# Unit Consistency in Solid Mechanics Simulations

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Ensuring unit consistency of input parameters in conducting a finite element analysis is critical, because mistakes in it will essentially alter the physical problem one’s trying to study, let along the rationality of outputs that would be produced. As an analyst, I always feel uncertain from multiple sources at the beginning of modeling a new problem, unit consistency is one of the first few certainties that can be built and stepped on, so I can safely move onto more essential parts of problem solving.

Taking unit consistency as a procedural step is sufficient for practicality, yet thoroughly knowing its theoretical root serves as the foundation of a good engineer because it reveals how our current framework of description of physical quantities is formed. I will review this part in the first section of this note. The second section summarizes the practical procedure to ensure unit consistency in solid mechanics simulation. You may comfortably skip to the second section directly for practical reasons.

Unit consistency is not special in simulation. When solving physical problems on paper (this note was written in 2017 when paper is still around) in high school or college, we list physical equations that relates the quantities in the problem and reduce them to a single algebra equation or a system of equations that can be solved mathematically. Before the final solution step we substitute numbers into equations, we almost subconsciously do unit conversion for each quantity so that each equation has the right unit on both sides of the equal sign.

Running simulations with general-purpose finite element software is quite different—we no longer derive and solve equations, instead, we set up the problem and let the program construct the system of equations and solve it. In the preprocess of modeling, analysts set up a playground, introduce a few objects with properties, interactions and loadings. Then the reduction and solution parts are passed over to the preprocessor and kernel of the finite element program. Overall, all the input parameters are scattered among different sections of the software interface, and we may have lost the insight of the relation between quantities, thus, overlook the unit consistency among them.

Fortunately, solving physical problems on paper has already given us practical experiences on describing and solving a simpler physical system, a reminder of definitions of concepts in a logical sequence would be enough to recall and systemize that experience and reveal the framework of description of physical quantities.

## Underlying Concepts

In this section, we list the definition of concepts in a logical sequence.

### Physical quantities

A physical quantity is a quantity that can be used in the mathematical equations of science and technology and is subject to the rule of changing units (see definition later).

Here I added the intrinsic property into the definition since I felt like things should be defined by their meaning (or purpose) and properties together rather than purely by their meaning, just like one way of defining tensors is tensors are quantities subject to the transformation rule of changing bases.

Physical quantities are categorized into base physical quantities and derived physical quantities, abbreviated as base quantities and derived quantities.

### Describing a physical event

A physical event in a general sense is any physical occurrence that happens in time and space (a more strict definition might be an occurrence with which a finite time and a finite location is associated 1).

Describing a physical event involves identifying physical quantities that participate in the event and listing physical equations relating those quantities. Those labor work is to prepare a system of physical quantities of the event (see next definition) for later investigation and solution.

### System of physical quantities (of a physical event)

A system of physical quantities (associated with a physical event) is a set of physical quantities together with the non-contradictory equations relating those quantities2.

### A base-quantity subset

As the system of physical quantities of one physical event is established, one can choose, from all physical quantities involved in the event, a few, to form a complete subset of independent physical quantities. This subset is called a base-quantity subset, of which the two requirements means:

• complete—all other physical quantities in the system of physical quantities of the studying event can be expressed in terms of quantities in this subset;
• independent—no physical quantity in the subset can be expressed in terms of others in the subset (this is also called dimensionally independent, which will be defined later).

The base-quantity subset is usually not unique in a physical event. It is, to some extent, made by our choice.

Violation of completeness makes the base-quantity subset fails to express some of the physical quantities in the system. Although this is a serious issue, it can be easily fixed by adding more base quantities into the subset until every quantities that do not belong to the subset can be expressed.

Violation of mutual independence makes derived quantities have multiple expressions. This also sounds serious, but as long as no contradiction occurs, it might be OK. In fact, the SI system of units does not use a base-quantity subset whose elements are strictly independent.

### Base quantities

Base quantities are the elements of the base-quantity subset.

Some literature defines base quantities prior of base-quantity subset acknowledging that base quantities have some special intrinsic properties that derived quantities don’t, details later.

### Derived quantities

Derived quantities are quantities in a system of quantities of a physical event that can be expressed in terms of base quantities. So derived quantities are quantities in the complement set of the base-quantity subset, where the universe is the set of all physical quantities in the event.

The expression of derived quantities on base quantities comes from two kinds of relations: definitions and physical laws. For example, area is defined as $$A=\int dxdy$$, while force is related with mass and acceleration by the physical law $$F=ma$$, detailed in Professor Ain A. Sonin’s lecture note 3.

Since there are freedom for the choice of base quantities, categorizing a physical quantity as either base or derived is not unique. It’s common that a derived quantity, under one choice of base-quantity subset, serves as a base quantity, under another choice of base-quantity subset. Professor Sonin actually called this kind of derived quantities the derived quantities of the second kind.

It is proven that all the derived quantities have a power law form of base quantities.

### Preferences of base quantities

There are actually two ways of defining base quantities in the literature. One seems to regard base quantities as ones that have some intrinsic speciality that derived don’t, as in Professor Ain A. Sonin’s lecture note. So base quantities are directly and first defined; base-quantity subset are thereby defined based on base quantities. This viewpoint implies not every physical quantity are qualified to serve as an element in base-quantity. The other is the one adopted in this note, which has a stricter and prior definition of the base-quantity subset. Then a base quantity is nothing special than an element of the subset.

The reason I preferred this viewpoint is that the purpose of base quantities and base-quantity subset is to express derived quantities in a physical event, and the base quantities subset is the direct tool to do the job. Thus, we should have a clear definition of this tool itself. Furthermore, I felt any quantity can serve as a base quantity as long as it along with its companion base quantities satisfies the requirement of the subset, just like in linear algebra, any vector (other than zero vector) has the chance to serve as a base vector as long as a specific number of mutually independent vectors can be found to form a base vector set. Yet indeed some physical quantities are more preferable than others in service, but this is due to the convenience rather than some intrinsic quantity.

Physical quantities that have tangible physical representations, for example, length, time, mass, force, etc., are preferred as candidates for base quantities, for such quantities evoke one’s intuition and are usually direct to measure.

### Kind (class) of physical quantities

A collection of mutually comparable quantities. For example, the quantities of diameters, circumference, and wavelength are considered as the same kind of quantities, namely, of the kind of quantities called length 4.

### Unit of physical quantities

Quantification of physical quantities is based on comparison. A unit is a standard sample of a particular physical quantity, defined and adopted by convention, with which other such quantities of the same kind are compared to express their value.

### Expressing a physical quantity

Expressing a physical quantity is comparing it to a selected unit, to yield a numerical count of the number of the whole and fractional units required to equate with the quantity. So the expression of physical quantity is the product of a numerical value and the chosen unit. For example, a physical quantity $$\boldsymbol{Q}$$ is expressed with the chosen unit $$\boldsymbol{q}$$ as

$$\boldsymbol{Q}=q\boldsymbol{q},$$

where $$q$$ is the numerical value of $$\boldsymbol{Q}$$ compared to a selected unit $$\boldsymbol{q}$$.

### Rule of changing unit

Since the choice of unit is arbitrary, a physical quantity can be expressed with different units, subject to the rule of changing unit, which is based on the invariant of physical quantity compared to different units.

Suppose $$\boldsymbol{Q}$$ is expressed with units $$\boldsymbol{q}$$ and $$\boldsymbol{q’}$$ with numerical value $$q$$ and $$q’$$, respectively

$$\boldsymbol{Q}=q\boldsymbol{q}=q’\boldsymbol{q’}.$$

If the unit $$\boldsymbol{q}$$ is compared to unit $$\boldsymbol{q’}$$ and yields a numerical value $$n$$, i.e.,

$$\boldsymbol{q}=n\boldsymbol{q’},$$

then the conversion between numerical values of $$\boldsymbol{Q}$$ with $$\boldsymbol{q}$$ and with $$\boldsymbol{q’}$$ can be obtained

$$q=\frac{1}{n}q’.$$

### Base units

Base units are the chosen units for base quantities in a system of physical quantities.

### Derived units

Since derived quantities can be expressed in terms of base quantities, as soon as the units for base quantities are chosen, we can substitute units of relevant base quantities into the expressions of derived quantities and derive the units for derived quantities.

### System of units (of a system of physical quantities)

A system of units attaches a unit to each physical quantity involved in a physical event. It is formed through three recipes

1. choice of a base-quantity subset,
2. choice of base units,
3. resulted derived units.

Meter, kilogram, and second (MKS) and centimeter, gram, and second (CGS) are examples of system of units.

If the system of units is determined, the base-quantity subset and thereby who are base quantities are all automatically determined since they are the recipes. In the above example of system of units, the base-quantity subset is {length, mass, time}.

### Class of system of units

The classification of systems of units is based on the base quantities (not their units). Those systems of units that use identical base quantities are called to be in the same class of systems of units. Same class of systems of units can use different units for those identical base quantities.

### Dimension of physical quantities

Dimension of a physical quantity in a system of quantities is the expression of the dependence of a quantity on the base quantities. Dimensions of base quantities are the quantities themselves since they are independent of other base quantities.

Dimension of a physical quantity is independent of units but depends on system of units because it stipulates the base-quantity subset and thus base quantities.

Different quantities can have same dimension for example work and torque both have dimension of force times length. Trivially, different dimensions must come from different quantities.

### $$[\cdot]$$ function and $$\{\cdot \}$$ function

For convenience, we define two functions that operates on physical quantities.

Function $$[\cdot]$$ operates on a quantity $$Q$$ and returns $$[Q]$$—the dimension of $$Q$$ in the system of units of the event:

$$\begin{eqnarray}[\cdot]: \text{physical quantities } &\rightarrow& \text{dimensions of physical quantities} \\ \text{quantity } Q &\rightarrow& \text{dimension of } Q, [Q]\end{eqnarray}$$
Function $$\{\cdot \}$$ operates on a physical quantity $$Q$$ and returns $$\{Q\}$$—the unit of $$Q$$ in the system of units of the event:

$$\begin{eqnarray}\{\cdot\}: \text{physical quantities } &\rightarrow& \text{units of physical quantities} \\ \text{quantity } Q &\rightarrow& \text{unit of }Q, \{Q\}\end{eqnarray}$$
Need to mention $$[\cdot]$$ is common and has been a convention in literature, $$\{\cdot \}$$ is more a self-devised function for the convenience in this note. We should be able to easily differentiate $$\{\cdot\}$$ with the use case { } are used to enclose a set definition.

### Notations

For effective communication, physicists have developed a notation system:

• notation for the kind of quantities

we usually use capital letter to represent the kind of quantities, for example,

• $$L$$ to indicate the kind of quantity, length
• $$M$$ to indicate the kind fo quantity, mass
• $$T$$ to indicate the kind of quantity, time instant
• $$F$$ to indicate the kind of quantity, force
• notation for the quantity

we may use the same symbol as for the kind of quantities for individual quantities as well, there might be a slight ambiguity, but it almost does not have any hurt. For example, for a quantity of kind force, we may use

• $$F$$ to stand for a general force quantify
• $$N$$ to stand for specific supporting forces
• $$f$$ to stand for friction forces

for a quantity of kind length, we may use

• $$L$$ to stand for the length of a rectangle
• $$W$$ to stand for the width of a rectangle
• $$D$$ to stand for the diameter of a circle
• $$\lambda$$ to stand for the wavelength of a color
• notation for quantity samples

samples or variables of quantity are instances of a physical quantity used in an event, to distinguish between each other, we may use uppercase, lowercase, subscripts, superscripts, etc., for example:

• $$F_1$$, $$F_2$$ to stand for two general force variables $$N_1$$, $$N_2$$ to stand for two supporting force variables
• $$l_1$$, $$l_2$$ to stand for length variables of two rectangle in a problem
• $$m_1$$, $$m_2$$ to stand for mass of two objects
• notation for dimensions

when a quantity serves as a base quantity, the dimension is the quantity itself, we usually use the kind of quantity it belongs to as the dimension, for example,

$$[L]=[W]=[D]=[D_1]=[\lambda]=L$$

$$[M]=[m_1]=M$$

notations for derived quantities are written as the power form of base quantities, for example, if mass, length, and time are selected as base quantities, then force is a derived quantity with dimension

$$[F]=MLT^{-2}$$

• notation for units

units are standard samples of quantities, for base units we assign special symbols to them, for example,

• $$\{L\}=m$$, $$\{L_1\}=cm$$, $$\{l_1\} =mm$$
• $$\{M\}=kg$$, $$\{M_1\}=g$$, $$\{m_1\}=lb_m$$
• $$\{F\}=N$$, $$\{F_1\}=lb_f$$

The notation system is mainly for the effectiveness and convenience of communication, thus is not rigorous, for example, a capital letter $$L$$ sometimes represents the kind of quantity of length, sometimes the length quantity, and even sometimes a length variable. But we shouldn’t be worried, we’re naturally conformable with their flexibility and able to differentiate the induced slight ambiguity.

### SI system of units

International system of units, abbreviated as SI (from French), adopts a base-quantity subset consisting of 7 base quantities and stipulate their units respectively. The units, their symbols, and corresponded base quantities are listed as follows:

1. { length $$L$$ } = meter (m)
2. { mass $$M$$ } = kilograms (kg)
3. { time $$T$$ } = second (s)
4. { temperature $$\Theta$$ } = kelvin (K)
5. { electric current $$I$$ } = ampere (A)
6. { amount of substance $$N$$ } = mole (mol)
7. { luminous intensity $$J$$ } = candela (cd)

We should be aware that the main purpose of SI system of units is to have a common reference conveniently for global industry use, the 7 base quantities are not strictly independent with each other: the temperature can be derived from energy, thus from the length, mass, and time, with the aid of Boltzmann constant.

Solid mechanics events mainly involve the first four quantities. A multiple-physics problem involves more quantities than the first four. In this post we mainly consider the pure mechanics problem so the temperature is also excluded. static problems, quasi-static, dynamic problems

## Choosing System of Units

When conducting finite element simulations of a solid mechanics event, we analysts should be aware of the system of units along the way starting from geometric modeling to results interpretation, although the software may not ask us to do so explicitly. Geometric dimensions, material properties, loading and boundary conditions, and results that would be produced should be all based on this one system of units consistently.

To set up a system of units, we need three recipes:

1. setting up a base-quantity subset,
2. choosing base units,
3. deducing derived units.

To set up a base-quantity subset, we first need to know the quantities that would appear in the physical problem we are simulating.

### List of quantities in solid mechanics problems

In a general dynamic solid mechanics problem, we can enumerate most (hard to list all) physical quantities participate in,

• strain $$\epsilon$$, Poisson’s ratio $$\nu$$
• angle $$\theta$$, length $$L$$, displacement $$U$$, area $$A$$, volume $$V$$
• time $$T$$
• velocity $$V$$ and $$\omega$$, acceleration $$a$$ and $$\alpha$$
• force $$F$$, moment $$M$$
• stress $$\sigma$$, modulus $$E$$ or $$G$$
• energy $$\Pi$$, power $$P$$

In a static problem, time is not involved, In a nonlinear static, or quasi-static problem, time is merely a mark to indicate the change of loading and boundary conditions, equilibrium equations (rather than equation of motion) are solved at each time increment.

### Size of base-quantity subset

The SI system of units gives us a sense of the size of the base-quantity subset in a solid mechanics problem. The first three base quantities of the SI system: length, mass, and time can form a base-quantity subset for a general mechanics problem. We may set up a different base-quantity subset but the whose size will still be 3.

### Following the modeling process

The decision of base-quantity subset can be made by following a typical modeling procedure.

In a typical modeling practice, analysts meet the geometry first, they either build the geometric model from scratch in or importing CAD model into the FE software. The dimension of all geometric quantities has a power law form of length, therefore, length should be in the base-quantity subset.

Then the analyst need to input the material properties, which are often some moduli in both static and dynamic analyses, and density specifically for dynamics and transient analyses. With aid of the length, which has already been in the base-quantity subset, we may put force which can derive modulus, or mass which can derive density in the base-quantity subset.

This is enough for a static analyses. In dynamic and transient analyses time duration for a physical event to happen, or in other words, how fast a event happens matters. Thus, time should be in the base-quantity subset.

In applying boundary conditions of the kind displacements, their dimension length is in the base-quantity subset. For other kinds of boundary conditions and loadings such as concentrated forces, tractions, and pressures, they all can be derived by the current base-quantity subset. Therefore, those subsets are complete. The two possible base-quantity subsets we currently have are:

• $$\color{green}{\{L,\, M,\, T\}}$$,
• $$\color{red}{\{L,\, F,\, T\}}$$.

We may exclude time by supplementing both force and mass to form another base-quantity subset:

• $$\color{blue}{\{L,\, F,\, M\}}$$.

Till this point, we finished the setup for a base-quantity subset. To further determine the system of units, we only need to decide the units for the base quantities in the base-quantity subset. And the units for derived quantities can be obtained by substituting units of base quantities into the dimension of derived quantities.

As a summary, we can have 3 categories of system of units based on the 3 base-quantity subsets. The decision process of a specific system of units may be divided into three steps: first, the unit of length is determined; second, select two quantities from force, mass and time, and assign their units based on the modeling preference; third, derive units for other quantities based on their respective dimensions. This process is shown in the following figure. The 3 categories of system of units are outlined by 3 colors. ## Frequently Used System of Units

Some frequently used specific systems of units are listed in the following. They are listed first by category of the base-quantity subset and then by specific units used for each base quantity. People usually prefer integers to fractions, and exponents of multiple of 3 to others, so these systems of units are more preferable.

### $$\{\text{length},\, \text{mass},\, \text{time} \}$$

• Meter, kilogram, second (MKS) • Foot, slug, second • Centimeter, gram, second • Millimeter, gram, second ### $$\{\text{length},\, \text{force},\, \text{time}\}$$

• Millimeter, kilonewton, millisecond • Inch, pound force, second • Foot, pound force, second ### $$\{\text{length},\, \text{force},\, \text{mass}\}$$

• Centimeter, kilonewton, gram • Inch, pound force, pound mass 1. Leo Sartori (1996). Understanding Relativity: a simplified approach to Einstein’s theories, University of California Press, ISBN 0-520-20029-2, p. 9
2. International Vocabulary of Metrology, p3
3. Ain A. Sonin, The physical basis of dimensional analysis, p24
4. International Vocabulary of Metrology, p3
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