應(yīng)力應(yīng)變的關(guān)系和行為外文文獻(xiàn)翻譯@中英文翻譯@外文翻譯

原文： Stress-Strain Relationships and Behavior 5.1 INRODUCTION 5.2 MODELS FOR DEFORMATION BEHAVIOR 5.3 ELASTIC DEFORMATION 5.4 ANISOTROPIC MATERIALS 5.5 SUMMARY OBJECTIVES Become familiar with the elastic, plastic, steady creep, and transient creep types of strain, as well as simple rheological models for representing the stress-strain-time behavior for each. Explore three-dimensional stress-strain relationships for linear-elastic deformation in isotropic materials, analyzing the interdependence of stresses or strains imposed in more than one direction. Extend the knowledge of elastic behavior to basic cases of anisotropy, including sheets of matrix-and fiber composite material. 5.1 INRODUCTION The three major types of deformation that occur in engineering materials are elastic, plastic, and creep deformation. These have already been discussed in Chapter 2 from the viewpoint of physical mechanisms and general trends in behavior for metals, polymers, and ceramics. Recall that elastic deformation is associated with the stretching, but not breaking, of chemical bonds. In contrast, the two types of inelastic deformation involve processes where atoms change their relative positions, such as slip of crystal planes or sliding if chain molecules. If the inelastic deformation is time dependent, it is classed as creep, as distinguished from plastic deformation, which is not time dependent. In engineering design and analysis, equations describing stress-strain behavior, called stress-strain relationships, or constitutive equations, are frequently needed. For example, in elementary mechanics of materials, elastic behavior with a linear stress-strain relationship is assumed and used in calculating stresses and deflections in simple components such as beams and shafts. More complex situations of geometry and loading can be analyzed by employing the same basic assumptions in the form of theory of elasticity. This is now often accomplished by using the numerical technique called finite element analysis with a digital computer. Stress-strain relationships need to consider behavior in three dimensions. In addition to elastic strains, the equations may also need to include plastic strains and creep strains. Treatment of creep strain requires the introduction of time as an additional variable. Regardless of the method used, analysis to determine stresses and deflections always requires appropriate stress-strain relationships for the particular material involved. For calculations involving stress and strain, we express strain as a dimensionless quantity, as derived from length change, = L/L. Hence, strains given as percentages need to be converted to the dimensionless form, = %/100, as do strains given as microstrain, = /106. In the chapter, we will first consider one-dimensional stress-strain behavior and some corresponding simple physical models for elastic, plastic, and creep deformation. The discussion of elastic deformation will then be extended to three dimensions, starting with isotropic behavior, where the elastic properties are the same in all directions. We will also consider simple cases of anisotropy, where the elastic properties vary with direction, as in composite materials. However, discussion of three-dimensional plastic and creep deformation behavior will be postponed to Chapters 12 and 15, respectively. 5.2 MODELS FOR DEFORMATION BEHAVIOR Simple mechanical devices, such as linear springs, frictional sliders, and viscous dashpots, can be used as an aid to understanding the various types of deformation. Four such models and their responses to an applied force are illustrated in Fig.5.1. Such devices and combinations of them are called rheological models. Elastic deformation, Fig.5.1(a), is similar to the behavior of a simple linear spring characterized by its constant k. The deformation is always proportional to force, x=P/k, and it is recovered instantly upon unloading. Plastic deformation, Fig.5.1(b), is similar to the movement of a block of mass m on a horizontal plane. The static and kinetic coefficients of friction are assumed to be equal, so that there is a critical force for motion P0= mg, where g is the acceleration of gravity. If a constant applied force P is less than the critical value, PP0, the block moves with an acceleration a =(P-P0)/m (5.1) When the force is removed at time t, the block has moved a distance a=at2/2, and it remains at this new location. Hence, the model behavior produces a permanent deformation, xp. Creep deformation can be subdivided into two types. Steady-state creep, Fig.5.1(c), proceeds at a constant rate under constant force. Such behavior occurs in a linear dashpot, which is an element where the velocity, dtdx/x 。 , is proportional to the force. The constant of proportionality is the dashpot constant c, so that a constant value of force P gives a constant velocity, cPx /。 , resulting in a linear displacement versus time behavior. When the force is removed, the motion stops, so that the deformation is permanent-that is, not recovered. A dashpot could be physically constructed by placing a piston in a cylinder filled with a viscous liquid, such as a heavy oil. When a force is applied, small amounts of oil leak past the piston, allowing the piston to move. The velocity of motion will be approximately proportional to the magnitude of the force, and the displacement will remain after all force is removed. The second type of creep, is called transient creep, Fig.5.1(d), slows down as time passes. Such behavior occurs in a spring mounted parallel to a dashpot. If a constant force P is applied, the deformation increases with time. But an increasing fraction of the applied force is needed to pull against the spring as x increases, so that less force is available to the dashpot, and the rate of deformation decreases. The deformation approaches the value P/k if the force is maintained for a long period of time. If the applied force is removed, the spring, having been extended, now pulls against the dashpot. This results in all of the deformation being recovered at infinite time. Rheological models may be used to represent stress and strain in a bar of material under axial loading, as shown in Fig. 5.2. The model constants are related to material constants that are independent of the bar length L or area A. For elastic deformation, the constant of proportionality between stress and strain is the elastic modulus, also called Youngs modulus, given by E= / (5.2) Substituting the definitions of stress and strain, and also employing P = k x, yields the relationship between E and k: E=kL/A (5.3) For the plastic deformation model, the yield strength of the material is simply 0=P0/A (5.4) For the steady-state creep model, the material constant analogous to the dashpot constant c is called the coefficient of tensile viscosity1 and is given by . (5.5) Where dtd /. is the strain rate. Substitution from Fig. 5.2 and P= cx. Yields the relationship between and c: AcL (5.6) Equations 5.3 and 5.6 also apply to the spring and dashpot elements in the transient creep model. Before proceeding to the detailed discussion of elastic deformation, it is useful to further to discuss plastic and creep deformation models. 5.2.1 Plastic Deformation Models As discussed in Chapter 2, the principal physical mechanism causing plastic deformation in metals and ceramics is sliding (slip) between planes of atoms in the crystal grains of the material, occurring in an incremental manner due to dislocation motion. The materials resistance to plastic deformation is roughly analogous to the friction of a block on a plane, as in the rheological model of Fig. 5.1(b). For modeling stress-strain behavior, the block of mass m can be replaced by a massless frictional slider, which is similar to a spring clip, as shown in Fig. 5.3(a). Tow additional models, which are combinations of linear springs and frictional sliders, are shown in (b) and (c). These give improved representation of the behavior of real materials, by including a spring in series with the slider, so that they exhibit elastic behavior prior to yielding at the slider yield strength o. In addition, model (c) has a second linear spring connected parallel to the slider, so that its resistance increases as deformation proceeds. Model (a) is said to have rigid, perfectly plastic behavior； model (b) elastic, perfectly plastic behavior； and model (c) elastic, linear-hardening behavior. Figure 5.3 gives each models response to three different strain inputs. The first of these is simple monotonic straining that is, straining in a single direction. For this situation, for models (a) and (b), the stress remains at o beyond yielding. For monotonic loading of model (c), the strain is the sum of strain 1 in spring E1 and strain 2 in the (E2, o) parallel combination: 21 , E11 (5.7) The vertical bar is assumed not to rotate, so that both spring E2 and slider o have the same strain. Prior to yielding, the slider prevents motion, so that strain 2 is zero: 02 , E1 ( o) (5.8) Since there is no deflection in spring E2, its stress is zero, and all of the stress is carried by the slider. Beyond yielding, the slider has a constant stress o, so that the stress in spring E2 is ( - o). Hence, the strain 2 and the overall strain are E 2 02 , EE 2 01 (5.9) From the second equation,the slope of the stress-strain curain curve is seen to be EE EEE edd 21 21 (5.10) Which is the equivalent stiffness Ee, lower than both E1 and E2, corresponding to E1 and E2 in series. Figure 5.3 also gives the model responses where strain is increased beyond yielding and then decreased to zero. In all three cases, there is no additional motion in the slider until the stress has changed by an amount 20 in the negative direction . For models (b) and (c) , this gives an elastic unloading of same slope E1 as the initial loading. Consider the point during unloading where the stress passes through zero, as shown in Fig. 5.4. The elastic strain, e, that is recovered corresponds to the relaxation of spring E1. The permanent or plastic strain p corresponds to the motion of the slider up to the point of maximum strain. Real materials generally have nonlinear hardening stress-strain curves as in (c), but with elastic unloading behavior similar to that of the rheological models. Now consider the response of each model to the situation of the last column in Fig. 5.3, where the model is reloaded after elastic unloading to = 0. In all cases, yielding occurs a second time when the strain again reaches the value 1 from which unloading occurred. It is obvious that the two perfectly plastic models will again yield at = 0. But the linear-hardening model now yields at a value = 1, which is higher than the initial yield stress. Furthermore, 1 is the same value of stress that was present at =1, when the unloading first began. For all three models, the interpretation may be made that the model possesses a memory of the point of previous unloading. In particular, yielding again occurs at the same - point from which unloading occurred, and the subsequent response is the same as if there had never been any unloading. Real materials that deform plastically exhibit a similar memory effect. We will return to spring and slider models of plastic deformation in Chapter 12, where they will be considered in more detail and extended to nonlinear hardening cases. 譯文： 應(yīng)力應(yīng)變的關(guān)系和行為 5.1 概述 5.2 變形的典型模式 5.3 彈性變形 5.4 各向異性材料 5.5 總結(jié) 目標(biāo) 熟悉彈性應(yīng)變，塑性應(yīng)變，穩(wěn)態(tài)蠕變和瞬態(tài)蠕變等應(yīng)變類型，以及每個(gè)用來(lái)表示應(yīng)力 應(yīng)變與時(shí)間相關(guān)的簡(jiǎn)單的流變類型。 探討在各向同性材料中線性彈性變形的三維的應(yīng)力應(yīng)變關(guān)系，分析應(yīng)力應(yīng)變?cè)诙鄠€(gè)方向上施加的相互作用力 擴(kuò)展在各向異性材料中以及一些基體纖維復(fù)合材料中彈性形變的基本情況的知識(shí)。 5.1 概述 工程材料發(fā)生變形的三種主要類型是彈性變形，塑性變形，蠕變變形。這些已經(jīng)在金屬聚合物和陶瓷行 為的物理機(jī)制和一般趨勢(shì)的觀點(diǎn)的第 2 章中被討論過(guò)了，記得彈性變形與拉伸相關(guān)，但是不打破化學(xué)鍵。相比之下，這兩種涉及原子的相對(duì)位置變化的過(guò)程類型的非彈性變形，比如晶面滑移和鏈分子滑動(dòng)。如果非彈性變形取決于時(shí)間，它被歸類為蠕變，區(qū)別于不取決于時(shí)間的的塑性變形。 在工程設(shè)計(jì)和分析中，應(yīng)力應(yīng)變行為的方程描述，稱為應(yīng)力應(yīng)變的關(guān)系或本構(gòu)方程是很必要的。比如，在基礎(chǔ)材料力學(xué)中，與線性應(yīng)力 -應(yīng)變相關(guān)的彈性行為是被假定和用來(lái)計(jì)算簡(jiǎn)單的構(gòu)件如梁和軸的應(yīng)力和變形的。在更復(fù)雜的幾何和加載情況下，可以由彈性理論的形式使用相同的基本假設(shè) 分析?，F(xiàn)在經(jīng)常利用被稱為與數(shù)字計(jì)算機(jī)相關(guān)的有限元分析的數(shù)字科技來(lái)完成。 應(yīng)力應(yīng)變關(guān)系需要考慮在三維中的行為，除了彈性應(yīng)變外，這個(gè)方程可能還需要包括塑性應(yīng)變和蠕變應(yīng)變。處理蠕變應(yīng)變要引入時(shí)間作為一個(gè)額外的變量。不管用什么方法，對(duì)于特定的材料分析確定應(yīng)力和變形總是需要適當(dāng)?shù)膽?yīng)力 -應(yīng)變關(guān)系。對(duì)于應(yīng)力和應(yīng)變的計(jì)算，我們把應(yīng)變作為一個(gè)無(wú)量綱的量表達(dá)，來(lái)自于長(zhǎng)度的變化， = L/L。因此，應(yīng)變給定的百分比需要被轉(zhuǎn)換成無(wú)量綱形式， = %/100，也可以把應(yīng)變百分比做為微應(yīng)變 = /106。 在本章中，我們將首先考慮一 維應(yīng)力應(yīng)變行為和一些相應(yīng)的彈性，塑性，蠕變變形的簡(jiǎn)單的物理模型。彈性變形的探討將擴(kuò)展到三個(gè)維度，從各向同性行為開(kāi)始，在所有的方向中的彈性性質(zhì)是相同的。我們也會(huì)考慮在復(fù)合材料中各向異性的簡(jiǎn)單情況，其中的彈性性質(zhì)隨方向而變化。但是，三維塑性變形和蠕變變形行為探討將分別推遲到第 12 章和 15 章。 5.2 變形的典型模式 簡(jiǎn)單的機(jī)械裝置，如線性彈簧、摩擦滑塊和粘滯阻尼器，可以用于幫助理解變形的各種類型。四個(gè)這樣的模型對(duì)于一個(gè)施加力的反應(yīng)展示在圖 5.1。這樣的裝置和它們的組合被稱為流變模型。 彈性變形，圖 .5.1(a)， 類似于一個(gè)簡(jiǎn)單的線性彈簧的行為，用常數(shù) K 表示，這種變形都是成比例的力， x=P/k,，并且當(dāng)它的力卸載時(shí)變形會(huì)恢復(fù)。塑性變形，圖 5.1(b),類似于一個(gè)質(zhì)量為 m 的塊在一個(gè)水平面上的運(yùn)動(dòng)，動(dòng)、靜摩擦系數(shù)被認(rèn)為是相等的，所以有一個(gè)臨界動(dòng)摩擦力 P0= mg，其中 g 是重力加速度。如果施加一個(gè)恒定的力 P 小于臨界值， P P0，這個(gè)塊做加速度運(yùn)動(dòng)， a =(P -P0)/m (5.1) 當(dāng)這個(gè)力作用時(shí)間為 t，這個(gè)塊移動(dòng)的距離 a= at2/2，它仍然是在這個(gè)新的位置。因此，該類型的行為產(chǎn)生永久變形， Xp。蠕變變形可分為兩種類型，穩(wěn)態(tài)蠕變，圖 5.1（ c），在恒力作用下進(jìn)行恒速運(yùn)動(dòng)，這種行為發(fā)生在一個(gè)線性阻尼情況下，一個(gè)恒定的速度單元dtdx/x 。 ，與力成正比。比例常數(shù)是阻尼常數(shù) C，因此，一個(gè)恒定的力 P值給出了一個(gè)恒定的速度， cPx /。 ，導(dǎo)致了一個(gè)線性位移與時(shí)間的關(guān)系。當(dāng)這個(gè)力撤去時(shí)，運(yùn)動(dòng)停止，所以這個(gè)變形是永久性的，不可恢復(fù) 的。一個(gè)阻尼器可以通過(guò)將一個(gè)活塞放在一個(gè)充滿粘性液體（如重油）的圓筒中構(gòu)造出來(lái)，當(dāng)施加一個(gè)力時(shí)，少量的油漏過(guò)活塞，使活塞移動(dòng)。運(yùn)動(dòng)的速度將與力的大小近似成比例，當(dāng)所有的力撤去時(shí)位移將會(huì)保持。 第二種蠕變類型，被稱為瞬態(tài)蠕變，圖 5.1（ d），隨著時(shí)間的推移減慢。這樣的情況是發(fā)生在一個(gè)裝有平行的阻尼和彈簧上，如果有一個(gè)恒定的力 P 作用，變形會(huì)隨時(shí)間增加。但隨著 x 的增加需要越來(lái)越多的一部分施加的力來(lái)拉彈簧，因此，這些小部分的力可以通過(guò)阻尼器獲得，并且變形速率降低。如果力是維持很長(zhǎng)一段時(shí)間變形值近似于 p / k，如果施加的力撤去，已經(jīng)被延伸的彈簧，現(xiàn)在拉回阻尼器。這導(dǎo)致所有的變形是在極長(zhǎng)的時(shí)間恢復(fù)。 流變模型可以用來(lái)表示軸向載荷下一塊材料的應(yīng)力和應(yīng)變，如圖 5.2 所示，該模型常數(shù)與材料常數(shù)有關(guān)，與材料的長(zhǎng)度或面積無(wú)關(guān)。對(duì)彈性形變來(lái)說(shuō)，應(yīng)力應(yīng)變之間的常數(shù)是彈性模量，也稱為楊氏模量，由 E= / （ 5.2） 得出，用應(yīng)力和應(yīng)變的定義，并采用 P = K X，可得出 E 和 K 之間的關(guān)系 E=k L/A (5.3) 對(duì)于塑性變形模 型，材料的