WYPiWYG Damage Mechanics for Soft Materials: A Data-Driven Approach

The conservative elastic behavior of soft materials is characterized by a stored energy function which shape is usually specified a priori, except for some material parameters. There are hundreds of proposed stored energies in the literature for different materials. The stored energy function may change under loading due to damage effects, but it may be considered constant during unloading–reloading. The two dominant approaches in the literature to model this damage effect are based either on the Continuum Damage Mechanics framework or on the Pseudoelasticity framework. In both cases, additional assumed evolution functions, with their associated material parameters, are proposed. These proposals are semi-inverse, semi-analytical, model-driven and data-adjusted ones. We propose an alternative which may be considered a non-inverse, numerical, model-free, data-driven, approach. We call this approach WYPiWYG constitutive modeling. We do not assume global functions nor material parameters, but just solve numerically the differential equations of a set of tests that completely define the behavior of the solid under the given assumptions. In this work we extend the approach to model isotropic and anisotropic damage in soft materials. We obtain numerically the damage evolution from experimental tests. The theory can be used for both hard and soft materials, and the infinitesimal formulation is naturally recovered for infinitesimal strains. In fact, we motivate the formulation in a one-dimensional infinitesimal framework and we show that the concepts are immediately applicable to soft materials.     

Efficient numerical integration of an elastic–plastic damage law within a mixed velocity–pressure formulation

This study focuses on numerical integration of constitutive laws in numerical modeling of cold materials processing that involves large plastic strain together with ductile damage. A mixed velocity–pressure formulation is used to handle the incompressibility of plastic deformation. A Lemaitre damage model where dissipative phenomena are coupled is considered. Numerical aspects of the constitutive equations are addressed in detail. Three integration algorithms with different levels of coupling of damage with elastic–plastic behavior are presented and discussed in terms of accuracy and computational cost. The implicit gradient formulation with a non-local damage variable is used to regularize the localization phenomenon and thus to ensure the objectivity of numerical results for damage prediction problems. A tensile test on a plane plate specimen, where damage and plastic strain tend to localize in well-known shear bands, successfully shows both the objectivity and effectiveness of the developed approach.     

Interval Finite Elements as a Basis for Generalized Models of Uncertainty in Engineering Mechanics

Latest scientific and engineering advances have started to recognize the need for defining multiple types of uncertainty. Probabilistic modeling cannot handle situations with incomplete or little information on which to evaluate a probability, or when that information is nonspecific, ambiguous, or conflicting [12], [47], [50]. Many interval-based uncertainty models have been developed to treat such situations. This paper presents an interval approach for the treatment of parameter uncertainty for linear static structural mechanics problems. Uncertain parameters are introduced in the form of unknown but bounded quantities (intervals). Interval analysis is applied to the Finite Element Method (FEM) to analyze the system response due to uncertain stiffness and loading. To avoid overestimation, the formulation is based on an element-by-element (EBE) technique. Element matrices are formulated, based on the physics of materials, and the Lagrange multiplier method is applied to impose the necessary constraints for compatibility and equilibrium. Earlier EBE formulation provided sharp bounds only on displacements [32]. Based on the developed formulation, the bounds on the system’s displacements and element forces are obtained simultaneously and have the same level of accuracy. Very sharp enclosures for the exact system responses are obtained. A number of numerical examples are introduced, and scalability is illustrated.     

Strong discontinuities in partially saturated poroplastic solids

This paper considers the analysis and numerical simulation of strong discontinuities in partially saturated solids. The goal is to study observed localized failures in such media like shear bands and similar. The developments consider a fully coupled partially saturated elastoplastic model for the (continuum) bulk response of the solid formulated in effective stresses, identifying the necessary mathematical conditions for the appearance of strong discontinuities (that is, discontinuities in the displacement field leading to singular strains) as well as the proper treatment for the fields characterizing the flow of the different fluid phases, namely, the fluid contents of these phases and their individual pore pressures. The geometrically linear range of infinitesimal strains is considered. These developments allow the formulation of multiphase cohesive laws along the strong discontinuity, capturing in this way the coupled localized dissipation observed in the aforementioned failures. Furthermore, the paper also presents the formulation of enhanced finite elements capturing all these discontinuous solutions in general unstructured meshes. In particular, the finite elements capture the strong discontinuity through the proper enhancements of the discrete element strains, allowing for a complete local resolution of these effects. This results in a particularly efficient computational approach, easily accommodated in an existing finite element code. Different representative numerical simulations are presented illustrating the performance of the proposed formulation, as well as its use in practical applications like the modeling of the excavation of tunnels in variably saturated media.     

A nine-node assumed-strain finite element for composite plates and shells

A new finite element for modeling fiber-reinforced composite plates and shells is developed and its performance for static linear problems is evaluated. The element is a nine-node degenerate solid shell element based on a modified Hellinger-Reissner principle with independent inplane and transverse shear strains. Several numerical examples are solved and the solutions are compared with other available finite solutions and with classical lamination theory. The results show that the present element yields accurate solutions for the test problems presented. Convergence characteristics are good, and the solution is relatively insensitive in element distortion. The element is also shown to be free of locking even for thin laminates.     

Viscoplasticity based on additive decomposition of logarithmic strain and unified constitutive equations: Theoretical and computational considerations with reference to shell applications

In contrast to multiplicative models of finite strain plasticity and viscoplasticity, a framework of additive nature is developed in this paper. The theory is based on the additive decomposition of the logarithmic strain tensor. The stress conjugate to the logarithmic strain then plays the role of the thermodynamically driving force. The approach in this paper is motivated by the search for numerically accessible structures which can be extended to incorporate anisotropy as well. Specifically in this region, multiplicative formulations become extremely tedious. The evolution equations are of the unified type due to Bodner and Partom, and are modified so as to fit into the theoretical framework adopted. The numerical treatment of the problem is fully developed. Specifically, the algorithmic aspects of the approach are discussed and various applications to shell problems are considered. A shell theory with seven degrees of freedom, together with a four-node enhanced strain finite element formulation, is used. A central feature of the shell formulation is its eligibility to the application of a three-dimensional constitutive law.     

Topology optimization of structures with gradient elastic material

Topology optimization of structures and mechanisms with microstructural length-scale effect is investigated based on gradient elasticity theory. To meet the higher-order continuity requirement in gradient elasticity theory, Hermite finite elements are used in the finite element implementation. As an alternative to the gradient elasticity, the staggered gradient elasticity that requires C ⁰-continuity, is also presented. The solid isotropic material with penalization (SIMP) like material interpolation schemes are adopted to connect the element density with the constitutive parameters of the gradient elastic solid. The effectiveness of the proposed formulations is demonstrated via numerical examples, where remarkable length-scale effects can be found in the optimized topologies of gradient elastic solids as compared with linear elastic solids.     

Multiscale computational analysis in mechanics using finite calculus: an introduction

The paper introduces a general procedure for computational analysis of a wide class of multiscale problems in mechanics using a finite calculus (FIC) formulation. The FIC approach is based in expressing the governing equations in mechanics accepting that the domain where the standard balance laws are established has a finite size. This introduces naturally additional terms into the classical equations of infinitesimal theory in mechanics which are useful for the numerical solution of problems involving different scales in the physical parameters. The discrete nodal values obtained with the FIC formulation and the finite element method (FEM) can be effectively used as the starting point for obtaining a more refined solution in zones where high gradients of the relevant variables occur using hierarchical or enriched FEM. Typical multiscale problems in mechanics which can be solved with the FIC method include convection–diffusion-reaction problems with high localized gradients, incompressible problems in solid and fluid mechanics, localization problems such as prediction of shear bands in solids and shock waves in compressible fluids, turbulence, etc. The paper presents an introduction of the treatment of multiscale problems using the FIC approach in conjunction with the FEM. Examples of application of the FIC/FEM formulation to the solution of simple multiscale convection–diffusion problems are given.     

A numerical method for non-linear water waves

A numerical method based on a complex potential formulation is developed for the computation on non-linear waves. It is applied to solve the problem of waves generated by a submerged doublet. Results for the dominant steady wave trains are presented. Good agreement with non-linear theory is obtained.     

An efficient algorithm for finite-difference modeling of mixed-boundary-value elastic problems

The paper describes an efficient numerical scheme for the solution of displacements and stresses in mixed-boundary-value elastic problems of solid mechanics. A new variable reduction scheme is used to develop the computational model. Solution of both two- and three-dimensional problems of linear elasticity is considered in the present paper. In the present approach, the problems are formulated in terms of a potential function, defined in terms of the displacement components. Compared to the conventional computational approaches, the present scheme is capable of providing numerical solution of higher accuracy with less computational effort. Application of the present finite-difference modeling scheme is demonstrated through the solutions of a number of practical stress problems of interest, and the results are compared with those obtained by the standard method of solution. The comparison of the results establishes the rationality as well as suitability of the present variable reduction scheme.     

Application of the concept of evolving structure tensors to the modeling of initial and induced anisotropy at large deformation

In this work, a thermodynamic approach to the modeling and simulation of induced elastic and inelastic material behaviour in the phenomenological realm as based on the concept of evolving structure tensors is discussed. From the constitutive point of view, these quantities determine the material symmetry properties. In addition, the stress and other dependent constitutive fields are isotropic functions of these by definition. The evolution of these during loading then results in an evolution of the anisotropy of the material. From an algorithmic point of view, the current approach leads to constitutive models which are quite amenable to numerical implementation. To demonstrate the applicability of the resulting constitutive formulation, we apply it to the cases of (i) metal plasticity with combined hardening involving both deformation- and permanently induced anisotropy relevant to the modeling of processes such as metal forming, and to (ii) deformation-induced anisotropy in an initially orthotropic pneumatic membrane consisting of a rubber matrix and nylon fibres.     

Modeling shear-induced diffusion force in particulate flows

In this paper, a novel approach is presented to mechanistically model the dispersion of solid particles due to shear-induced diffusion effect. The model is based on the interfacial force concept, so that it is compatible with the ensemble averaged multifield model of multicomponent flows. Although the new model has been developed with membrane filtration processes in mind, the proposed shear-induced diffusion force formulation can be utilized to model a variety of industrial processes where particulates are present. The model has been verified against the other phenomenological models currently in use.The first part of this paper is concerned with theoretical aspects of model derivation. Then, a numerical analysis is presented to illustrate the application of the new model to dilute liquid/particle two-phase flows in tube membrane systems similar to those that are used for micro- and nano-filtration processes. The numerical simulations have been performed using a state of the art multiphase/multicomponent CFD code, NPHASE [Antal SP, Ettorre SM, Kunz RF, Podowski MZ. Development of a next generation computer code for the prediction of multicomponent multiphase flows. In: International meeting on trends in numerical and physical modeling for industrial multiphase flow, Cargese, France; 2000; Kunz RF, Yu WS, Antal SP, Ettorre SM. An unstructured two-fluid method based on the coupled phasic exchange algorithm. AIAA Paper 2001; 2001:2672]. The numerical consistency of the results of computer calculations have been verified against simplified semi-analytical approximations.     

Constitutive and Geometric Nonlinear Models for the Seismic Analysis of RC Structures with Energy Dissipators

Nowadays, the use of energy dissipating devices to improve the seismic response of RC structures constitutes a mature branch of the innovative procedures in earthquake engineering. However, even though the benefits derived from this technique are well known and widely accepted, the numerical methods for the simulation of the nonlinear seismic response of RC structures with passive control devices is a field in which new developments are continuously preformed both in computational mechanics and earthquake engineering. In this work, a state of the art of the advanced models for the numerical simulation of the nonlinear dynamic response of RC structures with passive energy dissipating devices subjected to seismic loading is made. The most commonly used passive energy dissipating devices are described, together with their dissipative mechanisms as well as with the numerical procedures used in modeling RC structures provided with such devices. The most important approaches for the formulation of beam models for RC structures are reviewed, with emphasis on the theory and numerics of formulations that consider both geometric and constitutive sources on nonlinearity. In the same manner, a more complete treatment is given to the constitutive nonlinearity in the context of fiber-like approaches including the corresponding cross sectional analysis. Special attention is paid to the use of damage indices able of estimating the remaining load carrying capacity of structures after a seismic action. Finally, nonlinear constitutive and geometric formulations for RC beam elements are examined, together with energy dissipating devices formulated as simpler beams with adequate constitutive laws. Numerical examples allow to illustrate the capacities of the presented formulations.     

A multi-director formulation for nonlinear elastic-viscoelastic layered shells

A C⁰ finite element formulation for nonlinear analysis of multi-layered shells comprised of elastic and viscoelastic layers is presented for applications involving small strains but finite rotations. The elastic and viscoelastic layers may occupy arbitrary layer locations and the formulation is applicable to thick and thin shells. The formulation utilizes a three-dimensional variational approach in which the layered shell is represented as a multi-director field. The incorporated kinematic theory describes, within individual layers, the effects of transverse shear and transverse normal strain to arbitrary orders in the layer thickness coordinate. Stresses are computed through the three-dimensional constitutive equations and the usual “zero normal stress” shell hypothesis is not employed. Sufficiently general constitutive equations for the viscoelastic layers are proposed in objective rate form and a product algorithm, based on an operator split in the complete set of constitutive equations, is used for the temporal integration of the rate equations. The definition of the tangent operator, used in Newton's method for the solution of the nonlinear equations, is derived consistently from the product algorithm. Observations on the use of reduced/selective integration in the presence of high order kinematics are made and a number of numerical examples are presented to illustrate the capability of the formulation.     

Recovery of consistent stresses for compatible finite elements

In displacement based finite element models, stresses deduced directly from the constitutive relationship can show local erratic behaviour. This occurs in problems involving initial stresses or strains, or varying rigidities over the element domain, when local stresses do not meet a specific consistency requirement. In this context, an integrated procedure for recovering consistent stresses, that is stresses ridded of spurious outcomes, is proposed. The procedure is developed within a general weighted residual approach, suitably specialized for the purpose. The relationship between the proposed procedure and those based on the Hu–Washizu formulation is also elucidated. For illustration purpose, some numerical tests are included.     

Implicit dynamic three-dimensional finite element analysis of an inelastic biphasic mixture at finite strain: Part 1: Application to a simple geomaterial

Based on the mixture theory formulation for a fluid-saturated, inelastic, pressure-sensitive porous solid subjected to dynamic large strain deformation, a 3D finite element implementation with implicit time integration is presented. A recently published 2D implementation [Li et al. 2004, CMAME, v193, p3837–70] is extended to 3D, porosity-dependent permeability, and pressure-sensitive inelastic solid skeleton response at finite strain. The Clausius-Duhem inequality provides the form of the constitutive equations for the solid and fluid phases, as well as the dissipation function. A non-associative Drucker-Prager cap-plasticity model at finite strain is formulated based on a multiplicative decomposition of the deformation gradient, and numerically integrated semi-implicitly in the intermediate configuration to avoid questions of incremental objectivity. The elastic implementation is verified with available 1D analytical and 2D benchmark problems. New numerical solutions for 3D large strain dynamic behavior of saturated inelastic porous media are presented. The computational efficiency of the implemented formulation in achieving quadratic convergence is illustrated.     

Damage and progressive failure of concrete structures using non-local peridynamic modeling

Peridynamics (PD), a recently developed theory of solid mechanics, which employs a non-local model of force interaction and makes use of integral formulation rather than the spatial partial differential equations used in the classical continuum mechanics theory, has shown effectiveness and promise in solving discontinuous problems at both macro and micro scales. In this paper, the peridynamics theory is used to analyze damage and progressive failure of concrete structures. A non-local peridynamic model for ...     

Nonlinear finite element analysis of shells-part II. two-dimensional shells

A general nonlinear finite element formulation is given for two-dimensional problems. The formulation applies to the practically important cases of shells of revolution, tubes, rings, beams and frames. The approach is deduced from a corresponding three-dimensional formulation [4] and this enables a simplified implementation, especially with respect to constitutive software. Uniform reduced-integration Lagrange elements are employed and shown to be very effective for the class of problems considered.     

Variational Framework for FIC Formulations in Continuum Mechanics: High Order Tensor-Derivative Transformations and Invariants

This is part of an article series on a variational framework for continuum mechanics based on the Finite Increment Calculus (FIC). The formulation utilizes high order derivatives of the classical fields of continuum mechanics integrated over control regions to construct stabilizing modification terms. Fields may include displacements, body forces, strains, stresses, pressure and volumetric strains. To support observer-invariant FIC formulations, we have catalogued field transformation equations as well as sets of linear and quadratic invariants of fields and of their derivatives up to appropriate order. Attention is focused on the two-dimensional case of a body in plane strain under the drilling-rotation transformation group. Results are presented for displacement and body-force derivatives of orders up to 4, and for stress, strain, pressure and volumetric strain derivatives of order up to 3. The material assembled here is self-contained because this catalog is believed to be useful beyond FIC applications; for example gradient-based, nonlocal constitutive models of multiscale mechanics and physics that involve finite characteristic dimensions analogous to FIC steplengths.