Meccanica della frattura - Domande

EXPLAIN UNDER WHAT CIRCUMSTANCES A 3D PB. CAN BE SOLVED AS 2D PB.

AND OBTAIN THE CORRESPONDING FORMULATION IN THE DIFFERENT CASES.

Following we treat a 3D LINEAR ELAS. Problem by applying it in surfaces.

L Let's consider a 3D body V₀ where limit both body force F_i and surface force F.

The analyzed body can be analyzed mainly in two portions: one of them can be loaded

completely or partially under the effect the other one constrained on the ground. _Sup

• Volume
• Surface S = S_re+S_sup
• S_re = Constrained surface = S by Def.

... (rest of the page with equations and detailed explanations)

COMPLETELY

... (several technical details and equations)

COMPATIBILITY

• ε_yz = Y_z
• ε_yz = Y_z = ε

The ... (continuation with details on compatibility conditions)

CONSTITUTIVE LAW

• E_e = 1 ∝ (τ_e-∝τ_yy)
• ɛ_e = 1 ∝ (τ_e-∝τ_yy)

... (further technical notes)

Determine the new stress field from ... 2D problem as:

• σ_xx = λ(τ_x)
• τ_yy = λ(τ_y)

The ... (analysis continues with stress field determinations)

Your attempt solving a 3-body case... (many derivations)

Explain under what circumstances a 3D PB can be found as 2D PB and explain the corresponding formulation in the different cases.

Following we present a 3D linear elastic problem by applying it in the features set of a continuum body, on which physical load body forces F and surface forces f are applied, being the system supposed divided in two parts: one S₂ that can be loaded completely or partially with force f and the other one constituting all the ground. S₁

• Volume
• S = S₂ + S₁
• S = Continuous surface S = S² + S₁ + Sf

S₁ and S₂ that of surfaces are mutually orthogonal.

The BC which regulates the problem are:

• Small displacement and rotation
• Linear elastic material
• Tensor rotation (indices) 1,2,7 → i,j,k

Now we can set the equations that are necessary for a 3D problem and then we can separate them by introducing some specialized or intrinsic characteristics.

Equ eqt

1. _i,j = F in V
2. _j,n = F₁ on S₁

3 Eqt

1. (_ij,i) = ₁₁ i, j, k, 7, 8 indices

Later they can be written as:

1. ^x_x,x + ^x_yy + ^x_zz + f_x = 0

Compatibility

1. _ij = ( _ij,1 - _ij,j) or V
2. U_y = U_y or on S₁

Constitutive law

1. E_xx = 1/E ( ^x_xx - v^x_yy)

So we have 15 equations or 15 unknowns/problems, which problem can be solved in 3D. Now, considering the equations, we can specialize:

• (2D) also called plane problem. From the 2 types of problems they appear to restrain or a 3 problem of rotational symmetric.

We start by eliminating the sides x, y from the equations before written:

Eq:

1 ^x_xx + ^x_yy = 0 ➔ Only ^x_yy in y

^{9 5}

Compatibility:

1. E _xx = ^x_XX
2. ^y_yy = ^y_xy + ^y_yx
3. M = _yy, = - ^y_xy

Initial compatibility:

1. ^xy_yy = ^xy_xy

Constitutive law:

1. E_cc = 1/E ( ^xx+v^yy)
2. ^xy_yy = 2 (1+ ^xy) G; G

Other remarkable, the new stress field that can be instructed from the 2D problems are:

1. U_x = U_{x+y, x}
2. U_yy = U_yy(x-x)

To we start from 3D body and we try to find a solution for it:

Z = Z + Z

T_ar = circum. part of boundary

T_f = free surface

F = lat. surface = T_na+tf

➔ Extrm 1 to 2 axis to be now analyzed.

No loads on base surfaces x₁x₂

Tractions only on T₄

Unconstrained faces T₂ = T₃ = 0

Constrained only in z direction

Fan only…

External forces depends only on x₁x₂

In terms of stresses: from equilibrium

σ_xz = τ(x,y) T_z = 0 (_x, λ₃) not known

In terms of strain: from