Ethan Clarke
Elastic and plastic strain calculations are fundamental in understanding the mechanical properties of materials, particularly in engineering applications. The stress-strain relationship is quantified by the stress-strain curve, which describes the behaviour of materials under load. Elastic deformation is thickness-dependent, where changes in thickness are proportional to the original thickness. On the other hand, plastic deformation occurs when the applied stress surpasses the material's yield strength, resulting in permanent changes. The Ramberg-Osgood equation provides a relationship between total strain (elastic and plastic) and stress, with the plastic strain calculated using the strength coefficient and strain hardening exponent. Poisson's ratio is another factor that influences the elastic region of the stress-strain curve, typically observed in metals.
| Characteristics | Values |
|---|---|
| Calculating Elastic and Plastic Strain | σ = stress at the indicated point, ε = strain at the indicated point, E = elastic modulus |
| Plastic Strain at Failure | εf |
| Ultimate Strain | εu (total strain at failure) |
| Stress-Strain Curve | Ramberg-Osgood equation: σ = value of stress, E = elastic modulus, Sty = tensile yield strength, n = strain hardening exponent |
| Plastic Deformation | Occurs when stress reaches yield stress; deformation increases as stress is applied |
| Elastic Deformation | Thickness-dependent; change in thickness proportional to original thickness |
| Poisson's Ratio | Typically 0.3 for metals, maximum limit is 0.5 |
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What You'll Learn
Stress-strain curve
A stress-strain curve is used to determine a material's mechanical properties, such as strength, stiffness, and ductility. Strength refers to a material's ability to resist deformation under load, stiffness refers to its ability to resist deformation, and ductility measures the ability of a material to undergo plastic deformation without breaking.
The curve is typically divided into four regions: elastic, plastic, necking, and fracture. In the elastic region, the material deforms elastically, meaning it returns to its original shape when the load is removed. The stress is proportional to the strain, obeying Hooke's Law, and the slope is Young's modulus. In the plastic region, the material undergoes plastic deformation and does not return to its original shape when the load is removed. The necking region is characterised by the material narrowing as it deforms, and in the fracture region, the material ultimately breaks.
The yield strength of a material is determined by identifying the point on the stress-strain curve where the linear relationship deviates. This is the point at which plastic deformation begins. The ultimate tensile strength of a material is determined by identifying the highest point on the stress-strain curve, which represents the maximum stress a material can endure before fracturing.
The Ramberg-Osgood equation is a straightforward way to approximate the stress-strain curve for a material. It only requires the material's yield strength, ultimate strength, elastic modulus, and percent elongation.
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Poisson's ratio
> ν = transverse elongation / axial compression
Where ν is the Poisson ratio. Poisson's ratio is typically about 0.3 for most metals and ranges between 0.1 and 0.45 for rocks. For soft materials, such as rubber, where the bulk modulus is much higher than the shear modulus, Poisson's ratio is near 0.5. For open-cell polymer foams, Poisson's ratio is near zero, since the cells tend to collapse in compression. Many typical solids have Poisson's ratios in the range of 0.2 to 0.3.
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Strain hardening
Work hardening is a consequence of plastic deformation, which is a permanent change in shape, as opposed to elastic deformation, which is reversible. Most materials exhibit a combination of both types of deformation. Ductility, the ability of a material to undergo plastic deformation before fracture, decreases in work-hardened materials. This is because the increase in strength due to strain hardening is eventually outpaced by the reduction in load-carrying capacity due to the decrease in cross-sectional area.
The work hardenability of a material can be predicted by analyzing a stress-strain curve or by performing hardness tests before and after a process. The Ramberg-Osgood equation describes the total strain (elastic and plastic) as a function of stress and can be used to calculate the strain hardening exponent of the material. The number of dislocations in the lattice strain fields also plays a role in work hardening, with high levels of dislocations resulting in higher strength.
The final temper of a strain-hardened material is determined by the amount of cold work applied after the final anneal. Annealing must be done at a sufficiently high temperature and for a long enough period to destroy the old grains, while also preventing new grains from growing too large. A balance must be struck between the strength imparted to the metal through strain hardening and the loss of ductility that comes with it.
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Tensile testing
The stress versus strain curve illustrates the relationship between stress and strain in the material and is used to determine the elastic and plastic deformation regions. In the elastic region, the material returns to its original shape when the load is removed. Poisson's ratio, which is the ratio of transverse strain to axial strain, applies only within this region and is typically about 0.3 for most metals.
As the load increases, the material may enter the plastic deformation region, where it begins to experience a high rate of plastic deformation and will not return to its original shape when the load is removed. The yield strength, or stress required to produce a specified amount of plastic deformation, is an important value obtained from tensile testing and is commonly used for engineering purposes.
To calculate the elastic and plastic strain, the stress-strain curve can be used in conjunction with the Ramberg-Osgood equation, which relates total strain (elastic and plastic) to stress. The equation takes into account the stress at a given point, the elastic modulus of the material, the tensile yield strength, and the strain hardening exponent.
Additionally, tensile testing can provide information on ductility, or the ability of a material to withstand plastic strain before breaking. The percent elongation and reduction in area are common measures of ductility and can be calculated from the final length and initial length of the specimen after a tensile test.
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Ramberg-Osgood equation
The Ramberg-Osgood equation is a formula used to calculate the total strain (elastic and plastic) as a function of stress. It is used to approximate the stress-strain curve for a material, which is commonly needed when analysing an engineered component.
The equation is as follows: σ = P/A0, where σ is the value of stress, E is the elastic modulus of the material, Sty is the tensile yield strength of the material, and n is the strain hardening exponent of the material. The strain hardening exponent can be calculated from known material properties.
The Ramberg-Osgood model provides an explicit formula for obtaining strain and implies that plastic strain is present even for very low levels of stress. However, for low applied stresses and commonly used values of the material constants, the plastic strain remains negligible compared to the elastic strain.
The Ramberg-Osgood equation is particularly useful when stress-strain data is not readily available. It is also used to calculate the modulus of toughness by approximating the stress-strain curve and then integrating the area under the curve. This is important because the area under the plastic region of the stress-strain curve contributes significantly to the toughness of the material.
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Frequently asked questions
Elastic deformation is thickness dependent, meaning the change in thickness is proportional to the original thickness. Plastic deformation, on the other hand, causes "work hardening" or "strain hardening", which increases the yield stress of a material.
The stress-strain equation is given as σ=Eε, where strain ε=ΔL/L0. Solving for ΔL gives ΔL=σL0/E, indicating that the change in length is proportional to the original length.
The Ramberg-Osgood equation for total strain (elastic and plastic) as a function of stress is: σ=Eε(1+ε)^n, where σ is the value of stress, E is the elastic modulus of the material, ε is the strain at the indicated point, Sty is the tensile yield strength of the material, and n is the strain hardening exponent of the material.
