret previous measurements, the modified Zahalak model predicted higher values of the continuum stiffness of fibroblasts than earlier estimates. In this work, we further modify the Zahalak model to account for inhomogeneous strain fields in constructs whose cell orientations have a significant out-of-plane component. When applied to reinterpret results from the literature, the new model shows that estimates of continuum cell stiffness might need to be revised upward. As in this article’s companion, we updated the average cell strain by defining a correction factor (“strain factor”), based upon the elastic response. Three different cell orientation distributions were studied. We derived an approximate scaling model for the strain factor, and validated it against exact and self-consistent (mean-field) solutions from the literature for dilute cell concentrations, and Monte Carlo simulations involving three-dimensional finite element analyses for high cell concentrations.

A finite-strain quadrilateral shell element based on discrete Kirchhoff-Love constraints

This paper improves the 16 degrees-of-freedom quadrilateral shell element based on pointwise Kirchhoff–Love constraints and introduces a consistent large strain formulation for this element. The model is based on classical shell kinematics combined with continuum constitutive laws. The resulting element is valid for large rotations and displacements. The degrees-of-freedom are the displacements at the corner nodes and one rotation at each mid-side node. The formulation is free of enhancements, it is almost fully integrated and is found to be immune to locking or unstable modes. The patch test is satisfied. In addition, the formulation is simple and amenable to efficient incorporation in large-scale codes as no internal degrees-of-freedom are employed, and the overall calculations are very efficient. Results are presented for linear and non-linear problems. Copyright © 2005 John Wiley & Sons, Ltd.

Application of embedded discontinuities for softening solids

Additional, discontinuous functions are added to the displacement field of standard finite elements in order to capture highly localised zones of intense straining. By embedding discontinuities within an element it is possible to effectively model localisation phenomena (such as fracture in concrete) with a relatively small number of finite elements and without resorting to higher-order continuum theories. However, the application of continuum constitutive laws in combination with embedded discontinuities, several unique performance related issues arise. The source of these problems and some remedies are addressed here. Numerical results are compared with smeared type models for brittle fracture.

Finite element methods in fracture mechanics

Finite element methodology specific to the analysis of fracture mechanics problems is reviewed. Primary emphasis is on the important algorithmic developments which have enhanced the numerical modeling of fracture processes. Methodologies to address elasto-static problems in two- and three-dimensions, elasto-dynamic problems, elasto-plastic problems, special considerations for three-dimensional nonlinear problems and the modeling of stable crack growth are reviewed. In addition, the future needs of the fracture community are discussed and open questions are identified. The need for theoretical advancements in continuum mechanics and constitutive fracture laws coupled with improved numerical algorithms is emphasized. Extensive reference is made to the open literature base for further details.

On the theory of boundary conditions

The question which boundary conditions are appropriate for the given differential equations of a continuum theory, is treated by the method of nonequilibrium thermodynamics. In the first part the previous investigation1) which had been confined to ordinary hydro(aero-)dynamics is generalized by taking into account higher derivatives in the continuum constitutive laws. This kind of generalization becomes important when a rarefied gas with boundary is treated phenomenologically (wall influence, slip-flow regime). In the second part an even more general scheme is discussed. The underlying applications now are rarefied polyatomic gases within walls, perhaps in an external field. So, the starting point is a general set of transport-relaxation equations with more variables than the hydrodynamical ones. By considering the corresponding entropy production, especially its part due to the boundary, it is again possible to set up constitutive laws, i.e., matching or boundary conditions, at an interface or a surface.