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254 Ch a p t e r E i gh t

8.3.1 Special Strain Descriptions for Continuous Interface Elements
Occurrence of shear strain localization along an interface indicates the concentration of
shear deformation with high gradients within the narrow interface zone. High gradi-
ents occur only in the direction normal to the interface. In the direction parallel to the
interface, shear strains do not vary significantly. In conventional finite element analysis
with infinitesimal strain assumption, the strain-displacement relations for a 2D prob-
lem, also used for a continuous interface element, are written as:
⎧ u ∂ v ∂ ⎫
⎪ + ⎪
γ ⎧ ⎫ ⎪ n ∂ t ∂ ⎪
⎪ nt ⎪ ⎪ v ∂ ⎪
ε ⎨ nn ⎬ = ⎨ ⎬ (8-39)
⎪ ⎪ ⎪ ⎪ n ∂ ⎪
⎩ ε tt ⎭ ⎪ u ∂ ⎪
⎪ t ∂ ⎪
⎩ ⎭
Here, n denotes the direction normal to the interface, and t the direction parallel to
the interface, as shown in Figure 8.3(a). Displacements u and v are parallel and normal
to the interface, respectively. Displacement u has much higher gradient in n direction
than v has in t direction, which is mathematically expressed as:
∂u ∂v
(8-40)
∂n ∂t

So, it is accurate enough to keep the first term for the shear strain on the right-hand
side of Equation (8-39). Due to the very thin interface, γ nt and ε nn can be written as:
⎧Δ u⎫
⎪
γ ⎧ ⎪ nt ⎪ ⎫ ⎪ d ⎪
⎪
⎨ ⎬ ≈ ⎨ ⎬ (8-41)
⎩ ε ⎪ nn⎭ ⎪ ⎪ Δ v ⎪
⎩ ⎪ d ⎭ ⎪
Where Δu and Δv are tangential and normal relative displacements across the inter-
face, respectively, and d is thickness of the interface. The normal strain in the direction
parallel to the interface, e t , in any continuous interface element, takes the formulation
as in regular solid elements. Therefore, the mathematical expression for the strains in
continuous interface element is given by:
⎧Δ u⎫
⎪ ⎪
γ ⎧ ⎫ ⎪ d ⎪
⎪
⎪ nt ⎪ ⎪Δ v ⎪
ε ⎨ nn ⎬ = ⎨ ⎬
⎪ ε ⎪ ⎪ d ⎪ (8-42)
⎩ tt ⎭ ⎪ u ∂ ⎪
⎪ t ∂ ⎪
⎩ ⎭
There are two advantages to the special strain description in Equation (8-42). First,
the experimental relationship between stresses and relative displacements across an
interface can be used directly for interface elements. Second, it provides a fundamen-
tal approach to describe strains in three-node triangular and four-node quadrilateral