Linear Continuum Mechanics Research Articles

A novel continuum approach to gradient plasticity based on the complementing concepts of dislocation and disequilibrium densities

A geometrically linear continuum mechanics framework is proposed for gradient plasticity combining ‘strain gradients’ and, with a novel approach, ‘stress gradients’. Thereby the duality of kinematic and kinetic quantities is exploited in view of the ‘div-grad-curl orthogonality’ in continuum field theories. On the one hand the non-integrability of the plastic distortion results in the well-established dislocation density - often denoted as the geometrically-necessary-dislocation (GND) density - that enters the energy storage function. On the other hand - as entirely novel concept introduced in this contribution - the non-equilibrium of the plastic stress results in the disequilibrium density that parameterizes the dual dissipation potential within the convex analysis setting of plasticity. Consequently both, the dislocation density as well as the disequilibrium density contribute in modelling the size-dependent hardening state of a material in a continuum mechanics setting. The novel approach is eventually elucidated in much detail for the specific case of single crystal plasticity.

Axial buckling behavior of single-walled carbon nanotubes: Atomistic structural instability analysis

To reveal the atomistic scale mechanism of buckling of single-walled carbon nanotubes (SWCNTs) under axial compression, we carried out molecular dynamics (MD) simulation and atomistic structural instability (ASI) analysis. The ASI analysis enables us to reveal the development of ‘latent’ instability modes until buckling of structure, which cannot be clarified with MD alone. Euler-type buckling was found in relatively thin and long nanotube models, while buckling triggered by change of cross-sectional shape (i.e., radial buckling) was found in thick and short models. With growing radius, a crossover between the Euler-type and radial buckling modes was clearly found in the ASI analysis. Even when the deformation behavior was apparently the same, the buckling can be triggered by different instability modes. We also analyzed the structural instability using the Flügge theory based on linear continuum mechanics (LCM) to compare the results with the ASI analysis. Initial eigenvalues and development of instability modes were different between the ASI and LCM results, leading to a significant deviation of buckling strain and instability crossover points. We investigated the effect of chirality of CNT which is difficult to explain by LCM.

Minimum-resistance shapes in linear continuum mechanics

A necessary optimality condition for the problem of the minimum-resistance shape for a rigid three-dimensional inclusion displaced in an unbounded isotropic elastic medium subject to a constraint on the volume of the inclusion is obtained through Betti's reciprocal work theorem. It generalizes Pironneau's optimality condition for the minimum-drag shape for a rigid body immersed into a uniform Stokes flow and is specialized for axisymmetric inclusions in axisymmetric and transversal translations. In both cases of translation, the three-dimensional displacement field is represented in terms of generalized analytic functions, and the three-dimensional elastostatics problem is reduced to boundary-integral equations (BIEs) via the generalized Cauchy integral formula. Minimum-resistance shapes are found in the semi-analytical form of functional series from an iterative procedure coupling the optimality condition and the BIEs. They are compared with the minimum-resistance spheroids and with the minimum-resistance spindle-shaped and lens-shaped bodies. Remarkably, in the axisymmetric translation, the minimum-resistance shapes transition from spindle-like shapes to almost prolate spheroidal shapes as the Poisson ratio changes from 1/2 to 0, whereas in the transversal translation, they are close to oblate spheroidal shapes for any Poisson ratio.

An efficient stochastic and double-scale model to evaluate the effective elastic properties of 2D overlapping random fibre composites

This paper focuses on a stochastic and multi-scale finite element (FE) model to estimate the effective properties of 2D overlapping random fibre composites. We describe a new and efficient geometric approximation to solve different problems encountered when building the mesh in which the overlapping phenomenon between two or more fibres arises. The basic idea is to attain an approximation of the fibre geometry relative to quadrangular grid elements and an adaptive mesh refinement (AMR). Thus, the model allows one to obtain a fast, reliable and totally automated generation of a large number of representative volume elements (RVEs) of random fibre composites. Several numerical tests are performed to evaluate both the efficiency of the model and the reliability of the results. It can be noted that, here we only consider the mechanical response of the material in the framework of linear continuum mechanics.

Discontinuous Galerkin Methods for Friedrichs' Systems. Part III. Multifield Theories with Partial Coercivity

This paper is the third and last part of a work attempting to give a unified analysis of discontinuous Galerkin methods. The purpose of this paper is to extend the framework that has been developed in Part II for two-field Friedrichs' systems associated with second-order PDEs. We now consider two-field Friedrichs' systems with partial $L^2$-coercivity and three-field Friedrichs' systems with an even weaker $L^2$-coercivity hypothesis. In particular, this work generalizes the discontinuous Galerkin methods of Part II to compressible and incompressible linear continuum mechanics. We also show how the stabilizing parameters of the method must be set when the two-field Friedrichs' system is composed of terms that may be of different magnitude, thus accounting, for instance, for advection-diffusion equations at high Peclet numbers.

Discontinuous Galerkin Methods for Friedrichs’ Systems. Part II. Second‐order Elliptic PDEs

This paper is the second part of a work attempting to give a unified analysis of discontinuous Galerkin methods. The setting under scrutiny is that of Friedrichs’ systems endowed with a particular $2 \times 2$ structure in which one unknown can be eliminated to yield a system of second‐order elliptic‐like PDEs for the remaining unknown. A general discontinuous Galerkin method for approximating such systems is proposed and analyzed. The key feature is that the unknown that can be eliminated at the continuous level can also be eliminated at the discrete level by solving local problems. All the design constraints on the boundary operators that weakly enforce boundary conditions and on the interface operators that penalize interface jumps are fully stated. Examples are given for advection‐diffusion‐reaction, linear continuum mechanics, and a simplified version of the magnetohydrodynamics equations. Comparisons with well‐known discontinuous Galerkin approximations for the Poisson equation are presented.

A three-dimensional, higher-order, elasticity-based micromechanics model

The three-dimensional (3D) version of a new homogenization theory [A Two-Dimensional, Higher-Order, Elasticity-Based Micromechanics Model, in print] is presented. The 3D theory utilizes a higher-order, elasticity-based cell model (ECM) analysis for a periodic array of 3D unit cells. The unit cell is discretized into eight subregions or subcells. The displacement field within each subcell is approximated by a (truncated) eigenfunction expansion of up to fifth order. The governing equations are developed by satisfying the pointwise governing equations of geometrically linear continuum mechanics exactly up through the given order of the subcell displacement fields. The specified governing equations are valid for any type of constitutive model used to describe the behavior of the material in a subcell. The specialization of the theory to lower orders and to two-dimeinsions (2D) and to the exact one-dimensional (1D) theory is discussed. Since the proposed 3D homogenization theory correctly reduces to both 2D and 1D the validation process applied to the 2D theory [A Two-Dimensional, Higher-Order, Elasticity-Based Micromechanics Model, in print] directly applies to the current formulation. Additional comparisons of the predicted responses obtained from the 3D ECM theory with existing published results are conducted. The good agreement obtained shows that the current theory represents a viable 3D homogenization tool. The improved agreement between the current theory results and published results as compared to the comparison of the MOC results and the published results is due to the correct incorporation of the coupling effects between the local fields. Additional results showing the convergence behavior of the average fields as a function of the order of the analysis is presented. These results show that the 1st order theory may not accurately predict the local averages but that consistent and converged behavior is obtained using the higher order ECM theories. The proposed theory represents the necessary theoretical foundations for the development of exact homogenization solutions of generalized, three-dimensional microstructures.

A two-dimensional, higher-order, elasticity-based micromechanics model

A new, two-dimensional (2D) homogenization theory is proposed. The theory utilizes a higher-order, elasticity-based cell model (ECM) analysis. The material microstructure is modeled as a 2D periodic array of unit cells where each unit cell is discretized into four subregions (or subcells). The analysis utilizes a (truncated) eigenfunction expansion of up to fifth order for the displacement field in each subcell. The governing equations for the theory are developed by satisfying the pointwise governing equations of geometrically linear continuum mechanics exactly up through an order consistent with the order of the subcell displacement field. The formulation is carried out independently of any specified constitutive models for the behavior of the individual phases (in the sense that the general governing equations hold for any constitutive model). The fifth order theory is subsequently specialized to a third order theory. Additionally, the higher order analyzes reduce to a theory equivalent to the original 2D method of cells (MOC) theory when all higher order terms are eliminated. The proposed 2D theory is the companion theory to an equivalent 3D theory [T.O. Williams, A three-dimensional, higher-order, elasticity-based micromechanics model, Int. J. Solids Struc., in press]. Comparison of the predicted bulk and local responses with published results indicates that the theory accurately predicts both types of responses. The high degree of agreement between the current theory results and published results is due to the correct incorporation of the coupling effects between the local fields. The proposed theory represents the necessary theoretical foundations for the development of exact homogenization solutions of generalized, two-dimensional microstructures.

Advanced Mechanics of Solids

3R5. Advanced Mechanics of Solids. - OT Bruhns (Institut fur Mechanik, Ruhr-Univ Bochum, Universitatsstr 150, Bochum, D-44780, Germany). Springer-Verlag, New York. 2003. 206 pp. ISBN 3-540-43797-5. $49.95.Reviewed by P Puri (Dept of Math, Univ of New Orleans, 2000 Lakeshore Dr, New Orleans LA 70148).The author of this book is a well-known researcher in solid mechanics and has made some important contributions to the theory of elasticity and plasticity. In general, the material in this book is presented quite well and can be used as an introductory one-semester course in solid mechanics. Readers are expected to have the basic knowledge of ordinary and partial differential equations, mechanics, and some elementary knowledge of the theory of beams. The solved examples are well chosen and help the student understand the theory and procedure for solving problems given in the exercises. Useful exercises are given in the last section of every chapter, except Chapter 10. The reviewer likes the fact that answers to all exercises are given. The figures in the book are neat and clearly drawn. The book is not intended to be a comprehensive book on mechanics of solids like a recent book on elasticity by Hetnarski and Ignaczak. Chapter 1 presents the fundamental concepts, principles of linear continuum mechanics, and the derivation of the basic equations. A minor comment: Eq (1.84) is more similar to Eq (1.76) than to Eq (1.75). Chapter 2 contains the development of constitutive equations based on Hooke’s law. The expressions for strain and complementary energy are derived in Section 2. The three equilibrium equations in terms of displacements and the equivalent six Beltrami equations in terms of stresses are stated in Section 3. Section 4 discusses the influence of temperature. The rest of the sections are devoted to Hooke’s law in two dimensions, the strength criteria, and the principle of complementary virtual work. It appears to the reviewer that for the derivation of Eq (2.18) a reference to Eq (2.9) is more appropriate than Eq (2.6) and that it would be helpful to refer to Eq (1.53) for the derivation Eq (2.19). The theory of simple beams is developed in Chapter 3. Topics covered in this chapter are normal stresses, shearing stresses, shear center of thin-walled open sections, influence of distributed loads, and stresses in non-prismatic bars, and deflections of beams. In Chapter 4, the torsion of prismatic bars is explored. Various types of cross sections are considered in the first three sections, and the influence of restrained warping is discussed in Section 4. The next chapter is on curved beams. The four sections in this chapter analyze general static’s, beams with large curvature, beams with small curvature, and the deflection of curved beams. Reciprocity theorems, energy principles, theorems of Engesser and Castigliano, statically indeterminate systems, complementary and strain energy of the beams are explained in Chapter 6. Chapter 7 is concerned with two-dimensional problems of plane stress and plane strain, which are discussed in the first section. In the second section, the special case in which the body forces have potential is discussed. In the next section the equations of plane strain and plane stress are given in polar form. The theory of elastic plates and shells is presented in Chapter 8. General remarks are given in Section 1. Section 2 consists of the solved examples for discs. Discussions on thin plates, Kirchhoff’s theory, boundary conditions, axially symmetric bending of circular plates, and the elastic energy of plates are contained in Section 3. Membrane theory of shells of revolution is given in Section 4. The notion of stability, bifurcation with finite degrees of freedom, snap through buckling, and column buckling are the topics covered in Chapter 9. The principle of virtual work, Hamilton’s principle, Euler-Lagrange equations, and the Legrange multiplier method are given in Chapter 10. This is the only chapter, which has no solved examples or exercises. The next two chapters are concerned with vibration. The first two sections in Chapter 11 are on undamped and damped free vibrations. This material is generally covered in introductory differential equations. The third section consists of forced vibration with harmonic excitation, and the last section in this chapter is on isolators, which tend to dampen the vibrations. The last chapter covers the material on free and forced vibrations with several degrees of freedom. This chapter also includes answers to the exercises. Advanced Mechanics of Solids is recommended to people interested in learning the basics of solid mechanics and understanding the interaction of forces on solids. The book is particularly suitable for structural engineers.

Intrinsic Thermal Vibrations of Suspended Doubly Clamped Single-Wall Carbon Nanotubes

We report the observation of thermally driven mechanical vibrations of suspended doubly clamped carbon nanotubes, grown by chemical vapor deposition (CVD). Several experimental procedures are used to suspend carbon nanotubes. The vibration is observed as a blurring in images taken with a scanning electron microscope. The measured vibration amplitudes are compared with a model based on linear continuum mechanics.

Crack initiation behavior in magnetoelectroelastic composite under in-plane deformation

While the method of solution for crack problems in anisotropic elasticity and magnetoelectroelasticity is nearly identical, the prediction of crack initiation and/or growth behavior based on the asymptotic stress fields can differ widely depending on the chosen failure criterion. The latter has no haven to hide. Avoidance of contraction and satisfaction of the first law of thermodynamics are the rules that need to be observed. A negative crack tip energy release rate would infer the creation of energy and a negative strain energy density would imply the non-uniqueness of solution in classical linear continuum mechanics theories. Under these conditions, the crack initiation and growth behavior in a magnetoelectroelastic material will be examined. For the sake of notation and continuity, this work gives a brief account of the well-known solution for a line crack in a magnetoelectroelastic medium. Not to be underestimated is the derivation of the asymptotic form of the strain energy density function that was first given for the piezoelectroelastic crack problem. The equivalent expression for the magnetoelectroelastic case is given here. This sets the stage for a physical interpretation of how a crack would behave in a composite possessing the mixed properties of piezoelectric and piezomagnetic materials. The individual terms in the strain energy density function shows how the applied electric and magnetic field would affect the critical applied mechanical normal and shear stress to trigger crack initiation, respectively. When both normal and shear action prevail, the direction of crack initiation is no longer obvious and it needs to be determined since the energy stored ahead of the crack depends on the direction of prospective crack initiation. Numerical results are given for a BaTiO 3–CoFe 2O 4 composite. The inclusion is the piezoelectric BaTiO 3 material and the matrix is the magnetostrictive CoFe 2O 4 material. The proportion of the two phases can be varied by adjusting the volume fraction of the inclusions upon which the fluctuation of the energy density field near the crack depends and the material properties of the constituents. Directions of the applied electric and magnetic field can also be changed; their effects on crack initiation are discussed using the strain energy density criterion. Indeed, the additional magnetostrictive effect can have an influence on crack initiation as the applied field directions are altered. Possible behavior of crack initiation can be made for design-specific magnetoelectroelastic composites.

A consistent deduction of von K�rm�n-type plate theories from three-dimensional nonlinear continuum mechanics

Classical and refined plate theories derived from linear continuum mechanics lead to correct results only if the transverse deflection of the plate is small compared to its thickness. In the case of large deflections, geometrical nonlinearities have to be incorporated. For the classical Kirchhoff plate theory, a suitable extension for moderate rotations has been presented by von Karman in 1911. Starting from the three-dimensional equations of nonlinear continuum mechanics, a family of von Karman-type plate theories is deduced. For the derivation, the kinematical variables are replaced by a series representation and the principle of virtual displacements is used. It can be shown that most plate theories can be obtained from this type of theory and that the kinematical assumptions must fulfill certain conditions to obtain a solvable system of equations.

Renormalisation group theory for two problems in linear continuum mechanics

We consider the problems of: (1) the stress field in an infinite wedge when a moment is applied at the tip, and (2) the flow of an inviscid incompressible fluid past an infinite wedge. Both problems are linear, but nevertheless exhibit anomalous dimensions for wedge angles, α, larger than a critical value α c. In (1) it is the stress field and in (2) it is the velocity potential which can have anomalous power law behaviour in distance from the tip. We use these problems as simple examples to illustrate how partial differential equations can be solved using the renormalisation group.

On conservation theorems, Lagrangian functions and variational principles for linear continuum mechanics

It is shown that a general form of the bilinear functional in conjunction with classical concepts of thermodynamics leads to the results not admitted by the particular form commonly used in the literature. The conservation theorem for a wide class of continuous bodies is established. A generalized Lagrangian function for linear dissipative materials is introduced. Two specific examples: problem of elastodynamics and problem of thermoelasticity are selected in order to illustrate the proposed approach.

Crack propagation under variable load histories in linearly viscoelastic solids

The growth of a crack in a viscoelastic solid is considered within the limitations of linear continuum mechanics. Starting from the local stresses and deformation field at the tip of a crack and the first law of thermodynamics a non-linear differential equation is derived for the crack tip velocity in dependence on an arbitrary history of the crack tip stress intensity factor. Conditions for the simplification of this differential equation are discussed. The relevance of these calculations to the mechanical behavior of composite solids is pointed out. The case of cyclic loading (fatigue problem) is discussed in detail and sample calculations are given to illustrate the relative effect of maximum and minimum load during a cycle. Some experimental data is given for comparison with these calculations.

Analysis of Dynamic Systems Using the Mechanical Impedance Concept

The mechanical impedance concept is a powerful tool for the analysis of dynamical structural systems consisting of a single degree-of-freedom unit coupled to an infinite degree-of-freedom unit. Actual systems of interest that approximate this are (1) instrument components coupled through isolators to the vibrating shell of aircraft or missiles with the inherent problem of predetermining the dynamic forces acting upon the component, and (2) vibrating machinery coupled through isolators to the hulls of marine vessels with the inherent problem of minimizing the sound radiating into the sea. Formulas are presented for each case with damping included by rheological models. The quantities of the formulas can be measured if the continuous supporting unit is undefined. Simpler theoretical analyses result for defined continuous units than for the usual classical approaches. No approximations are made beyond those of classical linear continuum mechanics. (This research was supported by Armour Research Foundation.)