基坑工程弹性模型毕业论文中英文资料对照外文翻译文献综述

1、基坑工程弹性模型中英文资料对比外文翻译文献综述附录 1 外文翻译原文3.2.1Anisotropy 3.2Elastic models An isotropic material has the same properties in all directionswe 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

2、been deposited in some wayfor 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.

3、 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 directionsin other words to cause it to be anisotropi

4、c. We can write the stiffness relationship between elastic strain increment e and stress increment compactly as e D 3 . 36 1 whereD is the stiffness matrix and hence D is the compliance matrix. For a completely general anisotropic elastic material1 D1abcdef3 .37bghijkchlmnodimpqrejnqstfkortuwhereeac

5、hlettera,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,

6、1981. For example, for monoclinic symmetry z symmetry plane the compliance matrix has the form: D1abc00d3 .38bef00gcfh00i000jk0000kl0dgi00mand has thirteen elastic constants. Orthorhombic symmetry distinct x, y and z symmetry planes gives nine constants: D1abc0003 .39bde000cef000000g000000h000000iwh

7、ereas cubic symmetry identical x, y and z symmetry planes, together with planes joining opposite sides of a cube gives only three constants: D1abb0003 .40bab000bba000000c000000c000000c2 Figure 3.9: Independent modes of shearing for cross-anisotropic material bIf we add the further requirement that c

8、2aband set a1/Eand 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

9、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 an

10、isotropic 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 directi

11、ons look the same but horizontal sti.ness is expected to be di.erent from vertical stiffness. The form of the compliance matrix is now: D1abc0003 .41 bac000ccd000000e000000e000000fand we can write：a1/Eh,bvhh/Eh,cvvh/Ev,d1/Ev,e1/Gvh和f2 ab2 1vhh/Eh:3 D11/Ehhv hh/Ehv vh/Ev1/0vh1/0vh210/Eh3 .42vhh/E1/Eh

12、vv vh/Ev000vvh/Evvh/E1/E000vh000G000000G000000vhhThis is described as transverse isotropy or cross anisotropy with hexagonal symmetry. There are 5 independent elastic properties: E and E are Young moduli for unconfined compression in the vertical and horizontal directions respectively; G v h is the

13、shear modulus for shearing in a vertical plane Fig 3.9a.Poisson ratios V h h and V v h 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 anisotropi

14、c 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 Gvh1/E, since this would require controlled application of shear stresses to vertical and horizontal surfaces of the sampleand attendant rotation of princ

15、ipal 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：a1/E vv12v vh/Evva3 .43rvvh/Ev hh/ErT

16、he 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 determine E and Vvhthe only other elastic property that we can discover is the composite stiffnessEh/

17、1V hh.We are not able to separate E and VhhLings et al., 2022.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.4 D11/2v/

18、2v/021010v/21/2v/0001v/v/100021v/00E0000000v/00000E v02v/23 .44This is written in terms of a Young s modulusE,the Youngs modulus for loading in the vertical direction, a Poissons ratio V V hh ,together with a third parameter . The ratio of stiffness in horizontal and vertical directions isE h / E v

19、2and other linkages are forced：v vh v hh / ; G hv G hh / E / 2 1 v . For our triaxial stress and strain quantities, the compliance matrix becomes: p13GKJp3 .45qJqdetFigure 3.10: Effect of cross-anisotropy on direction of undrained effective stress path where det3 KGJ23 .46and the stiffness matrix is

20、 5 pK3Jp3 .47qqJGwhere KE19v4v223 .481v12 vGE262 v4v23 .491v12 vJE1vvv23 .503112 vThe stiffness and compliance matrices written in terms of correctly chosen work conjugate strain increment and stress quantities are still symmetricthe material is still elasticbut the non-zero off-diagonal terms tell

21、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 t

22、he slope of the effective stress path in an undrained test is, from 3.45, pJ321vvv223 .51 q3 G22v4From our definition of pore pressure parameter a 2.6.2 we find p3J3 .52qG6 Figure 3.11: Relationship between anisotropy parameter and pore pressure parameter a for different values of Poissons ratio . w

23、hich 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 ofa = 2/3 and a = -1/3 for very large（E hE ）and very small（EE ）repectively Fig 3.10, which in turn imply effective stress paths with constant axial e

24、ffective stress and constant radial effective stress respectively. The link between a and is actually slightly more subtle.In fact,for v 0 the relationship is not actually monotonic and the effective stress path direction overshoots the apparent limits Fig 3.11. The deduction of a value of and hence

25、 E hE v 2 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 3 a 3 a 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

26、Nonlinearity We will probably expect that the dominant source of nonlinearity of stress:strain response will come from material plasticityand we will go on to develop elasticplastic constitutive models in the next section. However, we also have an 7 expectation that some of the truly elastic propert

27、ies 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 materialsthe term hyperelasticity is used to describe such materialsmight make us a little cautious about plucking from the air arbitrary empirical

28、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 Poissons 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

29、that shown in Fig 3.12 energy would be created or lost creating a perpetual motion machine in violation of the first law of thermodynamicsthis 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 acc

30、eptable variation of moduli with stresses. Figure 3.12: Cycle of stress changes which should give zero energy generated or dissipated for conservative materialSuch a complementary energy function can be deduced from the nonlinear elastic model described by Boyce 1980: When K and G 1Vp n1n1K11q23 .53

31、16 Gpare reference values of bulk modulus and shear modulus and n is a nonlinearity parameter. The compliance matrix can then be deduced by differentiation ：8 n 1 n 2 n 2 1 nq pp n 1 K 11 6 Gn 1 3 G1 1 pq 3 . 54 3 G 1 3 G 1Where q / p .There is again as for the anisotropic model coupling between vol

32、umetric 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 pp0,q0the effective stres

33、s path is03 .55p 012npwhere 1nK1 6G 1；Contours of constant volumetric strainpare shown in 2. 17 values Fig 3.13 for n0 .2and Poissons ratiov0 .3implying K1G 1typical for the road sub-base materials being tested by Boyce for their small strain, resilient elastic properties. Similarly the path followe

34、d in a purely volumetric deformation q 0 will develop some change in distortional stress. For an initial state p p 0, q q 0 ,the effective stress path for such a test is qp0n13 .56q0pContours of constant distortional strain are also shown in Fig 3.13 for n = 0.2.9 Figure 3.13: Contours of constant v

35、olumetric 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 modulusof clays should be directly proportional to mean effective stress:Kp/k.Integration of this relationship shows th

36、at elastic unloading of clays produces a straight line response when plotted in a logarithmic compression planeplnv:lnp Fig 3.14 where v is specific volume. But s ratiowhat assumption should we make about shear modulus. If we simply assume that Poissonis constant, so that the ratio of shear modulus

37、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 Poissons rati

38、o for certa in 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: Uprep/kk323 .57q210 Figure3.14: Linear logarithmic relation

39、ship between and p for elastic material with bulk modulus proportional to pIncrementally this implies a stiffness matrix which, once again, contains off diagonal terms indicating coupling between volumetric and distortional elements of deformation: pp1/k/kp3 .58/kqqqIt can be deduced that q133 2kq23

40、 .59pqso that contours of constant distortional strain are lines of constant stress ratio Fig 3.15. Constant volume undrained stress paths are found to be parabolae Fig 3.15: q26kppp3 k/2.3 .60iiAll parabolae in this family touch the line11 Figure 3.15: Contours of constant volumetric strain solid l

41、ines and constant distortional strain dotted lines for nonlinear elastic model of Houlsby 1985The 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 a

42、ssumed that nonlinearity in soils comes exclusively from soil plasticityas will be discussed in the subsequent sectionswe 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 contai

43、n some element of elasticitywhich 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 princip

44、le 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 t

45、ricky. 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 12 provided the stress paths or strain paths to wh

46、ich 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 sca

47、le heterogeneityseasonal layeringmay 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 1.8glacial tills and residual soils often contain boulder-sized particles within an otherwise soil-like m

48、atrix. 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 veloc

49、ities 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 particleswhich will have a much higher stiffness and hence higher shear wave velocity than the surrounding soil Fig 3.16. The

50、receiver will show the travel time for the fastest wave which has taken this heterogeneous route. If the hard material occupies a proportion of the spacing between source and receiver, and the ratio of shear wave velocities is k and hence, 2 neglecting density differences, the ratio of shear moduli

51、is of the order of k , then the ratio of apparent shear wave velocity V s to the shear wave velocity of the soil matrix V s is V s k 3 . 61 V s k k13 Figure 3.16: a Soil containing boulders between boreholes used for measurement of shear wave velocity; b average stiffnesses deduced from interpretati

52、on of shear wave velocity and from matrix stiffness The deduced average shear modulus G the soil matrix G by the ratio is then greater than the shear stiffness of Gkk2112as k3 .62GkLaboratory testing of such heterogeneous materials is not easy because the test apparatus needs itself to be much large

53、r 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 2022 report tests to explore mechanical characteristics of mixtures of kaolin clay and a fine graveld50=2mm. They found that all the properties of the

54、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. For0.5 then, this implies a ratio of equivalent shear stiffness Gto soil matrix stiffn

55、ess G : G113 .63GThese two expressions, 3.62 and 3.63, are compared in Fig3.16 for a modulus ratio k21000014 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； 分层的范畴可能会特别小, 我

56、们不期望区分不同材料, 但在不 同方向的分层可能仍是足以转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De3 .36其中 D 是刚度矩阵,因此D1是柔度矩阵；对于一个完全整体各向异性弹性材料abcdefD1bghijkchlmno3 .37dimpqrejnqstfkortu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削减了（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：abc00dbef0

57、0gD1cfh00i3 .38000jk0000kl0dgi00m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：15 D1abc0003 .39bde000cef000000g000000h000000i然而,立方体对称性（同一的 一起）只给出三个常数：x 、 y 、 z 对称面,与立方体相反面结合的面D1caabba000v/E3 .40bab000bba000000c00假如我们进一步要求0000c0,那么我们发觉00000c2b和设1 /E和b（3.1）的各向同性弹性柔度矩阵；不过,假如岩土工程材料具有肯定的组构对称性,削减独立弹性参数的数量,明显是很便利的

58、,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,我们应当期望要求全部 21 个弹性参数独立；假如我们挑选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, 那么我们不得不辨论到这是对土体和岩石可能不 明白的建模结果；然而,很多土都在横向范畴区域内沉积,沉积的对称性基本上是垂直的；现在柔度矩阵 从全部水平方向看是一样的, 但横向刚度估计将不同于垂直刚度；的形式为 : 16 D1abc0003 .41 bac000ccd000000e000000e000000f并且我们可以写为：a1/Eh,bvhh/Eh,chvvh/Ev

59、,d1/Ev,e1/Gvh和f2 ab2 1vhh/Eh:1/Ehv hh/Ev vh/Ev000v hh / E h 1 / E h v vh / E v 0 0 01 v vh / E v v vh / E v 1 / E h 0 0 0D 3 . 42 0 0 0 1 / G v h 0 00 0 0 0 1 / G v h 00 0 0 0 0 2 1 v hh / E h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 个独立的弹性参数 : E 和 E 分别是垂直向和水平向不密闭压缩的杨氏模量；G v h 是一个垂直面上的剪切模量 图 3.9a；泊松比 V h h 及 V v

60、 h 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觉查实 G v h 1 / E,由于这要求掌握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；假如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,我们发觉：a1/Evv12 vvh/Evva.343 rvvh/Evhh/Er柔度矩阵不是对称的, 由于在三轴试验环境中, 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 E 和 V v h 时,我们