彈性模型外文翻譯

1、<p>　　附錄1 外文翻譯原文</p><p>　　Elastic models</p><p>　　Anisotropy</p><p>　　An isotropic material has the same properties in all directions—we cannot dis-tinguish any one direction

2、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 direc

3、tion 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 diffe</p><p>　　We can write the stiff

4、ness relationship between elastic strain increment and stress increment compactly as </p><p>　　whereis the stiffness matrix and henceis the compliance matrix. For a completely general anisotropic elastic m

5、aterial</p><p>　　whereeachlettera,b,... is,inprinciple,anindependentelasticpropertyandthe necessary symmetry of the sti?ness matrix for the elastic material has reduced the maximum number of independent prop

6、erties to 21. As soon as there are material symmetries then the number of independent elastic properties falls (Crampin, 1981).</p><p>　　For example, for monoclinic symmetry (z symmetry plane) the compliance

7、 matrix has the form:</p><p>　　and has thirteen elastic constants. Orthorhombic symmetry (distinct x, y and z symmetry planes) gives nine constants:</p><p>　　whereas cubic symmetry (identical x,

8、 y and z symmetry planes, together with planes joining opposite sides of a cube) gives only three constants:</p><p>　　Figure 3.9: Independent modes of shearing for cross-anisotropic material</p><p

9、>　　If we add the further requirement that and set and ，then we recover the isotropic elastic compliance matrix of (3.1).</p><p>　　Though it is obviously convenient if geotechnical materials have certain

10、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 pos

11、sess 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 </p>&

12、lt;p>　　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?eren

13、t from vertical stiffness. The form of the compliance matrix is now: </p><p>　　and we can write：</p><p>　　This is described as transverse isotropy or cross anisotropy with hexagonal symmetry. Th

14、ere are 5 independent elastic properties: andare Young’s moduli for unconfined compression in the vertical and horizontal directions respectively; is the shear modulus for shearing in a vertical plane (Fig 3.9a).Poisson’

15、s ratios and 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 (Fi</p><p>　　Testi

16、ng 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 ，since this would require controlled application of

17、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 stre

18、sses and strains for a sample with its vertical axis of</p><p>　　The compliance matrix is not symmetric because, in the context of the triaxial test, the strain increment and stress quantities are not proper

19、ly work conjugate. We deduce that while we can separately determineand the only other elastic property that we can discover is the composite stiffness.We are not able to separateand (Lings et al., 2000).</p><p

20、>　　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

21、anisotropic material. </p><p>　　This is written in terms of a Young’s modulus,the Young’s modulus for loading in the vertical direction, a Poisson’s ratio ,together with a third parameter . The ratio of stif

22、fness in horizontal and vertical directions is and other linkages are forced： </p><p><b>　　.</b></p><p>　　For our triaxial stress and strain quantities, the compliance matrix becomes

23、: </p><p>　　Figure 3.10: Effect of cross-anisotropy on direction of undrained effective stress path</p><p><b>　　where</b></p><p>　　and the stiffness matrix is</p>

24、<p><b>　　where</b></p><p>　　The stiffness and compliance matrices (written in terms of correctly chosen work conjugate strain increment and stress quantities) are still symmetric—the materi

25、al 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, ,distortio

26、nal strain during purely isotropic compression, ,and there will be change in mean effective stress in undrained tests, </p><p>　　In fact the slope of the effective stress path in an undrained test is, from (

27、3.45),</p><p>　　From our definition of pore pressure parameter a (§2.6.2) we find</p><p>　　Figure 3.11: Relationship between anisotropy parameter α and pore pressure</p><p>　　p

28、arameter a for different values of Poisson’s ratio .</p><p>　　which will, in the presence of anisotropy, not be zero.</p><p>　　A first inspection of (3.51) merely suggests that there are limits

29、 on the pore pressure parameter of a = 2/3 and a = -1/3 for very large（＞＞）and very small（＞＞）repectively (Fig 3.10), which in turn imply effective stress paths with constant axial effective stress and constant radial ef

30、fective stress respectively. The link between a and α is actually slightly more subtle.In fact,for the relationship is not actually monotonic and the effective stress path direction overshoots the apparent limits (</p

31、><p>　　Nonlinearity</p><p>　　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 elasticplasti

32、c 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

33、 thoughts about elastic materials as conservative materials—the term ‘hyperelasticity’ is used to describe such </p><p>　　Figure 3.12: Cycle of stress changes which should give zero energy generated or</p

34、><p>　　dissipated for conservative material</p><p>　　Such a complementary energy function can be deduced from the nonlinear elastic model described by Boyce (1980):</p><p>　　When and

35、are reference values of bulk modulus and shear modulus and n is a nonlinearity parameter. The compliance matrix can then be deduced by differentiation：</p><p>　　Where.There is again (as for the anisotropic m

36、odel) coupling between volumetric and distortional effects. The stiffnesses are broadly proportional to . </p><p>　　Because the compliances are now varying with stress ratio the effective stress path implie

37、d for an undrained (purely distortional) loading is no longer straight. In fact, for a reference state the effective stress path is </p><p>　　where。Contours of constant volumetric strain are shown in Fig 3.1

38、3 for and Poisson’s ratio implying —values typical for the road sub-base materials being tested by Boyce for their small strain, resilient elastic properties.</p><p>　　Similarly the path followed in a purely

39、 volumetric deformation will develop some change in distortional stress. For an initial state ,the effective stress path for such a test is</p><p>　　Contours of constant distortional strain are also shown in

40、 Fig 3.13 for n = 0.2.</p><p>　　Figure 3.13: Contours of constant volumetric strain (solid lines) and constant</p><p>　　distortional strain (dotted lines) for nonlinear elastic model of Boyce (1

41、980)</p><p>　　It is often proposed that the elastic volumetric stiffness—bulk modulus—of clays should be directly proportional to mean effective stress: .Integration of this relationship shows that elastic u

42、nloading of clays produces a straight line response when plotted in a logarithmic compression plane(Fig 3.14) where v is specific volume. But what assumption should we make about shear modulus? If we simply assume that P

43、oisson’s ratio is constant, so that the ratio of shear modulus to bulk modulus is const</p><p>　　Houlsby (1985) suggests that an acceptable strain energy function could be:</p><p>　　Figure3.14:

44、Linear logarithmic relationship between and for elastic material</p><p>　　with bulk modulus proportional to </p><p>　　Incrementally this implies a stiffness matrix which, once again, contains

45、off diagonal terms indicating coupling between volumetric and distortional elements of deformation:</p><p>　　It can be deduced that</p><p>　　so that contours of constant distortional strain are

46、lines of constant stress ratio η(Fig 3.15). Constant volume (undrained) stress paths are found to be parabolae (Fig 3.15):</p><p>　　All parabolae in this family touch the line.</p><p>　　Figure 3

47、.15: Contours of constant volumetric strain (solid lines) and constant</p><p>　　distortional strain (dotted lines) for nonlinear elastic model of Houlsby (1985)</p><p>　　The nonlinearity that ha

48、s been introduced in these two models is still associated with an isotropic elasticity. The elastic properties vary with deformation but not with direction.</p><p>　　Although it tends to be assumed that nonl

49、inearity 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 thermodynami

50、cally 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 plasti

51、c shearing will the</p><p>　　Heterogeneity</p><p>　　Anisotropy and nonlinearity are both possible departures from the simple assumptions of isotropic linear elasticity. A rather different depart

52、ure 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 m

53、an-made soils contain large ranges of particle sizes (§1.8)—glacial tills and residual soils often contain boulder-sized particles within </p><p>　　If we attempt to measure shear wave velocities in situ

54、, 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

55、 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

56、spacin</p><p>　　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 ma

57、trix stiffness</p><p>　　The deduced average shear modulus is then greater than the shear stiffness of the soil matrix by the ratio</p><p>　　as </p><p>　　Laboratory testin

58、g 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 sm

59、all scale, Muir Wood and Kumar (2000) report tests to explore mechanical characteristics of mixtures of kaolin clay and a fine gravel(=2mm). They found that all the properties of the clay/gravel system were controlled by

60、 the soil matrix until the volume fraction </p><p>　　These two expressions, (3.62) and (3.63), are compared in Fig3.16 for a modulus ratio </p><p><b>　　附錄2 外文翻譯</b></p><p

61、><b>　　3.2 彈性模型</b></p><p><b>　　3.2.1各向異性</b></p><p>　　各向異性材料在各個方向具有同樣的性質—我們不能將任何一個方向與任何其他方向區(qū)分開。從地下任何地方取出的試樣都表現出個性。然而，我們知道土已經以某種方式沉積—例如，沉積性土在垂直方向受重力作用而沉積。另外，沉積速度可能呈季節(jié)變化

62、，所以土體或多或少地包含了顆粒尺寸或可塑性略微相異的標志性土層。分層的范圍可能會非常小，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質—換句話說就是造成其各向異性。</p><p>　　我們可以將彈性應變增量和應力增量的剛度關系簡寫為</p><p>　　其中是剛度矩陣，因此是柔度矩陣。對于一個完全整體各向異性彈性材料</p><p>

63、　　其中，每個字母，，...是，在原理上是一個獨立的彈性參數，彈性材料剛度矩陣必要的對稱性已推導出獨立參數的最大值為21。一旦存在矩陣對稱性，獨立彈性參數的數量就減少了（克蘭平，1981）。</p><p>　　例如，對于單斜對稱（對稱面）柔度矩陣有形式如下：</p><p>　　有13個彈性常數。正交對稱（區(qū)分、、對稱面）給出9個常數：</p><p>　　然而，

64、立方體對稱性（同一的、、對稱面，與立方體相反面結合的面一起）只給出三個常數：</p><p>　　如果我們進一步要求和設和，那么我們發(fā)現（3.1）的各向同性彈性柔度矩陣。</p><p>　　不過，如果巖土工程材料具有一定的組構對稱性，減少獨立彈性參數的數量，顯然是很方便的，正如料想的那樣，受構造力、冰、或人推動的大部分材料，將不再擁有任何這類對稱性，只要有一個域的彈性反應，我們應該期望要

65、求全部21個彈性參數獨立。 如果我們選擇將這樣的材料建模成伴有某些限制對稱性的各向同性彈性或各向異性彈性，那么我們不得不分辨到這是對土體和巖石可能不了解的建模結果。</p><p>　　然而，許多土都在橫向范圍區(qū)域內沉積，沉積的對稱性基本上是垂直的。從所有水平方向看是一樣的，但橫向剛度預計將不同于垂直剛度?，F在柔度矩陣的形式為: </p><p><b>　　并且我們可以寫為：&

66、lt;/b></p><p>　　這被形容為橫向各向同性或六邊形對稱的交叉各向異性。有5個獨立的彈性參數: 和分別是垂直向和水平向不密閉壓縮的楊氏模量；是一個垂直面上的剪切模量(圖3.9a)。泊松比及分別是與發(fā)生在正交于壓縮的橫向方向和壓縮的垂直方向的水平方向上的橫向應變有關(圖3.9，)</p><p>　　主軸與儀器軸平行三軸儀的交叉各向異性土的試驗，并沒有給我們任何可能性發(fā)現查

67、實，因為這要求控制施加對試樣垂直和水平面上的剪應力。事實上，我們只能確定5個彈性參數中的3個。如果我們對于垂直軸與三軸儀主軸平行的試樣，就徑向和軸向的應力和應變書寫(3.42)，我們發(fā)現： </p><p>　　柔度矩陣不是對稱的，因為在三軸試驗環(huán)境中，應變增量和應力增量不是完全共軛的。我們推出：當我們可以分別確定和時，我們可以得到的僅有的另外一個彈性參數是一個復合剛度。</p><p>

68、　　我們不能將和分離開（林斯等，2000）。</p><p>　　另一方面，格拉漢姆和豪斯貝（1983）提出了（3.41）或（3.42）得特殊形式，只用了3個彈性參數，但對于此交叉各向異性材料，要求5個彈性參數是相互依賴的。 </p><p>　　這是書寫的楊氏模量，在垂直方向楊氏模量，泊松比， 連同第三個參數。在水平和垂直方向的剛度比是及其他約束關系： </p><p

69、><b>　　。</b></p><p>　　對于我們的三軸應力和應變量，柔度矩陣變?yōu)椋?</p><p><b>　　其中</b></p><p><b>　　并且，剛度矩陣是</b></p><p><b>　　其中</b></p>

70、<p>　　剛度和柔度矩陣（以正確選用工作共軛應變增量和應力增量方式書寫）依然是對稱的—材料依然是彈性的—但非零非對角線計算告訴我們體積作用和剪切作用之間是耦合的。進行純粹的各向同性壓縮試驗時，，當我們施加純剪力和剪應變時，將產生體積應變，，不排水試驗的平均有效應力將會改變，。</p><p>　　實際上，不排水試驗的有效應力路徑的斜率，形式（3.45）</p><p>　　從

71、我們對孔壓參數 (§2.6.2)的定義中,我們發(fā)現</p><p>　　在各向異性存在時，不會為零。</p><p>　　第一次研究(3.5.1)僅僅表明對于孔壓參數有限制， 非常大（＞＞）和非常?。ǎ荆荆r(圖3.10)分別為和，而這表示了依次施加恒定軸向有效應力和恒定徑向有效應力的有效應力路徑。和α之間的聯系實際是較為含蓄的。 事實上，對于，其關系其實并不單調，并且有效應力路

72、徑方向超出了明顯的界限 (圖3.11)。當在或2/3附近取值時(回憶介紹的數據圖2.51和2.49，§2.5.4)，從推導得到的α (因而)不是很可靠的。對于，或。這些關系符合和的預期范圍，但對于和， (3.51)有奇異的倒轉。</p><p><b>　　3.2.2非線形</b></p><p>　　我們大概預想的應力非線性的主要來源:應變反應將來自材料的

73、可塑性—并且下部分，我們將繼續(xù)發(fā)展彈塑性本構模式。 不過，我們也期待一些真正有彈性性質的土體將隨應力水平而變化，這可以看作彈性非線形的一個來源。我們把彈性材料作為保守材料—“超彈性”一詞是用來形容這種材料的-可能使我們在選取隨應力變化模量的任意經驗函數時更加謹慎。 舉例來說， 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，我們會發(fā)現,在圖3.12如示的一封閉的應力循環(huán)中，違反熱力學第一

74、定律創(chuàng)造一個永動機，能量將增加(或失去)，這不會是一個保守體系。我們必須找到一種應變能(3.7)或補充能量密度(3.11)函數，可以通過微分得到可接受的應力模量變量。 </p><p>　　這樣的補充能量密度函數能夠從鮑耶斯（1980）描述的非線形彈性模型中推導得到：</p><p>　　其中和是體積模量和剪切模量的參考值，是非線性參數。柔度矩陣然后可以通過微分導出：</p>

75、<p>　　其中。體積作用和剪切作用之間再次是共軛的（對于各向異性模型）。剛度是與廣泛成比例的。 </p><p>　　因為柔度現在是隨著應力比變化的，對于不排水（純剪切）加荷的有效應力路徑不再是直線。實際上，對于提到的情況，有效應力路徑為</p><p>　　其中。對于和泊松比，意味的常體積應變曲線，如圖3.13所示—由鮑耶斯對小應變回彈彈性參數測試的路基材料得到的典型值。&

76、lt;/p><p>　　同樣地，在純體積變形將使剪應力發(fā)生一些變化。對于初始狀態(tài) ，這個試驗的有效應力路徑為</p><p>　　的常剪切應變曲線如圖3.13所示。</p><p>　　經常提出的是粘土的彈性體積剛度—體積模量應當與平均有效應力直接成比例： 。當以對數壓縮平面（圖3.14 ）作圖時，其中是比容，這種關系的結合顯示粘土的彈性卸載形成一條直線反應。但我們要對

77、剪切模作什么假設呢？如果簡單地以為泊松比為常數，那么，剪切模量和體積模量的比值是常數, 那么我們將發(fā)現一個非保守物質(扎廷斯基等，1978)。 如果我們假定恒定的剪切模量值，獨立的應力水平，我們將獲得一個保守的材料，但也許會發(fā)現，我們泊松比在某種高或低應力水平呈現令人吃驚的值。再次，我們必須找到一種應變或補充能量函數，會給我們期望的基本模量變化。</p><p>　　豪斯柏（1985）建議一個可以接受的應變能函數

78、可為：</p><p>　　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角線量。</p><p><b>　　可以導出</b></p><p>　　所以常剪切變形的圖形為常應力比的直線（圖3.15）。常體積（不排水）應力路徑為拋物線（圖3.15）： </p><p>　　這組中所有的拋物線與線相切

79、。</p><p>　　這兩種模型中所引入的非線性依然與各向同性彈性相聯系。彈性參數隨應變而變化，而不是隨方向變化。</p><p>　　盡管它往往被猜想為土的非線性很多是來自土的可塑性-將在隨后一節(jié)中討論，我們從中看到謹慎地或許可以一種在熱力學上可以接受的方式來說明一些彈性非線性。同樣，大多數彈塑性模型將包含某種彈性-其中往往充滿塑性變形。但可以預料的是，其中伴有任何塑性剪切的結構變化將

80、導致土體彈性性質變化。這種剛度變化的方程式，原則上應當是基于微分偶然的發(fā)現彈性應變能量密度函數，因此彈性不應違反熱力學定律。很明顯的，允許各向異性彈性剛度演化的應變能函數的發(fā)展是棘手的。許多構模式，采取了務實的超彈性方式和單純定義應力狀態(tài)模量或不關心熱力后果的應變狀態(tài)。這可能不引起特定問題除非土體經歷的應力路徑或應變路徑不太反復循環(huán)。</p><p><b>　　3.2.3非均質性</b>&

81、lt;/p><p>　　各向異性和非線性均可能偏離簡單各向同性線彈性的假設。 一個頗為不同的偏離是與非均質性有關的。我們已經注意到小規(guī)模的非均質性-季節(jié)性分層—可能導致在典型樣本的范圍內剛度(和其他)性能的各向異性。許多天然及人工土中含有很大的粒徑變化范圍(§1.8)—冰磧和殘積土中往往在不同土樣基質中含有漂石顆粒。如果地質系統(tǒng)的規(guī)模比這些漂石的尺寸和空隙大，那么將這種材料看作基本上均勻，是理由充分的。但是

82、，我們仍希望確定其性能。 </p><p>　　如果我們試圖以地球物理勘探技術測定原位剪切波速，那么我們可以預料從發(fā)射源到接收器的最快波將利用大塊堅硬的巖樣顆粒的存在—將具有更高的剛度和因此比周圍土體(圖3.16)更高的剪切波速。接收機將顯示通過這種非均質路線最快波的旅行時間。如果堅硬材料在發(fā)射源和接收機間距中占有比例，剪切波速度比為(因此,忽略了密度的差異，剪切模量比是量級)，那么顯剪切波速與土基質剪切波速的比

83、值是 </p><p>　　推導出的平均剪切模量比土基質的剪切剛度大一些，比值為</p><p>　　非均質材料的實驗室試驗是不容易的，因為試驗儀器需求比典型的最大粒徑和間隔要大很多，以測得一個真實的平均參數。在一個小規(guī)模范圍內， 繆意爾·伍德及庫馬爾(2000)報告了探索高嶺土和細砂礫 (=2毫米) 混合物機械特性的試驗。他們發(fā)現：粘土/砂石系統(tǒng)的所有性能受土基質控制，直至該砂