【正文】 physically surprising values of implied Poisson’s ratio for certain high or low stress levels. Again we need to find a strain or plementary energy function that will give us the basic modulus variation that we desire.Houlsby (1985) suggests that an acceptable strain energy function could be: : Linear logarithmic relationship between and for elastic materialwith bulk modulus proportional to Incrementally this implies a stiffness matrix which, once again, contains off diagonal terms indicating coupling between volumetric and distortional elements of deformation: It can be deduced that so that contours of constant distortional strain are lines of constant stress ratio η(Fig ). Constant volume (undrained) stress paths are found to be parabolae (Fig ): All parabolae in this family touch the line.Figure : Contours of constant volumetric strain (solid lines) and constantdistortional strain (dotted lines) for nonlinear elastic model of Houlsby (1985)The nonlinearity that has been introduced in these two models is still associated with an isotropic elasticity. The elastic properties vary with deformation but not with direction.Although it tends to be assumed that nonlinearity in soils es exclusively from soil plasticity—as will be discussed in the subsequent sections—we have seen that with care it may be possible to describe some elastic nonlinearity in a way which is thermodynamically acceptable. Equally, most elasticplastic models will contain some element of elasticity—which may often be swamped by plastic deformations. It must be expected that the fabric variations which acpany any plastic shearing will themselves lead to changes in the elastic properties of the soil. The formulation of such variations of stiffness should in principle be based on the differentiation of some serendipitously discovered elastic strain energy density function in order that the elasticity should not violate the laws of thermodynamics. Evidently the development of strain energy functions which permit evolution of anisotropy of elastic stiffness is tricky. Many constitutive models adopt a pragmatic, hypoelastic approach and simply define the evolution of the moduli with stress state or with strain state without concern for the thermodynamic consequences. This may not provoke particular problems provided the stress paths or strain paths to which soil elements are subjected are not very repeatedly cyclic. HeterogeneityAnisotropy and nonlinearity are both possible departures from the simple assumptions of isotropic linear elasticity. A rather different departure is associated with heterogeneity. We have already noted that small scale heterogeneity—seasonal layering—may lead to anisotropy of stiffness (and other) properties at the scale of a typical sample. Many natural and manmade soils contain large ranges of particle sizes (167。對(duì)于一個(gè)完全整體各向異性彈性材料 其中，每個(gè)字母，...是，在原理上是一個(gè)獨(dú)立的彈性參數(shù)，彈性材料剛度矩陣必要的對(duì)稱性已推導(dǎo)出獨(dú)立參數(shù)的最大值為21?，F(xiàn)在柔度矩陣的形式為: 并且我們可以寫為： 這被形容為橫向各向同性或六邊形對(duì)稱的交叉各向異性。 這是書(shū)寫的楊氏模量，在垂直方向楊氏模量，泊松比， 連同第三個(gè)參數(shù)。 事實(shí)上，對(duì)于，其關(guān)系其實(shí)并不單調(diào)，并且有效應(yīng)力路徑方向超出了明顯的界限 ()。 舉例來(lái)說(shuō)， 如果我們假定土體體積彈性模量隨平均有效壓力變化，但泊松比(即剪切模量和體積模量的比值)是恒定的話，我們會(huì)發(fā)現(xiàn),，違反熱力學(xué)第一定律創(chuàng)造一個(gè)永動(dòng)機(jī)，能量將增加(或失去)，這不會(huì)是一個(gè)保守體系。對(duì)于和泊松比，意味的常體積應(yīng)變曲線，—由鮑耶斯對(duì)小應(yīng)變回彈彈性參數(shù)測(cè)試的路基材料得到的典型值。豪斯柏（1985）建議一個(gè)可以接受的應(yīng)變能函數(shù)可為： 更近一步地，這意味著剛度矩陣再次包含顯示變形的體積和剪切元素耦合的對(duì)角線量。這種剛度變化的方程式，原則上應(yīng)當(dāng)是基于微分偶然的發(fā)現(xiàn)彈性應(yīng)變能量密度函數(shù)，因此彈性不應(yīng)違反熱力學(xué)定律。)—冰磧和殘積土中往往在不同土樣基質(zhì)中含有漂石顆粒。他們發(fā)現(xiàn)：粘土/砂石系統(tǒng)的所有性能受土基質(zhì)控制。在一個(gè)小規(guī)模范圍內(nèi)， 繆意爾我們已經(jīng)注意到小規(guī)模的非均質(zhì)性季節(jié)性分層—可能導(dǎo)致在典型樣本的范圍內(nèi)剛度(和其他)性能的各向異性。同樣，大多數(shù)彈塑性模型將包含某種彈性其中往往充滿塑性變形。 如果我們假定恒定的剪切模量值，獨(dú)立的應(yīng)力水平，我們將獲得一個(gè)保守的材料，但也許會(huì)發(fā)現(xiàn)，我們泊松比在某種高或低應(yīng)力水平呈現(xiàn)令人吃驚的值。 因?yàn)槿岫痊F(xiàn)在是隨著應(yīng)力比變化的，對(duì)于不排水（純剪切）加荷的有效應(yīng)力路徑不再是直線。 不過(guò)，我們也期待一些真正有彈性性質(zhì)的土體將隨應(yīng)力水平而變化，這可以看作彈性非線形的一個(gè)來(lái)源。第一次研究()僅僅表明對(duì)于孔壓參數(shù)有限制， 非常大（＞＞）和非常?。ǎ荆荆r(shí)()分別為和，而這表示了依次施加恒定軸向有效應(yīng)力和恒定徑向有效應(yīng)力的有效應(yīng)力路徑。我們不能將和分離開(kāi)（林斯等，2000）。然而，許多土都在橫向范圍區(qū)域內(nèi)沉積，沉積的對(duì)稱性基本上是垂直的。分層的范圍可能會(huì)非常小，我們不期望區(qū)分不同材料，但在不同方向的分層可能還是足以改變不同方向的土的性