1、1,第七章 弹塑性断裂力学简介,7.1 裂纹尖端的小范围屈服,7.2 裂纹尖端张开位移,7.3 COD测试与弹塑性断裂控制设计,返回主目录,2,用线弹性材料物理模型,按照弹性力学方法,研究含裂纹弹性体内的应力分布,给出描述裂纹尖端应力场强弱的应力强度因子K,并由此建立裂纹扩展的临界条件,处理工程问题。,第七章 弹塑性断裂力学简介,线弹性断裂力学(LEFM),线弹性断裂力学给出的裂纹尖端附近的应力趋于 无穷大。然而,事实上任何实际工程材料,都不可能承受无穷大的应力作用。因此,裂尖附近的材料必然要进入塑性,发生屈服。,3,Linear elastic fracture mechanics pred
2、icts infinite stresses at the crack tip.In real materials,however,stress at the crack tip are finite because the crack tip radius must be finite.Inelastic material deformation,such as plasticity in metal,leads to further relaxation of the crack tip stress.,线弹性断裂力学预测裂纹尖端应力无穷大。然而 在实际材料中,由于裂尖半径必定为有限值,故
3、 裂尖应力也是有限的。非弹性的材料变形,如金 属的塑性,将使裂尖应力进一步松弛。,4,7.1 裂纹尖端的小范围屈服,1.裂尖屈服区,当r0时,s,必然要发生屈服。因此,有必要了解裂尖的屈服及其对K的影响。,无限大板中裂纹尖端附近任一点(r,)处的正应力x、y和剪应力xy的线弹性解为:,5,这里仅简单讨论沿裂纹线上屈服区域的大小。,线弹性断裂力学,裂尖附近任一点处的x、y xy,,6,对于平面问题,还有:yz=zx=0;z=0 平面应力 z=(x+y)平面应变,7,式中,ys为材料的屈服应力,为泊松比。对于金属材料,0.3,这表明平面应变情况下裂尖塑性区比平面应力时小得多。,8,虚线为弹性解,r
4、0,y。由于yys,裂尖处材料屈服,塑性区尺寸为rp。,当=0时(在x轴上),裂纹附近区域的应力分布及裂纹线上的塑性区尺寸如图。,与原线弹性解(虚线HK)相比较,少了HB部分大于ys的应力。,假定材料为弹性-理想塑性,屈服区内应力恒为ys,应力分布应由实线AB与虚线BK表示。,9,The simple analysis as above is not strictly correct because it was based on an elastic crack tip solution.When yielding occurs,stress must redistribute in ord
5、er to satisfy equilibrium.,上述简单分析是以裂纹尖端弹性解为基础的,故 并非严格正确的。屈服发生后,应力必需重分布,以满足平衡条件。,The region ABH represents forces that would be present in an elastic material but cannot be carried in the elastic-plastic material because the stress cannot exceed yield.The plastic zone must increase in size in order t
6、o carry these forces.,ABH区域表示弹性材料中存在 的力,但因为应力不能超过屈 服,在弹塑性材料中却不能承 受。为了承受这些力,塑性区 尺寸必需增大。,10,为满足静力平衡条件,由于AB部分材料屈服而少承担的应力需转移到附近的弹性材料部分,其结果将使更多材料进入屈服。因此,塑性区尺寸需要修正。,设修正后的屈服区尺寸为R;假定线弹性解答在屈服区外仍然适用,BK平移至CD,为满足静力平衡条件,修正后ABCD曲线下的面积应与线弹性解HBK曲线下的面积相等。,由于曲线CD与BK下的面积是相等的,故只须AC下的面积等于曲线HB下的面积即可。,11,于是得到:,12,依据上述分析,并
7、考虑到平面应变时三轴应力作用的影响,Irwin给出的塑性区尺寸R为:,上式指出:裂纹尖端的塑性区尺寸R 与(K1/ys)成正比;平面应变时的裂尖塑性区尺寸约为平面应力情况的1/3。,13,Most of the classical solution in fracture mechanics reduce the problem to two dimensions.That is at least one of the principal stresses or strains is assumed to equal zero(plane stress and plane strain res
8、pectively).,断裂力学中的大部分经典解都将问题减化为 二维的。即主应力或主应变中至少有一个被假设 为零,分别为平面应力或平面应变。,In general,the conditions ahead of a crack are neither plane stress nor plane strain,but are three-dimensional.There are,however,limiting cases where a two dimensional assumption is valid,or at least provides a good approximation
9、.,一般地说,裂纹前的条件既不是平面 应力,也不是平面应变,而是三维的。然 而,在极限情况下,二维假设是正确的,或者至少提供了一个很好的近似。,14,2.考虑裂尖屈服后的应力强度因子,曲线CD与线弹性解BK相同。假想裂纹尺寸由a增大到a+rp,则裂纹尖端的线弹性解恰好就是曲线CD。,对于理想塑性材料,考虑裂纹尖端的屈服后,裂尖附近的应力分布应为图中ACD曲线。,15,16,例7.1 无限宽中心裂纹板,受远场拉应力作用,试讨论塑性修正对应力强度因子的影响。,17,对于平面应力情况,=1;若(/ys)=0.2,=1%;若(/ys)=0.5,=6%;当(/ys)=0.8时,达15%。对于平面应变情况
10、,3,二者相差要小一些。,可见,(/ys)越大,裂尖塑性区尺寸越大,线弹性分析给出的应力强度因子误差越大。,18,3.小范围屈服时表面裂纹的K修正,前表面修正系数通常取为Mf=1.1;E(k)是第二类完全椭圆积分。,无限大体中半椭圆表面裂纹最深处的应力强度因子为:,19,可见,小范围屈服时,表面裂纹的 K计算只须用 形状参数Q代替第二类完全椭圆积分E(k)即可。,利用E(k)式的近似表达,可将形状参数Q写为:,20,例7.2 某大尺寸厚板含一表面裂纹,受远场拉应力 作用。材料的屈服应力为ys=600MPa,断裂韧 性K1c=50MPam1/2,试估计:1)=500MPa时的临界裂纹深ac。(设
11、a/c=0.5)2)a/c=0.1,a=5mm时的临界断裂应力c;,解:1)无限大体中半椭圆表面裂纹最深处的K最大,考虑小范围屈服,在发生断裂的临界状态有:,21,故得到:,22,不考虑屈服,将给出偏危险的预测。,23,一般地说,只要裂尖塑性区尺寸rp与裂纹尺寸a相比 是很小的(a/rp=20-50),即可认为满足小范围屈服条 件,线弹性断裂力学就可以得到有效的应用。,对于一些高强度材料;对于处于平面应变状态(厚度大)的构件;对于断裂时的应力远小于屈服应力的情况;小范围屈服条件通常是满足的。,24,Plasticity correcting can extend LEFM beyond its
12、 normal validity limits.One must remember,however,that Irwin correction are only rough approximate of elastic-plastic behavior.When nonlinear material behavior becomes significant,one should discard stress intensity and adopt a crack tip parameter(such as the crack tip opening displacement,CTOD)that
13、 takes the material behavior into account.,塑性修正可将LEFM延用至超过其原正确性限制。但必需记住Irwin修正只是弹塑性行为的粗略近似。当非线性材料行为为主时,应抛弃应力强度因子 而采用如CTOD的裂尖参数考虑材料的行为。,25,When Wells attempted to measure K1c value in a number of structural steels,he found that these materials were too tough to be characterized by LEFM.This discovery
14、 brought both good news and bad news:high toughness is obviously desirable to designers and fabricators,but Wells experiments indicated that existing fracture mechanics theory was not applicable to an important class of materials.,Wells试图测量结构钢材的K1c时,发现这些材料韧 性太大而不能用LEFM描述。这一发现带来的既有 好消息也有坏消息:高韧性显然是设计及
15、制造者所 希望的,但Wells的试验指出现有的断裂力学理论不 能用于这类重要的材料。,26,While examining fractured test specimens,Wells notice that the crack faces had moved apart prior to fracture;plastic deformation blunted an initially sharp crack.The degree of crack blunting increased in proportion to the toughness of material.This obser
16、vation led Wells to propose the opening at the crack tip as a measure of fracture toughness.Today this parameter is known as the crack tip opening displacement.,检查已断的试件,Wells注意到断裂前裂纹面已分开;塑性 变形使原尖锐的裂纹钝化。钝化的程度随材料的韧性而增 加。这一观察使Wells提出用裂尖的张开作为断裂韧性的度 量。此参数即现在的裂纹尖端张开位移。,27,习题:7-3,7-4,再 见,第一次课完请继续第二次课,返回主目录,28,第七章 弹塑性断裂力学简介,7.1 裂纹尖端的小范围屈服,7.2 裂纹尖端张开位移,7.3 COD测试与弹塑性断裂控制设计,返回主目录,29,7.2 裂纹尖端张开位移(CTOD-Crack Tip Opening Displacement),则塑性区将扩展至整个截面,造成全面屈服,小范围屈服将不再适用。,30,显然,COD是坐标x的函 数,且裂纹尺寸a越大,COD越大。裂尖张开位移(CTO
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