Computational Science Investigation of Material Mechanics

We can analyze it with the R-curve.

Important question: which way will the crack grow?

One possible approach:
If the material is isotropic and homogeneous,
the crack will propagate in such a way as
to maximize the energy release rate.

Challenges:

1. appropriate representation of near-tip stresses (singularity): infinity cannot be represented by computer (mesh)
2. crack advancement criterion (increment and orientation)
3. adapt discretization for crack increment (mesh)

Local stresses in grains (of microstructure) vary considerably and may be much larger than average stress.

Hence, small local slip accumulates over cycles of loading and unloading.

This is particularly true for grains close to the material surface, where constraints against slip are lower.

Fatigue is a material surface phenomenon. (at least initially)

Surface effects that are important for fatigue:

• Surface roughness (from production)
• Surface damages (scratches, dents, ...)
• Surface treatments

Generally, larger defects --> shorter fatigue life

S-N curve / Wöhler curve are stress-vs-endurance curve

Wöhler curve can provide a probability of failure after a certain number of cycles given applied stress

• some materials(e.g. steel, titanium) have a fatigue limit (endurance limit)
  --> an infinite number of cycles do not lead to fatigue failure
• other types of materials (e.g. aluminum) do not have a fatigue limit

Question:
how to account for the effect of mean stress?
Answer:
many alternative approaches exist

• Smith-Watson-Topper equation
• Morrow correction
• Goodman relationship
• Gerber relationship
• Soderbergrelationship

Strain energy density is a measure of the energy stored in a material as a result of deformation. Physically, it can be understood as the amount of energy required to deform a unit volume of a material by a unit amount.

Strain energy release rate is a measure of the rate at which energy is released from a material as a crack propagates. Physically, it can be understood as the amount of energy released per unit area of newly created crack surface.

When a crack propagates through a material, it creates new surfaces and the atoms and molecules on those surfaces are no longer in contact with each other. This separation of the surfaces requires energy, which is released from the material as strain energy.

from Linear Elastic Fracture Mechanics

A crack will propagate if the energy required to create new surfaces (i.e. the surface energy density) is less than the energy released by the crack as it grows.

\(G \geq G_c\)

1. Define the failure

2. Collect the evidence

3. Analyze the evidence

4. Establish possible root causes for the failure

5. Validate the hypothesis through structural analysis

6. Arrive at a conclusion regarding the cause(s)

7. Prepare the final report

Objective:

• Forensic: determine root cause for structural failure (Inverse problem)
• Design: make plans to build a structure (Forward problem)

Assumptions vs Evidence:

• Forensic: uses evidence to test various failure hypotheses
• Design: needs assumptions to design

Based on evidence --> tries to guess what happened
Based on assumptions --> tries to guess what is going to happen

(45-55%)   Design errors

(20-30%)   Construction defects

(5-10%)     Material deterioration / maintenance

Catastrophic events or overload

Human Factors for Failures:

• Negligence: Disregard of codes
• Incompetence: Failure to understand fundamental principles
• Ignorance: Failure to follow design and construction doc
• Greed: Intentional disregard of requirements and safe practice
• Disorganization: Failure to establish clear responsibilities
• Miscommunication: Failure to communicate between parties
• Group thinking: based on a common desire not to upset the balance of a group of people
• Misuse, abuse, neglect: Use and operation beyond its intent; lack of maintenance

Genua brücke

Use actual values for geometry, material, and load instead of design values.

For design, you aim to remain below a critical value --> which results in mostly linear behavior

For failure, you need to determine what happens beyond this critical value --> highly non-linear

1. Physical problem statement: describe the physical properties to model

2. Mathematical problem statement: translate the physics into mathematics to solve it

3. (Numerical) Solution of problem: find the exact or approximate solution to the problem

Then evaluate the model (e.g. compare it to actual data) and refine it if required.

Analytical or numerical (approximate) solutions.

Rounding errors: also called arithmetic errors are an unavoidable consequence of working in finite precision arithmetic.

Uncertainty in the data: It may arise in several ways: from errors in measuring physical quantities, from errors in storing the data on the computer (rounding errors), or, if the data is itself the solution to another problem, it may be the result of errors in an earlier computation.

Truncation or Discretization or Approximation errors: Many standard numerical methods can be derived by taking finitely many terms of a Taylor series. The terms omitted constitute the truncation error, and for many methods, the size of this error depends on a parameter (often called the stepsize e.g. \(\Delta t\)), whose appropriate value is a compromise between obtaining a small error and a fast computation.

• Use a suitable and accurate constitutive model for the material: The choice of a constitutive model that accurately describes the material behavior under different loading conditions is crucial for reducing the error in numerical results.

• Use a suitable mesh: A finer mesh will produce more accurate results but will also require more computational resources. The mesh should be fine enough to capture the behavior of the material near the crack or the region of interest.

• Use appropriate boundary conditions: Boundary conditions should be chosen to mimic the laboratory experiment as closely as possible, including the type of loading, the loading rate, and the support conditions.

• Validate the model using experimental data: Validate the FE model by comparing the numerical results with the experimental results obtained in the laboratory. This can help to identify any discrepancies and to adjust the model accordingly.

• Use appropriate numerical techniques: Use a suitable numerical algorithm that can accurately solve the equations of the FE model, such as the Newton-Raphson method, and use a suitable time-stepping scheme for dynamic problems.

• Use appropriate material properties: The material properties should be obtained from reliable sources, and it should be checked that they are appropriate for the intended application.

• Use of experimental data to adjust the model: If the model shows a significant deviation from experimental results, the model may be adjusted using experimental data.

• Stability: The discretization process can lead to stability problems, where the solution becomes unstable or oscillates. These problems can be mitigated by choosing a suitable time-step and/or spatial discretization.

• Accuracy: Discretization introduces errors into the solution, which can be mitigated by choosing a finer discretization. However, this can also lead to increased computational cost.

• Convergence: The discretization process can lead to problems with convergence, where the solution does not converge to the true solution as the discretization is refined. This can be mitigated by choosing a suitable discretization and numerical method.

• Ill-conditioning: Finite difference equations can lead to ill-conditioning, which can cause numerical instability and slow convergence of the method. This can be mitigated by choosing a suitable numerical method, such as using a multigrid technique.

• Dispersion: Discretization in time can lead to dispersion errors, where the solution deviates from the true solution due to the propagation of waves at different speeds. This can be mitigated by using a suitable numerical method, such as using a higher-order time-stepping scheme.

• Dissipation: Discretization in time can also lead to dissipation errors, which can cause the solution to be damped and lose energy. This can be mitigated by using a suitable numerical method, such as using an energy-preserving time-stepping scheme.

1. Principle of determinism

The current value of any physical variable can be determined from knowledge of the present
and past values of other variables.

2. Second law restrictions

Cannot violate the second law of thermodynamics. (entropy)
All balance equations must be fulfilled: conservation of mass, momentum, and energy.

3. Principle of material frame-indifference

Should not depend on whatever external frame (coordinate system) is used.

4. Material symmetry

Must respect any symmetries that the material possesses.

5. Principle of local action

The material response at a point depends only on an arbitrarily small region about that point.
There are also nonlocal continuum theories that reject this hypothesis.

6. Principle of fading memory

Values of constitutive variables further behind in time influence the current state of the
constitutive function not considerably. (not always)

7. Principle of causality

Displacement and temperature are the causes of the behavior of the body,
--> all other physical properties are dependent on them.

8. Principle of equipresence

A variable present as an independent variable in one constitutive equation should be so present in all.

Material point:

Representative Volume Element RVE:

• Smallest volume, whose effective properties do not depend on the extensions of the element anymore
  --> statistically homogeneous
• A structurally typical volume for the mean material system
  that includes a sufficient number of voids, defects, cracks...
• Identity of stored elastic energy U in volume and its represented continuum.
• The inherent structure of the RVE is arbitrary and can be disordered a.s.o.
  but has to self-repeat in space

Repeating Unit Cell RCU:

• Real structure is replaced by periodic phase arrangement
  --> discrete structure
• No limitation of phases per volume
• Identity of stored elastic energy U in volume and its represented continuum
• Discrete structure permits the use of fluctuation fields
• Periodic boundary conditions for the RUC