Analytical and numerical study of the mechanical behaviour of materials and structures in the theory of gradient elasticity

Some Aspects on the Modelling of Microstructure Deformation Mechanisms with Gradient Truss Model (GTM) Bar Elements

“…Similar equations can be obtained using the principle of virtual work Akintayo (2011) and the variational principle Tsepoura et al (2002). With q = 0, the solution of Eq.…”

“…Similar to the classical elastic truss model in Papadopoulos and Xenidis (1998) and Akintayo et al (1998Akintayo et al ( , 2000, the proposed gradient truss model simulates a material block by a plane truss consisting of a coarse grid of equal elementary square trusses including both of the diagonal bars Akintayo (2011) and Akintayo et al (2012). The relation between the strains and stresses of the bars connected to an interior node of the truss model and the strain and stress states at a point of the continuum are represented in Figure 1 by: εx = (εr + εl)/2; σx = (σr + σl)/2; εy = (εa + εu)/2; σy = (σa + σu)/2; γxy = (εra -εla + εlu -εru)sin45 0 /4; τxy = (σra -σla + σlu -σru)sin45 0 /4.…”

“…The relation between the strains and stresses of the bars connected to an interior node of the truss model and the strain and stress states at a point of the continuum are represented in Figure 1 by: εx = (εr + εl)/2; σx = (σr + σl)/2; εy = (εa + εu)/2; σy = (σa + σu)/2; γxy = (εra -εla + εlu -εru)sin45 0 /4; τxy = (σra -σla + σlu -σru)sin45 0 /4. As shown in Figure 1 the determination of the bar cross sections of the truss model reduces to the simulation of a square prism under biaxial stress by an elementary square truss and the square prism could be related to a representative volume of the material Akintayo (2011).…”

“…However, here the bar is assumed to be able to support elastic strain gradients along the bar; hence, the special theory of gradient elasticity proposed by Aifantis (1984), (1992) is used. As also used in Altan and Aifantis (1997), Akintayo (2011); Akintayo et al (2012); Akintayo (2014a, b, c, d), the one-dimensional gradient elasticity stress-strain constitutive equation is taken as…”

“…In search of an efficient and cost effective computational tool that considers the influence of microstructure in a very simple manner, Akintayo (2011) and Akintayo et al (2012) first presented the gradient truss model (GTM) bar elements. By incorporating the phenomenological gradient elastic theory of Aifantis (1984Aifantis ( , 1992 into the simple numerical process of the classical elasticity truss model, higher order strain gradient terms and an internal length parameter were included in the elastic constitutive stress-strain relation of a representative GTM bar element: Thus the microstructure deformation response is captured.…”