断裂力学——2Griffith 理论(1)

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Griffith理论
线弹性断裂力学的基本理论
线弹性断裂力学的基本理论包括：
Griffith理论，即能量释放率理论； Irwin理论，即应力强度因子理论。 断裂力学作为一门崭新的学科是在上个世纪50年代才建立和发展 起来的。但是Griffith在1920年建立的针对玻璃、陶瓷等脆性材 料的脆性断裂准则，成功地解释了这类材料的实际断裂强度远小 于理论强度这一客观事实。该理论仅适用于完全脆性材料，对于 绝大多数金属材料，在断裂前和断裂过程中裂纹尖端总存在塑性 区，裂纹尖端也因塑性变形而钝化。不能使用Griffith理论，这 就是该理论长期得不到重视和发展的主要原因。后来Irwin修正 了Griffith的理论，使得断裂力学成为一门学科。
Griffith理论
设想在板中沿垂直于载荷方向切开一条 长度为2a的贯穿裂纹，由于裂纹的长度 远小于板的面内尺寸，可以将此板视为 “无限大”板。由于设想切开了一条贯 穿裂纹，裂纹就形成了上下两个自由面， 原来作用于该表面位置的拉应力消失了， 与此同时，上下自由表面发生相对张开 位移，消失的拉应力对此张开位移做负 功，使得板内的应变能降低了。 Griffith根据Inglis（1913）对“无限 大”板内开了一个椭圆形圆孔后分析得 1 2 U a 2 2 B 到的应力场、位移场计算公式，得出当 E 椭圆孔短轴尺寸趋于零（理想尖裂纹） U 1 a 2 2 B E 时，弹性应变能的改变量为
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C. E. Inglis
Department of Engineering Head of Department 1919-43
He carried the largest teaching load, covering the subjects : statics, dynamics, theory of structures, materials and drawing, balancing engines, girder design and reinforced concrete.
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Griffith理论
Then he looked at other plates with notquite-elliptical holes, like these. From looking at these he realized that it's not really the shape of the hole that matters in cracking. What matters is the length of the crack that is perpendicular to the load and what the radius of curvature at the ends of the hole is. The longer the hole (or crack), the higher the stress, and the smaller the radius of curvature, the higher the stress.
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Griffith理论
C. E. Inglis
thin plate of glass with an elliptical hole in the middle He found that point A, at the end of the ellipse, was feeling the most pressure. He also found that as the ratio of a/b gets bigger (the ellipse gets longer and thinner) that the stress at A becomes greater and greater The maximum stress occurs at the ends of the major axis of the cavity and is given by Inglis’s formula
平面应变 平面应力
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Griffith理论
σ
σ +
σ
σ
= σ σ
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Griffith理论
σ σ
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Griffith理论
• 能量守恒定律：
能量守恒定律是自然界的一条普遍规律，它指出：系统能量 的增加等于输入的能量。对于热力学系统又可表述为：作用 于系统上功的增量δW加上系统接受的热的增量δQ等于系统 内能的增量ΔE加上动能的增量ΔK, 即
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Griffith理论
二、Griffith理论 1920年，Griffith研究玻璃与陶瓷材料脆性断裂问题 时，将Inglis解中的短半轴趋于0，得到Griffith裂纹。
Griffith研究了如图所示厚度为B的薄平板。 上、下端受到均匀拉应力作用，将板拉长 后，固定两端，构成能量封闭系统。
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"His personality was such that he not only taught the subject most delightfully but inspired the members of the audience with respect and affection."
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(1913), pp.219–230.
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C. E. Inglis
Sir Charles Edward Inglis (31 July 1875-19 April 1952) was a British civil engineer. Inglis spent much of his life as a lecturer and academic at King's College Cambridge and made several important studies into the effects of vibration and defects on the strength of plate steel. Inglis served in the Royal Engineers during the First World War and invented a lightweight, reusable steel bridge - the precursor and inspiration for the Bailey bridge of the Second World War . His military service was rewarded with an appointment as an Officer of the Order of the British Empire His mother died when he only is eleven days, he was brought up by his father’s unmarried sister.
Griffith理论
C. E. Inglis
A Mathematical Treatise on Vibrations in Railway Bridges. By C. E. Inglis. Cambridge, University Press, and New York, Macmillan, 1934. 203 pp. and 65 figures.
max
a 1 2 b
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Griffith理论
C. E. Inglis
He also found that pulling on the plate in a direction parallel to the ellipse does not produce a great stress at A. This leads to the fact that a load perpendicular, not parallel, to the crack will make it grow.
Royal Engineers constructing an Inglis Bridge Mk III at the School of Military Engineering, 18 June 1943
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Griffith理论
The International Union of Theoretical and Applied Mechanics was founded in 1946, during the Sixth Congress in Paris. Each of the first six congresses had been organized by a national committee of scientists from the country where the congress was to be held. Fourth C. E. Inglis Congress President 1934 Cambridge UK International Congresses on Theoretical and Applied Mechanics (ICTAM) Beijing， Bai Yilong 23rd
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Griffith理论
Inglis的论文1913年，Inglis研究了无限大板中含有一个穿透 板厚的椭圆孔的问题，得到了弹性力学精确分析解，称之为 Inglis解。1920年，Griffith研究玻璃与陶瓷材料脆性断裂问 题时，将Inglis解中的短半轴趋于0，得到Griffith裂纹。
Transactions of the Institution of Naval Architects,55
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Griffith理论
一、动机 两个矛盾的事实
The stress needed to fracture bulk glass is around 100 MPa. The theoretical stress needed for breaking atomic bonds is approximately 10,000 MPa experiments on glass fibers that Griffith himself conducted suggested that the fracture stress increases as the fiber diameter decreases. –尺寸相关性