In this paper the finite element method is used to explore the mechanics of the microindentation process. In the simulations discussed, aluminum and silicon are investigated both in their bulk forms and in thin film-substrate combinations. Among the quantities readily computed using this approach and given in this paper are hardness (computed using actual contact area), contact stiffness, effective composite modulus, and surface profile under load. Importantly, this investigation builds on previous work by providing a more critical examination of the amount of pileup (or sink-in) around the indenter in the fully loaded configuration, as well as the variation of the actual contact area during indenter withdrawal. A key conclusion of this study is that finite element simulations do not support the widely used assumption of constancy of area during unloading (for either bulk materials or thin film systems). Furthermore, the amount of pileup or sink-in can be appreciable. The implication of these findings is that for many situations the commonly used straight-line extrapolation of a plastic depth may render an estimate for the contact area that is quite distinct from the actual area. This assertion is demonstrated herein through comparison of hardnesses calculated using actual contact area with values calculated using the straight-line extrapolation of plastic depth.

Multi-invariants dependent isotropic yield function in terms of first invariant I1 of stress tensor, second J2 and third J3 invariants of deviatoric stress tensor is required for accurate description of plasticity of polycrystalline solids. In the context of multi-invariants dependent multiplicative hyperelasto-plasticity, two integration schemes (IS-1 and IS-2) are formulated and their computational costs are compared for the first time in this paper. IS-1 is fully-implicit in which backward Euler difference approximation of equivalent plastic strain rate is coupled with the exponential approximation of return mapping in principal Kirchhoff stress space and accordingly principal Kirchhoff stresses, equivalent plastic strain and plastic Lagrange multiplier need to be updated iteratively. IS-2 is semi-implicit in which plastic Lagrange multiplier evaluated at current time step and semi-implicit forward Euler difference approximation of equivalent plastic strain rate is coupled with logarithmic approximation of return mapping in Kirchhoff stress space and accordingly only plastic Lagrange multiplier needs to be updated iteratively. For J2 dependent plasticity, computation time involved in numerical simulations for both the integration schemes is nearly the same for same accuracy. For IS-1, computation time involved in simulation for multi-invariants dependent plasticity is significantly greater than that for J2 dependent plasticity due to the computation of the Jacobian matrix in return mapping involving lengthy terms of second order partial derivatives of multi-invariants dependent yield function. For IS-2, computation time involved in simulation for multi-invariants dependent plasticity is nearly the same as that for J2 dependent plasticity. This is due to the fact that the computation of the Jacobian matrix in IS-2 is not required and return mapping involves only first partial derivatives of yield function. Thus, IS-2 is computationally cost effective than IS-1.

Over 100 years ago, Wolff hypothesized that cancellous bone altered both its apparent density and trabecular orientation in response to mechanical loads. A mathematical counterpart of this principle is derived by adding a remodeling rule for the rate-of-change of the full anisotropic stiffness tensor (all 21 independent terms) to the density rate-of-change rule adapted from an existing isotropic theory. As a result, anisotropy and density patterns develop such that the local stiffness tensor is optimal for the given series of applied loadings. The method does not rely on additional morphological measures of trabecular orientation. Furthermore, assumptions of material symmetry are not required, and any observed regions of orthotropy, transverse isotropy, or isotropy are a result entirely of the functional adaptation of the bone and not the consequence of a modeling assumption. This approach has been implemented with the finite element method and applied to a two-dimensional model of the proximal femur with encouraging results.

This paper is concerned with the continuum formulation of a fully coupled thermomechanical constitutive model for highly deformable bodies with viscous dissipation. The threedimensional material law is applicable to the thermomechanical characterization of elastomeric (high-polymeric) solids in the finite strain domain under varying temperature. The description is based on the concept of internal state variables and rational thermodynamics. The main goal of the presentation is to develop consistent constitutive equations for the stress, entropy and independent internal variables such that the second law of thermodynamics, in the form of the Clausius-Duhem inequality, issatisfied. Motivated by the significant difference in bulk and shear response of elastomers, the model employs a local decomposition of the deformation into a dilatational and an isochoric part. The framework of nonlinear thermoviscoelasticity presented herein is formulated entirely in the reference configuration and provides a sound continuum basis for approximation techniques such as the Finite-Element method.Copyright © 1996 Elsevier Science Ltd.

This paper examines the long-term dissipativity and unconditional non-linear stability of time integration algorithms for the incompressible MHD equations. The results apply as a particular case to the uncoupled incompressible Navier-Stokes equations. The analysis is pet-formed in the setting defined by an abstract evolution equation possessing the same long-term dissipative structure as the above physical systems. A general class of rime-stepping schemes is analyzed in this context, showing the non-linear unconditional stability and long-term dissipativity of particular members of this family of algorithms. In particular, algorithms showing these strong properties which additionally involve the solution of a linear problem at each time step are identified. A particular linear algorithm is shown to exhibit optimal long-term dissipative properties and to possess an algorithmic global attractor. These abstract results apply rigorously to Galerkin mixed finite element discretizations of the problem. To this end, it is shown that such Galerkin-type projection inherits the dissipative properties of the continuum problem. Computational aspects involved in the numerical implementation of these methods are addressed and illustrated in representative numerical simulations.

This paper presents a detailed comparison of two implicit time integration schemes for a simple non-linear Hamiltonian system with symmetry: the motion of a particle in a central force field. The goal is to establish analytical and numerical results pertaining to the stability properties of the implicit mid-point rule (the proto-typical implicit symplectic method) and a particular energy-momentum conserving scheme, and to compare the two schemes with respect to accuracy. While all results presented herein are within the context of a simple model problem, the problem was constructed so as to exhibit key features typical of more complex systems with symmetry such as those arising in non-linear solid mechanics: namely, the presence of large (and relatively slow) overall motions together with high-frequency internal motions.

This paper presents a macroscopic continuum formulation and a numerical analysis of constitutive equations (for stress, entropy and heat flux) describing the thermoelastic behavior of amorphous cross-linked polymers above the glass transition temperature in which a specimen typically 'snap-back' with rubbery characteristics.The introduced coupled thermomechanical functional extends the classical strain energy function proposed by Ogden. Volume changes due to thermal expansion are regarded for a large temperature domain and play a remarkable role in thermoelastic materials.The non-linear thermoelastic problem is solved within a staggered (fractional-step) method which is based on a two-phase (isentropic) operator split. Each of the two symmetric sub-problems retains the characteristic dissipative structure of the implicit monolithic scheme which consequently results to an unconditionally stable product formula algorithm.The numerical analysis side shows in particular the capability of the theoretical framework reproducing the realistic physical stress-deformation-temperature relations of rubber. Distinctive attention is paid to the thermoelastic inversion phenomena, a remarkable property governing the class of rubber-like materials.

The present paper addresses modeling and numerical description of stress-induced phase transformations in solids. The study is developed in the setting of a generalized neo-Hookean elastic material under anti-plane deformations. It is assumed that the kinetics of phase transformation is governed by inequality constraints which lead to a dissipative behavior of the material. A thermodynamic analysis motivates the assumption of stress continuity across the surface of separation between phases. We propose a new finite element for the description of phase boundaries (characterized by strain discontinuities) in any position within the element. The inequality constraints of the problem are enforced by means of a return mapping algorithm. Dendritic formations are represented in the numerical results corresponding to the resulting model.

The long-term dynamic response of non-linear geometrically exact rods under-going finite extension, shear and bending, accompanied by large overall motions, is addressed in detail. The central objective is the design of unconditionally stable time-stepping algorithms which exactly preserve fundamental constants of the motion such as the total linear momentum, the total angular momentum and, for the Hamiltonian case, the total energy. This objective is accomplished in two steps. First, a class of algorithms is introduced which conserves linear and angular momentum. This result holds independently of the definition of the algorithmic stress resultants. Second, an algorithmic counterpart of the elastic constitutive equations is developed such that the law of conservation of total energy is exactly preserved. Conventional schemes exhibiting no numerical dissipation, symplectic algorithms in particular, are shown to lead to unstable solutions when the high frequencies are not resolved. Compared to conventional schemes there is little, if any, additional computational cost involved in the proposed class of energy-momentum methods. The excellent performance of the new algorithm in comparison to other standard schemes is demonstrated in several numerical simulations.

A framework for damage mechanics of brittle solids is presented and exploited in the design and numerical implementation of an anisotropic model for the tensile failure of concrete. The key feature exploited in the analysis is the hypothesis of maximum dissipation, which specifies a unique damage rule for the elastic moduli of the solid once a failure surface is specified. A complete algorithmic treatment of the resulting model is given which renders a useful tool for large-scale inelastic finite element calculations. A rather simple three-surface failure model for concrete, containing essentially no adjustable parameters, is shown to produce results in remarkably good agreement with sample experimental data.

A stability and convergence analysis is presented of a recently proposed variational formulation and finite element method for elasticity, which incorporates an enhanced strain field. The analysis is carried out for problems posed on polygonal domains in R(n), the finite element meshes of which are generated by affine maps from a master element. The formulation incorporates as a special case the classical method of incompatible modes. The problem initially has three variables, viz. displacement, stress, and enhanced strain, but the stress is later eliminated by imposing a condition of orthogonality with respect to the enhanced strains. Two other conditions on the choice of finite element spaces ensure that the approximations are stable and convergent. Some features of nearly incompressible and incompressible problems are also investigated. For these cases it is possible to argue that locking will not occur, and that the only spurious pressures present are the so-called checkerboard modes. It is shown that, as in the case of the Q(1) - P-0 element, the displacement and enhanced strain are convergent, and so is the pressure, after filtering out this mode.

This paper examines a new Galerkin method with scaled bubble functions which replicates the exact artificial diffusion methods in the case of I-D scalar advection-diffusion and that leads to non-oscillatory solutions; as the streamline upwinding algorithms for 2-D scalar advection-diffusion and incompressible Navier-Stokes. This method retains the satisfaction of the Babuska-Brezzi condition and, thus, leads to optimal performance in the incompressible limit: This method, when, combined with the recently-proposed linear unconditionally stable algorithms of Simo and Armero (1993), yields a method for solution of the incompressible Navier-Stokes equations ideal for either diffusive or advection-dominated flows. Examples from scalar advection-diffusion and the solution of the incompressible Navier-Stokes equations are presented.

Long bone structure occurs in two distinct forms. The bone mass near the joint is primarily found in a distributed, porous trabecular structure, while in the diaphyses a tubular cortical structure is formed. It seems likely that these two observed morphologies come about, at least in part, as a mechanical adaptation to the different mechanical demands in the two regions, Mathematical formulations of this dependency have been proposed, thus facilitating numerical simulations of bone adaptation. Recently two types of discontinuities have been observed in these simulations. The first type (near-field) appears in areas near distributed load application and is characterized by a 'checkerboard' pattern of density wherein adjacent remodeled elements alternate between low and high density. The second type of discontinuity (far-field) appears remote from the load application and is characterized by strut or column-like regions of elements which become fully compact bone while adjacent regions are fully resorbed. In fact, the far-field discontinuity is an accurate representation of bone physiology and morphology since it is consistent with the appearance of cortical bone in the diaphysis. On the other hand, the near-field discontinuity, appears in a region where continuous distributions of intermediate apparent densities (trabecular bone) are expected. This finding may cause some to question whether a single continuum formulation of bone remodeling can predict both discontinuous far-field behavior and continuous near-field behavior. We describe a node-based implementation of current continuum bone remodeling theories which eliminates the spurious near-field discontinuities and preserves the anatomically correct far-field discontinuities, thus indicating that a single biological process may be at work in forming and maintaining both far-field and near-field morphologies.

Three conservation laws are associated with the dynamics of Hamiltonian systems with symmetry: The total energy, the momentum map associated with the symmetry group, and the symplectic structure are invariant under the flow. Discrete time approximations of Hamiltonian flows typically do not share these properties unless specifically designed to do so. We develop explicit conservation conditions for a general class of algorithms on Lie groups. For the rigid body these conditions lead to a single-step algorithm that exactly preserves the energy, spatial momentum, and symplectic form. For homogeneous nonlinear elasticity, we find algorithms that conserve angular momentum and either the energy or the symplectic form.

This paper examines the long-term behavior, dissipativity and unconditional nonlinear stability properties of time integration algorithms for the incompressible Navier-Stokes equations, including both direct schemes and fractional step/projection methods. The algorithms are termed nonlinear unconditionally stable if, in the absence of a forcing term but for arbitrary initial conditions, the computed kinetic energy decreases for arbitrary step sizes (i.e., L2-stability). Such an a priori stability estimate is a characteristic feature of the Navier-Stokes system. Similarly, the algorithms are said to exhibit asymptotic δt-independent long-term dissipative behavior if, as the continuum flow generated by the Navier-Stokes equations, the algorithmic flow also exhibits an absorbing set (and a maximal attractor). Both direct and fractional step methods are described which inherit these fundamental properties of the Navier-Stokes system for any time step size. Moreover, a subclass of algorithms which retain these strong notions of nonlinear stability and long-term dissipative behavior is identified which, in addition, has the remarkable property of being linear within the time step. The key implication is that no iterations are required to compute the solution within a time step. A specific member of this class is shown to possess the same absorbing set as the continuum Navier-Stokes system. A second order accurate, unconditionally stable method is also identified which retains the property of linearity within a time step. Numerical analysis and computational aspects involved in the implementation of these methods are addressed in detail and illustrated in representative numerical simulations.

An algorithm designed for the determination of equilibrium shocks that appear in quasi-static evolution problems associated to elastic nonmonotonous stress-strain laws is presented in the context of one-dimensional media. Two basic procedures are involved in the proposed method: (i) enhancement of the finite element in order to describe the weak discontinuities in any point of its interior and (ii) implementation of a return mapping algorithm for the determination of the shocks, which have to satisfy the inequality constraints imposed by a maximally dissipative hypothesis. A rigorous proof of the unconditional stability property of the algorithm is also given. The present study is applied to the theoretical model presented by Abeyaratne and Knowles in the context of one-dimensional extensional deformations of bars. The numerical results are in complete agreement with the analytical ones.