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A review of continuum mechanics models for size-dependent analysis of beams and plates

Connections between the Hertz-type contact problems and depth-sensing indentation of materials are studied. Formulations of Hertz-type contact problems with various boundary conditions within the contact area are discussed in detail. The problems under investigations can be subdivided into two large groups: self-similar problems for anisotropic materials with various rheological properties and adhesive contact problems for arbitrary bodies of revolution or for power-law shaped blunt indenters. Specific features of indentation problems are described and the common methods for extracting elastic and adhesive properties of materials are briefly reviewed. The basic formulae are extended to the case of nonslipping boundary conditions between a probe and the material. The main formulae of the JKR theory of adhesion are extended to any material with rotational symmetry of the elastic properties. These materials include not only isotropic or transversely isotropic elastic solids but also homogeneously prestressed isotropic or transversely isotropic nonlinear elastic materials. The BG method introduced for extracting adhesive and elastic properties of isotropic elastic materials from depth-sensing diagrams of spherical indenters, is described and extended to linear or linearized materials with rotational symmetry of the elastic properties.

The Hertz-Type and Adhesive Contact Problems for Depth-Sensing Indentation

The size-dependent thermal bending and buckling analyses of composite laminate microplate based on new modified couple stress theory and isogeometric analysis

A new Bernoulli–Euler beam model is developed using a modified couple stress theory and a surface elasticity theory. A variational formulation based on the principle of minimum total potential energy is employed, which leads to the simultaneous determination of the equilibrium equation and complete boundary conditions for a Bernoulli–Euler beam. The new model contains a material length scale parameter accounting for the microstructure effect in the bulk of the beam and three surface elasticity constants describing the mechanical behavior of the beam surface layer. The inclusion of these additional material constants enables the new model to capture the microstructure- and surface energy-dependent size effect. In addition, Poisson’s effect is incorporated in the current model, unlike existing beam models. The new beam model includes the models considering only the microstructure dependence or the surface energy effect as special cases. The current model reduces to the classical Bernoulli–Euler beam model when the microstructure dependence, surface energy, and Poisson’s effect are all suppressed. To demonstrate the new model, a cantilever beam problem is solved by directly applying the general formulas derived. Numerical results reveal that the beam deflection predicted by the new model is smaller than that by the classical beam model. Also, it is found that the difference between the deflections predicted by the two models is very significant when the beam thickness is small but is diminishing with the increase of the beam thickness.

A new Bernoulli–Euler beam model incorporating microstructure and surface energy effects

A new Timoshenko beam model is developed using a modified couple stress theory and a surface elasticity theory. A variational formulation based on Hamilton’s principle is employed, which leads to the simultaneous determination of the equations of motion and complete boundary conditions for a Timoshenko beam. The new model contains a material length scale parameter accounting for the microstructure effect in the bulk of the beam and three surface elasticity constants describing the mechanical behavior of the beam surface layer. The inclusion of these additional material constants enables the new model to capture the microstructure-and surface energy-dependent size effect. In addition, both bending and axial deformations are considered, and the Poisson effect is incorporated in the current model, unlike existing Timoshenko beam models. The new beam model includes the models considering only the microstructure dependence or the surface energy effect as limiting cases and recovers the Bernoulli–Euler beam model incorporating the two effects as a special case. Also, the current model reduces to the classical Timoshenko beam model when the microstructure dependence, surface energy and Poisson’s effect are all suppressed. To demonstrate the new model, the static bending and free vibration problems of a simply supported beam are analytically solved by directly applying the general formulas derived. The numerical results for the static bending problem reveal that both the deflection and rotation of the simply supported beam predicted by the new model are smaller than those predicted by the classical Timoshenko beam model. In addition, the differences in both the deflection and rotation predicted by the two models are very large when the beam thickness is small, but they are diminishing with the increase of the beam thickness. Similar trends are observed for the free vibration problem, where it is shown that the natural frequency predicted by the new model is higher than that given by the classical model, with the difference between them being significantly large for very thin beams. These predicted trends of the size effect in beam bending at the micron scale agree with those observed experimentally.

A new Timoshenko beam model incorporating microstructure and surface energy effects

Analytical solutions for the problems of an elastic half-space and an elastic half-plane subjected to a distributed normal force are derived in a unified manner using the general form of the linearized surface elasticity theory of Gurtin and Murdoch. The Papkovitch–Neuber potential functions, Fourier transforms and Bessel functions are utilized in the formulation. The newly obtained solutions are general and reduce to the solutions for the half-space and half-plane contact problems based on classical linear elasticity when the surface effects are not considered. Also, existing solutions for the half-space and half-plane contact problems based on simplified versions of Gurtin and Murdoch’s surface elasticity theory are recovered as special cases of the current solutions. By applying the new solutions directly, Boussinesq’s flat-ended punch problem, Hertz’s spherical punch problem and a conical punch problem are solved, which lead to depth-dependent hardness formulas different from those based on classical elasticity. The numerical results reveal that smoother elastic fields and smaller displacements are predicted by the current solutions than those given by the classical elasticity-based solutions. Also, it is shown that the out-of-plane displacement and stress components strongly depend on the residual surface stress. In addition, it is found that the new solutions based on the surface elasticity theory predict larger values of the indentation hardness than the solutions based on classical elasticity.

Solutions of half-space and half-plane contact problems based on surface elasticity

A Nonclassical Model for Circular Mindlin Plates Based on a Modified Couple Stress Theory

General solutions for the problems of an elastic half-space and an elastic half-plane, respectively, subjected to a symmetrically distributed normal force of arbitrary profile are analytically derived using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter. Mindlin’s potential function method and Fourier transforms are employed in the formulation, and the half-space and half-plane contact problems are solved in a unified manner. The specific solutions for the problems of a half-space/plane subjected to a concentrated normal force or a uniformly distributed normal force are obtained by directly applying the general solutions, which recover the existing classical elasticity-based solutions of the Flamant and Boussinesq problems as special cases. In addition, the indentation problems of an elastic half-space indented by a flat-ended cylindrical punch, a spherical punch, and a conical punch, respectively, are solved using the general solutions, leading to hardness formulas that are indentation size- and material microstructure-dependent. Numerical results reveal that the displacement and stress fields in a half-space/plane given by the current SSGET-based solutions are smoother than those predicted by the classical elasticity-based solutions and do not exhibit the discontinuity and/or singularity displayed by the latter. Also, the indentation hardness values based on the newly obtained half-space solution are found to increase with decreasing indentation radius and increasing material length scale parameter, thereby explaining the microstructure-dependent indentation size effect.

Strain gradient solutions of half-space and half-plane contact problems

A unified treatment of axisymmetric adhesive contact problems is provided using the harmonic potential function method for axisymmetric elasticity problems advanced by Green, Keer, Barber and others. The harmonic function adopted in the current analysis is the one that was introduced by Jin et al. (2008) to solve an external crack problem. It is demonstrated that the harmonic potential function method offers a simpler and more consistent way to treat non-adhesive and adhesive contact problems. By using this method and the principle of superposition, a general solution is derived for the adhesive contact problem involving an axisymmetric rigid punch of arbitrary shape and an adhesive interaction force distribution of any profile. This solution provides analytical expressions for all non-zero displacement and stress components on the contact surface, unlike existing ones. In addition, the newly derived solution is able to link existing solutions/models for axisymmetric non-adhesive and adhesive contact problems and to reveal the connections and differences among these solutions/models individually obtained using different methods at various times. Specifically, it is shown that Sneddon's solution for the axisymmetric punch problem, Boussinesq's solution for the flat-ended cylindrical punch problem, the Hertz solution for the spherical punch problem, the JKR model, the DMT model, the M-D model, and the M-D-n model can all be explicitly recovered by the current general solution. (C) 2010 Elsevier Ltd. All rights reserved.

A unified treatment of axisymmetric adhesive contact problems using the harmonic potential function method

The generalized half-plane and half-space Cerruti problems with surface effects are analytically solved using Gurtin and Murdoch’s surface elasticity theory along with the Airy stress function method and the Papkovitch–Neuber potential function approach, respectively. The Fourier transform method is employed in the formulation. The newly derived solutions reduce to their classical elasticity-based counterparts if the surface effects are not considered. In addition, the current solution for the half-space Cerruti problem recovers that based on a simplified version of the surface elasticity theory as a special case. Further, specific solutions are obtained for the cases with loading by a concentrated tangential force and by a uniform traction acting over a circular region. The numerical results reveal that the predictions by the current solutions substantially deviate from those by their classical counterparts near the loading site, but the differences between the two sets of predictions diminish far away from the loading site. Moreover, the newly obtained solutions do not exhibit stress or displacement singularity at the loading point/boundary and predict smoother stress fields and smaller displacements than the classical solutions that do not incorporate the surface effects.

S.-S. Zhou

Papers

Solutions of half-space and half-plane contact problems based on surface elasticity

A Nonclassical Model for Circular Mindlin Plates Based on a Modified Couple Stress Theory

Strain gradient solutions of half-space and half-plane contact problems

A unified treatment of axisymmetric adhesive contact problems using the harmonic potential function method

Solutions of the generalized half-plane and half-space Cerruti problems with surface effects