Computational Science Investigation of Material Mechanics

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

Engineering stress is the applied load divided by the original cross-sectional area of a material. Also known as nominal stress.

True stress is the applied load divided by the actual cross-sectional area (the changing area with respect to time) of the specimen at that load

Engineering strain is the amount that a material deforms per unit length in a tensile test. Also known as nominal strain.

True strain equals the natural log of the quotient of the current length over the original length.

How a material deforms with respect to time since it doesn't change to the final deformation state instantaneously as the load is applied.

Creep (relaxation):

1. Primary creep:  \(\dot{\varepsilon }_f\) decays (dislocation hardening)
2. Secondary creep:  \(\dot{\varepsilon }_f\) is constant (equilibrium between hardening and dislocation motion)
3. Tertiary creep:  \(\dot{\varepsilon }_f\) increases exponentially --> creep rupture (void formation, grain boundary fractures)

Cauchy elastic materials

Hyper elastic materials (Green elastic materials)

Hypo elastic materials

Hypo elastic materials

• also called incremental formulation (hypo elastic means below elastic)
• on the increment the material behaves elastic reversible
  for larger deformations --> path dependence can be introduced
• stress increments depend on strain increments and other parameters \(\dot\sigma_{ij} = F(\dot\varepsilon_{ij}, \sigma_{mn} )\)
• strain-/stress induced anisotropy
• fully populated, not necessarily symmetric tangent stiffness matrix
  --> coupling of volumetric and deviatoric parts
• intrinsic failure at \(\dot\sigma_{ij} = 0\)
• can degenerate to a CAUCHY and GREEN material

Green elastic materials (Hyper elastic)

• The problem of energy production during loading-unloading cycles is unknown
• Uses alternative way to formulate material laws
  • via the elastic potential derived with respect to the strain components
    --> stress is calculated from the strain energy density \(\sigma_{ij} = \frac{\partial W}{\partial \varepsilon_{ij}}\)
    this always assures:
    • no energy production in loading-unloading cycles
    • full reversibility
    • path independence
• Non-linearity, dilatation or induced anisotropy can be well represented --> assures a symmetric stiffness matrix
• In general the material parameters have no physical meaning
  --> results in complicated parameter identification

Cauchy elastic material

A Cauchy-elastic material is one in which the Cauchy stress at each material point is determined only by the current state of w:deformation (with respect to an arbitrary reference configuration). Therefore, the Cauchy stress in such a material does not depend on the path of deformation or the history of deformation.

• Total stress strain model
  • stresses at a material point only depend on the present deformation state \(\sigma_{ij} = F(\varepsilon_{ij})\)
• \(F_{ij}\) is a tensor function called the response function
  • contains the operations needed on the deformation tensor to obtain the strain tensor
  • it is NOT dependent on path, history, evolution or rates
• Largest problem --> reversibility of \(U\) (inner energy) and \(\Omega\) (outer energy) is not automatically assured
  • energy might even be produced in loading-unloading cycles!
  • --> additional relation needed for uniqueness
  • the larger the non-linearity the larger the problem (cannot be ignored for high non-linearities)

Symmetry with respect to one plane

Symmetry with respect to two perpendicular planes

Rotational symmetry with respect to one axis

Rotational symmetry with respect to two axes

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.

Principal stress is the normal stress acting onto the principal plane that has no shear stress.

The hydrostatic stress/strain is related to volume change.

The deviatoric stress/strain is related to shape change.

\(\sigma=\sigma_{hyd} + \sigma_{dev}\)

Failure surfaces and constitutive laws formulated with invariants are invariant themselves.

Typically used sets are:

• Principal stresses: \(\sigma_1, \ \sigma_2, \ \sigma_3\)
• Invariants: \(I_1, \ J_2, \ J_3\)
• p-q-r invariants: \(p = I_1/3, \ q=\sqrt{3J_2}, \ r=3\sqrt{3}J_3 / 2\)
• \(\xi - \rho - \theta \ \text{ invariants: } \ \xi = \frac{I_1}{\sqrt{3}}, \\rho = \sqrt{2J_2}, \\cos{3\theta} = (r/q)^3\)

The scalar invariants of certain tensors play important roles in constitutive relations.

The Willam–Warnke yield criterion is a function that is used to predict when failure will occur in concrete and other cohesive-frictional materials such as rock, soils and ceramics.

5 parameters --> 5 experimental data points needed

Is valid for all stress combinations and gives in all practically relevant domains, including tension good agreement with reality.

Contains for different parameter choices different models (von Mises, Drucker-Prager, 3P-William-Warnke)

Uniqueness is important to assure that for every stress state exactly one and only one strain state is assigned and vice versa.

Stability

• Stability in small: when one requires that additional work done by additional forces has to be positive.
  --> if for decreasing strain, stresses would increase --> energy is produced --> contradicts thermodynamics
   
• Stability in cycle: requires if you load and unload in a cycle, the net work has to be non-negative

Normality says that the vector pointing outward normal to the tangent on a surface of equal \(\Omega\) is the strain.

Convexity means that each tangent plane to the surface \(\Omega\)=const never intersects the curve.
if not, --> stress increment with resulting strain would have a work increment <0

Solving an equation \(h(x) = g(x)\) is the same thing as finding the zeros for \(f(x)=h(x)-g(x)\)
--> which is termed root finding

Examples:

• Bisection method
• Secant method
• Newton-Raphson method (also called Newton method)
• Brent's method (combines other methods by selection of the best method for each iteration)

• Bisection method
  + it is guaranteed to converge if \(f(x)\) is continuous on \([a,b] \text{ and } f(a) \text{ and } f(b)\) have opposite signs
  + implementation is simple
  + derivatives are not required
  - at every iteration, the absolute error is halved --> convergence is linear (comparatively slow)
  - may be a problem if you have no clue where a potential root could be
   
• Secant method
  + the convergence rate is superlinear but not quadratic
  + no bracket required \(x_0, x_1\) are initial guesses, it also searches outside of \([x_0, x_1]\)
  + derivatives are not required
  - convergence is not guaranteed! (either it diverges or due to a division by zero occurs)
   
• Netwon-Raphson method (also called Newton method)
  + converges quadratic
  - derivatives are required
  - convergence not guaranteed (e.g. a stationary point --> leads to division by zero)

diverges

division by zero

choice of interval/initial points

There are several ways to determine whether a numerical solution algorithm was successful:

• Convergence: if it has converged to a solution
• Validity of the solution: comparing to experiments or analytical solutions (if they exist)
• Stability: small changes in the input parameters should not produce large changes in the output

They are used to find solutions to the equations of the problems where often there is no analytical solution.

Assumptions:

• models the material as continuous, rather than as a collection of discrete elements
  --> allows the use of continuum mechanics
• damage as a scalar variable (called the "damage variable")
  --> describes the degree of degradation
• Additivity: assumes that the overall damage is accumulative

CDM addresses the first two scales: study of the growth of defects such as micro-cracks or micro-voids and their effect on the mechanical behavior of the material. The third stage is usually studied using fracture mechanics.

Internal variables are variables that describe the state of a material and are not directly measurable.

for example:

• damage variables
   
• plastic strain

Moisture, heat, and mechanics can be coupled in building materials through a variety of physical processes.

• moisture-mechanics coupling
  swelling and shrinking --> material volume and stiffness
• moisture-heat coupling
  condensation and evaporation --> thermal conductivity and heat capacity
• heat-mechanics coupling
  thermal expansion and contraction --> material volume and stiffness

Fractures, or cracks, are a complex physical phenomenon because they involve a wide range of physical and mathematical concepts, as well as a great deal of uncertainty in their behavior.

LEFM: Linear Elastic Fracture Mechanics

CDM: Continuous Damage Model

DEM: Discrete Element Method

MD: Molecular Dynamics

The fracture process zone refers to the region around a crack tip where defects and variations in the material cause micro-crack nucleation, leading to crack growth and coalescence. This area is characterized by disorder in the material structure and is the result of the accumulation of tiny defects and variations.

Critical crack size for unstable (dynamic) growth