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### Max. shear stress theory-Maximum Shear Stress Theory ​ Maximum Distortional Strain Energy Theory​

• 1. Deformation And Fracture Theories Maximum Shear Stress Theory Maximum Distortional Strain Energy Theory Lecturer: Prof. Dr. Nihat Atmaca November 2023
• 2. Theories of Failure Failure theories for materials are often categorized into two main types based on the behavior of the materials: 1. Brittle materials: Brittle materials are those that exhibit little or no plastic deformation before they fail. When subjected to stress, brittle materials typically fracture without significant deformation. 2. Ductile materials: Ductile materials, in contrast to brittle materials, undergo significant plastic deformation before failure. Ductile materials can withstand higher levels of stress without fracturing.
• 3. The Importance Of Failure Theories • Safety and Reliability: Failure theories ensure the safety and reliability of structures and components by predicting material failure, crucial in aerospace, civil engineering, and manufacturing. • Design and Material Selection: Engineers employ these theories to design structures and select suitable materials, ensuring they can withstand expected loads and environmental conditions. • Cost Efficiency: Failure predictions via theoretical models help save costs by preventing over-engineering and excessive material usage, optimizing designs to meet safety standards economically.
• 4. Five Important Theories Of Failure 1. Maximum principal stress theory 2. Maximum shear stress theory 3. Maximum shear strain theory 4. Maximum strain energy theory 5. Maximum shear strain energy theory
• 5. Maximum Shear Stress Theory Definition 1. It is a fundamental concept in material science and structural engineering. 2. It provides insights into material behavior and failure under external forces. 3. It predicts that failure occurs when the maximum shear stress exceeds the material's shear strength. 4. The theory primarily centers on shear stresses, which are forces that cause materials to deform along a specific plane parallel to the applied force.
• 6. Shear Stress and Plastic Deformation • This theory is specifically tailored to ductile materials, which have the ability to undergo plastic deformation. • Plastic deformation refers to the permanent change in shape or structure of materials, and it occurs when a significant level of stress is applied. • The theory highlights that the maximum shear stress plays a crucial role in causing materials to deform or break.
• 7. Formula for Maximum Shear Stress Theory • The theory involves multiple formula designed for different stress scenarios. • These formula include parameters such as shear stress (τ), yield stress (σyield or σy), and the principal stresses (σ1, σ2, σ3) • The maximum shear stress equation: τmax = (σ1 - σ3) / 2 • According to this theory, yielding begins when τ_max surpasses a critical shear strength value (τc), which is specific to the material under consideration. In practical terms, this means that plastic deformation initiates when the maximum shear stress exceeds the material's critical shear strength.
• 8. Steps for using the Maximum Shear Stress Theory •Step 1: Determine the three principal stresses (σ1,σ2, and σ3). •Step 2: Find out the maximum (σ1) and the minimum (σ3) principal stresses. •Step 3: Determine the value of the maximum shear stress τmax=(σ1 -σ3 )/2. •Step 4: Find out the allowable stress value of the material; allowable stress = σsy /N or σy /2N (N=Factor of safety) •Step 5: Compare the value calculated in step 3 with the allowable value found in step 4. If the Value at step 3 is less than the allowable value at step 4, then the design is safe as per the maximum shear stress theory.
• 9. Maximum Distortional Strain Energy Theory • The Maximum Distortional Strain Energy Theory is a fundamental concept within materials engineering. • This theory serves as a pivotal tool for predicting the failure of materials, with a specific focus on ductile materials.
• 10. Concept of Distortional Strain Energy • The theory is based on the idea of distortional strain energy, which is a measure of the energy associated with deformation that occurs when a material is subjected to stress. • It specifically deals with the inelastic deformation of materials. • The theory introduces the critical concept that materials will undergo plastic deformation when the distortional strain energy reaches a certain critical value.
• 11. Formula for the Maximum Distortional Strain Energy Theory • The von Mises Theory introduces the concept of critical distortional strain energy (Uc). This is a key parameter in the theory and is crucial for predicting when material failure will occur. • The formula for calculating the critical distortional strain energy (Uc) is provided. It involves the difference between the principal stresses (σ1 and σ2) and the modulus of elasticity (E) of the material. • Uc = 0.5 * (σ1 - σ2) ^2 / E this formula quantifies the amount of energy associated with the material's deformation. • According to this theory, yielding and subsequent material failure occur when Uc exceeds the material's yield strength (σy).
• 12. Comparison with the Maximum Shear Stress Theory • In comparison to the Maximum Shear Stress Theory (Tresca Theory), the von Mises Theory is less conservative, offering a wider elastic range before failure prediction. This feature grants flexibility in material design and engineering, promoting efficient, cost-effective structures. Engineers must choose between these theories based on material behavior, safety margins, and design goals.

### Editor's Notes

1. We have 2 types of materials when it comes to failure: brittle and ductile. Brittle materials break without bending much, while ductile materials stretch and deform before giving in.
2. The term "yielding" is used to describe the point at which materials start to deform or break under the influence of maximum shear stress.
3. Begin with the introductory sentence: "The Maximum Shear Stress Failure Theory is a fundamental concept." Where: τ_max is the maximum shear stress. (TAU) σ1 is the maximum principal stress. (SİGMA) σ3 is the minimum principal stress.
4. Begin with the introductory sentence: "The Maximum Shear Stress Failure Theory is a fundamental concept." Where: τ_max is the maximum shear stress. (TAU) σ1 is the maximum principal stress. (SİGMA) σ3 is the minimum principal stress.
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