I Blechman Fig. I. Incremental strains in layered solid drm, de, -longitudinal strain of m-and a-layers: da leo -free Poisson extension of m-and a-layers; dam--gradient strain of tension in m-layer:de* gradient strain of compression in cf-layer: da-full extension. I Initial state, 2-comlpesseu state where c-free Poisson extension and &*transverse strain induced by gradient: do. de- the increments in longitudinal stress and strain. respectively The increment of the gradient forces can be expressed by the parameters of the layers. when their elastic moduli in the lateral direction are taken as equal to those in the longitudinal direction. (The layers are taken as isotropic in macro. dF= h, e, dag (4a) dFn=h, E,r dem where: E--modulus of elasticity. ha. hm-thicknesses of layers As follows from(2)and(4) hE h e The factor he resses the relationship between the stiffnesses of the two layers The increments in free Poisson extension of the layers are (7b Substituting the above equations in(1) yields
I. Blechman Fig. 1. Incremental strains in layered solid dr:,,,. dL,, -longitudinal strain of v- and a-layers : ds;; _ tk{:‘-free Poisson extension of 11~ and u-layers; ds,T, --gradient strain of tension in m-layer : d+ gradient strain of compression in u-layer; df:-+ full estension. I Initial state, 2 -compressed state. where Y--free Poisson extension and c*-transverse strain induced by gradient : do. dc+ the increments in longitudinal stress and strain. respectively. The increment of the gradient forces can be expressed by the parameters of the layers, when their elastic moduli in the lateral direction are taken as equal to those in the longitudinal direction. (The layers are taken as isotropic in macro.) dF<, = h,,E,, dc:. (4a) dF,,i = II,,, E,,, dG. (4b) where : E--modulus of elasticity, h,,, IT,,,--thicknesses of layers. As follows from (2) and (4) : The factor: expresses the relationship between the stiffnesses of the two layers. The increments in free Poisson extension of the layers are : (6) (7a) (7b) Substituting the above equations in ( 1) yields :
da (8) The expression in brackets is a combi-gradient 8, which includes the influence of Poisson ratios and elastic moduli We will also denote P Integrating( 8)between the limits 0 and o, with Poissons ratio taken independent of o. we obtain the gradient of transverse strain of tension, accumulated in the m layer 00 The opposite gradient strain of transverse compression in the a-layer is Equation(10a)can be also presented as Eu (10c) n the expression in brackets is a Poisson gradient in the" sandwich", corrected in the second term by the factor Eu Er E With a -aEa the cquation of gradient strain of local tension becomes In the compressed layer the gradient strain is e=P288 2=P,e* As can be seen from eqn(9)with some proportionality between two pairs of Poisson ratio and elastic moduli. the gradient 8, in the"sandwich "can be very slight. For example: if a-0.26、Vm-0.13, when E-40000MPa,Em-20,000 then d,=0 gradient factors at crystals level--dur (described helow in Section 5). will be higher than between the la. described below. not due to interlayer differences
Brittle solid under compressiorl-I 1569 (8) The expression in brackets is a combi-gradient 6,, which includes the influence of Poisson ratios and elastic moduli : We will also denote : I PC, pmi - 1 + PCS ’ and pn = i+Pn Integrating (8) between the limits 0 and C. with Poisson’s ratio taken independent of 0. we obtain the gradient of transverse strain of tension, accumulated in the nz layer: E,T, = p”, (5,G. (1W The opposite gradient strain of transverse compression in the n-layer is Equation (10a) can be also presented as (lob) (IOC) Then the expression in brackets is a Poisson gradient in the “sandwich”, corrected in the second term by the factor EJE,, : With E = a/E,, the equation of gradient strain of local tension becomes (9b) In the compressed layer the gradient strain is : As can be seen from eqn (9) with some proportionality between two pairs of Poisson ratio and elastic moduli. the gradient 6$ in the “sandwich” can be very slight. For example : if v,, = 0.26, v,,, = 0.13, when E, = 40.000 MPa, E,,, = 20,000 then ~5,~ =O! When the “sandwich” is built from crystalline brittle solids, it is quite possible that the gradient factors at crystals level& a,,,, (described below in Section 5). will be higher than between the layers. Then the “sandwich” will be split due to intercrystalline gradients described below. not due to interlayer differences