Bending problems in the theory of elastic materials with voids and surface effects

Abstract

In the present study, a comparison of pure, three-point, four-point and cantilever beam bending problems in the frame of the theory of elastic materials with voids (micro-dilatational elasticity) has been provided via analytical modelling and three-dimensional finite-element analysis. We consider the extended variant of the theory with surface effects using a variational approach. At first, we compare the known approximate semi-inverse analytical solution of the pure bending problem with a corresponding three-dimensional finite-element solution in the frame of micro-dilatational theory. It is demonstrated that in the numerical solution – unlike the analytical one – all boundary conditions are satisfied accurately, and there exist distortions of the cross-sections and lateral faces of the beam. The generalized analytical solution of the beam pure bending problem with surface effects is also established and compared with numerical simulations. The effective elastic properties of the beam with micro-dilatations are introduced by comparing its displacements and corresponding classical beam displacements. The influence of scale, coupling and surface parameters on the effective elastic moduli in different bending and simple tension tests is studied. It is shown that all considered types of bending experiments provide the determination of close values of the effective flexural modulus of the beams with different thickness. This means that it is possible to use any bending test with beams of different thickness for reliable identification of the material constants of the theory. It is also possible to use the simple analytical expression for an effective flexural modulus that follows from the analytical solution of the pure bending problem. It is also shown that stiffness of the beam with micro-dilatation in any bending tests should always be higher as compared with the stiffness of such a beam in simple tension. For thin beams, there are no scale effects, and its effective flexural modulus is equal to the material’s Young’s modulus without micro-dilatation. Possible rise of the negative size effects in the model with surface effects is also discussed.

On the dependence of standard and gradient elastic material constants on a field of defects

“…An increase in this function results in a decrease of the apparent elastic constants at a given point of the medium [10,13,35,42]. Apparent elastic stiffness is also reduced when increasing strain/micro-dilatation coupling effects [41]. In the present work, we show that the apparent elastic behavior of the considered gradient/micro-dilatation medium is more complex than those mentioned above: its apparent stiffness decreases, while the apparent length scale parameter increases, when increasing micro-dilatation effects.…”

“…It seems that the same results may be proven in the two-dimensional and three-dimensional problems by using generalized analytical and numerical approaches of strain gradient and micro-dilatation elasticity theories [36,41,58,59]. However, it might be necessary to consider more general models to describe the field of defects in materials with complex microstructure, such as micropolar, microstretch, or general micromorphic [9].…”

“…In [12, 13, 41] it has been shown that additional scalar field variables introduced in micro-dilatation elasticity can be treated as fields of defects. An increase in this function results in a decrease of the apparent elastic constants at a given point of the medium [10, 13, 35, 42].…”

On mechanically driven biological stimulus for bone remodeling as a diffusive phenomenon

“…In poromechanics (see, e.g., Lurie et al (2018); Khalili and Selvadurai (2003); Misra et al ( , 2013), and coherently to what done in the Biot model, we introduce the Lagrangian field ζ(X, t) defined as the change of the effective volume of the voids per unit volume in the transformation from Lagrangian to Eulerian configuration. In formulas…”

In-depth gaze at the astonishing mechanical behavior of bone: A review for designing bio-inspired hierarchical metamaterials

“…Since the bone tissue is a porous material, a first attempt to generalize its mechanical description involves adopting a micromorphic model in which to the standard kinematic variables is added a further quantity that takes into account the change of porosity of the tissue, as in the classical models developed by Biot and his followers [63][64][65][66][67][68][69] and by Cowin and Nunziato and their followers [70][71][72][73][74]. This approach is based on the energy density, and is quite common in poromechanics.…”