part_1_elastic_deformation

1、Stressed and Strained StatesLi ChenhuiStress Stress is the load applied to a body and related per unit area of the bodys section. 应力应力是和物体单位表面单位表面上受到的载荷载荷。A relative quantity; 相对量The dimension of stress is determined as the force active per unit area of the body section to which the force is applied

2、. 其大小由受力物体单位表面载荷大小决定。Usually measured as newtons per square metre (N/m2) or kgf/mm2; 通常单位 注： kgf/mm2表示的是每平方毫米的面积上施加1kg力的压力，这个压强大约相当于10Mpa。The units of stress express the principal mechanical properties (ultimate strength强度极限, resistance to plastic flow塑性变形抗力, resistance to indentation压痕阻力, fatigue s

3、trength强度疲劳, creep strength蠕变强度, etc.) 应力的单位反映了它的力学性质The case of axial tension of a cylindrical rod) 圆柱体受轴向载荷的情况SdFPif S= constant (uniform distribution of the stress over the cross section)应力在横截面上均匀分布。P=SF or S=P/FIn a more general case The normal stress (正应力)The shear stress(剪应力)单轴拉伸的莫尔圆 Thus, if

4、we know the tensile force P applied to the rod and the cross-sectional area F. we can determine the normal and shear stresses in any plane making an arbitrary angle with the rod axis. The distribution of normal and shear stresses in variously oriented planes of a tensioned specimen are illustrated i

5、n Fig. 4. Engineering/Actual (True) Stress工程应力和真实应力F :force applied作用力;A0: area before deformation 变形前的面积 The engineering stress is often employed for elastic stresses or stresses for components deformed to small plastic strains. 工程应力通常应用于弹性应力或者适用于微小塑性应变下的应力 At large strains, the change in cross-sec

6、tional area significantly alters the actual stresses. 在大的应变下，横截面的改变会显著改变真实应力 The true stress is :where A is the instantaneous 瞬时的area. Strain Strain is the ratio of the change in dimension to its initial value. 应变是材料尺寸的变化量和它初始尺寸的比值。Axial tension of a cylindrical rod as; 圆柱体受到的轴向拉力Load applied;拉力加载Ro

7、d deformed, the length increased from l0 to ln; 杆开始变形，长度由l0伸长到lnengineering strain00 n工程应变=杆长度改变量/杆的原始长度 The engineering strain should be used only if the deformation strains are small in magnitude (e.g., eeng E for a tensile test, a result intuitively deduced previously. In contrast, if the materia

8、l were compressed so that the cross-sectional area increased during deformation (with E 0), we would find T E相应，对于挤压时，截面积增加： T E Which shows that T E in a tension test (i.e., ln(l + x) a0 ,and u is positive. In compression, a a0 and u a0 ， u 0；受压时a a0 ， u 0。 The equilibrium condition 等效计算式can be wri

9、tten, as follows: where (u) is the bond（结合） energy on displacement（位移） u. 其中 (u)是距离为u时原子间的作用能。 By analysing the system of two atoms, it is also possible to derive Hookes law which establishes the relationship between the external force applied and the resulting displacement. 研究两个原子间的作用机制，可以从本质上来探寻Ho

10、okes law，来研究外作用力和它所造成的位移的关系。For Hookes law to be valid（有效）, the following three conditions must be satisfied:胡克定律的应用有三个条件： (1) the function（函数） (u) must be continuous; 作用能(u)必须是连续的 (2) the function (u) must have a minimum d/du = 0 at u = 0; and 当u = 0时， d/du = 0， (u)必须具有最小值 (3) the displacement u mu

11、st be much less than a0. 变形量u必须远小于原子间的初始距离a0。 The first condition makes it possible to expand the interaction energy function into a Taylor series: 第一个条件允许把方程展开成Taylor式In this equation, 0 is the interaction energy at u = 0 and, all the derivatives are obtained for the point u = 0. 在等式中， 0是位移量u = 0处的

12、原子间初始能量，该等式是在u = 0处展开的。 Since d/du is equal to zero at u = 0, and, the terms with the third and higher powers of u can be neglected (as u is small), we obtain: 因为在u = 0处d/du ，并且由于u很小，所以三次微分和更高次可以被忽略，得到：The second derivative (d2/du2)o is the curvature（曲率） of the function (u) in point u = 0, and, ther

13、efore, it does not depend on u and is a constant.二次微分项是函数(u)在u = 0处的变化率，因此它不依赖于U，它是一个常数。Thus, we obtainf = const u, i.e. the force is proportional to displacement (Hookes law). 力与形变量成比例。 这就解释了为什么应力和应变对应成比例关系。 It should be recalled that the region of a direct proportionality between the force and dis

14、placement is limited to slight deformations. 应当提醒的是：应力应变线性关系只适用于微量变形中。 With an appreciable magnitude of displacement u, the terms of higher powers of u cannot be neglected and, therefore, the (u) curve deviates from the straight line. 当位移量u很大时， u的高阶幂不能忽略，那(u)就不是直线了。 This phenomenon is never encounte

15、red in practice, since an irreversible plastic deformation begins in metal even at lower stresses. The law of direct proportionality is then disturbed but for different reasons.在实际中，这个理想情况不可能遇到，因为塑性变形在极小应力下就发生了。因为这个原因（这里面有位错的原因）， 胡克定律就不适用了。Perfect thread-shaped metal crystals of a diameter of around

16、 2 um (called whiskers（晶须）), in which plastic flow is impeded（阻碍）, can, however, be deformed elastically by a few per cent and, at high elastic deformations, a deviation from Hookes law can be observed experimentally 直径为2 um的针状金属，加载载荷，当变形为百分之几的时候尽管里面已经发生了塑性变形但仍符合为弹性变形规律。如果再超过一定的变形量，就不符合胡克定律。在实验中可能观测

17、到右图：In shear stress The shear stress is related with a corresponding shear deformation by similar expression: 切应力对应一个切变 量，有相同的表达式： where G is the shear modulus (or the modulus of elasticity in shear) G 是切变模量。 (1-3)gtG In hydrostatic compression (or tension) 在流体拉（压）中 Hookes law expresses a drec直接直接 p

18、roportionality between the hydrostatic pressure P and the volume change x : 胡克定律揭示了流体压力P和体积变化量x间的关系 where K is the modulus o f b u l k （ 体 积 ）（ 体 积 ） deformation. K称为体弹模量vv (1-4)PK Hookes law (3) Formulae (1-2), (1-3) and (1-4) express what is called Hooks law. (1-2), (1-3) 和(1-4)公式是胡克定律。 Determines

19、 the relationship between stress and strain acting in the same direction 用来决定同方向上的应力应变间的关系。 When deformation appear in a direction different from that of the stress action, it does not work. 不适用于不同方向上的应力应变。 Elementary form 基本形式 nomenclature (1) Poissons ratio Isotropic Anisotropic Moduli Coefficient

20、 Polymorphous transformation Phase transformation术语（1） 泊松比 各向同性的 各向异性的 modulus的复数 系数 多形态转变 相变nomenclature （2） Recrystallization Substantially Preferable orientation Texture Radiographic Heterophase Anomaly, （ anomalies, anomalous） Peculiar Magnetic effect Elinvar术语（2） 重结晶 充分地 择优取向 织构 辐射照相的 异质相 （名）不规

21、则，异常的人或物 罕见的、特殊的；特权 磁效应 恒弹性镍铬钢Poissons ratioA rod subjected to uniaxial tension not only increases in length ( a change in the size along the axis X) but also diminishes in diameter (compression along the two other axes). Thus, a uniaxial stressed state results in a tridimensional deformation.一个杆受到轴

22、向拉伸后，长度增加，同时直径减小，因此一个轴向载荷造成的是一个三维的变形。 The ratio of the sizes change in the lateral (横向的)direction to their change in the longitudinal direction is called Poissons ratio: 截面方向尺寸的变化和长度方向尺寸的变化比为泊松比 v is Poissons ratio and is a material elastic property; the negative sign in Eq. indicates that the sampl

23、e dimensions normal to the primary extension decrease (increase) as the axial length of the sample increases (decreases). v是泊松比，是一材料的弹性性能参数。上式中的负号(正号)表明当杆受拉(压)时，其截面积减小(增加)。 For metals, the value of v is often on the order of 1/3. 对金属来说，v大约在1/3左右。 The change in volume associated with the small strain

24、s of linear elastic deformation can be obtained by differentiating the expression for the volume (V =l1l2l3) and keeping terms only to first order. The result is 应变造成的体积方面的变化，可以由体积计算公式V =l1l2l3得到，如下：For uniaxial deformation, V/V = (l - 2 ). Given that = 1/3, an elastic uniaxial strain of 0.5% would

25、produce a volume change of ca. 0.2%. Since linear elastic strains are typically smaller than this, the volume change during this type of deformation is usually quite small. 对于轴向变形而言V/V = (l - 2 )，当 = 1/3时，一个0.5%的轴向变形在体积方面造成的变形为0.2%。因弹性变形体轴向变形明显小于0.5% ，因此其体积的变形往往很小。The elastic volume change decreases

26、 as increases. For an incompressible material, such as a plastically deforming metal for which the volume change is zero, the ratio of lateral to uniaxial strain is 1/2. Such a value does not imply that , an elastic property, has a value of 0.5 for a metal during plastic deformation. 当v增加，体积变形减小。对于一

27、个不可压缩的材料，例如体积变化量为0的塑性变形的金属，截面应变对轴向应变的比为-0.5，但它并不表示塑性变形中泊松比为0.5long-chain polymers typically have values of v greater than metals. Hence, and as noted in the previous section, these m a t e r i a l s d i f f e r substantially from other linear elastic materials.长链聚合物泊松比明显大于金属因此，这些材料的材质和线性弹性材料有明显的区别。F

28、our elastic constants of an isotropic body基本上与价位、熔点呈线性关系Refractory metal 难熔金属Strong carbide forming metal 强碳化物形成金属Effect of various factors on elastic moduli对弹性模量的几种影响因素 Temperature温度 Work hardening加工硬化 Alloying合金化 Anomalous异常现象Temperature effect Since elastic moduli are associated with interatomic

29、forces and the latter depend on the distances between atoms in the crystal lattice, elastic constants depend on temperature. 由于弹性模量和原子间的作用力有关，而原子间的作用力依靠晶体点阵中原子间的作用距离，所以弹性模量和温度有关。 The temperature dependence of elastic moduli is very weak; As may be seen, the magnitude of modulus decreases with increa

30、sing temperature, with the E (T) relationship being almost linear. On the average, the elastic modulus decreases by 2-4 per cent by every 100C. 弹性模量对温度的依赖是非常微弱的，由上图可以看出，弹性常数随温度的增加而减小， E (T)曲线几乎成线性关系。平均来说，温度每增加100度，弹性常数减小24个百分点。 The temperature coefficient of the elastic modulus of a metal depends on

31、 the melting point of that metal. For that reason it is sometimes convenient to consider the dependence of the modulus on homologous（） temperature. In this presentation, the temperature relationship of the modulus is nearly linear. 一块金属的弹性模量的温度因数取决于该金属的熔点。因此可明确相同温度下弹性模量的变化规律。表述之，温度和弹性模量呈近似线性关系。 Empi

32、rical（经验主义的） correlation indicates that the appropriate scaling constant is about 100 (when SI units are used; i.e., kTm in J and in m3). Thus, 经验公式表明近似比例常数是大约100.K=Boltzmann constant, 波尔兹曼常数Tm=absolute melting temperature,熔点温度 =volume per atom 单个原子的体积 The modulus decreases concurrent（一致的） with the

33、increased atomic separation. This decrease is essentially linear with temperature, and an approximate equation describing the modulus-temperature relationship is当原子间距离增加时，弹性系数减小，这个减小和温度成线性关系。相应弹性系数和温度间的关系式为：where E is the modulus at temperature T and E0 the modulus at 0 K. The proportionality consta

34、nt a for most crystalline（透明的，水晶般的） solids is on the order of 0.5. Thus, for such a typical material, the modulus decreases by about 50% as the temperature increases from 0 K to the materials melting point.上式中E是温度T时的弹性模量，E0是温度为0K时的弹性模量，比例常数a对多数晶体而言大约是0.5，因此对于一个典型材料，当温度由0K增加到材料的熔点时弹性模量减小50%Alloying (

35、1)Alloying (2) in Al The effect of alloying on elastic constants, like the effect of temperature, can be associated with variations in the interatomic distances and interatomic forces in the crystal lattice. 合金对弹性系数的影响，就像温度的影响一样，和晶体点阵内的内部原子间隔距离和作用力有关。 As has been demonstrated in radiographic studies

36、, the lattice parameter（参数） of a solvent（溶剂） varies almost linearly with the concentration of an alloying element. The dependence of the elastic modulus of an alloy on the concentration of an alloying element is also close to linear. 在多项晶体的研究中已经证实：溶剂点阵常数因合金成分浓度的不同而近乎成线性变化。合金弹性系数和合金成分的浓度也接近线性关系。 As m

37、ay be seen from the figure, alloying can increase the elastic modulus in some cases and decrease it in others, depending on the relationship between the bond forces of atoms of the solute（溶质） and solvent（溶剂）. 从上面的数据可以看出，合金有时增加弹性模量，有时减小弹性模量，是取决于溶剂、溶质原子间的相互作用力。 on the one hand, and the forces of atomi

38、c interaction in the solvent lattice. 1.如果溶质溶剂原子间的作用力小于溶剂点阵中溶剂原子间的相互作用力，那么合金将减小弹性模量。 on the other, If the former are greater than the latter, alloying will increase the elastic moduli. 2.如果溶质溶剂原子间的作用力大于溶剂点阵中溶剂原子间的相互作用力，那么合金将增加弹性模量。 Apart from the variations of the interatomic forces in the lattice o

39、f the base component, alloying can also cause certain structural changes which can influence appreciably the magnitude of the elastic constants. 除了改变基底点阵中原子间的作用力外，合金也可以引起其结构的改变，这将显著改变弹性模量常数的大小。 For instance, if alloying above a definite limit results in the formation of a second phase, the elastic m

40、odulus may change additionally compared with its value in a single-phase solid solution. 例如如果合金超过一个有限的度就可以形成第二相，那么其弹性系数和单相时相比会发生显著变化。 If the second phase has a higher modulus than that of the base metal, its presence will increase the modulus of the heterophase（异相质） alloy. 如果第二相的弹性系数比基底金属大，它的出现将增加此异

41、相合金的弹性系数。Work hardening Work hardening has no essential effect on elastic moduli. A slight decrease of elastic moduli (usually below 1 percent) on work hardening is usually associated with distortions of the crystal lattice of a metal or alloy. 加工硬化自身（冷塑性加工）对弹性模量没有什么本质影响。冷塑性加工导致弹性模量轻微减小，常伴随着金属或合金晶体点

42、阵的畸变。 Plastic deformation can also cause some other structural change in the material.Work hardening can result in the formation of preferable orientations（择优取向）, or textures（织构）, which make the material anisotropic （各向异性）and can change substantially（充分地） the elastic moduli. 塑性变形会导致金属结构的改变。加工硬化能使晶体的

43、形成晶面的择优取向或织构，从而导致材料内部晶体结构各向异向，从而大大改变弹性模量。 Recrystallization during heating of a deformed metal also forms textures and changes appreciably the elastic moduli. 变形金属在加热过程中的重结晶也能形成织构，从而明显改变材料的弹性模量。 Variations in elastic moduli and due to the formation and destruction of preferable orientations may reac

44、h a few tens per cent. 择优取向的形成或减小，导致部分弹性系数的改变可能达到几十个百分点。 In textured polycrystalline materials, the magnitude of an elastic modulus depends on the direction of measurement. 在已形成织构的多晶体材料中，弹性系数的大小和测量的方向有关。Anomalous异常现象 Elinvar（） 镍铬钢 Magnetic（有磁性的） effects compensate（补偿） the normal drop of moduli with

45、temperature. 磁效应补偿了由于温度而减小的弹性模量。 The range of climatic variations of temperature. 气温（-50 50）变化范围下的异常现象。Review Stress (relative / engineering or actual /true) Strain (relative / engineering or actual /true) Hookes law Youngs modulus （Stiffness) Shear modulus Bulk modulus Shear strain Bulk Strain elas

46、tic moduli nomenclature (1） Anelasticity（） Hysteresis （） Microscopic Macroscopic Coordinates Thermodynamic Linearity Quasi-术语（1） n.滞弹性 n.滞后现象 微观的 宏观的 坐标 热力学的 线性 准、伪，类似nomenclature （2） Instantaneously Reciprocity Microplastically Macroplastically Hysteresis loop Elastic aftereffects Stress relaxation

47、 Internal friction Dissipate术语（2） 即时地，瞬时地 互惠 微观塑性（地） 宏观塑性（地） 滞后环 弹性后效 应力松弛 内摩擦、内耗 消耗Ideal elastic bodies理想弹性体 A unique relationship between stress and strain in the elastic region 弹性范围内应力和应变有精确关系。 Assumption: the load is increased infinitely slow so that the state of the system has the time to follo

48、w load variations. 假定：载荷无限慢地加载，体系状态能有足够的时间来产生应变。 Or: a change in the state of a system occurs instantaneously with a change in the load. 或者：载荷每一个点变化系统中都有一个实时的应变和它对应。 The process of loading and unloading can be regarded energetically reversible. 加载和卸载过程在能量上可认为是可逆的。Anelasticity滞弹体 In real bodies, the

49、direct relationship between stress an strain is disturbed and a hysteresis loop appears on the Stress-Strain diagram 在实际受力体中，应力和应变间的直接关系被破坏了，应力应变图中出现了一个滞后环。Stress-strain diagram in cyclic loading and unloading循环加载卸载中应力应变图 Anelasticity An irreversible dissipation of energy during the processes of loa

50、ding and unloading; 在加载和卸载中产生一个不可回复的能量损失。 The energy dissipated in one cycle is determined as the area of the hysteresis loop in the - coordinates and is the measure of internal friction in the material. 在一个循环中损失的能量由应力应变图中后滞环的面积来确定。其面积也是材料内耗的一个度量。 在弹性极限内应变落后于应力的现象称为滞弹性。Three different meanings of an

51、elastic deformation:滞弹性变形的三种情况 Anelastic deformation is possible without participation of dislocations;(below microscopic elastic limit) 1.滞弹性可能没有位错的参与。例如：它能在弹性极限下的应力发生。它的大小不符合胡克定律，滞弹性变形需要一段时间间隔才发生，并不是及时发生的。就这而言滞弹性现象和塑性变形有一点相似性。例如，刚卸载时，材料的直径和初始直径有一定的差别。但和塑性变形不同的是，过一段时间后，这种不同便会逐渐消失，最终没有任何残余变形被观察到。这种变

52、形应该称作“伪滞弹性”。 Anelastic deformation can be due to mechanically irreversible movement of dislocation;(between microscopic elastic limit and macroscopic elastic limit) 2.弹性后滞也可以解释为位错不可回复的运动而造成的。例如，当应力没有达到弹性极限时，有些位错就开始运动，但在到达晶体表面之前，就在晶体内被阻塞了。当卸载时，阻碍位错运动的力消失，它们就可以回到原来的位置，所以没有残余变形发生。 但任何位错的运动都会消耗能量，所以滞弹性现

53、象在能量上是不可回复的。就这种解释而言，滞弹性变形可以理解为可以回复的塑性变形。 At still higher stresses, movement of dislocations ceased（中止） to be mechanically reversible. 3.当应力增大到一定程度时，位错就不可回复，卸载后位错就不回到它原来的位置，一个可测量的变形出现了。无论经过多长时间弹性滞后环都不会在 =0，=0处合拢。这种情况，滞弹性变形在变形机制和卸载体积残余变化上都类似于塑性变形。我们应该知道的是在实际中，几种不同的滞弹性效应是同时发生的。在应力超过弹性极限时，伪滞弹性变形可以忽略，因为它

54、和总的滞弹性变形相比来说很小。Elastic aftereffects and stress relaxation弹性后效和应力松弛 把应力和应变的时效差异考虑在内的话，应描述为： (t)=M(t)where M is the static modulus of elasticity 其中M是弹性模量的状态参数.Relaxation at constant stress (a) and constant strain (b)Elastic aftereffects and stress relaxation (2) The gradual rise of strain in loading a

55、nd gradual disappearance upon unloading are called respectively the direct and the reverse elastic aftereffect. 加载时逐渐增加，卸载时逐渐减小的应变，称为直接可回复的弹性后效 The gradual variation of the stress to the value corresponding to Hookes law is called stress relaxation 应力逐渐变化到依胡克定律理论计算的应力大小称为应力松弛。Elastic and plastic strain in stress relaxation开始时，残余塑性变形为0，所以0=e1。塑性变形随时间而增加，而弹性变形随时间而减小。因为应力和弹性变形相伴出现，弹性减小时，应力减小，因此应力松弛出现了。 nomenclature （1） Bauschinger effect Inhomogeneuos Damping Precipitation Dissolution Amplitude Resonance Acoustic术语（1） 包申格效应