Discussion On The Typical Problems Of Modified Gradient Elasticity

When describing the mechanical behaviors which the microstructures play a dominant role,the size effects cannot be captured by the mechanical/mathematical models which are derived from the classical continuum theory.In order to describe the mechanical behavior of materials at micro and meso scales more reasonably,the Bernoulli-Euler beam model and the analytical stress solution for mode ? crack are deduced in the framework of modified gradient elasticity(MGE).Firstly,a new Bernoulli-Euler beam model is developed based on modified gradient elasticity theory.The governing equation and boundary conditions,which contain two internal length scales(i.e.,lx and lz),are derived by the variational principle.The new model can be simplified to the classical beam theory when the two internal length scales vanish.The numerical examples of cantilever beams subjected to two typical loadings are presented.The results show that the size effect can be captured by the new model,and the deflection decreases with the internal length scales increasing.The influence of lz(the internal length scale along the beam thickness direction)on deflection is greater than that of lx(the internal length scale along the beam length direction),and the increment of beam stiffness is main controlled by lz.The new beam model is convenient for engineering applications and designs.Secondly,the strain gradient and the internal length scales are introduced into the basic equations of mode ? crack by the modified gradient elasticity(MGE).By using a complex function approach,the analytical solution of stress fields for mode ? crack problem is derived within MGE.When the internal length scales vanish,the stress fields can be simplified to the stress fields of classical linear elastic fracture mechanics.The results show that the singularity of the shear stress is made up of two parts,r-1/2part and r-3/2part,and the sign of the stress ?yz changes.With the increase of lx,the peak value of ?yz.decrease and its location moves farther from the fracture vertex.The strain gradient exists near a crack tip significantly influence the near-tip stress field,and the influence of strain gradient for mode? crack problem cannot be ignored.