Combining Equilibrium, Smooth Deformation, and Linear Elastic Behavior (§ 2.4.0)

The table below summarizes all the equations we have derived in previous sections of this tutorial. If they are still unfamiliar, the numbers correspond to the equation number. Return to the section and review the derivation of the equations before continuing the tutorial.

Equilibrium	δσxx δx + δτxy δy = 0 (1)		δτyx δx + δσyyδy = 0 (2)
Smooth Deformation	δ2εxxδy2 + δ2εyyδx2 - δ2γxyδxδy = 0 (11)
Linear Elastic Behavior	εxx = 1E (σxx - νσyy) (22)	εyy = 1E (σyy - νσxx) (23)	γxy = τxyG (24)	G = E 2 (1 + ν) (25)

Our first step is to combine the constitutive relations (from § 2.1.3: Linear Elastic Behavior) with the continuity expression (from § 2.1.2: Smooth Deformation) and equilibrium expressions (from § 2.1.1: Equilibrium). Starting with the continuity expression (Equation 11),

δ2εxxδy2 + δ2εyyδx2 - δ2γxyδxδy = 0

(26)

and inserting the constitutive relations (Equations 22-25) into Equation 26, we get:

δ2 δy2 ( 1E (σxx - νσyy)) + δ2δx2 ( 1E (σyy - νσxx)) - δ2δxδy ( (2+2ν)E τxy) = 0

(27)

Multiplying through by E and expanding the products leads to:

δ2σxxδy2 + δ2σyyδx2 - ν ( δ2σyyδy2 + δ2σxxδx2 ) - 2 δ2τxyδxδy - 2 ν δ2τxyδxδy = 0

(28)

Our next step is to take the equilibrium expressions (Equations 1-2) and put them in a form where they can be substituted into Equation 28 above. To do this, take the partial derivatives of the equilibrium expressions, as shown below.

Equilibrium Expression

Partial Derivative

x-direction

δσxxδx = - δτxyδy

(29)

δ2σxxδx2 = - δ2τxyδxδy

(30)

y-direction

δτyxδx = - δσyyδy

(31)

δ2σyyδy2 = - δ2τxyδxδy

(32)

Plugging these new expressions for equilibrium (Equations 29 - 32) into our partial differential equation for smooth deformation in a linear elastic solid (Equation 28) and simplifying leads to the following expression:

δ2σxxδy2 + δ2σyyδx2 -ν( - δ2τxyδxδy + - δ2τxyδxδy ) - 2 δ2 τxyδxδy - 2 ν δ2τxyδxδy = 0

(33)

δ2σxxδy2 + δ2σyyδx2 -2 δ2τxyδxδy = 0

(34)

This single partial differential equation describes the stresses, σ_xx, σ_yy, and τ_xy in any plane strain problem in a linear elastic solid in which deformations are smooth, satisfying our three assumptions mentioned in the Introduction to this chapter.

Remember that Equation 34 was derived in the same way that most mechanics problems are solved: by combining separate expressions developed from equilibrium, continuity, and material properties. Our next step is to fine-tune the equation to the specific problem setup of fracture mechanics by means of defining the boundary conditions.

Exercise 2: For the following questions, match each value to its variable.

1) Shear stress: σ x ε τ

2) Young's Modulus: G γ E σ