# Thermal Calculations

Thermal calculations in MPM include stresses and strains induced by temperature differentials and conduction induced by thermal gradients and boundary conditions.

## Introduction

NairnMPM can do coupled elasticity-thermal conductivity calculations, provide thermal boundary conditions, convert various material processes into heat, and/or or can apply a constant temperature difference to all particles. In the presence of these temperature changes and heat input or output, the calculations will find thermal stresses and tracks heat flow for valid thermodynamics or, when needed, for tracking thermodynamic quantities. This section is for MPM only; see separate help on thermal calculations when doing FEA calculations with NairnFEA.

## Stress Free Temperature

Thermal strains and stresses are always calculated relative to the stress free temperature, which is set in the `StressFreeTemp` command. In scripted input files, the command is:

StressFreeTemp (temp)

In `XML` input files, the command, which must be in the `<MPMHeader>` block is

<StressFreeTemp>(temp)</StressFreeTemp>

where `(temp)` is the stress free temperature in degrees. The default value is 0. Note that some Material Models and Traction Laws depend on the stress free temperature on an absolute temperature scale and therefore calculations that use those models should always set this temperature in degrees Kelvin.

Note that setting initial particle temperature different than the stress free temperature will cause strains to immediately evolve toward the changed state. The net effect will be an instantaneous "impact" of temperature change that might cause undesirable dynamic effects.

## Conduction

The `Conduction` command activates thermal conductivity calculations and also activates various optionals involving heat flow in the simulations or other mechanisms that cause temperature changes. These settings can change and select the thermodynamics mode of the simulation. By default, conduction uses update methods analogous to FLIP methods used in mechanics. This update, however, sometimes results in temperature oscillations on particles within one cell. Conduction simulations with oscillations can be improved by using periodic FMPM(k) for conduction updates.

### Scripted Input Commands

In scripted input files, the command is

Conduction (isOn),<(option1)>,<(option2)>...

where

`(isOn)`is "`Yes`" or "`No`" to include or to not include thermal conductivity calculations in the MPM analysis.

The optional parameters `(option1)`, `(option2)`, *etc*. can activate various physical mechanisms as heat sources that may cause temperature rises and may change the thermodynamics of the analysis. The first two are complementary options (*i.e*., `Adiabatic` and `Isothermal` are complementary options, which means only one should be used. If both are omitted, the default setting is `Isothermal`.) are:

- "
`Adiabatic`" (used to be`Mechanical energy`, which is still allowed) to be locally adiabatic, which means any elastic thermal energy or dissipated energy in a material results in a corresponding change in the particle temperature, the magnitude of which is determined by the material's heat capacity. This option can be used with conduction on or off. If conduction is off, the particle temperature rise will remain on the particle or the particle will be locally adiabatic. When conduction is on, the temperature rise can conduct as heat to neighboring particles meaning with time, the particle will not be locally adiabatic. The entire problem, however, will be globally adiabatic unless there are thermal boundary conditions. Finally, the available mechanics to generate thermal energy depend on material model being used. - "
`Isothermal`" to be locally isothermal, which means the material response will never cause a temperature rise. Implicitly all dissipated energy is expelled to the exterior. The problem will be globally isothermal unless there are thermal boundary conditions.

The following options activate conversion of various physical mechanisms into heat. Note that these mechanisms inject heat into the problem though heat flux term in the conduction equation. In other words, they have no affect unless conduction is activated by setting `(isOn)` is "Yes":

- "
`Crack Tips`" to have cracks convert total energy released into heat at the crack tips. For crack-tip heating to work correctly, you have to set the crack thickness to define its thickness. - "
`Friction`" to convert work of frictional sliding between materials in multimaterial mode into heat. - "
`Crack Friction`" to convert work of frictional sliding between crack surfaces into heat.

### XML Input Commands

In `XML` input files, conduction is activated with a `<Thermal>` block:

<Thermal> <Conduction/> <EnergyCoupling/> <CrackTipHeating/> <CrackContactHeating/> <ContactHeating/> <Isothermal time="5" start="2">-200</Isothermal> </Thermal>

where

- <Conduction>
- This tag activates condition calculations. Its omission turns conduction off. When conducton is activated, you should set all needed thermal material properties.
- <EnergyCoupling/>
- The presence of this tag is equivalent to the
`"Adiabatic"`option while its omission is equivalent to the`"Isothermal"`option (see above). - <CrackTipHeating/>
- The presence of this tag is equivalent to the
`"Crack Tips"`option (see above). - <CrackContactHeating/>
- The presence of this tag is equivalent to the
`"Crack Friction"`option (see above). - <ContactHeating/>
- The presence of this tag is equivalent to the
`"Friction"`option (see above). - <Isothermal>
- This command sets up a thermal ramp (see next section).

### Material Support for Thermal Energy

Mechanics calculations of thermal conduction are coupled phenomena, but the degree of coupling depends on the analysis method. The two ways they are coupled are:

- Thermal expansion - a change in temperature causes most material the develop thermal strains that will change the particle stress.
- Thermal energy - various mechanical process can generate thermal energy that changes the particle temperature

Most analytical methods (especially for elastic solids) include thermal expansion, but ignore thermal energy. This approximation is done because it decouples the thermal conduction equation for the mechanics equation and thereby simplifies the analysis. It is a good approximation for elastic solids because small-stain, elastic deformation does not generate much thermal energy, But when modeling large strains or modeling materials that generate or dissipate more thermal energy (plastic or viscoelastic materials, liquids, and gases), thermal analyses should always be coupled, which involves adding heat source terms to the thermal conduction equation. Furthermore, when solving the two equations simultaneously by numerical methods, there is virtually no performance penalty to including all coupling terms (just the slight work to calculate the thermal energy).

Nevertheless, proper modeling of coupled mechanical and thermal problems requires that the material model implemented in `NairnMPM` correctly calculates all thermal energy terms. That calculation is not part of generic MPM because it depends on the material model being used. The following lists explain which materials support various types of thermal coupling. Materials not in these lists may or may support these features as could be checked by looking at source code.

- Thermal Expansion - All materials
- Dilational Energy - By thermodynamics, all materials heat when compressed and cool when expanded (gases a lot; solids very little). The follow materials support dilational energy:
- Isotropic
- IsoPlasticity
- IsoSoftening
- IsoPlasticSoftening
- Viscoelastic
- Mooney
- Neohookean
- HEIsotropic
- HEMGEOSMaterial
- TaitLiquid
- IdealGas (heat of expansion and compression, which is analog of elastic thermal energy in solids)

- Dissipated Energy - Some materials dissipate energy and calculations can cause that energy to be converted into heat. The following materials calculate their dissipated energy:
- IsoPlasticity (from plastic energy)
- HillPlastic (from plastic energy)
- Viscoelastic (from viscoelastic dissipation and artificial viscosity)
- IsoSoftening
- IsoPlasticSoftening
- TransIsoSoftening 1 (2)
- Mooney (from artificial viscosity)
- Neohookean (from artificial viscosity)
- HEIsotropic (from plastic energy and artificial viscosity)
- HEMGEOSMaterial (from plastic energy and artificial viscosity)
- TaitLiquid (from shear work)

## Thermal Ramp

To avoid thermal shocks, it is usually better to apply initial thermal fields using a PropertyRamp custom task.

The scripting commands `ThermalRamp` and `RampStart` and the `XML` command `<Isothermal>` have been deleted and replaced by the new PropertyRamp custom task.

## Thermodynamics Modes in MPM

By using settings for conduction calculations along with optional thermal boundary conditions, NairnMPM can simulate various thermodynamics conditions, and it is important to be aware of the thermodynamics that is being modeled. The two basic thermodynamics modes refer to instantaneous temperature change on an isolated particle and are the "Adiabatic" and "Isothermal" modes (set using a conduction option). Formally, the "Adiabatic" model is modeling the full dynamic problem while "Isothermal" mode is used to emulate quasi-static simulations. These modes control how each particle handles localized energy flow. The full thermodynamics, however, also depends on interactions between particles through conduction and on exchange of heat with the exterior through boundary conditions. These options are:

- Conduction: when conduction is on, heat can flow between particles; when it is off no heat flows between particles.
- Thermal Boundary Conditions: an "Isolated" system means there are no thermal boundary conditions that can heat or change temperature of particles; a "Nonisolated" system has thermal boundary conditions

The following grid explains all combinations of these settings and describes the behavior of some thermodynamics quantities. In the table **dq** is cumulative heat flow on a single particle, **dQ** is sum of heat flow for all particles, **dS** is total entropy change, and **dT** is change in average temperature.

Adiabatic Mode | Isothermal Mode | ||||||||
---|---|---|---|---|---|---|---|---|---|

System (Thermal BCs) | Conduction (Particles) | dq | dQ | dS | dT | dq | dQ | dS | dT |

Isolated | No (Isolated) | 0^{1} | 0 | 0 | ≠0 | ≠0 | ≠0 | ≠0 | 0 |

Isolated | Yes (Nonisolated) | ≠0^{1} | 0 | ≥0^{2} | ≠0 | ≠0^{3} | ≠0^{3} |
≠0^{3} | 0^{3} |

Nonisolated | No (Isolated) | ≠0 | ≠0 | ≠0 | ≠0^{4} |
≠0 | ≠0 | ≠0 | ≠0^{4} |

Nonisolated | Yes (Nonisolated) |

- If any particles start with a temperature that is different then the stress free temperature, the first time step will add dq = C
_{V}(dT_{i}−dT_{SF}) to the particle heat energy. The above conditions will hold thereafter, but a constant will be added to dS and dQ. - The dq will be due to conduction only and entropy will increase because conduction is irreversible.
- This mode is identical to system isolated with no conduction if the initial temperature of all particles is equal to the stress-free temperature. Thus, such a simulation should turn off conduction for efficiency because it is not needed.
- When conduction is "No", the only possible thermal boundary condition is a thermal ramp that applies uniform temperature change to all particles. If there is no thermal ramp, then the system is isolated and refer to first mode instead.

### Tracking Thermodynamic Quantities

In any thermodynamics mode, you can easily track various thermodynamics quantities:

- Total entropy (S) in a global archive.
- Temperature (T) in a global archive and on each particle.
- Work Energy (w) in a global archive and on each particle where: [math]\displaystyle{ w = \int \sigma\cdot d\varepsilon }[/math]
- Heat (q) in a global archive and on each particle.
- Total Internal Energy (U = w + q) in a global archive.
- Total Helmholz Free Energy (A = U - TS) in a global archive.
- Strain Energy (W) in a global archive and on each particle where: [math]\displaystyle{ W = \int \sigma\cdot d(\varepsilon-\varepsilon_{res}) }[/math]
- Kinetic Energy (K) in a global archive (with "Grid Kinetic Energy" being the preferred option for archiving)

## Thermal Boundary Conditions

The possible thermal boundary conditions are:

- You can set temperature on the grid.
- Rigid particles can provide moving temperature boundary conditions.
- You can set a heat flux on particle surfaces.
- A thermal ramp can ramp of the temperature of all particles

## Thermal Material Properties

The results of thermal calculations will depend on the thermal properties of the materials in the model, which should all be set when doing such calculations. The relevant thermal properties are thermal expansion coefficients, thermal conductivity, and heat capacity. See the material definitions for how to set these properties. A few material models and hardening laws also depend on thermal transitions, such as a melting point, and on the stress free temperature.