Explicit, implicit, and hybrid methods

Time integration methods can be separated into two groups: explicit and implicit. Methods which do not involve the solution of any algebraic equations are called explicit, while those that require the solution of equations are called implicit. The relative advantages and disadvantages of explicit and implicit methods are summarized. The major trend in the past decade of research was to use hybrization methods to take advantage of the complementary nature of the positive attributes of explicit and implicit integration. These trends are briefly discussed

On the stability of a class of implicit algorithms for nonlinear structural dynamics

Stability in energy for the Newmark beta-family of time integration operators for nonlinear material problems is examined. It is shown that the necessary and sufficient conditions for unconditional stability are equivalent to those predicted by Fourier methods for linear problems

Efficient linear and nonlinear heat conduction with a quadrilateral element

A method is presented for performing efficient and stable finite element calculations of heat conduction with quadrilaterals using one-point quadrature. The stability in space is obtained by using a stabilization matrix which is orthogonal to all linear fields and its magnitude is determined by a stabilization parameter. It is shown that the accuracy is almost independent of the value of the stabilization parameter over a wide range of values; in fact, the values 3, 2, and 1 for the normalized stabilization parameter lead to the 5-point, 9-point finite difference, and fully integrated finite element operators, respectively, for rectangular meshes and have identical rates of convergence in the L2 norm. Eigenvalues of the element matrices, which are needed for stability limits, are also given. Numerical applications are used to show that the method yields accurate solutions with large increases in efficiency, particularly in nonlinear problems

An adaptive continuum/discrete crack approach for meshfree particle methods

A coupled continuum/discrete crack model for strain softening materials is implemented in a meshfree particle code. A coupled damage plasticity constitutive law is applied until a certain strain based threshold value - this is at the maximum tensile stress of the equivalent uniaxial stress strain curve - is reached. At this point a discrete crack is introduced and described as an internal boundary with a traction crack opening relation. Within the frame-work of meshfree particle methods it is possible to model the transition from the continuum to the discrete crack since boundaries and particles can easily be added and removed. The EFG method and an explicit time integration scheme is used. The integrals are evaluated by nodal integration, an integration with stress points and also a full Gauss quadrature. Some results are compared to experimental data and show good agreement. Additional comparisons are made to a pure continuum constitutive law

Corotational velocity strain formulations for nonlinear analysis of beams and axisymmetric shells

Finite element formulations for large strain, large displacement problems are formulated using a kinematic description based on the corotational components of the velocity strain. The corotational components are defined in terms of a system that rotates with each element and approximates the rotation of the material. To account for rotations of the material relative to this element system, extra terms are introduced in the velocity strain equations. Although this formulation is incremental, in explicitly integrated transient problems it compares very well with formulations that are not

Probabilistic fracture finite elements

The Probabilistic Fracture Mechanics (PFM) is a promising method for estimating the fatigue life and inspection cycles for mechanical and structural components. The Probability Finite Element Method (PFEM), which is based on second moment analysis, has proved to be a promising, practical approach to handle problems with uncertainties. As the PFEM provides a powerful computational tool to determine first and second moment of random parameters, the second moment reliability method can be easily combined with PFEM to obtain measures of the reliability of the structural system. The method is also being applied to fatigue crack growth. Uncertainties in the material properties of advanced materials such as polycrystalline alloys, ceramics, and composites are commonly observed from experimental tests. This is mainly attributed to intrinsic microcracks, which are randomly distributed as a result of the applied load and the residual stress

Continuum mechanics modeling and simulation of carbon nanotubes

The understanding of the mechanics of atomistic systems greatly benefits from continuum mechanics. One appealing approach aims at deductively constructing continuum theories starting from models of the interatomic interactions. This viewpoint has become extremely popular with the quasicontinuum method. The application of these ideas to carbon nanotubes presents a peculiarity with respect to usual crystalline materials: their structure relies on a two-dimensional curved lattice. This renders the cornerstone of crystal elasticity, the Cauchy–Born rule, insufficient to describe the effect of curvature. We discuss the application of a theory which corrects this deficiency to the mechanics of carbon nanotubes (CNTs). We review recent developments of this theory, which include the study of the convergence characteristics of the proposed continuum models to the parent atomistic models, as well as large scale simulations based on this theory. The latter have unveiled the complex nonlinear elastic response of thick multiwalled carbon nanotubes (MWCNTs), with an anomalous elastic regime following an almost absent harmonic range

Nonlinear mechanical response and rippling of thick multi-walled carbon nanotubes

The measured drop of the effective bending stiffness of multiwalled carbon nanotubes (MWCNTs) with increasing diameter is investigated by a generalized local quasicontinuum method. The previous hypothesis that this reduction is due to a rippling mode is confirmed by the calculations. The observed ripples result from a complex three-dimensional deformation similar to theYoshimura buckling pattern. It is found that thick MWCNTs exhibit a well-defined nonlinear moment-curvature relation, even for small deformations, governed by the interplay of strain energy relaxation and intertube interactions. Rippling deformations are also predicted for MWCNTs subject to torsion, resulting in an effective torsional modulus much smaller than that predicted by linear elasticity

Finite element methods for the non-linear mechanics of crystalline sheets and nanotubes

The formulation and finite element implementation of a finite deformation continuum theory for the mechanics of crystalline sheets is described. This theory generalizes standard crystal elasticity to curved monolayer lattices by means of the exponential Cauchy–Born rule. The constitutive model for a two‐dimensional continuum deforming in three dimensions (a surface) is written explicitly in terms of the underlying atomistic model. The resulting hyper‐elastic potential depends on the stretch and the curvature of the surface, as well as on internal elastic variables describing the rearrangements of the crystal within the unit cell. Coarse grained calculations of carbon nanotubes (CNTs) are performed by discretizing this continuum mechanics theory by finite elements. A smooth discrete representation of the surface is required, and subdivision finite elements, proposed for thin‐shell analysis, are used. A detailed set of numerical experiments, in which the continuum/finite element solutions are compared to the corresponding full atomistic calculations of CNTs, involving very large deformations and geometric instabilities, demonstrates the accuracy of the proposed approach. Simulations for large multi‐million systems illustrate the computational savings which can be achieve