Abstract
Nonlocal hereditariness in Bernoulli–Euler beam is investigated in this paper. An approach to solve that problem is proposed and some analytical solutions are provided. To this aim, time-dependent hereditary behavior is modeled by means of non-integer order operators of the fractional linear viscoelasticity. While, space-dependent nonlocal phenomena are simulated through the integral stress-driven formulation. These two approaches are combined providing a new model able to simulate nonlocal viscoelastic bending problem. Several application samples of the proposed formulation and a thorough parametric study are presented showing the influences of hereditariness and nonlocal effects on the mechanical bending response. Proposed formulation can be useful for design and optimization of structures used in advanced applications when local elastic theory cannot be adopted.
Highlights
Small-scale devices [1,2,3,4], biological and bioinspired materials [5,6,7], advanced porous nanostructures [8, 9], self-healing matter [10], hierarchical and periodic structures [11, 12], new-generation of complex composites [13,14,15] require sophisticated methodologies and advanced models to predict their mechanical behavior
This manuscript deals with the simulation of two phenomena which occur in some mechanical behaviors providing a new approach to solve the bending problem of hereditary nonlocal beam
In the integral formulation of hereditariness the stress–strain relation is given as an integral Volterra-Boltzamann relation where the kernel is a time-dependent function which takes into account the memory of the material
Summary
Small-scale devices [1,2,3,4], biological and bioinspired materials [5,6,7], advanced porous nanostructures [8, 9], self-healing matter [10], hierarchical and periodic structures [11, 12], new-generation of complex composites [13,14,15] require sophisticated methodologies and advanced models to predict their mechanical behavior. Among the various model of linear viscoelasticity, hereditary model based on the fractional calculus is able to represent the real time-dependent behavior of a wide variety of materials, like polymers [18], biological tissues [19], clays [20], non-newtonian fluids [5], rubbers [21], bones [22], and so on [23, 24]. These models are recently used to model nonlinear time-dependent behavior [25,26,27] For these capabilities viscoelastic models based upon fractional calculus will be used in this paper to obtain a versatile timedependent stress–strain relation useful to model several advanced bending problems where hereditary effects cannot be neglected. 4 providing a new formulation of bending behavior of a nonlocal hereditary beam Such a section contains analytical solution of this time-dependent problem where hereditariness and nonlocality can be modeled varying two specific mechanical parameters.
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