2.2 Balance Relations
2.2.3 Entropy inequality
20 2 Fundamentals of the Theory of Porous Media Table 2.6: Local balance equations for constituent
mass: (⇢↵)0↵+⇢↵divx0↵ = ⇢ˆ↵
momentum: ⇢↵x00↵ = divT↵+⇢↵b↵+ ˆp↵
m.o.m: 0 = I⇥T↵+ ˆm↵
energy: ⇢↵("↵)0↵ = T↵·L↵ divq↵+⇢↵r↵+ ˆ"↵ entropy: ⇢↵(⌘↵)0↵ = div ↵⌘ + ↵⌘ + ˆ⇣↵
As an opposite case againstCauchy type materials, granular materials are so-called micro-polar materials, which, at the micro scale, are defined as rotatable particles. As a result, an additional degree of freedom, the rotation angle must be accounted for leading to an asymmetrical stress tensor. The interested reader might refer to the work of Cosserat brother [50, 51]. For theCosserat model in the framework of the TPM, compare the work of Diebels & Ehlers [57], Diebels [58], Ehlers [68], Ehlers & Volk [64, 65] and Scholz [186].
2.2 Balance Relations 21
which is also known as the Clausius-Duhem15 inequality. For a purely mechanical model, where the thermal e↵ect does not play a role, the temperature is assumed to be spatially and temporally constant:
✓ =✓↵ ⌘const. (2.57)
Taking into account that the absolute Kelvin temperature is always a positive number helps to reduce the above-mentioned inequality to the so-called Clausius-Planck
inequal-ity: X
↵
[T↵·L↵ ⇢↵( ↵)0↵ pˆ↵·x0↵ ⇢ˆ↵( ↵+1 2
x0↵·x0↵)] 0. (2.58) This inequality provides a necessary condition for the thermodynamical consistency and will be further discussed in Chapter 4.
15Pierre Maurice Marie Duhem (1861-1916): French physicist, mathematician and philosopher of sci-ence.
Chapter 3:
Fundamentals of Fracture Mechanics
This chapter aims to give a brief introduction to the basic theory of fracture mechanics. It starts with an explanation of the physical mechanism from both macro- and nanoscopic points of view and follows an elementary knowledge of several classical theories such the Stress Intensity Factor, Griffith’s theory, and the J-integral. In the last section, a phase-field model applied to fracture is presented, which in the following context is chosen as an approach for the crack phenomena in porous media. For a detailed depiction of fracture mechanics, compare the books of Gross & Seelig [98], Anderson [10], and Kanninen & Popelar [128]. Note that the discussion within this chapter is limited to the failure in a pure solid. For porous media, special considerations need to be paid owing to the heterogeneity of the micro-structure. For example, the failure mechanism of porous materials under compression is discussed in Salje et al. [183].
3.1 What is Fracture?
Fracture is one of the most common and important reasons for structural failure. How-ever, it is not that easy to find an exact definition to conclude these phenomena. From a macroscopic point of view, a fracture can be straightforwardly interpreted as the sep-aration of an object or material into two or more pieces. When one fracture occurs, it must be accompanied by the generation of two new surfaces, forming additional internal boundaries within the body. These new boundaries are the so-called crack surface in three dimensions or the crack lips in two dimensions, the joint curve or point of which is known as crack front or crack tip correspondingly. According to the relative movement of the crack surface, three independent fundamental fracture modes are defined by Irwin [119], cf. Figure 3.1. These basic fracture modes are usually known as Mode I (Opening Mode/Tension), where the two crack surfaces only move in e2 direction symmetrically
Mode I Mode II e1 Mode III
e2
e3
Figure 3.1: Three fundamental fracture modes.
23
24 3 Fundamentals of Fracture Mechanics
with respect to the undeformed crack plane (e1 e3 plane); Mode II (Sliding Mode/In-plane Shear), where the crack surfaces slide against each other with the same magnitude ine3 direction; and Mode III (Tearing Mode/Out-of-plane Shear), where the crack parts proceed in the opposite directions parallel to the crack front. In particular, an arbitrary fracture mode can be described as one of these three modes, or their combinations.
On the other hand, from a nanoscopic point of view, a fracture is directly caused by a break of bonds that hold atoms together. These bonds are formed, for example, when the liquid-state metal is cooled down until a polycrystalline structure is accomplished. The bonds are stable owing to the electromagnetic forces between the electrons of neighbour atoms. Note that the inter-atomic distance is reduced with the decrease of temperature, leading to an equilibrium state at the critical distance x0 of a bond creation. Assume an opposite condition, when a sufficient stress is applied under a tensile test, these bonds will be broken if the atoms are far away from each other and the mutual attractive forces are too small. Following an interpretation in Anderson [10], the bond energy is proposed to be a function of the separation distancedthat reaches a minimum at the critical distance x0, cf. Figure 3.2. In order to force this distance greater than x0, an external tension force needs to be applied while, on the other hand, the bond shrinks under a compression force. In particular, the work done by the external force has to be equal to the di↵erence of the bond energy. So as to compute the critical strength in this setup, the dependency of the inter-atomic force ¯f between atoms on their distance is ideally approximated by a sine wave as shown in Figure 3.2 with a dashed curve,
f¯=fcsin [⇡(x x0)
] for x2(x0, x0+ ). (3.1) Herein, is chosen to be the two-time value of the distance between the moments when the force turns from compression to tension and when the maximum tensile force fc is achieved. Since sin(⇡x) ⇡ (⇡x) for a small x, the elastic sti↵ness around x0 can be approximated by
k =fc
⇡. (3.2)
Notice thatk measures the sti↵ness of each bond and the more common elastic modulus, Young’s1 modulus E, is referred to a unit area. Therefore, if the number of bonds per unit area is known and expressed by Nb, the Young’s modulus can be computed by
E = kNb
x0
. (3.3)
In addition, the multiplication of fc and Nb yields the critical force per unit area, which is usually named after the critical stress and expressed by c. Thus, this stress takes the form of
c = E
⇡x0. (3.4)
As the bond energy measures the work done by the external force, it can be approximated under the above assumptions by
Eb = Z 1
x0
fdx⇡ Z x0+
x0
fcsin [⇡(x x0)
] dx. (3.5)
1Thomas Young (1773-1829): English polymath.
3.1 What is Fracture? 25
+ +++
-
+-
--
-+++ ++
--
--
-Distancex Repulsion
Attraction
d
x0
Distancex Tension
Compression
Equilibrium Spacing
Bond Energy
k
Bond Energy Cohesive
Force
Figure 3.2: Schematic illustration of potential energy and force with respect to the distance between two atoms (originated from Anderson [10]).
Correspondingly, the surface energy per unit surface area s is defined as one half of the multiplication of Eb and Nb by noticing two surfaces with the same areas are created when a material fractures. In some references, s is also called the surface tension of the material.
s ⇡ 1 2
Z x0+ x0
csin [⇡(x x0)
] dx= c
⇡ (3.6)
In order to rewrite the critical stress in terms of the surface energy, insertion of (3.17) into (3.5) yields
c =
rE s x0
. (3.7)
It is observed that for most materials, the surface energy s is usually on the order of
26 3 Fundamentals of Fracture Mechanics
2
2
2
2
(a) (b)
Figure 3.3: (a) straight stress trajectories (lines of force) in an unbroken plate (b) curved stress trajectories in a plate with an elliptical hole in the middle.
0.01E x0, cf. Cottrel [52]. Hence, a practical estimation of the theoretical strength is given by
c = E
10. (3.8)
However, this result has a large deviation from the one discovered by experiments whereas the experimental result is much lower. The main reason for such a discrepancy is the stress concentration caused by the inevitable flaws in the practical-size bulk material. Details in this regard will be discussed in the following section.