12.9
Problem Solving on Stress and Strain
Stress is a quantity that describes the magnitude of a force that causes deformation, generally defined as internal force per unit area. When forces p…
Consider a piece of box-shaped gelatin dessert of known dimensions, whose bottom surface is fixed.
If a shearing force of 0.50 N is applied to its top surface, the upper surface displaces by 0.5 cm relative to the bottom surface.
What is the resulting shear stress, measured shear strain, and shear modulus of the gelatin?
An object under shear stress experiences two anti-parallel forces of equal magnitude that are applied tangentially to the object's opposite parallel surfaces.
So, the ratio of this tangential force to the cross-sectional area gives the shear stress.
Since there is no change in the direction transverse to the applied forces, the transverse length remains unchanged.
The shear strain is defined by a gradual shift of the layers in the direction tangential to the force.
By substituting the obtained quantities in the expression, the gelatin's shear stress and strain are calculated.
The ratio of shear stress to shear strain gives the shear modulus. A higher shearing force increases the shear stress in the gelatin, causing the material to collapse.
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Q1: How is shear stress calculated in a material?
Shear stress is calculated as the ratio of the tangential force applied to an object's surface divided by the cross-sectional area perpendicular to that force. When antiparallel forces of equal magnitude act tangentially on opposite surfaces, the resulting shear stress quantifies the intensity of deformation. This measurement is essential for understanding how materials respond to forces that cause lateral displacement rather than compression or tension.
Q2: What is the difference between shear strain and shear stress?
Shear stress describes the magnitude of tangential force per unit area causing deformation, while shear strain measures the resulting deformation itself. Shear strain is defined by the ratio of the largest displacement in the direction parallel to applied forces to the transverse distance over which this displacement occurs. Together, stress and strain characterize how a material deforms under shearing forces.
Q3: What does the shear modulus tell you about a material?
The shear modulus is the ratio of shear stress to shear strain and indicates a material's resistance to shear deformation. A higher shear modulus means the material is stiffer and resists lateral displacement more effectively. This property helps predict how materials like gelatin, buildings, or tectonic plates will respond when subjected to tangential forces.
Q4: Why does transverse length remain unchanged during shear deformation?
During shear deformation, antiparallel forces act tangentially to opposite surfaces, causing layers to shift parallel to the applied forces. Because the forces are applied tangentially rather than perpendicular to the surface, there is no compression or extension in the transverse direction. This characteristic distinguishes shear deformation from tensile or compressive stress, where dimensional changes occur perpendicular to the applied force.
Q5: What happens to a material when shearing force increases?
As shearing force increases, shear stress on the material increases proportionally. Higher shear stress causes greater layer displacement and larger shear strain. If the shearing force becomes excessive, the material may exceed its elastic limit and collapse, transitioning from elastic deformation to permanent damage or failure.
Q6: How do you calculate shear strain from displacement measurements?
Shear strain is calculated by dividing the largest displacement of the upper surface by the transverse distance between the fixed bottom surface and the displaced top surface. For example, if a gelatin dessert's top surface displaces 0.5 cm relative to its bottom surface over a known height, the shear strain equals this displacement divided by the original height, yielding a dimensionless ratio.
Q7: What are real-world examples of objects experiencing shear stress?
Buildings and tectonic plates commonly experience shear stress. When you hold a book between your palms and pull opposite covers in opposite directions, the book experiences shear stress. Submarines subjected to pressure from ocean depths and any solid material deformed by tangential forces demonstrate shear deformation in practical applications.