Displacement Discontinuity Method (DDM)

Introduction

The displacement discontinuity method (DDM) uses the physical analog of a slit-like crack and a series of distribution of dislocations (Figure 1). Consequently, it only needs to discretize the crack surfaces, making itself perhaps the most efficient method for crack problems [1]. The DDM is applied for crack problems based on the fundamental dislocation theory.

../_images/ddm-scheme.png

Figure 1: Scheme of the DDM

E and v represent the Young’s modulus and Poisson ratio.

Fundamental dislocation theory in DDM

The DDM uses the physical analog of a slit-like crack and a series distribution of edge dislocation dipoles in a homogeneous medium (Figure 2). Each dipole represents a crack element and a constant opening over it.

../_images/ddm-dipole.png

Figure 2: Dislocation dipoles in DDM (only by is present due to visual simplification)

The stress components due to two pairs of dislocation dipoles ( Figure 3) are:

(1){σyy=bxky(A4−A3)+byk[(x+h)A6−(x−h)A5]σxy=bxk[(x+h)A4−(x−h)A3]+byky(A4−A3)

where,

(2){A3=(x−h)2−y2[(x−h)2+y2]2A4=(x+h)2−y2[(x+h)2+y2]2A5=(x−h)2+3y2[(x−h)2+y2]2A6=(x+h)2+3y2[(x+h)2+y2]2

and

(3)k=G2π(1−v)

and G is the shear modulus.

../_images/ddm-dipole-pair.png

Figure 3: Two dislocation dipole pairs

The bx and by are the burger’s vectors analog to the constant shear opening Dx and normal opening Dy over a crack element, respectively.

Influence coefficients

Influence between two elements

The influence between two crack elements, say the stress at the center of the i th element (field element) due to the dislocation dipole pairs over the j th element (source element), is illustrated in Figure 4 (a)

../_images/ddm-influence.png

Figure 4: Influence in DDM

and calculated as Figure 5:

../_images/ddm-coefficients.png

Figure 5: DDM influence coefficients

where,

(4){¯A3=(ˉxi−h)2−ˉy2i[(ˉxi−h)2+ˉy2i]2¯A4=(ˉxi+h)2−ˉy2i[(ˉxi+h)2+ˉy2i]2¯A5=(ˉxi−h)2+3ˉy2i[(ˉxi−h)2+ˉy2i]2¯A6=(ˉxi+h)2+3ˉy2i[(ˉxi+h)2+ˉy2i]2

and,

  • (ˉxi,ˉyi) are the center coordinate of the i th element under the local coordinate defined by the j th element ( Figure 4 (b))
  • h is the half size of the j th element
  • The coefficients B etc. are the influence coefficients.

Note

From Eqn (4), the influence between two elements is proportional to r−2, where r represents the distance between the two elements.

The total influence on an element

Then the stress at the center of the field (i th) element due to all the dislocation dipoles (say N source elements, as illustrated in Figure 6) over the crack surface is obtained as Eqn (6):

../_images/all-influence.png

Figure 6: The total influence on the field (i th) element

(5){σin=N∑j=1BijnsDjs+N∑j=1BijnnDjn=pinσis=N∑j=1BijssDjs+N∑j=1BijsnDjn=pis⇓
(6){N∑j=1BijnsDjs+N∑j=1BijnnDjn=pinN∑j=1BijssDjs+N∑j=1BijsnDjn=pis

where,

  • pis and pin are the shear and normal tractions applied on the field (i th) element.
  • Dis and Din are the shear and normal openings (displacement discontinuities) over the field (i th) element.

Coefficient matrix

The coefficient matrix is assembled by ranging the index of the field (i th)element from 1 to N.

(7)[B11ssB11sn……B1jssB1jsn……B1NssB1NsnB11nsB11nn……B1jnsB1jnn……B1NnsB1Nnn⋮⋮⋱⋱⋮⋮⋱⋱⋮⋮⋮⋮⋱⋱⋮⋮⋱⋱⋮⋮Bi1ssBi1sn……BijssBijsn……BiNssBiNsnBi1nsBi1nn……BijnsBijnn……BiNnsBiNnn⋮⋮⋱⋱⋮⋮⋱⋱⋮⋮⋮⋮⋱⋱⋮⋮⋱⋱⋮⋮BN1ssBN1sn……BNjssBNjsn……BNNssBNNsnBN1nsBN1nn……BNjnsBNjnn……BNNnsBNNnn][D1sD1n⋮⋮DjsDjn⋮⋮DNsDNn]=[p1sp1n⋮⋮pispin⋮⋮pNspNn]

According to the Einstein notation, Eqn (7) can be written in a more concise form:

(8)KijDj=pi

where,

(9)Kij=[BijssBijsnBijnsBijnn]Dj=[DjsDjn]pi=[pispin]

Then, the crack opening is obtained by:

(10)Dj=K−1ijpi

Reference

[1]Peirce, A. and Siebrits, E: Uniform asymptotic approximations for accurate modeling of cracks in layered elastic media, International Journal of Fracture 110(3), 205-239 (2001)