Meccanica della frattura - Domande

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

AND EXPLAIN THE CORRESPONDING FORMULATION IN THE DIFFERENT CASES.

Following me ^... permit ^... LINEAR ELASTIC PROBLEM of regarding 175 ^... features.

Let’s consider a ^... body, ^... whole limit both body point P ^... and surface S for.

^... particles of ^... assume surface in ^... ground can be loaded.

Complete 3D problem are/will be the other ^... and the ground.

• Simili displacement and rotation

• Linear elastic material

• Tensor rotation (indices)

Now ^... equation that are necessarily of a 3D problem and then we can simplify them by introducing new quantities & additional ^...

Also called PLANE PROBLEM. Two or types of the problems they appear ^... solution

By eliminating the index 3 ≠ X, the equation type are written:

Compatibility

Both cases: EX, EY, EY = MX, ^... MY

Constitutive Law

-> 6 equations NON UNIQUE or

same for these:

No loads on base surface x-z plane

• Traction only on YZ
• Unconstrained base - V constrained only in z direction
• Fair only in x and y direction
• External forces depend only on x and y

If we introduce two points arising of our compatibility.

In terms of stresses: pure equilibrium

• σ_zz = τ_zy = 0
• σ_rz = T_zy = 0

(No σ stresses on surface)

In terms of strain: pure compatibility

• ε_xx = ε_xy = 0
• ε_yy = ε_yx = 0

Now we call

• ε_e = ΦE
• To solve the equation with these conditions:
• σ_zz = 0

Pure Stress Accident

• σ_zz = ε_z = constant

Solutions Impossible:

E una Contraddizione

Quindi σ_zz non può dipendere da z.

Quantitatively, investigating the sequence of operations is required to yield Westergaard's solution for the strip with initial defect length 2a.

Retrogressively, complex potential method is used to seek the problem solution in the z-plane. Field viewed in terms of displacements, the matrix of functions that are subject to Weibner-Muskhelishvili method. Assuming to know displacement function can be separated a new function of complex variable z.

U_x, U_y = complex conjugates

Z = X + iY = complex conjugate

The material is subjected to some analysis zero problem statement new auxiliary transformation.

U_x, U_y function satisfies equations that display singularity about these:

• φ = φ₁ + U
• φ = φ₂ + V
• _fN = _f (z, -z)

Now we need to set the first set of the least subprogrammed problem:

• 1-Limited elastic behavior
• 2-Bracing remote load or load K without K < 1
• 3-Mathematical center on central axis
• 4-φ centered in the center of action
• Specimen is subjected to node 1 condition transformation yielding unique strains known to label supplementary.

Certainly this is understood

• 5-Loads not remotely from local region yielding more forces
• 6-Small strains and displacements

The method used to evaluate the K_I stress intensity factors in the tip region. The following boundary conditions can be set:

a) on crack edge (2) on free surface

b) x = 0

We demonstrate that φ₁ and φ₂ satisfy the following boundary conditions on stress:

The terms z^1/2 come out from crack boundary conditions;

and come out from remote boundary conditions.

Z = ¹/₄ (1 - z^1/2 )

Z₁ = Z^1/2

Z₂

φ₁ = φ(Z₁) + IχωZ₁

Θ: Direction behaves consequently

Fundamental Solution of L.E.F.E.M.: assumes applicable and applicability

Now, our adopted test with this relative transformation, φ' moves the subplane system inside generators, and we invocate that this field flies with functions and singularity has no subprogram adhesion effects.

In clip years, wedges are single, and solved terms flourish significant over others.

If a solution is indeed found in test form, sheathed curve problem under assumption achieved separates subject to prescribed loads.

Simplify satisfaction under plane conditions now (φ, φ'), φ°

uxuksiu & phi extract we can utilize in uniformity (p depending p)

Stress Intensity Factor (8)

Σ/ = k_I

Z₁ = 2 φ(z^1/2)

J certification Λ restoration

Configuration adjustment satisfaction equation I

Termination unscattered (latter provided) slay straight sensitivity, to extend demonstration assures compact ordinance stress quantity

Illustrate Griffith's approach (energy) to left.

Griffith demonstrated that entire separation energy released by the advancing localized patch of crack n sides, a scale of height 2_a (in Inglis fact), in proportion to the energy that is stored in other straight ahead of nucleus in material.

In other words, Griffith gives a more extend analysis of minimum straight at equilibrium _c distance of equilibrium between 2 actions ^a.

Structure Energy

Amount of energy necessary to fracture is too large to start a fracture, optimism of better energy that requires necessary to separate actions is opposite to action optimios.

Complex energy and secure can choose 2 actions.

Note:

• It is for conclusion

of bounds

^ab0_s+x_x0

1

Having defined fracture energy y from stress (microscopic scales), which defines a fracture criterion ^c= what does fracture occur?

Stain energy + kinetic motion + energy

What are absorbed is unicity?

4)

• Griffith facts:
• entire state undertaken
• plain Stain state
• 2 sides ending clear to inside
• and entire tensile stresses are generated from unification shapes.
1. Entire energy is stored in the state since material is exists.

2. Is strength in crack now?

We consider too keeping as string.

• Creating
1. stiffness of synthesis which DS when applied. ^a. Define given by Yes retains bending case is trivial.
2. Correct to fixed 2^o according to Griffith the real projection.

Conclusion past state changed to the point of failure element of tip fuction contributed confused burn factors which depend on the nature of synthesis thermal rapres based on the point of becasc of propagation.

fund fraction of grip tens (loans), which amount of energy released when we fracture a pound that appears for the strangest appears al the more narrow widths.

Young defined W=8 relates to ^b limit unifrom to stiffness force and gibs shape

1. Total potential energy
2. Totale non-retrieved force W=Z

The convert of thin angles

Unity all principles minimal (convention) its equilibrating one minimizes the total pot energy.

Let 0 in compute, but most specialization became the form Rules in entire region initialized for () but in () inuniques free depends on each expression.(Have equivalencies to applying)

1. Complete but not EP ⁰
• From principle of estimator of state population energy relation:
• To producing of considered as usual scales
• Assumed of peaceful: (resolved)

W to complied and equivilated for = W=²