A simple model for the small-strain behaviour of soils

Transcription

1 A simple moel for the small-strain behaviour of soils José Jorge Naer Department of Structural an Geotechnical ngineering, Polytechnic School, University of São Paulo , São Paulo, Brazil, Keywors: constitutive moels; elasticity; soil mechanics; soil eformation. Abstract: The constitutive equation iscusse in this note eliminates some efects of linear elasticity in the escription of the small-strain behaviour of soils. It is capable of representing volume changes in pure shear an ifferent values of bulk moulus in compression an expansion. The new equation provies a simple escription of soil behaviour at small strains in that it prouces linear stress-strain relations that can approximate the initial part of experimental stress-strain curves. 1. Introuction Linear elasticity is not a goo moel for soils even at the small-strain level. onsier, for instance, ientical isotropic soil samples subject to ifferent stress paths in the triaxial cell. Although the beginning of each particular stress-strain curve can be approximate by a tangent line through the origin, experience shows that ifferent pairs of elastic parameters (say, bulk an shear moulus) woul be necessary to represent the small-strain response in the various stress paths. The simplest example of this fact occurs in the isotropic compression or expansion of a normally consoliate sample: the bulk moulus for a subsequent virgin loaing is smaller than the bulk moulus for unloaing. Another shortcoming of the linear elastic moel is that it cannot preict volume changes in pure shear (e.g., volume increase in ense sans, volume ecrease in loose sans) as well as values of Skempton s A parameter ifferent from 1/3 in the unraine compression of a saturate soil (see, e.g., Lambe an Whitman 1979, for experimental results). This note presents a constitutive moel for the small-strain behaviour of soils that, unlike linear elasticity, is able to preict the occurrence of volume changes in pure shear, the existence of ifferent values of bulk moulus in isotropic compression an expansion an ifferent values of Young s moulus an Poisson s ratio in axial compression an axial extension. On the other han, like linear elasticity, the

2 propose moel yiels linear stress-strain relations in straight stress-paths an so provies a relatively simple approximate representation of the behaviour at small strains. Throughout the text, we use the soil mechanics sign convention for stresses an strains (compressive stresses an strains are positive etc). 2. The new moel As a starting point, consier the isotropic linear elastic equation, as use in soil mechanics: T = KI ( + G. (1) 1 ) I 2 T, an I are, respectively, the effective stress increment tensor, the strain tensor an the ientity tensor; is the eviatoric part of, I ( ) is the first invariant of, K is the bulk moulus an G is the 1 shear moulus. Soils are always subject to initial effective stresses, whether in the fiel or in laboratory. q. 1 thus yiels the increment T that shoul be ae to the initial effective stress tensor to obtain the final effective stress tensor (applications of linear elasticity to soil mechanics problems is a classical subject; see, e.g., Terzaghi 1943; Davis an Selvaurai 1996). Now the first term on the right-han sie of q. 1 will be replace by a more general function of to prouce the new moel. The resulting equation reas T = α( ) I + 2G, (2) where 1 ( ) / 2 + ( K K ) I1( ) / 2 + ξg I 2 ( ) α ( ) = ( K + K ) I. I ) enotes the secon 2 ( invariant of. The parameters K (compression bulk moulus), K (expansion bulk moulus) an G (shear moulus) are positive whereas ξ (ilatancy coefficient) may be negative, positive or null. Their physical meaning will become clear in the examples below. Note that, for I ( ) 0, 1 K 1 ( ) + ξg I 2 ( ) α ( ) = I while, for I ( ) 0, 1 K 1 ( ) + ξg I 2 ( ) α ( ) = I. Of course, q. 2 reuces to q. 1 if K = K an ξ=0. In the next section the behaviour of the propose equation will be investigate in typical situations (isotropic compression an expansion, pure shear, axial compression an extension, unraine axial compression). For the analysis of cases in which T is given an must be foun, it is convenient to invert the stress-strain relation expresse by q. 2. As = I ( +, it suffices to write I ( ) an 1 ) I / 3 1

3 in terms of T. learly, = / 2G (in which T is the eviator stress increment tensor). T Further, it can be conclue that, if I ( ) 0, then I ( ) ) / K while, if I ( ) 0, K 1 1 T 1 I ( ) ) /, where ρ( T) = I ( T) / 3 ξ I ( ) / 2. Hence, the following equation hols: 1 T 1 2 T 1 = β( T) I + T, (3) 2G in which β( T) = (1/ 6K + 1/ 6K ) ρ( T) + (1/ 6K 1/ 6K ) ρ( T). Therefore, for ρ( T ) 0, β ( T) T) / 3K, whereas, for ρ( T ) 0, β ( T) T) / 3K. In any case, the volumetric strain is given by ε = I ( ) = 3β( ) v 1 T. 3. Moel behaviour in simple cases a. Isotropic compression an expansion The response of q. 2 to an isotropic strain ( = εi ) is an isotropic stress increment ( T = σi ). The relation between the volumetric strain ( ε v = 3ε ) an σ epens on the sign of ε. It is σ = K εv in case of compression ( ε > 0 ) an σ = K ε v in case of expansion ( ε < 0 ). Thus, if K K, the moel preicts a ifferent linear stress-strain relation in each case. b. Pure shear For a pure shear efine by the stress increment matrix [ T ] = 0 0 τ (4) 0 τ 0 ( τ > 0 ), q. 3 gives the following strain matrix: ε 0 0 [] = 0 ε γ / 2. (5) 0 γ / 2 ε

4 Thus the shear strain γ is relate to the shear stress τ through γ = τ / G, as in linear elasticity. By introucing ρ( T ) = ξτ / 2 in the expression for β ( T), the relation between the volumetric strain ( ε v = 3ε ) an τ can be obtaine. If ξ < 0, then ε v = ξτ / 2K, a positive number, inicating volume ecrease (contraction). On the other han, if ξ > 0, then ε v = ξτ / 2K, which is negative, inicating volume increase (expansion). Therefore, unlike linear elasticity, the present moel preicts volume changes in pure shear. c. Axial compression an extension Now the only non-zero stress increment component is, say, T = σ (axial compression: σ > 0 ; axial 33 extension: σ < 0 ). Accoring to q. 3, the corresponing strain matrix is iagonal, with 11 = 22. In compression ρ( T ) = mσ, with m = 1 3ξ / 2 3, whereas in extension ρ( T ) = nσ, with n = 1 + 3ξ / 2 3. Table 1 gives the obtaine expressions for the Young s moulus ( σ / 33 ) in compression ( ) an in extension ( ) in terms of the basic parameters, G, K, K an ξ. The corresponing values of the Poisson s ratio ( 22 / 33 ) in compression ( ν ) an in extension ( ν ) are given in all cases by ν = / 2G 1 an ν = / 2G 1. Note that there are two ifferent expressions to compute both an, each expression associate to an interval of values of ξ. In aition, Table 1 gives expressions for the volumetric strain an, in the last column, restrictions that must be impose, if any, on the basic parameters in orer that the values of Young s moulus an Poisson s ratio are positive, as expecte on physical grouns. The meaning of the symbols H, L, L, H appearing in the first column of Table 1will now be explaine. They constitute a classification base on the value of the ilatancy coefficient that helps unerstan the behaviour of the material efine by q. 2 in regar to volume changes. The symbols an stan for two main material classes relate to volume changes in pure shear. lass ( ξ < 0 ): materials that contract in pure shear. lass ( ξ 0 ): materials that expan ( ξ > 0 ) or suffer no volume change ( ξ = 0 ) in pure shear. lasses an are further ivie into two subclasses accoring to the behaviour in axial compression an extension. Subclass H ( ξ < 2 3 / 3; high contractibility): materials in class that contract both in axial compression an extension. Subclass L ( 2 3 / 3 ξ < 0 ; low contractibility;): materials in class that contract in axial compression an expan ( 2 3 / 3 < ξ ) or suffer no volume

5 change ( 2 3 / 3 = ξ ) in axial extension. Subclass L ( 0 ξ 2 3 / 3: low expansibility): materials in class that expan in axial extension an contract ( 0 ξ < 2 3 / 3) or suffer no volume change ( ξ = 2 3 / 3 ) in axial compression. Subclass H ( ξ > 2 3 / 3 ; high expansibility): materials in class that expan both in axial compression an extension. 1. The appropriate expressions for, an ε v in each subclass can be obtaine irectly from Table Table 1: Young s moulus an volumetric strain in axial compression an extension. Axial compression ( σ > 0 ; m = 1 3ξ / 2 3 ) ξ 2 3 / 3 ( m 0 ) H, L, L = 3 K + mg ε v = mσ / 3K (7) (6) ( ε v 0 ) > 0. ν > 0, if ξ > 2 3 / 3 3K / G. ξ > 2 3 / 3 ( m < 0 ) H = 3 K + mg ε v = mσ / 3K (9) (8) ( ε v < 0 ) > 0 an > 0, if ν ξ < 2 3G / K. Axial extension ( σ < 0 ; n = 1 + 3ξ / 2 3 ) ξ < 2 3 / 3 ( n < 0 ) H = 3 K + ng ε v = nσ / 3K (11) (10) ( ε v > 0 ) > 0 an > 0, if ν ξ > 2 3G / 3 2 3K. ξ 2 ( n 0 ) 3 / 3 L, L, H = 3 K + ng ε v = nσ / 3K (13) (12) ( ε v 0 ) > 0. ν > 0, if ξ < 2 3 / 3 + 3K / G.

6 . Unraine axial compression Here the strain matrix reas: ε / [] = 0 ε / 2 0, (14) 0 0 ε where ε > 0 is the axial compressive strain. From q. 2, the non-zero stress increment components rea (recall they are effective stresses): 3 T 11 = T22 = (1 ξ) Gε, (15) 2 3 T 33 = (2 + ξ) Gε. (16) 2 In the traitional unraine compression test the total lateral stress remains constant. Hence the pore pressure increment u is the negative of the lateral effective stress increment: 3 u = ( 1 ξ) Gε. (17) 2 It is positive if ξ < 2 3 / 3 (H, L, L). Only soils in the class H generate negative pore pressures. The corresponing Skempton s pore pressure parameter is A = 1/ 3 3ξ / Final remarks learly, avance constitutive moels can correct the mentione efects of linear elasticity an, in aition, are able to escribe soil behaviour in complex stress paths (for funamentals of elastoplastic moels see Davis e Selvaurai 2005; as for hypoplastic moels, Kolymbas 2000, Naer 2003, 2010). But, ue to its simplicity, the propose moel may be avantageous if we are intereste only in moelling small-strain behaviour in monotonic short stress-paths, far from failure. Besies, the moel may have a iactic value: it can be an intermeiate stage between linear elasticity an more complex theories.

7 References Davis, R.O. an Selvaurai, A.P.S. (1996). lasticity an Geomechanics. ambrige University Press. Davis, R.O. an Selvaurai, A.P.S. (2005). Plasticity an Geomechanics. ambrige University Press. Kolymbas, D. (2000). Introuction to Hypoplasticity. Balkema. Lambe, T.W. an Whitman, R.V. (1979). Soil Mechanics. Wiley. Naer, J.J. (2003). Analysis of a new hypoplastic equation. International Journal of Plasticity 19, Naer, J.J. (2010). A new constitutive theory extening Hypoplasticity. Mechanics Research ommunications 37, Terzaghi, K. (1943). Theoretical Soil Mechanics. Wiley.