Lesson 14 Course Notes

Mechanical Properties of Materials

Objectives

Define stress, strain, stiffness, and compliance.
Describe the stress-strain curves for ductile, brittle and elastic materials.
Define viscoelasticity, viscosity, plasticity, and elasticity.
Define Young’s modulus.
Define hysteresis, energy absorbed, energy returned, creep, and force relaxation.
Describe the effects of temperature and velocity of loading on the mechanical response of viscoelastic materials.

Strength of Materials

The strength of a material can be defined by the ability of the material to withstand forces without breaking or failing.  Several factors such as: microstructure, age, temperature, fluid content, type, velocity and direction of loading can alter the strength of a material.    The figure below shows the types of loads that can be applied to a material.   Compression and tension are both axial loads since the forces are directed along the long axis of the material.  Torsion and bending are known as shear loads since the forces act in a tangential or parallel direction.

When a load is applied to a material the material will develop internal resistance to the applied load.  The magnitude of this internal resistance to applied loads is dependent upon the stiffness of the material.  For example the internal resistance developed by compressing a tennis ball is considerably less than the internal resistance developed by compressing a bowling ball. 

Axial Stress
The internal resistance of a material to an applied axial load is called axial stress (σ).   The Greek symbol sigma (σ) is typically used for axial stress. Stress is a normalized variable where the force applied is divided by the area over which the force is distributed.  Axial stress is

Where F is force in N, A is the cross-sectional area in m2.  Axial stress is in units of pascal (Pa), where 1 Pa = 1 N/m2.

Shear Stress

In the figure above the force for both torsion and bending loads is applied tangential or parallel to the long axis of the material.  When the force is applied tangential or parallel to the long axis of the material the internal stress created is termed shear stress (τ), denoted by the Greek letter (τ) tau.  The formula to calculate sheer stress is

Where F is force in N, A is the cross-sectional area in m2.  Shear stress is in units of pascal (Pa), where 1 Pa = 1 N/m2.

Strain

When a load is applied to a material it will undergo a change in shape.  This change in shape or deformation of the material is described by the strain (ε).  The Greek symbol (ε) epsilon is used for strain.  Strain can be measured in absolute or relative units see the figure below.  Strain is usually given in relative units or percent change.  In the figure below, the compression force causes 10 mm of absolute strain.  The relative strain for this example is the absolute strain 10 mm divided by initial dimension of 100 mm, which results in a relative strain of 0.1 or 10%.  Relative strain is a normalized variable.

Stress-Strain Relationship

The relative strength of a material can be quantified by graphing its stress-strain relationship.    There is a direct relationship between the strength of a material and its stress-strain curve.  The figure below shows an example of a stress-strain curve for a soft metal like copper.

The maximum amount of deformation that a material can withstand and still return to its original shape is defined as the yield point.  When the material is loaded beyond the yield point permanent damage occurs to the material and it can no longer return to its original shape.  If the force is continues the material may break into two or more pieces when the failure point is reached.  The region of the stress – strain curve up to and including the yield point is termed the elastic region.   In the elastic region the material will deform during loading and return to its original shape when the load is removed.  The region of the stress – strain curve beyond the yield point is termed the plastic region.   In the plastic region the material will deform during loading, when the load is removed the material does not return to its original shape.  A plastic deformation results in permanent change of the material.  If the magnitude of loading is just above the yield point the permanent deformation may be microscopic and imperceptible.

The stiffness of a material is defined as the slope of its stress – strain curve.  Stiffness is the ratio of stress to strain (σ/ε), in other words stiffness is stress divided by strain.  Compliance is the inverse of stiffness, so a material that deforms easily is said to be compliant and a material that is resistant to deformation is said to be compliant.

Young’s Modulus

During axial loading many materials show a linear stress – strain response in the elastic region of the stress – strain curve.  Young’s modulus, also called the modulus of elasticity, describes the stiffness of a material in the elastic region during axial loading.  If the material is uniform, such as a solid copper or aluminum rod, Young’s modulus is essentially constant.  Uniform materials are known as isotropic materials.  Isotropic is defined as having identical mechanical properties in all directions.  Some materials do not exhibit a consistent mechanical response when the direction of loading is changed, these materials are said to be anisotropic.  Anisotropic materials are directionally dependent.  Impurities within the material or the direction of fibers within the material can produce an anisotropic mechanical response.  Young’s modulus is not constant for an anisotropic material.  

Stress – Strain Curves for Different Types of Materials

The stress – strain graph below shows the typical mechanical response for loading of a ductile, brittle and elastic material.  A ductile material can be deformed beyond its yield point without causing a fracture of the material.  If you bend a thin plastic rod beyond its yield point it will remain bent, permanently changed.  Aluminum cans, and other soft metals will also exhibit this behavior.  Ductile materials are stiffer up to and including the yield point.  A ductile material becomes more compliant when it is deformed beyond its yield point.  The stress – strain curve for the elastic material indicates that elastic materials are initially compliant and they become progressively stiffer as the load is increased.  This mechanical behavior is clearly felt when stretching a rubber band, initially it is very easy to deform, but it becomes much stiffer as the elongation increases.  Some materials such as a piece of chalk or a thin glass rod will exhibit the stress – strain curve for a brittle material shown in the graph below.  When a brittle material is subjected to loading it will fail (break) with little deformation.  The yield point and the failure point are the same for a brittle material, thus it is not possible to bend a piece of chalk and produce a small crack in the chalk without breaking it.

Viscoelastic Materials

When an elastic material containing fluid is deformed the return of the material to its original shape is delayed in time and it is slower to restore to its original position.  Viscoelastic materials exhibit both viscous damping and an elastic response during deformation.  A simple example of viscous damping can be seen when dropping a basketball and observing the height of successive bounces of the ball.  Each time the ball impacts the ground it absorbs some of the energy which results in a progressive decrease in the bounce height. A shock absorber on a car is an example of a viscoelastic element.  The figure shown below is an example of a model used to describe the behavior of a viscoelastic material.  The spring models the elastic response of the material.  The elastic response of the spring is linear and the force necessary to deform the spring is described by Hooke’s law:

F = -kx

Where F is the force to compress or extend the spring in N, x is the amount of compression or extension of the spring in m, and k is the spring constant or stiffness of the spring in N/m.  The dashpot in the figure below models the viscosity.  The dashpot is a piston in an enclosed or sealed cylinder that has been filled with a fluid.  The thickness or viscosity of the fluid in the cylinder will alter the damping response provided by the dashpot element.  Viscosity is a measure of the internal resistance of a fluid to flow.  The dashpot or viscous element will cause a time delay in the mechanical response of the material. The viscous response of the fluid will be altered by both velocity and temperature.  Honey or molasses are much more viscous (behave stiffer) when cooled than they are when heated.   A viscoelastic material is stiffer when loaded fast and less stiff when loaded slow. 

When the spring in the viscoelastic model above is loaded it will deform in a linear fashion directly related to the applied force and the stiffness of the spring.  When the load is removed the spring returns elastic energy that was stored in the spring during deformation.  The figure below shows a typical stress – strain curve for an elastic element, notice that the stress – strain curve traces along the same line for the unload phase as it did for the load phase.

Unlike the spring, a viscoelastic material will absorb some energy during deformation.  Energy is lost during the deformation of a viscoelastic material primarily to heat.  Since energy is absorbed by the viscous fluid in a viscoelastic material the stress – strain curve for the unload phase of the motion is different that the stress – strain for the loading phase. In the figure below the loading of the material occurs along the black line and unloading of the material occurs along the blue stress – strain line.  The amount of energy absorbed by the material is equal to the area between the load and unload curves, shown in light blue.  The area between the load and unload curves is called a hysteresis.  The energy absorbed is the area under the unload portion of the stress – strain curve, shown by the vertical red lines. 

Viscoelastic materials show the following properties: hysteresis, creep, and force relaxation.  Hysteresis was defined above energy absorbed by the material. 

Creep Response of a Viscoelastic Material

The creep response for a viscoelastic material can be observed by applying a constant load to the material and measuring the deformation of the material over time.  In the figure below a constant force of 30 lbs is applied to the material.  When the load is applied the material deforms quickly until the internal stress in the material reaches a relatively stable stiffness level, shown in blue as the initial deformation.  After attaining this initial change in length the material undergoes creep where it slowly continues to deform over time. 

Force Relaxation Response of a Viscoelastic Material

The force relaxation response of a viscoelastic material is shown in the figure below.  In force relaxation the material is deformed to a fixed length and the change in force over time is measured.  As shown below when the material is initially deformed the force rises rapidly (A to B).  The material is stiff during this initial loading.   If the material is held at the new position the fluid in the viscoelastic material will settle and cause the internal resistance of the material to decrease.  This decrease in stiffness is known as force relaxation.

The dashpot element in the viscoelastic material is responsible for the hysteresis, creep, force relaxation.  All viscoelastic materials will be stiffer when loaded fast than they are when loaded slow.  This velocity dependent response is attributed to the fluid in the material, at high rates of loading the fluid is compressed as it attempts to flow from one region to another region of the material.  The stress – strain response of viscous fluids is also altered by temperature such that increasing the temperature lowers the stiffness and decreasing the temperature increases the stiffness.