Static And Dynamic Responses Of Graded Porous Beams Under The Follower Force In A Hygrothermal Environment

As a lightweight composite material,gradient porous material has been widely used in modern industrial fields such as material science,biomedical engineering,and energy storage.Driven by these applications,this material has attracted much attention.In addition,because the gradient porous structure is often exposed to extreme environments such as high temperature,it is often affected by the thermal gradient environment and the mechanical load changing with time,which leads to significant changes in the structure’s material properties.The gradient porous structure can adjust the macroscopic mechanical strength of the structure by controlling the volume fraction,microstructure,and porosity of material components in the manufacturing process.To this end,a comprehensive parameter study is proposed in this paper to evaluate the effects of material properties,external load,temperature,and humidity on the static deformation and dynamic response of gradient porous structures.The concrete research contents are as follows:(1)Based on the cosine pore distribution model,the fundamental equation of the gradient porous simply supported beam subjected to the follower force in a hygrothermal environment is established.Firstly,it is assumed that the material beam has two types of porosity distribution.The performance of the beam varies with temperature and humidity,while the environmental field changes continuously with the beam section height.By establishing the large deformation mathematical model of the gradient porous material beam,the thermal coupling constitutive relation is given by using the strain energy power series function form,and the governing equations of static deformation motion and free vibration are derived.(2)The multivariable partial differential equation is transformed into a two-point boundary value problem by using the shooting method based on the differential governing equation of the motion of a gradient porous beam under the follower force.According to the error between the boundary conditions and the approximate solution,the iterative method is used for numerical optimization,and the initial conditions are constantly improved until the exact solution satisfying the boundary conditions is solved to realize the numerical solution of the governing equation.Then,the calculated results are compared with the existing references to confirm the feasibility of the study.Finally,the influence of porosity size and distribution,temperature,and humidity on the static response of the structure is discussed emphatically.(3)Based on the nonlinear geometry theory of axially extendable beams,the dynamic response of graded porous beams under the follower force in a hygrothermal environment is discussed.In order to make the discrete nodes of the dynamic response problem match with the discrete nodes of the static response problem,the two boundary value problems of the static problem and the dynamic problem are solved simultaneously.By eliminating the time variable in equations with dynamic unknowns,the partial differential equation is transformed into an ordinary differential equation.Finally,the numerical results are calculated by using Matlab and Fortran programs,and the influence of various parameters(temperature,humidity,porosity,slenderness ratio,etc.)on the dynamic response of the graded porous material beam are analyzed and discussed.In this paper,the mechanical behavior of the gradient porous beam under the action of the follower force is analyzed and summarized,which provides a new idea and theoretical reference for the future engineering and research.