Mechanics of Materials Assignment Help

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Problems:
Instrumented Nano-Indentation: Instrumented nano-indentation is a new technique in materials
science and engineering to determine material strengths at very fine scales. The test consists in a
penetration of a needle-type indenter in a continuous material system (see experimental setup in figure
(a) below). The force required to penetrate is then related to the strength of the material — by means
of mechanical modeling
In this exercise, we propose to develop a simplified triaxial stress—strength model of the
nanoindentation test. To simplify the problem, we consider that the indenter is a rigid cylinder of radius
r0, situated on the surface of a horizontal half-space composed of a homogeneous material, as
sketched in figure (b) below. A vertical force F is exerted on the cylinder in the direction of the cylinder
axis Oz, until it penetrates into the half-space. The value of the force F at this moment is noted max F,
and the material property that is reported from the test is known as micro-hardness:
where A is the contact area of the indenter with the material. We suppose that the contact of the
cylinder with the half-space (at z = 0; r ≤ r0) is without friction. Aim of this exercise is to relate the
micro-hardness measurement to the strength properties of the material composing the half-space.
Throughout this exercise we will assume quasi-static conditions (inertia effects neglected), and we will
neglect body forces.
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Nano-Indentation test: (a) Experimental Setup; (b) Simplified Mechanical Model.
1. Statically Admissible Stress Field: For purpose of analysis, we separate the halfspace Ω in two
subdomains, noted respectively Ω1 and Ω2. In these domains, we consider the following stress fields:

(a) Specify precisely ALL conditions which statically admissible stress fields in Ω1 and Ω2 need to satisfy.
(b) Determine the constants q , q, q and σ, so that the stress field σ is statically admissible in Ω = Ω1∪
Ω2.
(c) In the Mohr Plane (σ × τ ), give a graphical representation of the stress field σ for 2 Ω1 and Ω2, by
considering that F > qπr0. In both Mohr Plane and material plane, determine the surface and the
corresponding stress vector, where the shear stress is maximum in Ω.
2. Mohr-Coulomb Strength Criterion: The material we consider is a Mohr-Coulomb material, for which
the strength domain is defined by:
where |τ| = T2 − σ2, σ = n·σ·n; tan ϕ is the friction coefficient, and c is the cohesion. Alternatively, the
Mohr-Coulomb criterion can be written in terms of the principal stresses σI ≥ σII ≥ σIII :

(a) Display the Mohr-Coulomb criterion in the Mohr Plane (σ × τ);
(b) Determine the relation between micro-hardness H and the strength material properties of the
Mohr-Coulomb criterion.
(c) In the material plane, represent the orientation of the critical material surfaces, on which the
Mohr-Coulomb criterion is reached.
3. Refined Approach: By considering that the stress field in Ω2 was constant, determine a second
relation between the micro-hardness H and the Mohr-Coulomb model parameters. Which of the two
solutions is closer to the ‘real’ maximum micro-hardness value at failure of the Mohr-Coulomb material
system. Say why (HINT: Sketch your response in the MohrPlane)?

1 Statically Admissible Stress Fields A statically admissible stress field is a stress field which satisfies (i)
the force boundary conditions, (ii) the stress vector continuity condition on any surface in the material;
(iii) the symmetry of the stress tensor; (iv) the momentum balance.
1.1 Boundary Conditions
For the nanoindentation test, the boundary conditions are:
Solutions:

— From a combination of (1) and (2), it follows:
where σij A = 2 σijda stands for the stress average of σij over the surface A= 0 πr A = πr0 2, and H = F/A
is the micro-hardness measured in the nanoindentation test. Note that it cannot a priori be concluded
that σzz = −F/A, since σzz may not be constant over the contact area. • In Ω2:
1.2 Continuity of Stress Vector
On the interface between domain Ω1 and Ω2:

1.3 Form of the Stress Tensor Given the rotational symmetry of the problem, σθr = 0 in Ω. The stress
tensor, therefore, is of the diagonal form:
which satisfies the symmetry condition, σ = t σ, and the boundary conditions (3) and (4), and the
continuity condition (5)
1.4 Momentum Balance
Neglecting body forces, the stress tensor σ in Ω must satisfy the following momentum balance
equations (cylinder coordinates):

1.5 Application
• In Ω1
From (7)1:
From (7)3 and (3)3:
• In Ω2: The stress field,

satisfies the boundary condition (4), the momentum balance equations (7)1:
The stress continuity (5) is satisfied for:

Figure 1: Stress Representation of the Nano-Indentation stress model: (a) material plane; (b) Mohr
plane.
1.6 Mohr Representation
The stress field are:
The Mohr circles are displayed in Figure 1b, for H>q. The maximum shear occurs in Ω1 and the
corresponding stress vector has the components:
The maximum shear stress occurs on the surface oriented by (see Figure 1a):

2 Mohr-Coulomb Criterion
The unknown of the problem is the stress quantity q. It is determined through application of the Mohr-
Coulomb criterion:
Use in the Mohr-Coulomb Criterion reads:
• In Ω2:
Use in the Mohr-Coulomb criterion reads:

From (20), we obtain:
Use in (19) gives the sought relation between the Hardness measurement and the MohrCoulomb
model parameters:
The representation of this limit state is shown in Figure 2, in both the physical plane and the Mohr-plane.
The critical stress state is reached in Ω1 on a material surface oriented by:
3 Refined Approach
We consider that the stress state in Ω2 was constant. From the momentum balance (7)1 we find:

This stress field is statically admissible provided that the stress continuity along r = r0 is ensured:
It follows:
• In Ω1:
Use in the Mohr-Coulomb Criterion reads:

Figure 2: Stress State at strength limit: (a) material plane; (b) Mohr plane.
• In Ω2:
Use in the Mohr-Coulomb criterion yields:

From (30) we obtain:
Use in (28) yields:
This improves stress-strength solution is displayed in the Mohr-Plane in Figure 4. It is obtained by shifting
the Mohr-circle in domain Ω2 along the normal stress axis into the compression domain. This yields a
higher micro-hardness than the first solution (see Figure 3). Since the stress field is statically admissible
and since it satisfies the strength criterion, this higher value is closer to the ‘real’ micro-hardness of the
material. The material planes along which the material realizes the strength criterion are still the same as
before (see Figure 2a), but extends now also in domain Ω2. This is displayed in Figure 5

Figure 3: Comparison of the normalized micro-hardness values versus friction angle.

Figure 4: Improved Stress-Strength Solution in the Mohr-Plane. The stress state is statically admissible,
and satisfies the Mohr-Coulomb strength criterion.
Figure 5: Improved Stress-Strength solution: Display of normal planes on which the strength criterion
is achieved.

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