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## Important Announcement

2 December, 2020 at 1:14 PM

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## Mechanics of Materials: Strain

#### Strain

So far, we’ve focused on the stress within structural elements. When you apply stress to an object, it deforms. Think of a rubber band: you pull on it, and it gets longer – it stretches. Deformation is a measure of how much an object is stretched, and strain is the ratio between the deformation and the original length. Think of strain as percent elongation – how much bigger (or smaller) is the object upon loading it.

Just like stress, there are two types of strain that a structure can experience: 1. Normal Strain and 2. Shear Strain. When a force acts perpendicular (or “normal”) to the surface of an object, it exerts a normal stress. When a force acts parallel to the surface of an object, it exerts a shear stress.

Let’s consider a rod under uniaxial tension. The rod elongates under this tension to a new length, and the normal strain is a ratio of this small deformation to the rod’s original length.

Strain is a unitless measure of how much an object gets bigger or smaller from an applied load. Normal strain occurs when the elongation of an object is in response to a normal stress (i.e. perpendicular to a surface), and is denoted by the Greek letter epsilon. A positive value corresponds to a tensile strain, while negative is compressive. Shear strain occurs when the deformation of an object is response to a shear stress (i.e. parallel to a surface), and is denoted by the Greek letter gamma.

#### Mechanical Behavior of Materials

Clearly, stress and strain are related. Stress and strain are related by a constitutive law, and we can determine their relationship experimentally by measuring how much stress is required to stretch a material. This measurement can be done using a tensile test. In the simplest case, the more you pull on an object, the more it deforms, and for small values of strain this relationship is linear. This linear, elastic relationship between stress and strain is known as Hooke’s Law. If you plot stress versus strain, for small strains this graph will be linear, and the slope of the line will be a property of the material known as Young’s Elastic Modulus. This value can vary greatly from 1 kPa for Jello to 100 GPa for steel. For most engineering materials, the linear region of the stress-strain diagram only occurs for very small strains ( Generalized Hooke’s Law

In the last lesson, we began to learn about how stress and strain are related – through Hooke’s law. But, up until this point we’ve only considered a very simplified version of Hooke’s law: we’ve only talked about stress or strain in one direction. In this lesson, we’re going to consider the generalized Hooke’s law for homogenous, isotropic, and elastic materials being exposed to forces on more than one axis.

First things first, even just pulling (or pushing) on most materials in one direction actually causes deformation in all three orthogonal directions. Let’s go back to that first illustration of strain. This time, we will account for the fact that pulling on an object axially causes it to compress laterally in the transverse directions:

So, pulling on it in the x-direction causes it to shrink in the y & z directions. This property of a material is known as Poisson’s ratio, and it is denoted by the Greek letter nu, and is defined as:

Or, more mathematically, using the axial load shown in the above image, we can write this out as an equation:

Since Poisson’s ratio is a ratio of two strains, and strain is dimensionless, Poisson’s ratio is also unitless. Poisson’s ratio is a material property. Poisson’s ratio can range from a value of -1 to 0.5. For most engineering materials, for example steel or aluminum have a Poisson’s ratio around 0.3, and rubbers have a Poisson’s ratio around 0.5, which are referred to as “incompressible”. Incompressible simply means that any amount you compress it in one direction, it will expand the same amount in it’s other directions – hence, its volume will not change.

There has been some very interesting research in the last decade in creating structured materials that utilize geometry and elastic instabilities (a topic we’ll cover briefly in a subsequent lecture) to create auxetic materials – materials with a negative Poisson’s ratio. Physically, this means that when you pull on the material in one direction it expands in all directions (and vice versa):

This principle can be applied in 3D to make expandable/collapsible shells as well:

Through Poisson’s ratio, we now have an equation that relates strain in the y or z direction to strain in the z direction. We can in turn relate this back to stress through Hooke’s law. This is an important note: pulling on an object in one direction causes stress in only that direction, and causes strain in all three directions. So, sigmay = sigmaz = 0. Let’s write out the strains in the y and z direction in terms of the stress in the x direction.

Remember, up until this point, we’ve only considered uniaxial deformation. In reality, structures can be simultaneously loaded in multiple directions, causing stress in those directions. A helpful way to understand this is to imagine a very tiny “cube” of material within an object. That cube can have stresses that are normal to each surface, like this:

So, applying a load in the x direction causes a normal stress in that direction, and the same is true for normal stresses in the y and z directions. And, as we now know, stress in one direction causes strain in all three directions. So now we incorporate this idea into Hooke’s law, and write down equations for the strain in each direction as:

These equations look harder than they really are: strain in each direction (or, each component of strain) depends on the normal stress in that direction, and the Poisson’s ratio times the strain in the other two directions. Now we have equations for how an object will change shape in three orthogonal directions. Well, if an object changes shape in all three directions, that means it will change its volume. A simple measure for this volume change can be found by adding up the three normal components of strain:

Now that we have an equation for volume change, or dilation, in terms of normal strains, we can rewrite it in terms of normal stresses.

A very common type of stress that causes dilation is known as hydrostatic stress. This is just simply a pressure that acts equally on the entire material. Since it is acting equally, that means:

So, in the case of hydrostatic pressure we can reduce our final equation for dilation to the following:

This final relationship is important, because it is a constitutive relationship for how a material’s volume changes under hydrostatic pressure. The prefactor to p can be rewritten as a material’s bulk modulus, K.

Finally, let’s get back to the idea of “incompressible” materials. What happens to K – the measure of how a material changes volume under a given pressure – if Poisson’s ratio for the material is 0.5?

### Hooke’s Law in Shear

In the previous section we developed the relationships between normal stress and normal strain. Now we have to talk about shear. Let’s go back to that imaginary cube of material. In addition to external forces causing stresses that are normal to each surface of the cube, the forces can causes stresses that are parallel to each cube face. And, as we know, stresses parallel to a cross section are shear stresses

Now that cube of material looks a lot more complicated, but it’s really not too bad. On each surface there are two shear stresses, and the subscripts tell you which direction they point in and which surface they are parallel to. For instance, take the right face of the cube. Stresses normal to this face are normal stresses in the x direction. There are two stresses parallel to this surface, one pointing in the y direction (denoted tauxy) and one pointing in the z direction (denoted tauxz). In order for the cube to be in equilibrium, tauxy = tauyx (otherwise, the cube would rotate). Therefore, there are now six stresses (sigmax, sigma y , sigma z , tauxy, tauyz, tauxz) that characterize the state of stress within a homogenous, isotropic, elastic material.

So, how do these shear stresses relate to shear strains? Hooke’s law in shear looks very similar to the equation we saw for normal stress and strain:

In this equation, the proportionality between shear stress and shear strain is known as the shear modulus of a material. That’s the equation in its general form, but we can rewrite it more explicitly in terms of its components of x,y, and z. Doing so will give us the generalized Hooke’s law for homogenous, isotropic, elastic materials.

In our generalized Hooke’s law we have our six components of stress and strain, and three material properties. A natural question to as is how do these three material properties relate to each other? That relationship is given by the following equation:

### Summary

We’ve introduced the concept of strain in this lecture. Strain is the deformation of a material from stress. It is simply a ratio of the change in length to the original length. Deformations that are applied perpendicular to the cross section are normal strains, while deformations applied parallel to the cross section are shear strains. For linear, elastic materials, stress is linearly related to strain by Hooke’s law. The proportionality of this relationship is known as the material’s elastic modulus. Using Hooke’s law, we can write down a simple equation that describes how a material deforms under an externally applied load.

Additionally, we learned about multiaxial loading in this section. In particular, we learned that stress in one direction causes deformation in three directions. This occurs due to a material property known as Poisson’s ratio – the ratio between lateral and axial strains. The strains occurring in three orthogonal directions can give us a measure of a material’s dilation in response to multiaxial loading. In particular, a material can commonly change volume in response to changes in external pressure, or hydrostatic stress. This lead to a definition of a materials resistance to volume change under hydrostatic stress – the bulk modulus. By inspecting an imaginary cubic element within an arbitrary material, we were able to envision stresses occurring normal and parallel to each cube face. This gave us six stresses and six strains (three normal and three shear) that we related to each other using a generalized Hooke’s law for homogenous, isotropic, and elastic materials. These components of multiaxial stress and strain are related by three material properties: Young’s elastic modulus, the shear modulus, and Poisson’s ratio.

This material is based upon work supported by the National Science Foundation under Grant No. 1454153. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Important Announcement 2 December, 2020 at 1:14 PM Visit Back2BU for the latest updates and information on BU’s response to COVID-19. Students can find additional information in the

## Which Weed Strains Are Highest in THC?

It’s hard to pinpoint which marijuana strain is highest in THC because strains aren’t an exact science. They can vary across sources, and new ones are popping up constantly.

Then there’s the issue of THC and CBD, two of the most well-known compounds in marijuana.

THC is the psychoactive compound responsible for the high marijuana produces. When people say that a particular strain of weed is very strong, it’s likely a high-THC strain.

High-THC strains will produce strong psychoactive effects and may be beneficial for:

• reducing nausea
• increasing appetite
• reducing pain
• decreasing inflammation
• improving muscle control problems

We’ve rounded up the strains that tend to have the most THC, according to Leafly’s strain explorer.

They’re broken down into three groups, depending on their effects:

• sativas (energetic)
• indicas (relaxing)
• hybrids (a combination)

Keep in mind that there’s some debate around whether sativa and indica strains are all that different from each other.

Sativas usually have higher levels of THC and lower levels of CBD. They tend to produce a stimulating or invigorating effect, making them better for daytime use.

#### Lemon Meringue

This sativa strain is around 21 percent THC. It is considered to have an uplifting effect. People tend to use it for:

Users of this strain report feeling:

• happy
• uplifted
• energetic

Some also say it boosts creativity.

#### Laughing Buddha

Laughing Buddha is an award-winning sativa strain that’s 21 percent THC. And its name is fitting. Users report it has the power to make you feel happy and cause giggling, even when you’re feeling depressed.

It’s sought out by people dealing with:

• stress
• anxiety
• depression
• fatigue
• pain

Along with feelings of happiness, it can also make you feel euphoric and energetic.

#### Hawaiian

Hawaiian is apparently the strain of choice for those looking to feel happy and relaxed, much like when you’re on vacay. It’s 22 percent THC. Users report feeling equally relaxed and uplifted.

As with other high-THC sativa strains, people use Hawaiian in an attempt to relieve stress and anxiety, as well as depression, pain, and fatigue.

Feelings associated with this strain include:

• happiness
• creativity
• relaxation
• energy
• euphoria

Thai is a popular strain with 22 percent THC that’s associated with feeling uplifted and focused.

Users say it helps relieve:

• stress
• depression symptoms

Based on user reviews, this strain is reported to leave you feeling happy, energetic, and relaxed.

#### Silver Haze

Silver Haze packs a lot of punch at 23 percent THC. Incidentally, the THC is where this strain gets its name. It has a copious amount of glistening THC glands that cover the buds.

People use Silver Haze for:

• stress
• anxiety
• depression
• poor appetite
• pain

User reviews say it produces feelings of:

• happiness
• euphoria
• relaxation

#### Memory Loss

This one is technically a hybrid, but it’s still mostly sativa. The name is fitting, given that this strain is 26 to 31 percent THC. It’s fast acting and capable of producing some intense mental effects.

People use this strain mainly for:

• stress
• depression symptoms
• fatigue
• nausea

Indica strains tend to have more CBD than THC, though this isn’t always the case. As a result, you won’t find as many pure indica strains with percentages of THC.

While sativa strains are said to produce more invigorating effects, indica strains are linked to relaxing effects that make them best for night-time use (or days where you don’t have a ton on your plate).

They’re typically recommended for folks dealing with:

• sleep issues
• pain
• nausea
• low appetite

#### Kosher Kush

Kosher Kush originated as a clone-only strain in Los Angeles. It’s 21 percent THC and associated with major relaxation and pain relief.

It has a tendency to put you to sleep, which may be why people often seek it out to treat insomnia.

It may also help with:

• stress
• anxiety
• depression

According to user reviews, you can expect to feel:

• relaxed
• sedated
• happy
• euphoric
• hungry

#### Triangle Kush

This strain has an average THC level of 23 percent. It seems to be a favorite with creative types and artists for boosting creativity.

People also seek it out to relieve:

• chronic pain
• stress
• depression symptoms

Users report feeling especially:

• creative
• euphoric
• chilled out after using it

Hybrids are the result of crossbreeding sativa and indica strains, often resulting in what may be considered to be the best of both worlds.

The effects of specific hybrid strains depend on the ratio of indica to sativa, and combination of strains that make up the hybrid.

#### Death Star

Death Star is an indica-dominant hybrid that comes in at 21 percent THC. Its effects are said to come on slowly at first. But eventually they lead to a powerful state of relaxation and euphoria.

Users attest to its ability to relieve:

• stress
• anxiety symptoms
• depression symptoms
• insomnia

#### Ghost OG

If you’re looking for a balance between mind and body effects, this indica-dominant strain may be the way to go.

It contains up to 23 percent THC and is sought out by people looking to manage:

• stress
• pain
• insomnia
• depression
• anxiety

Users report that it produces a calming, sleepy effect.

With up to 24 percent THC, this indica-dominant strain, sometimes called Garlic Gookies, has a sedative effect and can make you incredibly sleepy.

Medicinally speaking, it’s mainly used to relieve:

• chronic pain
• stress
• anxiety symptoms
• insomnia

Another indica-dominant strain, this one offers 23 percent THC. Some dispensaries say the THC level can be as high as 30 percent.

People use it for:

• pain
• inflammation
• stress
• insomnia

Word around user forums is that it also has a mild aphrodisiac effect and can make you feel relaxed, euphoric, happy, and sleepy.

#### Banana OG

Yet another indica-dominant hybrid, Banana OG clocks in at 23 percent THC. It’s referred to as a “creeper” because using too much can leave you in a major stupor before surprising you with intense munchies and sleepiness.

People use it for:

• muscle pain
• poor appetite
• insomnia

Its other reported effects include:

• relaxation
• euphoria
• hunger

#### Lemon Kush

This is a 50/50 hybrid that averages around 22 percent THC.

People mostly use it for relaxation, feeling euphoric, and boosting appetite.

Other reported effects include:

• increased creativity
• happiness
• stress relief
• hunger

#### Gorilla Glue

Another 50/50 hybrid, Gorilla Glue — also called GG for legal reasons — hits hard at 23 percent THC.

This potent strain is known for its cerebral and physical effects that come on quickly and last longer than those of other strains.

It’s used mostly for its relaxing and sedating effect, which is helpful for stress relief and insomnia. People also use it for pain, including menstrual cramps, according to online reviews.

#### The White

Coming in at around 23 percent THC, The White is a potent indica-dominant hybrid.

Many user reviews mention its ability to relieve:

• pain
• insomnia
• nausea
• stress
• depression symptoms

Its effects include:

• relaxation
• sedation
• feelings of euphoria and happiness

#### Bruce Banner

At around 25 percent THC, this hybrid hits strong and fast, eventually settling into a state of euphoria and heightened creativity according to users.

It’s used to relieve:

• stress
• depression symptoms
• pain

THC can cause temporary side effects, which can be more pronounced in higher doses or if you’re new to marijuana.

• increased heart rate
• decreased blood pressure
• dry mouth
• coordination problems
• slower reaction times
• short-term memory loss
• panic
• paranoia
• hallucinations

Experts still don’t know the full health impact of high-THC strains that have popped up in recent years. Some research suggests a potential link between high-THC marijuana and long-term mental health effects, including psychosis, especially in regular users and young people.

You may also have a higher risk for addiction when exposed to higher THC levels, according to the National Institute on Drug Abuse.

If you’re going to use cannabis, especially high-THC strains, consider these harm reduction tips:

• Start off with a low THC strain and gradually work your way up to avoid severe side effects.
• Look into nonsmoking methods, such as edibles or oils, to protect your lungs.
• If you do smoke, avoid deep inhalation and holding your breath to limit exposure to harmful by-products in the smoke.
• Limit your use of marijuana, especially high-THC strains, to lower your risk for long-term health risks, including addiction.
• Don’t drive for at least 6 hours after using cannabis — longer if you’re still feeling its effects.
• Avoid marijuana entirely if you’re pregnant or breastfeeding.

Though many states have legalized cannabis for medical and recreational purposes, it isn’t legal everywhere and is still considered illegal under federal law.

It’s important to know the laws for your state before you attempt to purchase or use marijuana to avoid facing legal consequences.

Check your local law if you’re not in the United States, as laws may be different.

High-THC strains are among the more potent marijuana products you can find. While they can be useful for treating certain health conditions, they also tend to have strong psychological effects.

If you’re new to marijuana, consider starting with low-THC strains and working your way up. Even if you’re a seasoned consumer, go slow when using high-THC products.

Last medically reviewed on October 29, 2019

Looking for high-THC strains? We've rounded up 17 sativas, indicas, and hybrids containing at least 20 percent THC.