By Saouma S.E.
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Additional info for Lecture Notes in Fracture Mechanics
The enunciation of the preceding three conservation laws plus the second law of thermodynamics, constitute what is commonly known as the fundamental laws of continuum mechanics. 2 Fluxes 58 Prior to the enunciation of the ﬁrst conservation law, we need to deﬁne the concept of ﬂux across a bounding surface. The ﬂux across a surface can be graphically deﬁned through the consideration of an imaginary surface ﬁxed in space with continuous “medium” ﬂowing through it. If we assign a positive side to the surface, and take n in the positive sense, then the volume of “material” ﬂowing through the inﬁnitesimal surface area dS in time dt is equal to the volume of the cylinder with base dS and slant height vdt parallel to the velocity vector v, Fig.
13 The following rules deﬁne indicial notation: 1. If there is one letter index, that index goes from i to n (range of the tensor). 2) assuming that n = 3. 2. A repeated index will take on all the values of its range, and the resulting tensors summed. 3) a1i xi = a11 x1 + a12 x2 + a13 x3 3. 4) other ﬁrst order tensors aij bj = ai1 b1 + ai2 b2 + ai3 b3 , Fikk , εijk uj vk • Second order tensor (such as stress or strain) will have two free indices. 5) other examples Aijip , δij uk vk . • A fourth order tensor (such as Elastic constants) will have four free indices.
In 1976, Hillerborg (Hillerborg, A. and Mod´eer, M. E. 1976a) introduced the ﬁctitious crack model in which residual tensile stresses can be transmitted across a portion of the crack. Thus a new meaning was given to cracks in cementitious materials. 12. P. and Cedolin, L. 1979) showed that for the objective analysis of cracked concrete structure, fracture mechanics concepts must be used, and that classical strength of materials models would yield results that are mesh sensitive. This brief overview is designed to make detailed coverage of subsequent topics better understood when put into global perspective.