8.1 Elastic and plastic behavior of materials

Materials can behave elastically or plastically when stressed. Elastic deformation is temporary, with the material returning to its original shape when the load is removed. Plastic deformation is permanent, occurring when stress exceeds the material's yield point.

Understanding elastic and plastic behavior is crucial for engineering design. Stress-strain curves illustrate these properties, showing the transition from elastic to plastic deformation. The yield strength and ultimate tensile strength are key parameters derived from these curves, guiding material selection and component design.

Elastic vs Plastic Deformation

Types of Deformation

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• Elastic deformation is a temporary deformation where the material returns to its original shape and size once the load is removed
  • The material undergoes no permanent change and follows a linear stress-strain relationship in the elastic region
  • Examples of elastic deformation include stretching a rubber band or compressing a spring, which return to their original shape when released
• Plastic deformation is a permanent deformation where the material does not return to its original shape and size after the load is removed
  • The material undergoes permanent changes in its internal structure and does not follow a linear stress-strain relationship in the plastic region
  • Examples of plastic deformation include bending a metal paperclip or deforming a clay sculpture, which retain their new shape after the force is removed

Yield Point and Material Behavior

• The transition point between elastic and plastic deformation is called the yield point, which is the stress at which a material begins to deform plastically
  • The yield point represents the onset of permanent deformation and is a critical material property for design purposes
  • The yield point can be determined from a stress-strain curve by identifying the point where the curve deviates from linearity or by using the offset method (typically 0.2% strain offset)
• Ductile materials, such as metals, exhibit both elastic and plastic deformation, while brittle materials, such as ceramics, exhibit little to no plastic deformation before fracture
  • Ductile materials can undergo significant plastic deformation before fracture, allowing them to be shaped and formed (steel, aluminum, copper)
  • Brittle materials have little to no capacity for plastic deformation and fracture with little warning once their yield strength is exceeded (glass, concrete, cast iron)

Stress-Strain Curves Analysis

Stress-Strain Curve Characteristics

• A stress-strain curve is a graphical representation of the relationship between the applied stress and the resulting strain in a material during a tensile or compressive test
  • Stress is plotted on the y-axis and strain on the x-axis, with the curve showing how the material responds to increasing loads
  • The shape of the stress-strain curve provides valuable information about the material's mechanical properties and behavior under load
• The slope of the linear portion of the stress-strain curve in the elastic region is called Young's modulus (or elastic modulus), which is a measure of the material's stiffness
  • A steeper slope indicates a higher Young's modulus and a stiffer material (diamond, tungsten, steel)
  • A shallower slope indicates a lower Young's modulus and a more flexible material (rubber, low-density polyethylene)

Key Material Properties from Stress-Strain Curves

• The yield strength is the stress at which the material begins to deform plastically, and it is identified as the point where the stress-strain curve deviates from linearity
  • The yield strength represents the upper limit of the elastic region and the onset of permanent deformation
  • Materials with higher yield strengths can withstand greater loads before experiencing plastic deformation (high-strength steels, titanium alloys)
• The ultimate tensile strength (UTS) is the maximum stress a material can withstand before fracture, and it is the highest point on the stress-strain curve
  • The UTS represents the maximum load-carrying capacity of the material
  • Materials with higher UTS values can withstand greater loads before fracture (carbon fiber, Kevlar, spider silk)
• The area under the stress-strain curve represents the energy absorbed by the material during deformation, which is related to the material's toughness
  • Tougher materials can absorb more energy before fracture and are more resistant to crack propagation (metals, composites, some polymers)
  • Brittle materials have low toughness and absorb little energy before fracture (ceramics, glass, some plastics)
• The strain at fracture, or elongation, is the maximum strain the material can withstand before breaking, and it is a measure of the material's ductility
  • Ductile materials have high elongation values and can undergo significant plastic deformation before fracture (copper, aluminum, gold)
  • Brittle materials have low elongation values and fracture with little plastic deformation (concrete, glass, cast iron)

Hooke's Law Application

Hooke's Law Formula

• Hooke's law states that the stress is directly proportional to the strain in the elastic region of deformation, with the proportionality constant being Young's modulus (E)
  • Hooke's law describes the linear relationship between stress and strain in the elastic region
  • The proportionality constant, Young's modulus, is a material property that quantifies the stiffness of the material
• The mathematical expression for Hooke's law is $\sigma = E\epsilon$, where $\sigma$ is the stress (force per unit area), $E$ is Young's modulus, and $\epsilon$ is the strain (change in length per unit original length)
  • Stress is measured in units of pascals (Pa) or megapascals (MPa), strain is dimensionless, and Young's modulus is measured in the same units as stress
  • The equation can be rearranged to solve for stress, strain, or Young's modulus, depending on the given information

Calculating Stress and Strain using Hooke's Law

• To calculate stress using Hooke's law, multiply the strain by Young's modulus: $\sigma = E\epsilon$
  • Example: If a steel bar with a Young's modulus of 200 GPa experiences a strain of 0.001, the stress in the bar is $\sigma = (200 \times 10^9 \text{ Pa})(0.001) = 200 \times 10^6 \text{ Pa} = 200 \text{ MPa}$
• To calculate strain using Hooke's law, divide the stress by Young's modulus: $\epsilon = \sigma/E$
  • Example: If an aluminum rod with a Young's modulus of 70 GPa is subjected to a stress of 100 MPa, the strain in the rod is $\epsilon = (100 \times 10^6 \text{ Pa})/(70 \times 10^9 \text{ Pa}) = 0.00143$
• Hooke's law is valid only in the elastic region of deformation, where the stress-strain relationship is linear. It does not apply to the plastic region or to materials that exhibit non-linear elastic behavior
  • Non-linear elastic materials, such as rubber or biological tissues, have stress-strain curves that are not straight lines and require more complex models to describe their behavior
  • In the plastic region, the relationship between stress and strain is no longer linear, and permanent deformation occurs, invalidating Hooke's law

Yield & Tensile Strength Determination

Yield Strength Determination Methods

• The yield strength is the stress at which a material begins to deform plastically, and it is a critical material property for design purposes
  • The yield strength represents the upper limit of the elastic region and the onset of permanent deformation
  • Designing components to operate below the yield strength ensures that they will not experience permanent deformation under normal loading conditions
• The yield strength can be determined from a stress-strain curve by identifying the point where the curve deviates from linearity (the proportional limit) or by using the offset method (typically 0.2% strain offset)
  • The proportional limit method involves identifying the point where the stress-strain curve deviates from a straight line, indicating the end of the linear-elastic region
  • The offset method involves drawing a line parallel to the linear portion of the stress-strain curve at a specified strain offset (usually 0.2%) and determining the stress at which this line intersects the curve
• Factors such as material composition, heat treatment, strain rate, and temperature can influence the yield strength
  • Alloying elements, heat treatment processes (quenching, tempering), and cold working can increase the yield strength of metals
  • Higher strain rates and lower temperatures generally increase the yield strength, while lower strain rates and higher temperatures decrease it

Ultimate Tensile Strength (UTS) Determination

• The ultimate tensile strength (UTS) is the maximum stress a material can withstand before fracture, and it is another important material property for design and material selection
  • The UTS represents the maximum load-carrying capacity of the material
  • Designing components to operate well below the UTS ensures a safety factor against fracture
• The UTS is determined from a stress-strain curve as the highest point on the curve, which corresponds to the maximum load divided by the original cross-sectional area of the specimen
  • The UTS is calculated by dividing the maximum load (in newtons) by the original cross-sectional area of the specimen (in square meters)
  • Example: If a tensile test specimen with a cross-sectional area of 100 mm² fails at a maximum load of 10 kN, the UTS is $\text{UTS} = (10 \times 10^3 \text{ N})/(100 \times 10^{-6} \text{ m}^2) = 100 \times 10^6 \text{ Pa} = 100 \text{ MPa}$
• Ductile materials typically have a distinct yield point and a large difference between yield strength and UTS, while brittle materials often have a small difference between yield strength and UTS or no distinct yield point at all
  • Ductile materials, such as most metals, have a clear yield point and can sustain significant plastic deformation between the yield strength and the UTS (mild steel, aluminum alloys)
  • Brittle materials, such as ceramics and some plastics, often have a small difference between yield strength and UTS or fracture before yielding (glass, concrete, acrylic)