Introduction to Polymer Physics

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1 Introduction to Polymer Physics
Prof. Dr. Yiwang Chen School of Materials Science and Engineering, Nanchang University, Nanchang

2 Chapter 7 Mechanical Properties 7
Chapter 7 Mechanical Properties 7.1 Methods of Testing and Mechanical properties Static Testing Static tests refer to those for which the deformation rate is ready in time. While tensile, compressive, or shear modes may be employed, tensile testing is the most common. In a typical tensile test, a polymer sample, in the form of a dogbone, is clamped at one end and pulled at a constant rate of elongation at the other clamped end. The thinner portion of the tensile specimen encourages the sample to fail at the center of the bar, where the stress is the highest, and not a the grip sites, where stress concentration may otherwise result in premature failure.

3 The initial length of a central section contained within the narrow region of the tensile specimen is called the initial gage length, l0. During deformation, force, F, is measured as a function of elongation at the fixed end by means of a transducer. Usually, the tensile response is plotted as engineering (nominal) stress, , versus engineering (nominal) strain, , where and

4 A0 is the original (undeformed) cross-sectional area of the gage region and l is the change in sample gage length (l-l0) due to the deformation. Sample length can be determined from instrumental setting of the mechanical-testing instrument or by an extensometer which is a strain gage that is attached to the gage-length region of the tensile specimen.

5 Alternately, the stress-strain response of a sample may be reported in terms of true stress and true strain. The true stress is defined as the ratio of measured force to the actual cross-sectional area, A, at a given elongation Since the actual cross-sectional area decreases as the sample is elongated, the true stress will always be larger than the engineering stress. Assuming that the volume of the sample remains constant during deformation, it can be shown that the true stress is simply related to the engineering stress as The true strain, ’, is defined as

6 With the exception of elastomers, the assumption of a constant volume during deformation is not strictly correct, because the volume of glassy polymers increases, or dilates, during extension. This change in volume, V, at a given stain may be calculated from the relation Where V0 is the initial (unstrained) volume,  is true strain, and  is called Poisson’s ratio which is defined as the ratio of true strain in the transverse direction, T, to the true strain in the longitudinal direction, L, and is calculated as

7 For the majority of glassy polymers, 0
For the majority of glassy polymers, 0.4, as shown by data for polystyrene, poly(methyl methacrylate), and poly(vinyl chloride). For completely incompressible materials, for which the term (V/) within brackets is zero,  obtains its maximum value of 0.5 as approached by natural rubber and low-density polyethylene (=0.49).

8 Hooke’s law for an ideal elastic solid provides a relationship between stress and strain for tensile deformation as Where the proportionality factor, E, is called the tensile (or Young’s) modulus. Conversely, the strain and stress are related by the tensile compliance, D, defined by For tensile deformation, the compliance is therefore the reciprocal of the modulus

9 The point at which stress begins to deviate from a linear stress-strain relation is called the proportional limit. This normally occurs before 1% strain. Therefore, to designate a value for the modulus, a convenient procedural definition must be adopted. Consequently, the initial slope of the stress-strain curve is called the initial modulus. The modulus, or the compliance, is a material property that is a function of both temperature and the time scale of the deformation.

10 In addition to tensile deformation, samples may be compressed or sheared. In the case of compression, the sample is typically prepared in the form of a disk. The measured parameters are the bulk modulus, B, and compliance, K.

11 The engineering shear stress, s, is defined as
Where A0 is the area of the surface on which the shear force acts. The shear strain, , is given by the angle of deformation, , as Hooke’s law for shear deformation is given as Where G is the shear modulus, while the shear compliance is normally designated as J (1/G).

12 Moduli obtained in different deformation modes (tensile, compression, and shear) may be interrelated through Poisson’s ratio, . For isotropic materials, the following relationships hold: And In the limit of incompressibility (i.e., =0.5), the equations reduce to and

13 弹性模量 对于理想的弹性固体，应力与应变成正比，比例常数称为弹性模量。 衡量材料抵抗变形能力 三种基本形变类型对应的弹性模量为： ⑴. 简单拉伸: 杨氏模量 (Younger’s Modules) ⑵. 简单剪切: 剪切模量 (Shear Moldules) ⑶. 均匀压缩: 体积模量 (Volume Modules) 拉伸柔量 剪切柔量 可压缩度

14 大多数材料在变形时，有体积变化，拉伸时发生体积膨胀：
各向同性的材料，三种模量的关系为：  为泊松比，在拉伸实验中，材料横向单位宽度的减小与纵向单位长度的增加的比值。 理想不可压缩变形时，体积不变： 大多数材料在变形时，有体积变化，拉伸时发生体积膨胀：

15 对于各向异性的材料，其在各个方向上有不同的性质，因而有不止两个独立的弹性模量，其数目决定于体系的对称性。

16 机械强度 (Mechanical strength)
所受外力超过了材料的承受能力，材料就要发生破坏，机械强度就是材料抵抗外力破坏的能力。 在各种实验应用中，机械强度是材料力学性能的重要指标。对于各种不同的破坏力，有不同的强度指标。

17 几种常用的力学性能指标 1.拉伸强度: (Tensile Strength) 在规定温度、湿度和一定速度的情况下，在标准试样上沿轴向施加拉伸载荷，直到试样断裂为止。 拉伸强度： 拉伸模量：

18 压缩强度: (Compressible Strength)
在试样上施加压缩载荷至其破裂（脆性材料）或产生屈服现象（非脆性材料）时，原单位横截面上所能承受的载荷称为压缩强度。 压缩强度： 压缩模量：

19 弯曲强度: (Flexural Strength)
在两支点间的试样上施加集中载荷，使试样变形直至破裂时的载荷称为弯曲强度。 弯曲强度： 弯曲模量： P b d P/2 P/2

20 试样弯曲变形时的杨氏模量表式

21 冲击强度: (Impact Strength)
在高速冲击状态下,标准试样在断裂时,单位面积所吸收的能量。衡量材料韧性指标 冲击强度： 摆锤式、落重式、高速拉伸

22 硬度: (Hardness) 硬度的大小与材料的抗张强度和弹性模量有关，其是衡量材料表面抵抗机械压力能力的一种指标。

23 Mechanical Behavior of Polymers
Polymers exhibit a wide range of mechanical behavior depending upon temperature and rate of deformation. Typical stress-strain curves covering this range of behavior are illustrated in Figure. At normal use temperatures, brittle polymers (e.g., polystyrene) exhibit a rapid increase in stress with increasing strain (i.e., high modulus) up to the point of sample failure (curve 1). The stress at failure is called the ultimate stress (u) or stress-at-break (b).

24 Unlike modulus, ultimate stress, resulting from large and irreversible deformation, is a sample rather than material property and is strongly influenced by sample defects and processing history. For this reason, a sufficiently large number of samples must be evaluated and their values averaged in order to get a statistically meaningful value.

25 Ductile polymers, including many engineering thermoplastics, polyamides, and toughened (rubber-modified) plastics, exhibit stress-strain behavior represented by curves 2 and 3. As shown, the stress reaches a maximum value, which is called its yield stress, y, at a certain strain, y. As strain is further increased, stress at first decreases. This process is called strain softening, which usually occurs at strains between 5% and 50%. A minimum in stress reached during strain softening is called the draw stress. At this point, the sample may either fail (curve 2) or experience orientation hardening (curve 3) prior to failure.

26 During orientation hardening, polymer chains are stretched locally in the tensile direction. Chain extension causes a resistance to further deformation; stress is, therefore, observed to increase. Accompanying the molecular processes that are occurring during orientation are macroscopic changes in the shape of the tensile specimen. Above the yield point, a portion of the tensile dogbone begins to locally decrease in width or neck within the gage region. If orientation hardening occurs before sample failure, the neck is said to stabilize. This means that no further reduction in a cross-sectional area occurs and the neck propagates along the length of the gage region until the sample finally breaks. This process of neck propagation is called cold drawing.

27 The initial slope of the stress-strain plot for ductile polymers (curves 2 and 3) is smaller than that observed for polymers that fail in a brittle mode (curve 1). In other words, the modulus of ductile polymers is lower. On the other hand, the energy required to deform the sample to the point of failure is much higher for ductile polymers, as indicated by comparison of the areas under the stress-strain curves for a brittle (curve 1) and for a ductile polymer (curve 3). This means that ductile polymers are able to absorb more energy upon impact.

28 Rubbery polymers follow stress-strain behavior similar to that of curve 4. Modulus is low, but ultimate extension can be very high, on the order of several hundred percent. Before failure, the rubber may experience an increase in stress as a consequence of strain-induced crystallization caused by molecular orientation in the stretch direction.

29 The exact nature of the tensile response (modulus, yield strength (stress-at-yield), ultimate strength (stress-at-break), and elongation-to-break) of a polymeric material depends upon the chemical structure of the polymer, conditions of sample preparation, molecular weight, molecular-weight distribution, crystallinity, and the extent of any crosslinking or branching. The mechanical response also depends in a very significant way on temperature and the rate of deformation. Any amorphous polymer can exhibit the entire range of tensile behavior, from brittle to rubbery response, by increasing the testing temperature from room temperature to above the Tg of the polymer or (to a lesser extent) by decreasing the rate of deformation.

30 玻璃态高聚物的强迫高弹形变 玻璃态高聚物，外力作用的松弛时间与应力的关系: 影响强迫高弹形变的因素: 外力的大小 温度的影响 作用速度的影响 结构因素的影响 分子量的大小也有影响 脆化温度：其是一个特征温度，用 Tb 表示，当温度低于 Tb 时，玻璃态高聚物不能发生强迫高弹形变，而必定发生脆性断裂，因此称 Tb 为脆化温度。 (Brittle Temperature) 玻璃态高聚物只有在 Tb ~ Tg 之间的温度范围内，才能在外力作用下实现强迫高弹形变，而强迫高弹形变又是塑料具有韧性的原因，因此 Tb 是塑料使用的最低温度。

31 结晶高聚物的拉伸 曲线形状 对拉伸过程的解释 结晶高聚物与非晶玻璃态高聚物在拉伸情况下的表现相似，本质上都是高弹形变。 差别：可被冷拉的温度范围及聚集态结构的变化不同。 结晶聚合物的拉伸和玻璃态聚合物拉伸： 相似之处： 经历弹性变形、屈服（成颈）、发展大变形、应变硬化 后阶段都呈现强烈各向异性，断裂前大形变在室温下不能自发回复，加热回复 本质上大形变属于高弹形变，统称“冷拉” 不同之处： 玻璃态高聚物冷拉区间Tb~Tg, 只发生分子链取向，无相变 结晶高聚物冷拉温度在Tg~Tm, 还包括结晶破坏、取向和再结晶

32 硬弹性材料的拉伸 (Hard Elastic Materials)

33 应变诱发塑料—橡胶转变 (Strain-induced plastics-to-rubber transition)
第一次拉伸大形变可立即基本回复，无需加热到Tg或Tm附近

34

35 高聚物的屈服 脆性和纫性高聚物材料的受力情况：

36 1. 高聚物单轴拉伸的应力分析： 在横截面上： 任意斜截面上： 结论：试样受到拉力时，其内部任意截面上的法应力和切应力只与试样的正应力和截面切角有关。

37 当  = 45时，切应力出现最大值； 当  = 0 时，法应力出现最大值。

38 对倾角为 =  + 90 的另一个截面，同样有：
即：两个互相垂直的斜截面上的法向应力之和等于正应力；切应力的数值相等，方向相反，它们是不能单独存在的，总是同时出现，这种性质称为双生互等定律。

39 纫性和脆性高聚物材料的受力情况的解释：

40 2. 真应力—应变曲线及其屈服判据： 如试样变形时体积不变，则 Ao lo = A l，再定义伸长比= l/l0=1+ 。 实际受力截面积： 真应力： 在 对 曲线上,  达最大值时，试样开始成颈，习用应力下降，最后试样在细颈的最窄部位断裂。而在 ’ 对 曲线上, ’ 却随 的增加单调的升高，试样成颈时， ’ 并不出现极大值。

41 真应力—应变曲线的类型： 屈服点判据：在屈服点，曲线出现拐点d /d = 0。 表示在  = -1 处向 ’ -  曲线作切线，切点就是屈服点。

42 真应力—应变曲线有三种类型： 表示在 = -1 点，不可能向 ’-   曲线作切线，这种高聚物在冷拉时不能成颈，高聚物随拉力的增大而均匀伸长。 表示在 = -1 点，可以向 ’-  曲线作切线，这种高聚物有屈服和成颈，高聚物均匀伸长到这点成颈，细颈逐渐变细负荷下降，直至断裂。 表示在 = -1 处，可以向 ’- 曲线作两条切线，D 为屈服点，E 点之后高聚物出现冷拉现象。

43 高聚物的破坏和理论强度 1. 高聚物的破坏 化学键的破坏 分子间的滑脱 分子间力的破坏

44 第一种情况： 共价键键能一般约350 KJ/mol, 键能可看作成键原子从平衡位置移开距离d，克服相互吸引力 f 需要作的功，d=0.15 nm 每根高分子链截面积0.2 nm2, 则每平方米截面积上有51018根高分子链，理想拉伸强度为 第二种情况： 每0.5 nm链段的摩尔内聚能20 KJ/mol, 高分子链总长100 nm, 则总的内聚能为4000 KJ/mol，若仅有范德华力，每0.5 nm链段内聚能5 KJ/mol, 则总的内聚能为1000 KJ/mol 第三种情况： 氢键解离能20 KJ/mol，作用范围0.3 nm; 范德华力解离能8 KJ/mol, 作用范围0.4 nm, 则拉断一个氢键或范德华力需要的力为：110－10 N和310-11N. 每0.25 nm2上有一个氢键或范德华力，则拉伸强度为400 MPa和120 MPa.

45 2. 理论强度 (1). 化学键的破坏：实际强度比理论值小几十倍。 (2). 分子间的滑脱：无单纯由分子间滑脱引起的断裂。 (3). 分子间力的破坏：理论强度与实际强度属同数量级。 高聚物的理论强度大于实际强度，是由于材料内部存在杂质及某种缺陷，造成应力集中的原因。

46 影响高聚物实际强度的因素 1.分子结构本身的影响 增加高分子的极性或产生氢键可使强度提高，极性基团或氢键的密度越大，则强度越高。 主链含有芳杂环的高聚物：其强度和模量都比脂肪族高，引入芳环侧基时，强度和模量也会提高。 分子链有支化的高聚物：支化使分子间距离增加，支化程度增大，高聚物的拉伸强度降低，冲击强度升高。 适度交联的高聚物：适度交联可以提高抗张强度。 高聚物分子量的影响：强度随分子量的增大而升高。但当分子量足够大时，拉伸强度与分子量无关，冲击强度继续增大。 分子量分布的影响：其中低分子量部分使强度降低。

47 2.结晶和取向的影响 结晶度影响：结晶度增加，对提高拉伸强度、弯曲强度和弹性模量有利；但如果结晶度太高，则会使冲击强度和断裂伸长率降低。 球晶结构的影响：大球晶通常使冲击强度和断裂伸长率降低；小球晶使拉伸强度、模量和断裂伸长率降低。 结晶形态的影响：由伸直链组成的纤维状晶体，其抗张性能较由折叠链组成的晶体优越得多。 取向的影响： 取向使聚合物材料产生各向异性和在取向方向上的强度增加。在取向方向上，抗张强度、屈服应力和模量比未取向时高 2 ~ 5 倍， 但垂直方向的强度却低于未取向时的 2 ~ 3 倍。

48 3. 应力集中物的影响 应力集中：材料在加工过程中存在的缺陷，使其在受力时内部应力分布不平均，缺陷附近范围内的应力急剧增加，远远大于平均值，这一现象称应力集中。 缺陷的形状不同，应力集中的程度也不同。锐口的小裂缝比钝口的大裂缝更有害。 缺陷包括：裂缝、空隙、缺口、银纹、杂质

49 4.增塑剂的影响 增塑剂的加入，对聚合物起到了稀释作用，使高分子链之间的作用力减小，带来强度的降低。一般加入增塑剂会使材料的抗张强度、模量等降低，而断裂伸长和冲击强度则随之升高。 5.填料的影响 填料有两种—即惰性填料和活性填料。 惰性填料可以降低成本，但强度也降低。 活性填料的增强程度与填料本身的强度及其与高聚物的亲和力大小有关。 ⑴.粉状填料：碳黑、石墨 ⑵.纤维状填料： 天然纤维如棉、麻、丝，碳纤维、玻璃纤维、硼纤维

50 6.共聚和共混的影响 共聚可以综合两种以上均聚物的性能。 共混也是一种很好的改性方法，通过共混可以得到某些特定性能的高聚物。共混材料有两相存在：即塑料相和橡胶相。 两相相容性过分好，形成均相体系，便得到基本保持塑料的模量和硬度。 相容性太差，两相结合力太差，受到冲击时界面易于发生分离，起不到增韧。 不管是用接枝共聚得到的高抗冲聚苯乙烯和ABS树脂，还是用共混得到的改性聚苯乙烯和ABS，它们都具有两相结构，橡胶以微粒分散于连续的塑料相之中，由于塑料连续相的存在，使材料的弹性模量和硬度不致过分降低，而分散的橡胶微粒作为大量应力集中物，当材料受到冲击时，它们引发大量裂纹，从而吸收冲击能量。同时大量裂纹之间应力场互相干扰，阻止裂纹进一步发展。

51 7.外力作用速度和温度的影响 由于高聚物是粘弹性材料，所以其破坏过程是一种松弛过程。因此外力作用速度和温度对高聚物的强度有影响。 提高拉伸速度与降低温度对应力－应变曲线影响效果相似。 低于Tg愈远，不同品种之间的差别愈大，主要取决于脆点的高低。 对于结晶高聚物，如果Tg在室温以下，则必然有较高的冲击强度，非晶态部分在室温处在高弹态，起到增韧作用。

52 7.2 Introduction to Rubber Elasticity
In 1805, John Gough observed that the temperature of a rubber band would increase as it is stretched adiabatically. Also, if a weight is hung from the end of the rubber band and heat is applied, the rubber band will decrease in length rather than stretch, as observed for other materials such as common metals and gases. It was not until the 1930s that an explanation for this behavior, known as the Gough-Joule effect, was provided by means of classical thermodynamics.

53 橡胶的使用温度范围 扩大橡胶的使用温度范围可以从两方面着手： 1.改善高温耐老化性能，提高耐热性 ⑴. 改变橡胶的主链结构 双键易老化；非碳原子有利于稳定，聚硫胶、硅橡胶 ⑵. 改变取代基的结构 供电子易氧化，吸电子有 氯丁橡胶、氟橡胶 ⑶. 改变交联链的结构 硫磺和促进剂，缩短硫桥；ZnO硫化、过氧化物硫化、辐射交联 2.降低Tg ,避免结晶，改善耐寒性 ⑴. 加入增塑剂 消弱分子间作用，增加链活动性，降低链规整排列避免结晶；考虑增塑剂用量和Tg ⑵. 采用共聚的方法 丁苯橡胶、丁氰橡胶；交替聚合降低氰基相互作用、交替结构乙丙橡胶

54 高弹性的特点 1. 弹性模量小，弹性形变大 橡胶类物质的弹性模量随温度的增加而增加。分子拉直后力图回复卷曲状态，形成回缩力，促使自发回复。 2. 变形需要时间 橡胶的形变需要时间，整链分子运动或链段运动需要克服分子间作用力和内摩檫力。形变落后于外力，其过程是蠕变和应力松弛过程。 3. 变形时有热效应 橡胶在伸长时会放热，在回缩时会吸热。 伸长的热效应随伸长率而增加，称热弹效应 橡胶伸长变形时，熵值减小；分子间内摩檫产生热量；分子规则排列而结晶放热

55 Thermodynamics The first and second law of thermodynamics applied to a reversible, equilibrium process provides the relationship between internal energy (U), entropy (S), and work (W) as For deformation of a rubber band, the work is a combination of pressure-volume expansion and the work due to the (tensile) force (f) applied to the rubber band, given as Where the convention is that work done by the system (i.e., pressure-volume work) is positive and the work done on the system (i.e., force-displacement work) is negative.

56 Rearrangement of equations gives
Since elastomers are nearly incompressible, the equation reduce to At constant temperature and volume, rearrangement of equation in terms of force and taking the (partial) derivative with respect to length at constant temperature and volume gives Equation shows that elastic force has both an enthalpic and entropic component.

57 In order to obtain an expression for the temperature dependence of force, we make use of the relation

58 橡胶弹性的热力学分析 使长度为 l0 的橡皮试样在拉力 f  作用下伸长 dl 。 体系的内能变化为：  dU = Q -  W 等温可逆过程： Q = TdS 体系对外所做的功： W = pdV - fdl 所以： dU = TdS - pdV + fdl 等温拉伸过程体积不变，则：dU =TdS + fdl 作用力 f  引起弹性体内能和熵的变化。

59 在试样的长度 l 和体积 V 维持不变的情况下，试样张力 f 随温度 T 的变化。

60 高聚物的高弹形变主要是由于单键内旋转导致的链段运动，在理想条件下：
拉伸过程内能几乎不变，则：fdl = -TdS = - Q 表明，高弹形变过程中，外力所做的功全部转为高分子长链构象熵的减少。 外力作用下，橡胶分子链由原来卷曲变为伸直，熵值减小，终态不稳定，当外力除去就会自发回复，这说明橡胶高弹形变是可回复的 。 等温形变过程中内能保持不变的弹性体称为理想高弹体；理想高弹体拉伸时只引起熵变，也称熵弹性。

61 7.3 Viscoelasticity of Polymers
At the extremes, a polymer may exhibit mechanical behavior characteristic of either an elastic solid or a viscous liqid. The actual response depends upon temperature, in relation to the glass-transition temperature (Tg) of the polymer, and upon the time scale of the deformation. Under usual circumstances, the mechanical response of polymeric materials will be intermediate between that of an ideal elastic or viscous liquid. In other words, polymers are viscoelastic.

62 This behavior is readily illustrated by Silly Putty, which is a silicon rubber having a low Tg. If rolled up into a ball and dropped to the floor, it will bounce with the resilience of a rubber ball. This is a manifestation of elastic behavior resulting from rapid deformation. If left on a desk for several days, the ball will flatten-it will flow like a viscous liquid over long time. If pulled at a moderate rate of strain, the ball will elongate with high extension and eventually fail.

63 高聚物的力学松弛现象 理想弹性体：平衡形变与时间无关。 理想粘性体：形变随时间线性发展。 高分子材料的形变性质与时间有关，其界于理想弹性体和理想粘性体之间。 高聚物的性质随时间的变化统称为力学松弛。 高分子材料受到外部作用的情况不同，有不同的力学松弛现象。

64 ⒈ 蠕变(Creep)： 蠕变：在一定的温度和较小的恒定外力作用下，材料的形变随时间的增加而逐渐增大的现象。

65 The response of different materials-ideal elastic, ideal viscous, and viscoelastic-to a step change in stress (or load) is illustrated. For an ideal (Hookean) elastic material, the resulting strain is instantaneous and constant during the duration of the applied stress. When the load is removed at t=t’, the strain instantaneously drops to zero. This follows from Hook’s law applied to the case of a constant stress:

66 In the case of an ideal (Newtonian) viscous fluid, the strain response is obtained by integration of Newton’s law of viscosity arranged as Here, the strain increases linearly with increasing time until the load is removed at t=t’, at which time the strain remains constant (t’/) in time unless an additional load is applied. The deformation is said to be permanent since no elastic recovery is possible.

67 蠕变过程包括三种形变： 1.普通形变 2.高弹形变 松弛时间与链段运动粘度2和高弹模量E2有关，=2/E2 3.不可逆形变

68 高聚物在受到外力作用时，往往三种形变一起发生，材料的总形变为：
温度对三者形变的影响 蠕变与温度高低和外力大小有关，温度过低，外力太小，蠕变很小而且很慢，短时间不易觉察； 温度过高，外力过大，形变发展过快，也感觉不出蠕变； 只有适当外力作用下，通常在Tg以上不远，链段在外力下可以运动，但受到内摩檫又较大，只能缓慢运动，则可观察到明显的蠕变现象。 含芳杂环的刚性链高聚物有较好抗蠕变性能 硬聚氯乙烯可用作化工管道，但易蠕变，需增加支架防止蠕变 聚四氟乙烯自润滑性好，蠕变严重，不能做机械零件，却是很好密封材料 橡胶硫化防止蠕变而造成不可逆形变

69 Instrumentation for creep testing can be a simple laboratory setup whereby a plastic film or bar is clamped at one end to a rigid support (normally enclosed in a temperature-controlled chamber) and a weight is added to the opposite end. The deformation of the loaded specimen is then measured by following the relative movement of two marks, inscribed on the sample, by means of a cathetometer, or traveling microscope. Typical creep curves of a low-Tg polymer measured over four decades of time and at several temperatures are shown. As illustrated, creep compliance increases with increasing temperature. As discussed later, individual creep curves obtained at different temperatures over short time periods can be shifted horizontally to yield a master curve at a given temperature and covering a wider range of temperature.

70 2. 应力松弛(Stress-relaxation)
应力松弛：在恒定温度和形变保持不变地情况下，高聚物内部的应力随时间增加而逐渐衰减的现象。 当温度很高，远超过Tg，象常温下的橡胶，链段运动内摩檫很小，应力很快松弛掉，甚至觉察不到 当温度太低，比Tg低很多，如常温下的塑料，虽然链段受到很大的应力，但内摩檫很大，链段运动能力很弱，应力松弛极慢，也不易觉察到 只有在Tg附近，应力松弛明显，例如柔软塑料片，交联使分子间不能滑移，应力不会松弛到零，只能松弛到某一数值，橡胶经过交联正是因为如此。

71 3．滞后现象(Phase angle) 高聚物在交变应力作用下，形变落后于应力的变化的现象称为滞后现象。 滞后现象与化学结构有关，一般刚性分子的滞后现象小，柔性分子滞后严重 滞后还受外界条件影响，外力作用频率低，滞后小；频率高，滞后也小 温度也有类似影响

72 4．力学损耗(Loss modulus) 形变的变化落后于应力的变化，发生滞后现象，则每一循环过程变化中就要消耗功，称为力学损耗，又称内耗。 拉伸对外做功，回缩时需克服链段间的内摩檫阻力，一部分功被损耗掉，转化为热 称力学损耗角，常用Tan表示内耗大小 内耗与分子结构有关，顺丁橡胶内耗小；丁苯和丁氰内耗大；丁基橡胶取代基数目多，内耗很大；内耗大吸收冲击能量大，回弹性差 内耗与频率和温度关系

73 Calculation of Modulus
As mentioned earlier, two types of transient mechanical tests are creep and stress-relaxation measurements, which can be used to characterize the dimensional stability of a material. A creep test measures the elongation of a specimen subjected to a rapid application of a constant load, 0, at constant temperature. By contrast, a stress-relaxation test records the stress required to hold a specimen at a fixed elongation, 0, at constant temperature.

74 Creep tests can be made in shear, torsion, flexure, or compression modes, as well as in tension.
Results of these tests are particularly important for selecting a polymer that must sustain loads for long periods. Usually, the parameter of interest is the tensile compliance, D, which is the ratio of strain to stress. While the stress, 0, is held constant during a creep test, the strain depends upon the time during which the load has been applied and, therefore, the compliance also becomes a function of time as

75 Stress-relaxation experiments can be conveniently performed with the same commercial instruments used in tensile tests. Since deformation must be as close as possible to being instantaneous, the preferred instrumentation is hydraulically-driven, rather than screw-driven, tensile testing machines. A rapid extension is applied to the sample and the stress on the sample is measured as a function of time by means of a force transducer. In a stress-relaxation experiment, stress is a function of time and, therefore, the stress-relaxation modulus, Er, is also time dependent.

76 Dynamic-Mechanical Analysis
For tensile strain that is a sinusoidal function of time, t, the strain function may be expressed as In this expression, 0 is the amplitude of the applied strain and  is the angular frequency of oscillation (units of radians per second). The angular frequency is related to frequency, f, measured in cycles per second (Hz), as =2f. The stress resulting from the applied sinusoidal strain will also be a sinusoidal function, which may be written in the most general form as Where 0 is the amplitude of the stress response and  is the phase angle between the stress and the strain.

77 In the case of an ideal elastic solid, the stress is always in phase with strain (i.e., =0). This can be shown from Hooke’s law Where 0 is the magnitude (i.e., 0=E0) of the resulting stress function.

78 In contrast, the stress of an ideal viscous fluid is always 90 out of phase (i.e., =/2) with the strain. This can be shown to result from Newton’s law of viscosity, given as The derivative of  with respect to time is Substitution of this expression, and noting that the cosine function is 90 out of phase with the sine function, gives where

79 At temperature below Tg, polymeric materials behave more as Hookean solids at small deformations, but at higher temperatures (i.e., in the vicinity of Tg) their behavior is distinctly viscoelastic. Over these temperatures,  will have a (temperature-dependent) value between 0 (totally elastic) and 90 (totally viscous). An alternative, and perhaps more useful, approach to discuss the dynamic response of a viscoelastic material to an applied cyclical strain is by use of complex number notation by which a complex strain, *, can be represented as where i=(-1)1/2.

80 The resulting complex stress, *, can be written as
It follows from Hooke’s law that a complex modulus, E*, can be defined as the ratio of complex stress to complex strain as This complex modulus can be resolved into two components-one that is in phase (i.e., E’) and one that is out of phase (i.e., E’’) with the applied strain. Substitution of Euler’identity into equation gives

81 Equation may be written in the form given as
where E’ is called the (tensile) storage modulus given as and E’’ is the (tensile) loss modulus: The ratio of loss and storage moduli defines another useful parameter in dynamic-mechanical analysis called tan, where

82 As in static testing where the tensile modulus is the inverse of the compliance, the dynamic tensile-modulus is inversely related to the dynamic tensile-compliance, D*, as Substitution of Euler’s identity in the form of exp(-i)=cos-isin into equation gives Equation may be written in the form Where D’ is called the storage compliance D’’ is the (tensile) loss compliance,

83 As in the case of dynamic modulus, tan is related to the components of the complex dynamic-compliance as It is possible to obtain values of dynamic storage and loss moduli from corresponding values of compliance, and vice versa, by manipulation of their complex conjugates. The complex conjugate of the dynamic modulus is The magnitude of the dynamic modulus, E*, is obtained from its complex conjugate as

84 Then, we can write Comparison of the form of equation with the expression for D* gives the interrelationships between the components of dynamic modulus and dynamic compliance: and Similar relationships may be written for E’ and E’’ as functions of D’ and D’’.

85 在交变的应力、应变作用下发生的滞后现象和力学损耗，称为动态力学松弛。
当应力和应变都是时间的函数，弹性模量的计算： 复数模量： 内耗： 动态模量：

86 粘弹性固体的E’和E” 与频率的关系：

87 Mechanical Modes of Viscoelastic Behavior
As insight into the nature of the viscoelastic properties of polymers can be obtained by analyzing the stress or strain response of mechanical models using an ideal spring as the Hookean element and a dashpot as the viscous element. A dashpot may be viewed as a shock absorber consisting of a piston in a cylinder filled with a Newtonian fluid of viscosity . The elemental models are a series combination of a spring and dashpot, the Maxwell element, and a parallel combination of a spring and dashpot, the Voigt element.

88 Maxwell Element In the case of a series combination of a spring and dashpot, the total strain (or strain rate) is a summation of the individual strains (or strain rates) of the spring and dashpot. From Hooke’s law (=E), the strain rate of an ideal elastic spring can be written as while the strain rate for the dashpot is obtained by rearranging Newton’s law of viscosity as

89 Therefore, the basic equation for strain rate in the Maxwell model is the summation of the strain rates for the spring and dashpot as This differential equation can be solved for creep, stress relaxation, and dynamic response by applying the appropriate stress or strain function.

90 In a creep experiment, a constant stress, 0, is applied instantaneously. Equation then reduces to
Rearrangement and integration gives Where 0 represents the instantaneous (i.e., t=0) strain response of the spring element.

91 The creep compliance function, D(t), is then obtained as
Where D (=0/0) is the instantaneous compliance (of the spring). An alternative form may be obtained by defining a relaxation time, , as Equation can then be represented in normalized form as Maxwell 模型用于蠕变过程并不成功，它的蠕变相当于牛顿流体的黏性流动。同样也不能模拟交联高聚物的应力松弛过程。

92 In a stress-relaxation experiment, the strain (0) is constant and, therefore, the strain rate is zero. Equation then reduces to the first-order ordinary differential equation Rearrangement and introduction of  gives Integration yields the stress response as Where 0 is the instantaneous stress response of the spring.

93 The stress relaxation modulus, E(t), is then obtained as
where E (=0/0) is the (Young’s) modulus of the spring element.

94 To obtain an expression for the dynamic-mechanical response (i. e
To obtain an expression for the dynamic-mechanical response (i.e., complex compliance or modulus), the expression for complex stress, =0exp(it), is substituted into the Maxwell equation to give Integration from time t1 to t2 gives

95 Since the corresponding stress increment can be written as
division of both sides by this expression, and making use of , gives the complex compliance as Therefore, the storage compliance obtained from the Maxwell model is simply the compliance of the spring which is independent of time or frequency, while the loss compliance is

96 The corresponding expression for complex modulus, E
The corresponding expression for complex modulus, E*, is obtained by recalling that E* is the reciprocal of D* and utilizing the complex conjugate of D* as Performing the multiplication, rearranging, and using the inverse relation E=1/D gives where and It follows from equations that tan for a Maxwell model is simply

97 Voigt Element. For a parallel combination of a spring and dashpot, the Voigt model shown in Figure, the strain on each element must be equal while the stress is additive. The fundamental relation for the Voigt model is, therefore,

98 Making use of the relaxation time, the Voigt equation for creep deformation becomes a linear differential equation: which can be solved by using an integrating factor (et/). Solving for (t) gives the compliance function as where D(t)1/E(t). In contrast to the Maxwell model, the Voigt equation cannot be solved in any meaningful way for stress relaxation because the dashpot element cannot be deformed instantaneously.

99 当除去应力时，＝0 当 t=0 时，＝() 这就是模拟蠕变回复过程的方程

100 The response of a Voigt model to dynamic strain gives the relationships for the storage and loss compliance as and The forms of these relationships for compliance are the same as those for dynamic moduli given by the Maxwell model.

101 Comparison of Simple Models
The two-element models approximate some of the viscoelastic properties of real polymers. In general, modulus is modeled best by the Maxwell element, while compliance is better represented by the Voigt element. The most significant limitation of these two models is that they employ only a single relaxation time. This results in the prediction of only a single transition in modulus or compliance, whereas high-molecular-weight polymers exhibit both a glass-to-rubber and rubber-to-liquid transition. As would be expected, improvement is achieved by adding more elements.

102 Multi-element Models A multi-element model, particularly suited for modulus, is called the Maxwell-Wiechert model, which is a parallel combination of multiple Maxwell elements. In this model, the strains on each Maxwell element are equal and the stresses experience by each Maxwell element are additive, as they are in the case of single spring and dashpot in the simple Voigt model. For the two-element Maxwell-Wiechert model, the stress-relaxation modulus is given as where 1=1/E1 and 2=2/E2.

103 四元件模型：模拟线性高聚物蠕变过程

104 Further improvement in terms of realistic slopes in the transition regions is obtained by adding additional Maxwell elements in the model and, therefore, providing a distribution of relaxation times as would be expected for a high-molecular-weight polymer with a broad distribution of molecular weights. In general, the stress-relaxation modulus of a Maxwell-Wiechert model consisting of N Maxwell elements can be written as the summation

105 An alternate approach for a multi-element model is a Voigt-Kelvin model which consists of a series arrangement of Voigt elements. For N elements, the creep compliance can be written as a summation

106 Time-Temperature Superposition
Often, it is important to know how a material will behave (e.g., creep or stress relaxation) at a fixed temperature, but over a long time period (perhaps years) that may not be realistically accessible. Fortunately, long-time behavior can be evaluated by measuring stress-relaxation or creep data over a shorter period of time but at several different temperatures. Information from each of these different temperature curves may then be combined to yield a master curve at a single temperature by horizontally shifting each curve along the log time scale. This technique is called time-temperature superposition and is a foundation of linear viscoelasticity theory.

107 In this procedure, the master curve is plotted as stress-relaxation modulus or creep compliance versus reduced time, t/T. The shift factor, T, is defined as the ratio of (real) time to reach a particular value of modulus at some temperature to the reference-scale time coordinate, tr, corresponding to the same value of modulus in the master curve at the reference temperature, Tr,

108 The dependence of the shift factor, T, on temperature is given by the Williams-Landel-Ferry (WLF) relationship where C1 and C2 are constants for a given polymer and Tr is the reference temperature. When Tr is taken to be the polymer Tg, as it often is, C1 and C2 may be approximated by the “universal” values of and 51.6, respectively.

109 在绘制叠合曲线时，各条实验曲线在时间坐标上的平移量是不同的。
由移动量与温度的关系曲线可以找到所需温度下的 logT 值。

110 The measurement of a full range of stress-relaxation (or creep-compliance) behavior at a given temperature can take years. Fortunately, it is possible to shift data taken over shorter time periods but at different temperatures to construct a master curve covering a longer time scale at some reference temperature. The principle that allows horizontal shifting of data is called time-temperature superposition.

111 Boltzmann Superposition Principle
Tensile (or shear) modulus and tensile (or shear) compliance are inversely related. The same is true for complex modulus and complex compliance, but creep-compliance and stress-relaxation moduli are not as simply related; that is This is because creep-compliance and stress-relaxation moduli are obtained by distinctly different experimental procedures.

112 As discussed before, a fixed load is applied instantly at t=0 in a creep experiment, while a fixed strain is applied instantaneously at t=0 for stress-relaxation measurements. Fortunately, a relationship between the creep-compliance and stress-relaxation moduli can be established by application of another fundamental statement of linear viscoelasticity, the Boltzmann superposition principle. This Boltzmann superposition principle states that the total strain is a linear function of total applied stress.

113 In the case of a creep experiment, the strain is a function of the total load (stress) history in which each load makes an independent contribution to the creep. If a load, 0, is instantaneously applied at time t=0 and a second load, 1, is applied at time t=u1, then the strain at some time t’, where t’>u1, is given by the Boltzmann superposition principle as In general case of the application of a series of discrete loads, 1, 2, …n, applied at times t=u1, u2, …un, the total resulting strain response in creep deformation is calculated from the tensile compliance function, D(t), as

114 Boltzmann 叠加原理指出，高聚物的力学松弛行为是其整个历史上诸松弛过程线性加和的结果。
蠕变过程：总的蠕变是各个负荷引起的蠕变的线性加和。 应力松弛过程：高聚物的总应力等于历史上诸应变引起的应力松弛过程的线性加和。

115 Work in Dynamic Deformation
As is the case for other forms of mechanical deformation, work is expended during dynamic-mechanical testing. The amount of work, or power, consumed depends upon the viscoelastic properties of the polymer. The work per unit of simple volume, W, may be expressed in terms of stress and strain as Where f is the force, dl is the differential extension of the sample, and V is the sample volume. Per cycle, (i.e., 2 radians) of oscillation, the dynamic work is then obtained as

116 Integration gives the work per cycle per unit volume for dynamic oscillation of a viscoelastic solid as which can be substituted into above equation to obtain

117 From a consideration of the sin contribution to W, it is obvious that the work of oscillation for an ideal elastic solid (i.e., =0 and sin=0) is zero, while the maximum work is expended during deformation of an ideal viscous material (i.e., =/2, sin=1, and W=00). The power consumed per unit volume is then obtained by multiplying the work per cycle (per unit volume) or by the number of cycles per unit time (i.e., f=/2) as

118 Experimental Techniques.
The usual way to report the viscoelastic response of a polymer is by means of a semilog plot of storage modulus and loss modulus (or tan) as a function of temperature at one or more frequencies. Maxima in loss modulus, or tan, occur both at Tg and at low temperatures, where small-scale molecular motions can occur (i.e., secondary relaxation). In the case of semicrystalline polymers, an additional peak in tan corresponding to Tm will occur above Tg. Traditionally, the highest-temp peak is designated the  relaxation (i.e., the glass transition in amorphous polymers and the crystalline-melting transition in semicrystalline polymers), the next highest is the  relaxation, then the  relaxation, and so on.

119 A number of different instruments have been used to obtain dynamic-mechanical data. The earliest designs used a free-vibration approach (torsion pendulum and torsional braid) in which a polymer sample in the form of a bar or cylinder is set to oscillate and the damping of the oscillation is recorded as a function of time at a controlled temperatures. More recent and versatile instrumentation uses a forced-oscillation method whereby a sinusoidal strain (tensile, flexure, compression, shear, or torsional) is applied to a sample and the stress response is recorded as a function of temperature at different frequencuies.

120 Free-Vibration methods
The simplest and earliest dynamic-mechanical technique, the torsion pendulum, is a free-vibration technique. In this experiment, a polymer film is rigidly clamped at one end, while the other end is attached to an inertia disk that is free to oscillate. Once the disk is set in oscillation, the viscous response of the polymer sample causes the amplitude of the oscillation to decay. The time required for one complete oscillation is called the period of oscillation (P).

121 The value of tan can be determined from the ratio of amplitudes (A) of any two consecutive peaks as
where  is called the log decrement.

122 An important variation of the torsion-pendulum technique is called torsional-braid analysis (TBA).
The TBA sample is supported on a braid, typically made from glass fibers. The braid is coated with a dilute solution of the sample and dried in a vacuum to remove all solvent. The advantage of this technique is that the support provided by the braid enables the dynamic-mechanical characterization of low-molecular-weight and liquid samples as well as high-molecular-weight polymer samples near and above Tg, where softening would preclude the use of unsupported-sample techniques.

123 Forced-Vibration Methods
Today, the most frequently used commercial instruments utilize a forced-(rather than free) vibration method by which a dynamic tensile, compression, torsional, or flexural strain is applied to the sample, which is cut or molded in the form of a thin film or bar. One important advantage of the forced-vibration mode is that frequency can be controlled precisely over a wider range than is possible for free-vibration methods.

124 思考题 可以将WLF方程写成适用于任意便利的温度做参考温度，方程保留原来形式但常数C1 和C2值必须改变。利用C1 和C2的普适值，计算以 为参考温度的C1 和C2值。 作业： P333: (11)