Stress-Strain Relationship
Strain is a measure of how much an object has deformed. Axial strain, \varepsilon, is defined by comparing the amount of axial length change, \delta, against the original unstressed object length, L.
\varepsilon=\dfrac{\delta}{L}
By plotting the applied stress and resulting strain a relationship between stress and strain appears.
Ductile materials undergo several transformations as when being stretched to the point of breaking. When initially being stressed the ductile material will undergo elastic (reversible) elongation. Once the yield stress, \sigma_y, is reached the material will undergo plastic (permanent) deformation. Strain loading or strain hardening occurs after yielding and will increase the strength of the material. Once the ultimate stress, \sigma_u, is reached the ductile material will begin necking, experiencing a rapid decrease in it’s cross-section until it breaks. Brittle materials undergo the same initial elastic and plastic deformations as a ductile material but will abruptly break without necking.
Hooke’s law describes the relationship between stress and strain in the elastic region of the stress strain curve. The relationship between stress and strain is linear in the elastic region, as such Hooke’s law states:
\sigma_{ij}=E\varepsilon_{ij}
Where E is Young’s modulus, a constant property dependent on the material in question. Combining Hooke’s law with the definitions of stress and strain we can obtain a direct relationship between axial deformation and loading:
\delta=\dfrac{FL}{AE}
Deformation Caused by Temperature Change
Changes in temperature cause deformation in materials as molecules and atoms become more energetic. The strain caused by temperature changes do not cause stress unless elongation is restrained in some way. The deformation caused by a temperature change is directly related to the temperature change by a coefficient of thermal expansion, \alpha.
\delta_T=\alpha L\Delta T
Poisson’s Ratio
When a material is elongated in a given direction there will be some corresponding elongation in directions perpendicular to the elongation. The ratio of lateral to axial strain is referred to as Poisson’s ratio, \nu, and is a property of materials.
\nu=-\left|\dfrac{\text{lateral strain}}{\text{axial strain}}\right|
Generalized Hooke’s Law
For small deformations, normal strain can be determined using the principle of superposition.
\varepsilon_x=\dfrac{\sigma_x}{E}-\nu\dfrac{\sigma_y}{E}-\nu\dfrac{\sigma_z}{E}
\varepsilon_y=\dfrac{\sigma_y}{E}-\nu\dfrac{\sigma_x}{E}-\nu\dfrac{\sigma_z}{E}
\varepsilon_z=\dfrac{\sigma_z}{E}-\nu\dfrac{\sigma_x}{E}-\nu\dfrac{\sigma_y}{E}
Bulk Modulus
Dilation, e, is a measure of the change in volume relative to the object volume, essentially the volumetric equivalent of strain.
e=\varepsilon_x+\varepsilon_y+\varepsilon_z
e=\dfrac{1-2\nu}{E}(\sigma_x+\sigma_y+\sigma_z)
Subjecting a material to a hydrostatic (uniform) pressure, P, results in the following dilation:
e=-\dfrac{3(1-2\nu)}{E}P
The bulk modulus, k, describes how easily a material’s volume will change when subjected to a uniform load. Bulk modulus is defined by the above-mentioned hydrostatic loading condition.
k=-\dfrac{P}{e_{hydrostatic}}
k=\dfrac{-P}{-\dfrac{3(1-2\nu)}{E}P}
k=\dfrac{E}{3(1-2\nu)}
Shearing Strain
Shearing strain describes angular deformation. Shearing strain will cause a square piece of material to deform into a rhombus. Shear strain, \gamma, is related to the angle between the initial and deformed state, \theta.
\gamma=\tan\theta\approx\theta
Similar to normal stress, shear stress is directly proportional to shearing strain. The modulus of rigidity, G, is a material property relating shearing stress and strain to one another by the following:
\tau_{ij}=G\gamma_{ij}
Bringing it All Together
Poisson’s ratio, Young’s modulus, and the modulus of rigidity are all related to each other. As such, knowing only two of the material properties allows the third to be derived.
G=\dfrac{E}{2(1+\nu)}
To summarize, the following equations fully describe the relationship between stess and strain for small deformations:
\varepsilon_x=\dfrac{\sigma_x}{E}-\nu\dfrac{\sigma_y}{E}-\nu\dfrac{\sigma_z}{E}
\varepsilon_y=\dfrac{\sigma_y}{E}-\nu\dfrac{\sigma_x}{E}-\nu\dfrac{\sigma_z}{E}
\varepsilon_z=\dfrac{\sigma_z}{E}-\nu\dfrac{\sigma_x}{E}-\nu\dfrac{\sigma_y}{E}
\gamma_{xy}=\dfrac{2(1+\nu)}{E}\tau_{xy}
\gamma_{xz}=\dfrac{2(1+\nu)}{E}\tau_{xz}
\gamma_{yz}=\dfrac{2(1+\nu)}{E}\tau_{yz}
Stress Concentrations
Saint-Venant’s principle states that the effect of loading specifics becomes less important the further you get from the load. In other words, the method of loading is most important close to the point of loading and becomes less important and more distributed and uniform the further the load travels from the loading point. Saint-Venant’s principle is the reason why average stress is so often used as a metric for determining structural integrity.
The distribution of stress can change drastically in objects with abrupt geometry changes. These distribution changes will be most drastic around the geometry change in question, and become less severe further away from these features. These abrupt stress distribution changes are referred to as stress concentrations, and describe the typically heightened stress at the locations of features. More abrupt cross-sectional area changes will result in greater stress concentrations. As such, fillets and round corners mitigate the effects and magnitude of stress concentrations. The stress concentation factor, K, is a constant determined by the object geometry which can be used to find the maximum stress in the stress concentration distribution. The stress concentration factor is empirically measured for several common features such as fillets and holes, and is usually tabulated in relevant mechanics textbooks.
K=\dfrac{\sigma_{max}}{\sigma_{avg}}
The maximum stress acting on the object is used to determine whether or not the object will meet failure criteria. Therefore, it is of extreme importance to analyze stress concentrations when determining if a design can handle given loading criteria.