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.

_SupVolumeSurface S = S_re+S_supS_re = Constrained surface = S by Def.... (rest of the page with equations and detailed explanations)

Completely... (several technical details and equations)

Compatibility

Compatibilityε_yz = Y_zε_yz = Y_z = εThe ... (continuation with details on compatibility conditions)

Constitutive law

Constitutive lawE_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.

S₁VolumeS = S₂ + S₁S = Continuous surface S = S² + S₁ + SfS₁ and S₂ that of surfaces are mutually orthogonal.

Boundary conditions

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_i,j = F in V_j,n = F₁ on S₁

3 Eqt (_ij,i) = ₁₁ i, j, k, 7, 8 indices

Later they can be written as:

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

Compatibility

Compatibility_ij = ( _ij,1 - _ij,j) or VU_y = U_y or on S₁

Constitutive law

Constitutive lawE_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

Compatibility:E _xx = ^x_XX^y_yy = ^y_xy + ^y_yxM = _yy, = - ^y_xy

Initial compatibility:^xy_yy = ^xy_xy

Constitutive law

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

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

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

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

Z = Z + ZT_ar = circum. part of boundaryT_f = free surfaceF = 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 equilibriumσ_xz = τ(x,y) T_z = 0 (_x, λ₃) not known

In terms of strain: from