Dynamic void nucleation and growth in solids: A self-consistent statistical theory

We present a framework for a self-consistent theory of spall fracture in ductile materials, based on the dynamics of void nucleation and growth. The constitutive model for the material is divided into elastic and “plastic” parts, where the elastic part represents the volumetric response of a porous elastic material, and the “plastic” part is generated by a collection of representative volume elements (RVEs) of incompressible material. Each RVE is a thick-walled spherical shell, whose average porosity is the same as that of the surrounding porous continuum, thus simulating void interaction through the resulting lowered resistance to further void growth. All voids nucleate and grow according to the appropriate dynamics for a thick-walled sphere made of incompressible material. The macroscopic spherical stress in the material drives the response in all volume elements, which have a distribution of critical stresses for void nucleation, and the statistically weighted sum of the void volumes of all RVEs generates the global porosity. Thus, macroscopic pressure, porosity, and a distribution of growing microscopic voids are fully coupled dynamically. An example is given for a rate-independent, perfectly plastic material. The dynamics of void growth gives rise to a rate effect in the macroscopic material even though the parent material is rate independent.     

A new damage model for microcrack-weakened brittle solids

In the present paper, a micromechanically based damage model for microcrack-weakened solids is developed. The concept of the domain of microcrack growth (DMG) is defined and used to describe the damage state and the anisotropic properties of brittle materials. After choosing an appropriate fracture criterion of microcrack, we obtain the analytical expression of DMG under a monotonically increasing proportional plane stress. Under a complex loading path, the evolution equation of DMG and the overall effective compliance tensor of damaged materials are given. The project supported by National Natural Science Foundation of China     

A pressure-sensitive yield criterion under a non-associated flow rule for sheet metal forming

Spitzig and Richmond [Acta Metall. 32 (1984) 457] proposed that plastic yielding of both polycrystalline and single crystals of steel and aluminum alloys shows a significant sensitivity to hydrostatic pressure. They further showed that under the associated flow rule, this pressure sensitivity leads to a plastic dilatancy, i.e. permanent volume change, that is at least an order of magnitude larger than observed. Indeed, the plastic dilatancy for most materials is on the order of the measurement error and must be zero in the absence of phase change and significant void nucleation during plastic deformation. A non-associated flow rule based on a pressure sensitive yield criterion with isotropic hardening is proposed in this paper that is consistent with the Spitzig and Richmond data and analysis. The significance of this work is that the model distorts the shape of the yield function in tension and compression, fully accounting for the strength differential effect (SDE). This capability is important because the SDE is sometimes described through kinematic hardening models using only pressure insensitive yield criteria.     

A new yield stress scaling function for electrorheological fluids

A new scaling function capable of modeling the yield stress behavior of electrorheological (ER) fluids through the full range of electric fields is proposed. In spite of its simple form, a comparison of the model predictions with experimental data for both ac and dc fields and the polarization model shows that the proposed model correctly predicts the yield stress behavior both quantitatively as well as qualitatively.     

A first-order strain gradient damage model for simulating quasi-brittle failure in porous elastic solids

In order to simulate quasi-brittle failure in porous elastic solids, a continuum damage model has been developed within the framework of strain gradient elasticity. An essential ingredient of the continuum damage model is the local strain energy density for pure elastic response as a function of the void volume fraction, the local strains and the strain gradients, respectively. The model adopts Griffith’s approach, widely used in linear elastic fracture mechanics, for predicting the onset and the evolution of damage due to evolving micro-cracks. The effect of those micro-cracks on the local material stiffness is taken into account by defining an effective void volume fraction. Thermodynamic considerations are used to specify the evolution of the latter. The principal features of the model are demonstrated by means of a one-dimensional example. Key aspects are discussed using analytical results and numerical simulations. Contrary to other continuum damage models with similar objectives, the model proposed here includes the effect of the internal length parameter on the onset of damage evolution. Furthermore, it is able to account for boundary layer effects.     

A void nucleation and growth based damage framework to model failure initiation ahead of a sharp notch in irradiated bcc materials

A continuum damage framework is developed and coupled with an existing crystal plasticity framework, to model failure initiation in irradiated bcc polycrystalline materials at intermediate temperatures. Constitutive equations for vacancy generation due to inelastic deformation, void nucleation due to vacancy condensation, and diffusion-assisted void growth are developed. The framework is used to simulate failure initiation at dislocation channel interfaces and grain boundaries ahead of a sharp notch. Evolution of the microstructure is considered in terms of the evolution of inelastic deformation, vacancy concentration, and void number density and radius. Evolution of the damage, i.e., volume fraction of the voids, is studied as a function of applied deformation. Effects of strain rate and temperature on failure initiation are also studied. The framework is used to compute the fracture toughness of irradiated specimens for various loading histories and notch geometries. Crack growth resistance of the irradiated specimens are computed and compared to that of virgin specimens. Results are compared to available experimental data.     

A quest for a model of non-colloidal suspensions with Newtonian matrices

It would be convenient to have a model, albeit approximate, of particle-laden materials (suspensions) that would not need large amounts of computing and/or experimentation to implement for design purposes. There are now adequate models of the pure matrix fluid behaviour, but there are no such models for suspensions with large particles (non-colloidal suspensions). One of the obstacles has been the single-minded devotion to shearing motions of suspensions; experience with the matrix modelling has shown that it is not possible to formulate widely usable models if only shear is considered. Here some new results of axially symmetric elongational tests on suspensions are compared with shearing data. Some suggestions for modelling these and other observations based on using strain rate and strain in a modified Reiner-Rivlin constitutive equation are presented. The model generally works quite well, but it does not predict the positive storage modulus seen in small and medium amplitude oscillatory shear flows.     

A micromechanical model for porous materials with a reinforced matrix

In this study a micromechanical model is proposed for ductile porous material whose matrix is reinforced by small inclusions. The solid phase is described by a pressure sensitive plastic model. Based on works of Maghous et al. [6], a macroscopic plastic criterion is firstly obtained by using a two-step homogenization procedure. The effect of porosity at the mesoscale and the influence of inclusions at the microscale are taken into account simultaneously by this criterion. With a non-associated plastic flow rule, the micro-macro model is applied to modeling of mechanical behavior of a cement paste. In particular, we have considered at the microscopic scale the formation of calcite grains by carbonation process in the solid matrix. The studied cement paste is then seen as a reinforced matrix–pore system. Comparisons between numerical results and experimental data show that the proposed model is able to capture the main features of the mechanical behavior of the studied material.     

A damage-mechanics model for fracture nucleation and propagation

In this paper, a composite model for earthquake rupture initiation and propagation is proposed. The model includes aspects of damage mechanics, fiber-bundle models, and slider-block models. An array of elements is introduced in analogy to the fibers of a fiber bundle. Time to failure for each element is specified from a Poisson distribution. The hazard rate is assumed to have a power-law dependence on stress. When an element fails it is removed, the stress on a failed element is redistributed uniformly to a specified number of neighboring elements in a given range of interaction. Damage is defined to be the fraction of elements that have failed. Time to failure and modes of rupture propagation are determined as a function of the hazard-rate exponent and the range of interaction.     

A Gurson-type criterion for porous ductile solids containing arbitrary ellipsoidal voids—II: Determination of yield criterion parameters

The aim of this paper is to fully determine the parameters of the approximate homogenized yield criterion for porous ductile solids containing arbitrary ellipsoidal cavities proposed in Part I. This is done through improvements of the limit-analysis of some representative hollow cell presented there. The improvements are of two kinds. For hydrostatic loadings, the limit-analysis is refined by performing micromechanical finite element computations in a number of significant cases, so as to replace Leblond and Gologanu (2008)'s trial velocity field representing the expansion of the void by the exact, numerically determined one. For deviatoric loadings, limit-analysis is dropped and direct use is made of some general rigorous results for nonlinear composites derived by Ponte-Castaneda (1991), Willis (1991) and Michel and Suquet (1992) using the earlier work of Willis (1977) and the concept of “linear comparison material”. This hybrid approach is thought to lead to the best possible expressions of the yield criterion parameters. The criterion proposed reduces to (variants of) classical approximate criteria proposed by Gurson (1977) and Gologanu et al., 1993, Gologanu et al., 1994, Gologanu et al., 1997 in the specific cases of spherical or spheroidal, prolate or oblate cavities. An overview of the validation of this criterion through micromechanical finite element computations is finally presented.     

A physically motivated anisotropic tensorial representation of damage with separate functions for void nucleation,growth, and coalescence

A phenomenological anisotropic damage progression formulation for porous ductile metals with second phases is described through mechanisms of void nucleation, growth and coalescence. The model is motivated from fracture mechanisms and microscale physical observations. To describe the creation of new pores, the decohesion at the particle–matrix interface and the fragmentation of second phase particles, the void-crack nucleation equation is related to several microstructural parameters (fracture toughness, length scale parameter, particle size, volume and fraction of second phase), the plastic strain level, and the stress state. Nucleation is represented by a general symmetric second rank tensor, and its components are proportional to the absolute value of the plastic strain rate components. Based on the Rice and Tracey model, void growth is a scalar function of the trace of damage tensor and the positive triaxiality. Like nucleation, coalescence is a second rank tensor governed by the plastic strain rate tensor and the stress state. The coalescence threshold is related to the void length scale for void impingement and void sheet mechanisms. The coupling of damage with the Bammann–Chiesa–Johnson (BCJ) plasticity model is written in the thermodynamic framework and derives from the concept of effective stress assuming the hypothesis of energy equivalence. A full-implicit algorithm is used for the stress integration and the determination of the consistent tangent operator. Finally, macroscale correlations to cast A356 AL alloy and wrought 6061-T6 AL alloy experimental data are completed with predictive void-crack evolution to illustrate the applicability of the anisotropic damage model.     

A model reduction technique for laminated solids of revolution with a curved cross-section

This paper presents an extension of the numerical reduction method, which has been proposed in Lejeunes et al. (Arch Appl Mech, 76:311–326, 2006), for modeling curved laminated structures of revolution such as for instance rubber bearings. This method based on high-order finite elements is developed in the context of nearly incompressible hyperelastic behavior. The displacement is approximated with a sum of independent functions, leading to a separation of variables. Therefore, a one-dimensional finite element can be formulated, which represents a 3-dimensional solid in a general loading case. Comparisons with classical finite element models are provided and show the reliability of this model reduction. An important decrease in the model size and a greatly reduced computing time, compared to standard models, is observed.     

A constitutive model for plastically anisotropic solids with non-spherical voids

Plastic constitutive relations are derived for a class of anisotropic porous materials consisting of coaxial spheroidal voids, arbitrarily oriented relative to the embedding orthotropic matrix. The derivations are based on nonlinear homogenization, limit analysis and micromechanics. A variational principle is formulated for the yield criterion of the effective medium and specialized to a spheroidal representative volume element containing a confocal spheroidal void and subjected to uniform boundary deformation. To obtain closed form equations for the effective yield locus, approximations are introduced in the limit-analysis based on a restricted set of admissible microscopic velocity fields. Evolution laws are also derived for the microstructure, defined in terms of void volume fraction, aspect ratio and orientation, using material incompressibility and Eshelby-like concentration tensors. The new yield criterion is an extension of the well known isotropic Gurson model. It also extends previous analyses of uncoupled effects of void shape and material anisotropy on the effective plastic behavior of solids containing voids. Preliminary comparisons with finite element calculations of voided cells show that the model captures non-trivial effects of anisotropy heretofore not picked up by void growth models.     

A zonal noise prediction method for trailing-edge noise with a porous model

In this contribution, a hybrid zonal simulation tool with volumetric inflow turbulence forcing is applied to trailing-edge noise of a NACA0012 airfoil with and without a porous insert at representative Mach and Reynolds number of 0.1118 and 1.0 × 10⁶, respectively. The governing equations constitute the non-linear perturbation equations with viscous terms (i.e., the full Navier–Stokes equations), in which the porous material is modelled by a volume-averaged approach. Generic simulations with a single vortex passing the trailing edge revealed the expected noise reduction as well as an additional noise source. This new source originates from the turbulent flow passing the transition from solid to porous surface and was shown to increase significantly with increasing permeability. 3D simulations with a solid and porous trailing-edge showed good agreement with experimental aerodynamic and aeroacoustic validation data. The application of porous material to trailing-edge noise confirms the results reported in literature and underlines the validity of the porous model as well as it illustrates possible applications.     

A semi-implicit finite volume scheme for a simplified hydrostatic model for fluid-structure interaction

Simulating fluid-structure interaction problems usually requires a considerable computational effort. In this article, a novel semi-implicit finite volume scheme is developed for the coupled solution of free surface shallow water flow and the movement of one or more floating rigid structures. The model is well-suited for geophysical flows, as it is based on the hydrostatic pressure assumption and the shallow water equations. The coupling is achieved via a nonlinear volume function in the mass conservation equation that depends on the coordinates of the floating structures. Furthermore, the nonlinear volume function allows for the simultaneous existence of wet, dry and pressurized cells in the computational domain. The resulting mildly nonlinear pressure system is solved using a nested Newton method. The accuracy of the volume computation is improved by using a subgrid, and time accuracy is increased via the application of the theta method. Additionally, mass is always conserved to machine precision. At each time step, the volume function is updated in each cell according to the position of the floating objects, whose dynamics is computed by solving a set of ordinary differential equations for their six degrees of freedom. The simulated moving objects may for example represent ships, and the forces considered here are simply gravity and the hydrostatic pressure on the hull. For a set of test cases, the model has been applied and compared with available exact solutions to verify the correctness and accuracy of the proposed algorithm. The model is able to treat fluid-structure interaction in the context of hydrostatic geophysical free surface flows in an efficient and flexible way, and the employed nested Newton method rapidly converges to a solution. The proposed algorithm may be useful for hydraulic engineering, such as for the simulation of ships moving in inland waterways and coastal regions.     

Plastic potential and yield function of porous materials with aligned and randomly oriented spheroidal voids

Based on an energy approach, the plastic potential and yield function of a porous material containing either aligned or randomly oriented spheroidal voids are developed at a given porosity and pore shape. The theory is applicable to both elastically compressible and incompressible matrix and, it is proved that, in the incompressible case, the theory with spherical and aligned spheroidal voids also coincides with Ponte Castaneda's bounds of the Hashin-Shtrikman and Willis types, respectively. Comparison is also made between the present theory and those of Gurson and Tvergaard, with a result giving strong overall support of this new development. For the influence of pore shape, the yield function and therefore the stress-strain curve of the isotropic porous material are found to be stiffest when the voids are spherical, and those associated with other pore shapes all fall below these values, the weakest one being caused by the disc-shaped voids. The transversely isotropic nature of the yield function and stress-strain curves of a porous material containing aligned pores are also demonstrated as a function of porosity and pore shape, and it is further substantiated with a comparison with an exact, local analysis when the void shape becomes cylindrical.     

A linear dynamic model for a saturated porous medium

A linear isothermal dynamic model for a porous medium saturated by a Newtonian fluid is developed in the paper. In contrast to the mixture theory, the assumption of phase separation is avoided by introducing a single constitutive energy function for the porous medium. An important advantage of the proposed model is it can account for the couplings between the solid skeleton and the pore fluid. The mass and momentum balance equations are obtained according to the generalized mixture theory. Constitutive relations for the stress, the pore pressure are derived from the total free energy accounting for inter-phase interaction. In order to describe the momentum interaction between the fluid and the solid, a frequency independent Biot-type drag force model is introduced. A temporal variable porosity model with relaxation accounting for additional attenuation is introduced for the first time. The details of parameter estimation are discussed in the paper. It is demonstrated that all the material parameters in our model can be estimated from directly measurable phenomenological parameters. In terms of the equations of motion in the frequency domain, the wave velocities and the attenuations for the two P waves and one S wave are calculated. The influences of the porosity relaxation coefficient on the velocities and attenuation coefficients of the three waves of the porous medium are discussed in a numerical example.     

A finite strain constitutive model for metal powder compaction using a unique and convex single surface yield function

Constitutive modelling of metal powder compaction processes is a challenge in view of realistic simulations. To this end, the article under consideration has two objectives: the first goal is to present a new unique and convex single surface yield function for pressure dependent materials, which is also applicable to other areas of granular materials such as soils or concrete. The flexibility is shown at various materials. The yield function is based on a log-interpolation of two known simple yield functions. A convexity proof of the new yield function is provided. The second objective is to propose a new rate-independent finite strain plasticity model for metal powder compaction, which is based on the multiplicative decomposition of the deformation gradient into an elastic and a plastic part with evolution equations for internal variables representing the basic behaviour of powder materials under compaction conditions. These variables are used for the evolution of the yield function in order to represent the compressible hardening behaviour of powder materials. On the basis of the constitutive model, the material parameters are identified at experimental data of copper powder.     

A unified approach to quasi-static shakedown problems for elastic-plastic solids with piecewise linear yield surface

Summary The paper concerns shakedown analysis of elastic-plastic bodies subjected to quasi-statically varying loads within a given domain. Using a perturbation method, a general inequality is given, from which, by simply specializing the perturbing terms, the generalized Melan theorem as well as bounds on various deformation parameters (such as displacements or plastic strain intensities) are derived. The solution of the «perturbed» shakedown problem in finite or holonimic terms permits the bound to be the most stringent and expressible in «local» terms instead of integral terms. A simple application concludes the paper.

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Sommario La memoria considera problemi di analisi a shakedown (o adattamento) relativamente a solidi elastoplastici sottoposti a carichi quasi-statici i quali variano restando all'interno di un dato dominio. Mediante l'uso di un metodo di perturbazione si fornisce una disuguaglianza generale dalla quale — particolarizzando opportunamente i termini della perturbazione — si deducono il teorema di adattamento di Melan nonché delimitazioni a priori su alcuni parametri della deformazione (come spostamenti o intensità delle deformazioni plastiche). La soluzione del problema perturbato, espressa in termini olonomi, consente di rendere tali delimitazioni stringenti al massimo possibile ed esprimibili in termini locali anziché in termini integrali. Una semplice applicazione conclude il lavoro.

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The results presented in this paper were obtained in the course of a research project sponsored by the National (Italian) Research Council, C.N.R., PAdIS Committee.