外文翻译(土木工程专业毕业设计)

3.2

Elastic models

3.2.1

Anisotropy

An isotropic material has the same properties in all directions—we cannot dis-tinguish any one direction from any other. Samples taken out of the ground with any orientation would behave identically. However, we know that soils have been deposited in some way—for example, sedimentary soils will know about the vertical direction of gravitational deposition. There may in addition be seasonal variations in the rate of deposition so that the soil contains more or less marked layers of slightly different grain size and/or plasticity. The scale of layering may be suffciently small that we do not wish to try to distinguish separate materials, but the layering together with the directional deposition may nevertheless be suffcient to modify the properies of the soil in different directions—in other words to cause it to be anisotropic.

We can write the stiffness relationship between elastic strain increment e and stress increment  compactly as

De (3.36)

whereDis the stiffness matrix and henceD1is the compliance matrix. For a completely general anisotropic elastic material

abcdefijkbghchlmno1 (3.37) Ddimpqrejnqstfkortuwhereeachlettera,b,... is,inprinciple,anindependentelasticpropertyandthe necessary

symmetry of the sti?ness matrix for the elastic material has reduced the maximum number of independent properties to 21. As soon as there are material symmetries then the number of independent elastic properties falls (Crampin, 1981).

For example, for monoclinic symmetry (z symmetry plane) the compliance matrix has the form:

D1abc00dbef00gcfh00i000jk0000kl0dgi (3.38) 00m

and has thirteen elastic constants. Orthorhombic symmetry (distinct x, y and z symmetry planes) gives nine constants:

c000abbde000cef0001 (3.39) D000g000000h000000iwhereas cubic symmetry (identical x, y and z symmetry planes, together with planes joining opposite sides of a cube) gives only three constants:

abb000bab000bba0001 (3.40) D000c000000c000000c

Figure 3.9: Independent modes of shearing for cross-anisotropic material

If we add the further requirement that c2(ab)and set a1/E and

bv/E，then we recover the isotropic elastic compliance matrix of (3.1).

Though it is obviously convenient if geotechnical materials have certain fabric symmetries which confer a reduction in the number of independent elastic properties,

it has to be expected that in general materials which have been pushed around by tectonic forces, by ice, or by man will not possess any of these symmetries and, insofar as they have a domain of elastic response, we should expect to require the full 21 independent elastic properties. If we choose to model such materials as isotropic elastic or anisotropic elastic with certain restricting symmetries then we have to recognise that these are modelling decisions of which the soil or rock may be unaware.

However, many soils are deposited over areas of large lateral extent and symmetry of deposition is essentially vertical. All horizontal directions look the same but horizontal sti?ness is expected to be di?erent from vertical stiffness. The form of the compliance matrix is now:

abc000bac000ccd000000e000000e0000 (3.41) 00fD1and we can write：

a1/Eh,bvhh/Eh,cvvh/Ev,d1/Ev,e1/Gvh和f2(ab)2(1vhh)/Eh:

1/Ehvhh/Ehv/Evhv1D000vhh/Eh1/Ehvvh/Ev000vvh/Evvvh/Ev1/Eh0000001/Gvh0000001/Gvh000 (3.42) 0021vhh/Eh0

This is described as transverse isotropy or cross anisotropy with hexagonal symmetry. There are 5 independent elastic properties: EvandEhare Young’s moduli for

unconfined compression in the vertical and horizontal directions respectively; Gvhis the shear modulus for shearing in a vertical plane (Fig 3.9a).Poisson’s ratios Vhh and Vvh relate to the lateral strains that occur in the horizontal direction orthogonal to a horizontal direction of compression and a vertical direction of compression respectively (Fig 3.9c, b).

Testing of cross anisotropic soils in a triaxial apparatus with their axes of anisotropy aligned with the axes of the apparatus does not give us any possibility to

discover Gvh1/E，since this would require controlled application of shear stresses to vertical and horizontal surfaces of the sample—and attendant rotation of principal axes. In fact we are able only to determine 3 of the 5 elastic properties. If we write (3.42) for radial and axial stresses and strains for a sample with its vertical axis of symmetry of anisotropy aligned with the axis of the triaxial apparatus, we find that：

a1/Evrvvh/Ev2vvh/Eva' (3.43) 1vhh/Evr'The compliance matrix is not symmetric because, in the context of the triaxial test, the strain increment and stress quantities are not properly work conjugate. We deduce that while we can separately determineEvand Vvhthe only other elastic property that we can discover is the composite stiffnessEh/(1Vhh).We are not able to separateEhand Vhh(Lings et al., 2000).

On the other hand, Graham and Houlsby (1983) have proposed a special form of

(3.41) or (3.42) which uses only 3 elastic properties but forces certain interdependencies among the 5 elastic properties for this cross anisotropic

material.

1/2v/21v/E000v/21/2v/000v/v/10000021v/00000021v/00000021v2/D10 (3.44)

This is written in terms of a Young’s modulusEEv,the Young’s modulus for loading in the vertical direction, a Poisson’s ratio VVhh,together with a third parameter . The ratio of stiffness in horizontal and vertical directions is

Eh/Ev2and other linkages are forced：

vvhvhh/;GhvGhh/E/2(1v).

For our triaxial stress and strain quantities, the compliance matrix becomes:

p3G1detJqJp' (3.45) Kq

Figure 3.10: Effect of cross-anisotropy on direction of undrained effective stress path

where

det3KGJ2 (3.46)

and the stiffness matrix is

p'KqJJp (3.47) 3Gq

where

1v4v22 (3.48) KE91v12v22v4v2 (3.49) GE61v12v1vv2 (3.50) JE31v12vThe stiffness and compliance matrices (written in terms of correctly chosen work conjugate strain increment and stress quantities) are still symmetric—the material is still elastic—but the non-zero off-diagonal terms tell us that there is now coupling between volumetric and distortional effects. There will be volumetric strain when we apply purely distortional stress, p'0,distortional strain during purely isotropic

compression, q0,and there will be change in mean effective stress in undrained tests,

p0.

In fact the slope of the effective stress path in an undrained test is, from (3.45),

p'J21vv2 (3.51) q3G322v4v2From our definition of pore pressure parameter a (§2.6.2) we find

p'J (3.52) q3G

Figure 3.11: Relationship between anisotropy parameter α and pore pressure

parameter a for different values of Poisson’s ratio  .



which will, in the presence of anisotropy, not be zero.

A first inspection of (3.51) merely suggests that there are limits on the pore pressure parameter of a = 2/3 and a = -1/3 for  very large（Eh＞＞Ev）and  very

small（Ev＞＞Eh）repectively (Fig 3.10), which in turn imply effective stress paths with constant axial effective stress and constant radial effective stress respectively. The link between a and α is actually slightly more subtle.In fact,for v0the relationship is not actually monotonic and the effective stress path direction overshoots the apparent limits (Fig 3.11). The deduction of a value

2of (and henceEh/Ev) from a is not very reliable when a is around -1/3 or 2/3 (recall

the data presented in Figs 2.51 and 2.49, §2.5.4). For v0.5,a(12)/3(1) or

(13a)(3a2).These relationships satisfy the expected limits for 0 and but

there are singularities in the inversion of (3.51) for1 and v0.5.

3.2.2

Nonlinearity

We will probably expect that the dominant source of nonlinearity of stress:strain response will come from material plasticity—and we will go on to develop elasticplastic constitutive models in the next section. However, we also have an expectation that some of the truly elastic properties of soils will vary with stress level and this can be seen as a source of elastic nonlinearity. Our thoughts about elastic materials as conservative materials—the term ‘hyperelasticity’ is used to describe such materials—might make us a little cautious about plucking from the air arbitrary empirical functions for variation of moduli with stresses. For example, if we were to suppose that the bulk modulus of the soil varied with mean effective stress but that Poisson’s ratio (and hence the ratio of shear modulus to bulk modulus) were constant then we would find that in a closed stress cycle such as that shown in Fig 3.12 energy would be created (or lost) creating a perpetual motion machine in violation of the first law of thermodynamics—this would not be a conservative system. We need to find a strain energy (3.7) or complementary energy density (3.11) function which can be differentiated to give acceptable variation of moduli with stresses.

Figure 3.12: Cycle of stress changes which should give zero energy generated or

dissipated for conservative material

Such a complementary energy function can be deduced from the nonlinear elastic model described by Boyce (1980):

211qn1 (3.53) Vp'n1K16Gp'When K1and G1 are reference values of bulk modulus and shear modulus and n is a

nonlinearity parameter. The compliance matrix can then be deduced by differentiation：

1n2n2n1pK6G1n1p'1nq3G11n3G1p' (3.54) 1q3G1Whereq/p'.There is again (as for the anisotropic model) coupling between volumetric and distortional effects. The stiffnesses are broadly proportional to p'1n. Because the compliances are now varying with stress ratio  the effective stress path implied for an undrained (purely distortional) loading is no longer straight. In fact, for a reference state p'p0,q0the effective stress path is

p'0n12 (3.55) p'where(1n)K1/6G1。Contours of constant volumetric strainp0 are shown in

Fig 3.13 for n0.2and Poisson’s ratiov0.3 implying K1/G12.17—values

typical for the road sub-base materials being tested by Boyce for their small strain, resilient elastic properties.

Similarly the path followed in a purely volumetric deformationq0 will

develop some change in distortional stress. For an initial state p'p0,qq0,the effective

stress path for such a test is

qp'0q0p'n1 (3.56)

Contours of constant distortional strain are also shown in Fig 3.13 for n = 0.2.

Figure 3.13: Contours of constant volumetric strain (solid lines) and constant distortional strain (dotted lines) for nonlinear elastic model of Boyce (1980)

It is often proposed that the elastic volumetric stiffness—bulk modulus—of clays should be directly proportional to mean effective stress: Kp'/k.Integration of this

relationship shows that elastic unloading of clays produces a straight line response when plotted in a logarithmic compression plane(plnv:lnp')(Fig 3.14) where v is specific volume. But what assumption should we make about shear modulus? If we simply assume that Poisson’s ratio is constant, so that the ratio of shear modulus to bulk modulus is constant, then we will emerge with a non-conservative material (Zytynski et al., 1978). If we assume a constant value of shear modulus, independent of stress level, we will obtain a conservative material but may find that we have physically surprising values of implied Poisson’s ratio for certain high or low stress levels. Again we need to find a strain or complementary energy function that will give us the basic modulus variation that we desire.

Houlsby (1985) suggests that an acceptable strain energy function could be:

Up'rep/k32kq (3.57)

2

Figure3.14: Linear logarithmic relationship between  and

with bulk modulus proportional to

p' for elastic material

p'

Incrementally this implies a stiffness matrix which, once again, contains off diagonal terms indicating coupling between volumetric and distortional elements of deformation:

1/kp'qp'/k/kp (3.58) /qqIt can be deduced that

3qq2 (3.59) p'13q2kso that contours of constant distortional strain are lines of constant stress ratio η(Fig 3.15). Constant volume (undrained) stress paths are found to be parabolae (Fig 3.15):

q26kpi'p'pi' (3.60) All parabolae in this family touch the line3k/2.

Figure 3.15: Contours of constant volumetric strain (solid lines) and constant distortional 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 comes 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 elastic-plastic models will contain some element of elasticity—which may often be swamped by plastic deformations. It must be expected that the fabric variations which accompany 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.

3.2.3

Heterogeneity

Anisotropy 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 man-made soils contain large ranges of particle sizes (§1.8)—glacial tills and residual soils often contain boulder-sized particles within an otherwise soil-like matrix. If the scale of our geotechnical system is large by comparison with the size and spacing of these boulders then it will be reasonable to treat the material as essentially homogeneous. However, we will still wish to determine its mechanical properties.

If we attempt to measure shear wave velocities in situ, using geophysical techniques, then we can expect that the fastest wave from source to receiver will take advantage of the presence of the large hard rock-like particles—which will have a much higher stiffness and hence higher shear wave velocity than the surrounding soil (Fig 3.16). The receiver will show the travel time for the fastest wave which has taken this heterogeneous route. If the hard material occupies a proportion λ of the spacing between source and receiver, and the ratio of shear wave velocities is k(and hence, neglecting density differences, the ratio of shear moduli is of the order of k2 ), then the ratio of apparent shear wave velocity Vs to the shear wave velocity of the soil matrix Vs is

Vsk (3.61) Vskk

Figure 3.16: (a) Soil containing boulders between boreholes used for measurement of shear wave velocity; (b) average stiffnesses deduced from interpretation of shear wave velocity and from matrix stiffness

The deduced average shear modulus G is then greater than the shear stiffness of the soil matrix G by the ratio

Gk1 as k (3.62) 2Gkk12Laboratory testing of such heterogeneous materials is not easy because the test apparatus needs itself to be much larger than the typical maximum particle size and spacing in order that a true average property should be measured. At a small scale, Muir Wood and Kumar (2000) report tests to explore mechanical characteristics of mixtures of kaolin clay and a fine gravel(d50=2mm). They found that all the properties of the clay/gravel system were controlled by the soil matrix until the volume fraction of the gravel was about 0.45-0.5. At that stage, but not before, interaction between the ‘rigid’ particles started rapidly to dominate. For0.5 then, this implies a ratio of equivalent shear stiffness G to soil matrix stiffness G:

G1 (3.63) G1These two expressions, (3.62) and (3.63), are compared in Fig3.16 for a modulus ratio k210000

附录2 外文翻译

3.2 弹性模型

3.2.1各向异性

各向异性材料在各个方向具有同样的性质—我们不能将任何一个方向与任何其他方向区分开。从地下任何地方取出的试样都表现出个性。然而，我们知道土已经以某种方式沉积—例如，沉积性土在垂直方向受重力作用而沉积。另外，沉积速度可能呈季节变化，所以土体或多或少地包含了颗粒尺寸或可塑性略微相异的标志性土层。分层的范围可能会非常小，我们不期望区分不同材料，但在不同方向的分层可能还是足以改变不同方向的土的性质—换句话说就是造成其各向异性。

我们可以将弹性应变增量e和应力增量的刚度关系简写为

De (3.36)

其中D是刚度矩阵，因此D1是柔度矩阵。对于一个完全整体各向异性弹性材料

abcdefbghijkchlmno1 (3.37) Ddimpqrejnqstfkortu 其中，每个字母a，b，...是，在原理上是一个独立的弹性参数，弹性材料

刚度矩阵必要的对称性已推导出独立参数的最大值为21。一旦存在矩阵对称性，独立弹性参数的数量就减少了（克兰平，1981）。

例如，对于单斜对称（z对称面）柔度矩阵有形式如下：

c00dabbef00gcfh00i1 (3.38) D0jk000000kl0dgi00m 有13个弹性常数。正交对称（区分x、y、z对称面）给出9个常数：

D1abc000bde000cef000000g000000h0000 (3.39) 00i然而，立方体对称性（同一的x、y、z对称面，与立方体相反面结合的面一起）只给出三个常数：

abb000bab000bba000000c000000c0000 (3.40) 00c

D1

如果我们进一步要求c2(ab)和设a1/E和bv/E，那么我们发现（3.1）的各向同性弹性柔度矩阵。

不过，如果岩土工程材料具有一定的组构对称性，减少独立弹性参数的数量，显然是很方便的，正如料想的那样，受构造力、冰、或人推动的大部分材料，将不再拥有任何这类对称性，只要有一个域的弹性反应，我们应该期望要求全部21个弹性参数独立。 如果我们选择将这样的材料建模成伴有某些限制对称性的各向同性弹性或各向异性弹性，那么我们不得不分辨到这是对土体和岩石可能不了解的建模结果。

然而，许多土都在横向范围区域内沉积，沉积的对称性基本上是垂直的。从所有水平方向看是一样的，但横向刚度预计将不同于垂直刚度。现在柔度矩阵的形式为:

abc1D000并且我们可以写为：

bac000ccd000000e000000e0000 (3.41) 00fa1/Eh,bvhh/Eh,cvvh/Ev,d1/Ev,e1/Gvh和f2(ab)2(1vhh)/Eh:

1/Ehvhh/Ehv/Evhv1D000vhh/Eh1/Ehvvh/Ev000vvh/Evvvh/Ev1/Eh0000001/Gvh0000001/Gvh000 (3.42) 0021vhh/Eh0

这被形容为横向各向同性或六边形对称的交叉各向异性。有5个独立的弹性

Gvh是一个垂直面参数: Ev和Eh分别是垂直向和水平向不密闭压缩的杨氏模量；

上的剪切模量(图3.9a)。泊松比Vhh及Vvh分别是与发生在正交于压缩的横向方向和压缩的垂直方向的水平方向上的横向应变有关(图3.9c，b) 主轴与仪器轴平行三轴仪的交叉各向异性土的试验，并没有给我们任何可能性发现查实Gvh1/E，因为这要求控制施加对试样垂直和水平面上的剪应力。事实上，我们只能确定5个弹性参数中的3个。如果我们对于垂直轴与三轴仪主轴平行的试样，就径向和轴向的应力和应变书写(3.42)，我们发现：

a1/Evrvvh/Ev2vvh/Eva' (3.43) 1vhh/Evr'柔度矩阵不是对称的，因为在三轴试验环境中，应变增量和应力增量不是完全共轭的。我们推出：当我们可以分别确定Ev和Vvh时，我们可以得到的仅有的另外一个弹性参数是一个复合刚度Eh/(1Vhh)。

我们不能将Eh和Vhh分离开（林斯等，2000）。

另一方面，格拉汉姆和豪斯贝（1983）提出了（3.41）或（3.42）得特殊形式，只用了3个弹性参数，但对于此交叉各向异性材料，要求5个弹性参数是相互依赖的。

1/2v/21v/E000v/21/2v/000v/v/10000021v/00000021v/00000021v2/D10

这是书写的杨氏模量EEv，在垂直方向杨氏模量，泊松比VVhh， 连同第三个参数。在水平和垂直方向的刚度比是Eh/Ev2及其他约束关系：

vvhvhh/;GhvGhh/E/2(1v)。

对于我们的三轴应力和应变量，柔度矩阵变为：

p3G1detJqJp' (3.45) Kq

其中

det3KGJ2 (3.46)

并且，刚度矩阵是

p'KqJJp (3.47) 3Gq

其中

1v4v22 (3.48) KE91v12v22v4v2 (3.49) GE61v12v1vv2 (3.50) JE31v12v刚度和柔度矩阵（以正确选用工作共轭应变增量和应力增量方式书写）依然是对称的—材料依然是弹性的—但非零非对角线计算告诉我们体积作用和剪切作用之间是耦合的。进行纯粹的各向同性压缩试验时，q0，当我们施加纯剪力p'0和剪应变时，将产生体积应变，，不排水试验的平均有效应力将会改变，

p0。

实际上，不排水试验的有效应力路径的斜率，形式（3.45）

p'J21vv2 (3.51) 2q3G322v4v从我们对孔压参数a (§2.6.2)的定义中,我们发现

p'J (3.52) q3G

在各向异性存在时，不会为零。

第一次研究(3.5.1)仅仅表明对于孔压参数有限制， a非常大（Eh＞＞Ev）

和非常小（Ev＞＞Eh）时(图3.10)分别为a2/3和a1/3，而这表示了依次施加恒定轴向有效应力和恒定径向有效应力的有效应力路径。a和α之间的联系实际是较为含蓄的。 事实上，对于v0，其关系其实并不单调，并且有效应力路径方向超出了明显的界限 (图3.11)。当a在1/3或2/3附近取值时(回忆介绍的数据图2.51和2.49，§2.5.4)，从a推导得到的α (因而Eh/Ev2)不是很可靠的。对于v0.5，a(12)/3(1)或(13a)(3a2)。这些关系符合0和

的预期范围，但对于1和v0.5， (3.51)有奇异的倒转。

3.2.2非线形

我们大概预想的应力非线性的主要来源:应变反应将来自材料的可塑性—并且下部分，我们将继续发展弹塑性本构模式。 不过，我们也期待一些真正有弹性性质的土体将随应力水平而变化，这可以看作弹性非线形的一个来源。我们把弹性材料作为保守材料—“超弹性”一词是用来形容这种材料的-可能使我们在选取随应力变化模量的任意经验函数时更加谨慎。 举例来说， 如果我们假定土体体积弹性模量随平均有效压力变化，但泊松比(即剪切模量和体积模量的比值)是恒定的话，我们会发现,在图3.12如示的一封闭的应力循环中，违反热力学第一定律创造一个永动机，能量将增加(或失去)，这不会是一个保守体系。我们必须找到一种应变能(3.7)或补充能量密度(3.11)函数，可以通过微分得到可接受的应力模量变量。

这样的补充能量密度函数能够从鲍耶斯（1980）描述的非线形弹性模型中推导得到：

211qn1 (3.53) Vp'n1K16Gp'其中K1和G1是体积模量和剪切模量的参考值，n是非线性参数。柔度矩阵然后可以通过微分导出：

1n2n2n1pK6G1n1p'1nq3G11n3G1p' (3.54) 1q3G1其中q/p'。体积作用和剪切作用之间再次是共轭的（对于各向异性模型）。刚度是与p'1n广泛成比例的。

因为柔度现在是随着应力比变化的，对于不排水（纯剪切）加荷的有效应力路径不再是直线。实际上，对于提到的p'p0,q0情况，有效应力路径为

p'0n12 (3.55) p'其中(1n)K1/6G1。对于n0.2和泊松比v0.3，意味K1/G12.17的常体积应变p0曲线，如图3.13所示—由鲍耶斯对小应变回弹弹性参数测试的路基材料得到的典型值。

同样地，在纯体积变形q0将使剪应力发生一些变化。对于初始状态

p'p0,qq0，这个试验的有效应力路径为

qp'0q0p'n1 (3.56)

n2的常剪切应变曲线如图3.13所示。

经常提出的是粘土的弹性体积刚度—体积模量应当与平均有效应力直接成比例： Kp'/k。当以对数压缩平面(plnv:lnp')（图3.14 ）作图时，其中v是比容，这种关系的结合显示粘土的弹性卸载形成一条直线反应。但我们要对剪切模作什么假设呢？如果简单地以为泊松比为常数，那么，剪切模量和体积模量的比值是常数, 那么我们将发现一个非保守物质(扎廷斯基等，1978)。 如果我们假定恒定的剪切模量值，独立的应力水平，我们将获得一个保守的材料，但也许会发现，我们泊松比在某种高或低应力水平呈现令人吃惊的值。再次，我们必须找到一种应变或补充能量函数，会给我们期望的基本模量变化。

豪斯柏（1985）建议一个可以接受的应变能函数可为： Up'rep/k32kq (3.57)

2

更近一步地，这意味着刚度矩阵再次包含显示变形的体积和剪切元素耦合的

对角线量。

1/kp'p'q/k/kp (3.58) /qq可以导出

3qq2 (3.59) 3p'12kq所以常剪切变形的图形为常应力比的直线（图3.15）。常体积（不排水）应力路径为抛物线（图3.15）：

q26kpi'p'pi' (3.60) 这组中所有的抛物线与线3k/2相切。

这两种模型中所引入的非线性依然与各向同性弹性相联系。弹性参数随应变而变化，而不是随方向变化。

尽管它往往被猜想为土的非线性很多是来自土的可塑性-将在随后一节中讨论，我们从中看到谨慎地或许可以一种在热力学上可以接受的方式来说明一些弹性非线性。同样，大多数弹塑性模型将包含某种弹性-其中往往充满塑性变形。但可以预料的是，其中伴有任何塑性剪切的结构变化将导致土体弹性性质变化。这种刚度变化的方程式，原则上应当是基于微分偶然的发现弹性应变能量密度函数，因此弹性不应违反热力学定律。很明显的，允许各向异性弹性刚度演化的应变能函数的发展是棘手的。许多构模式，采取了务实的超弹性方式和单纯定义应

力状态模量或不关心热力后果的应变状态。这可能不引起特定问题除非土体经历的应力路径或应变路径不太反复循环。

3.2.3非均质性

各向异性和非线性均可能偏离简单各向同性线弹性的假设。 一个颇为不同的偏离是与非均质性有关的。我们已经注意到小规模的非均质性-季节性分层—可能导致在典型样本的范围内刚度(和其他)性能的各向异性。许多天然及人工土中含有很大的粒径变化范围(§1.8)—冰碛和残积土中往往在不同土样基质中含有漂石颗粒。如果地质系统的规模比这些漂石的尺寸和空隙大，那么将这种材料看作基本上均匀，是理由充分的。但是，我们仍希望确定其性能。

如果我们试图以地球物理勘探技术测定原位剪切波速，那么我们可以预料从发射源到接收器的最快波将利用大块坚硬的岩样颗粒的存在—将具有更高的刚度和因此比周围土体(图3.16)更高的剪切波速。接收机将显示通过这种非均质路线最快波的旅行时间。如果坚硬材料在发射源和接收机间距中占有比例，剪切波速度比为k(因此,忽略了密度的差异，剪切模量比是k2量级)，那么显剪切波速vs与土基质剪切波速vs的比值是

Vsk (3.61) Vskk

推导出的平均剪切模量G比土基质的剪切刚度G大一些，比值为

Gk1 当 k (3.62) 2Gkk12非均质材料的实验室试验是不容易的，因为试验仪器需求比典型的最大粒径和间隔要大很多，以测得一个真实的平均参数。在一个小规模范围内， 缪意尔·伍德及库马尔(2000)报告了探索高岭土和细砂砾 (d50=2毫米) 混合物机械特性的试验。他们发现：粘土/砂石系统的所有性能受土基质控制，直至该砂石的容积率约为0.45-0.5。在这个阶段，而不是之前，‘硬’颗粒之间的相互作用开始迅速占支配地位。对于＜0.5，那么，这意味着等效剪切刚度G与土的刚度矩阵比G的比值:

G1 (3.63) G1这两个表达式，（3.62）和（3.63），对于模量比k210000，在图3.16中进行了比较。

附录3 简化算法求最危险滑动面程序

程序如下：

C 简化算法求最危险滑动面 PARAMETER (A=3.141593)

REAL N,K,K1,H,I,F,G,Q,C,Z1,Z2,B,W,M,T1,T2,D,SH,LB1,LB2,R,E,l1,l2, * H1,H2,P1,P2,R0,Fy,d0 OPEN (3,FILE='IN.DAT')

READ (3,*) H,I,G,F,C,Q,D,SH,R,E,L1,L2,H1,H2,Fy,d0 M=A/180 K=3.0 Z2=1.0

DO Z1=10,70,0.5

B=H/TAN(Z1*M)-H/I W=G*H*B/2

LB1=L1-H1/SIN(ATAN(I))*SIN((ATAN(I)-Z1*M)/SIN((Z1+R)*M) LB2=L2-H2/SIN(ATAN(I))*SIN((ATAN(I)-Z1*M)/SIN((Z1+R)*M) R0=1.1*1000*A*(d0*d0)*fy/4 P1=A*D*LB1*E P2=A*D*LB2*E T1=P1 T2=P2

IF(T1>R0) THEN T1=R0 ENDIF

IF(T2>R0) THEN T2=R0 ENDIF

N=(W+Q*B)*COS(Z1*M)+(T1+T2)*SIN((Z1+R)*M)/SH K1=(N*TAN(F*M)+C*H/SIN(Z1*M)+(T1+T2)*COS((Z1+R)*M * )/SH)/((W+Q*B)*SIN(Z1*M)) IF (K1<=K) THEN K=K1 Z2=Z1 ENDIF ENDDO

WRITE(*,*) K,Z2 END

主要变量说明：K,K1——安全系数；I——墙的坡比；F——土体内摩擦角； T1,T2——土钉拉力; R——土钉的水平倾角；SH——土钉水平间距；H1,H2——土钉距离墙底距离；T1,T2——滑动面倾角；D——孔径；C——土体内聚力