# Griffith 1921 In 1921 Griffith determined experimentally the

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Griffith的�� （1921） In 1921 Griffith determined experimentally the fracture stress σb of glass fibers as a function of their diameter. For d > 20 μm the bulk strength of 170 MPa was found. However, σb approached theoretical strength of 14000 MPa in the limit of zero thickness.

Inglis的理� （1913） 对于无限大板 A点应力分布： Inglis C E. Stresses in a plate due to the presence of cracks and sharp corners. Transactions of the institute of naval architects, 1913, 55(219 -241): 193 -198.

Kirsh的研究 （1898） The 2 -D stress field in a large body under uniform remote tensile load and containing a circular hole is given by (Kirsch, 1898) Y r r X 2 a The general stress field in dimension less form has the structure: For a linear elastic material, the stress field is always linearly proportional to the applied load

Kirsh的研究 （1898） 理�推�：一般情况 平面极坐标下的应力分量： 双调和函数： Kirsch E G. Die Theorie der Elastizit t und die Bed rfnisse der Festigkeitslehre. Zeitshrift des Vereines deutscher Ingenieure, 1898, 42: 797 -807.

Kirsh的研究 （1898） 理�推� 边界条件： Kirsch E G. Die Theorie der Elastizit t und die Bed rfnisse der Festigkeitslehre. Zeitshrift des Vereines deutscher Ingenieure, 1898, 42: 797 -807.

Kirsh的研究 （1898） 一些特殊情况 Several aspects of this stress field are worth our attention: 1) As =0 Y yy X xx 2 a As x , xx 0 and yy

Kirsh的研究 （1898） 一些特殊情况 a=0. 5 =1 xx Y yy X xx As x , xx 0 and yy 2 a

Kirsh的研究 （1898） 一些特殊情况 Y yy a=0. 5 =1 X xx 2 a yy Stress concentration As x=a, yy/ =3 Independent of circular hole size a As x , xx 0 and yy

Kirsh的研究 （1898） 一些特殊情况 3) As =0, a 0 Y X 2 a If the circular hole changes to an elliptical hole, then what happens on the stress field?

��形孔洞： Inglis的理� （1913） The 2 -D elastic solution for a large body containing an elliptical cavity under uniform remote tensile load was given by (Inglis, 1913) 2 b ym yy xm xx X 2 a Further discussion: 1)When a=b, then the stress concentration factor of three is recovered for a circular defect; 2) When 0 (the ellipse flattens and become a crack-like defect, ym , no material can withstand infinite stress, so material at ellipse tip must become inelastic, plastic yielding or other inelastic processes

Griffith能量平衡

Griffith �力分析：材料柔度的�化 固定荷载 能量变化 Griffith能量平衡： The energy release rate can be calculated from the change in compliance and that the result for the fixed grip approach is exactly the same as that for the constant load method.

Griffith �力分析：材料柔度的�化 The energy release rate can be calculated from the change in compliance and that the result for the fixed grip approach is exactly the same as that for the constant load method.

�力�度因子的�算校核 Calibration of K, when a full field is know, either analytically or numerically (e. g. FEM), the stress intensity factor (SIF) can be extracted. Unit: MPam 1/2 and MNm-3/2

�力�度因子的�算校核 应力强度因子与材料宏观物理量之间的关系 Case I: through crack Case II: edge crack The 12% increase in KI for the edge crack is caused by different boundary conditions at the free edge. The edge crack opens more because it is less restrained than the through crack, which forms an elliptical shape when loaded.

�力�度因子的�算校核 应力强度因子与材料宏观物理量之间的关系 Case III: penny crack Case IV: randomly oriented through crack

�力�度因子的�算校核 应力强度因子计算实例 已知：材料尺寸和荷载情况见右图， y=0 处的应力场可以表示为 问题： (1) Derive the stress intensity factor (KI) for the given configuration using the above stress field. (2) Give the crack tip approximate stress field ( yy) using above stress intensity factor (KI). (3) Assume the acceptable tolerance in the stress field yy is 10%, i. e. solve the outer limit of the zone (K- dominant zone) in the form of x/a.

�力�度因子的�算校核 解： (1)Shift the co-ordinate to the crack tip, then: x’=x-a and x=x’+a (2) (3) Approximate solution of yy Exact solution of yy

�力�度因子的�算校核 裂缝和试件尺寸的影响 When the crack dimensions are small compared to the size of the plate; the cracktip conditions are not influenced by external boundaries. As the crack size increases, or as the plate dimensions decrease, the outer boundaries begin to exert an influence on the crack tip.

K与G的关系 We now know tow important parameters: the energy release rate G and the stress intensity factor K, The energy release rate describes global behavior, while K is a local parameter Is one related to the other? For linear elastic materials, YES Plane stress Plane strain G-K relations apply only to a through crack in an infinite plate, are they the general relationships that apply to all configurations?

Home work 3 • Calculate and plot the maximum external load P for three point bending specimen with crack length up to 70% beam height. B=100 mm, W=100 mm, S=4 W. KIC=0. 2 MPa(m 1/2). P a S W