Stress concentration is a phenomenon where localized stress suddenly increases at points of abrupt changes in a part's shape or material discontinuities.
In actual part structures, functional requirements often result in notches such as holes, grooves, keyways, threads, and shoulders, causing sudden changes in the part's cross-sectional dimensions or shape, thus exacerbating stress concentration at these notches. The more drastic the change in cross-sectional dimensions, the more severe the stress concentration.
Properly designing notch structures is crucial for improving the fatigue strength of parts. When the part's structure allows, minimizing changes in cross-sectional dimensions is the primary measure (Figure 4.3-41 shows the stress concentration of plates or shafts with different notch shapes under tension).
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Stress Concentration in Shaft Parts and Reduction Measures
1. Stress Concentration in Shaft Parts:
Shafts subjected to bending moment and torque will experience bending and shear stress concentration at points of localized changes in cross-sectional shape and dimensions (Figure 4.3-42). The magnitude of these concentrations depends on the shape, size, and stress type of the notch.
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2. Stress Concentration Factor:
The ratio of the maximum local stress at a stress concentration point to the nominal stress is called the theoretical stress concentration factor.
The influence of material properties and load type on stress concentration is considered to characterize the true reduction in fatigue strength. When the material, load conditions, and absolute dimensions are the same, the effective stress concentration factor is equal to the ratio of the fatigue limit of a smooth specimen to that of a specimen with stress concentration, i.e.:
[Image] If there are several different stress concentration sources on the same calculation section, the maximum value is taken in the strength calculation. The stress concentration factor values for common notch shapes are shown in the table below (Table 4.3-4 Values of bending stress concentration factor and shear stress concentration factor):
[Image] [Image] 3. Structural measures to reduce stress concentration in shaft parts:
Shoulders: Various fillet transition forms can be used (Figure 4.3-43), such as fillets of the largest possible size or composed of straight lines (Figure a), fillets made according to elliptical curves (Figure b), fillets composed of several arcs (Figures c, d), and concave fillet structures (Figures e, f); adding or removing grooves near the fillets can more effectively reduce the stress concentration factor.
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**Screen Keyway on Shaft:** The stress concentration factor of a keyway machined with a disc milling cutter is approximately 20% lower than that machined with a finger milling cutter (Figure 4.3-44, Figure a is unreasonable, Figure b is reasonable).
**Image:** Shaft-Hub Interference Fit Connection: When the shaft is longer than the hub, the portion of the shaft outside the hub hinders compression of the portion inside the hub, resulting in uneven radial pressure distribution along the contact length (Figure 4.3-45), causing stress concentration on the shaft.
**Image:** The following structural measures can be taken to reduce stress concentration (Figure 4.3-46): Make the shaft diameter of the non-fitting part smaller than the fitting shaft diameter, typically (Figure a: stepped shaft); add unloading grooves to the enclosed part (Figure b); machine unloading grooves on the enclosing part (Figure c).
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Content Source: Wen Bangchun, *Mechanical Design Handbook*, 6th Edition, Volume 1, Section 4: Structural Design of Mechanical Components, Chapter 3: Structural Design to Meet Working Capacity Requirements, 1.3.2 Reducing Stress Concentration (pp. 4-24)
Further Reading:
Stress concentration in engineering is not entirely a "negative phenomenon." By actively utilizing its principles, specific goals can be achieved in material processing, structural design, and functional devices. Its core application logic is: by designing local structures (such as notches, sharp corners, and holes), stress is concentrated in a predetermined area, thereby controllably guiding material deformation, fracture, or achieving functionality, avoiding structural failure due to stress concentration in unexpected locations. The following are its main application scenarios and principles:
I. Material Processing and Forming: Achieving "Controllable Fracture" through Stress Concentration
During material cutting, separation, or shaping, stress concentration can reduce processing difficulty, achieving precise and efficient material handling, avoiding the complex procedures of traditional machining.
1. Glass Cutting (Most Typical Application)
Principle: Glass is a brittle material, easily cracking along stress concentration areas under external force. During cutting, a tiny notch is first made on the glass surface using a diamond cutter. The stress at the notch will concentrate dramatically (extremely high stress concentration factor). Then, a slight bending force is applied along the notch. The molecular bonds in the stress concentration area break preferentially, allowing the glass to separate precisely along the notch, resulting in a clean cut without excessive fragmentation.
Application Scenarios: Cutting mobile phone screens, architectural glass, and optical lenses, replacing traditional abrasive wheel cutting (which easily produces burrs and damages the glass surface).
2. Notched Tensile Testing and Specimen Preparation for Metallic Materials
Principle: In the mechanical property testing of metallic materials (such as fracture toughness and fatigue strength), specimens with standard notches (such as V-notches or U-notches) need to be prepared. The stress concentration at the notch simulates the weak points in the actual structure, causing the specimen to fracture preferentially at the notch under tension or fatigue loading. This allows for accurate measurement of the material's fracture resistance under stress concentration, providing data support for structural design.
Application Scenarios: Mechanical property testing of aerospace titanium alloys and high-strength steel, ensuring the safety of materials in actual structures (such as bolt holes and welds).
3. Stamping and Blanking
Principle: In sheet metal stamping (e.g., making gaskets, housings) or blanking (separating part blanks), the die cutting edge is designed with sharp corners or local notches to concentrate stress in the localized area where the sheet metal contacts the cutting edge. When the stress exceeds the material's yield strength, the sheet metal will precisely separate or deform along the cutting edge contour, reducing material waste and improving processing efficiency.
Application Scenarios: Mass production of automotive body stamping parts and electronic component housings.
II. Structural Design: Optimizing "Function and Safety" Using Stress Concentration
In structural design, by actively setting stress concentration areas, "directional protection" or "functional triggering" can be achieved, preventing the overall structure from failing due to uncontrollable stress concentration.
1. Safety Structure: Fusible Plugs and Rupture Discs (Pressure Vessel Protection)
Principle: Pressure vessels (such as boilers and gas cylinders) need to prevent explosions caused by excessive internal pressure. Fusible plugs (made of low-melting-point alloys) or rupture discs (thin metal sheets) are designed into localized weak areas of containers (such as areas with reduced thickness or pre-cracked sections), where the stress concentration factor is much higher than in other areas. When the internal pressure exceeds a safe value, the stress in the weak area first reaches the material's fracture limit, causing the fusible plug to melt or the rupture disc to rupture, releasing the pressure and protecting the container from explosion.
Application Scenarios: Chemical reactors, automotive air conditioning pipes, safety devices in fire extinguishers.
2. Mechanical Connections: "Anti-Loosening Design" for Bolts and Rivets
Principle: The root and head transitions of the bolt or rivet threads are designed with rounded corners (rather than sharp corners), but in some scenarios, a slight "stress concentration feature" (such as a small radius arc at the thread root) is intentionally retained. This design allows the stress concentration area to undergo slight plastic deformation when the bolt is subjected to vibration loads, thereby increasing the friction between the threads and preventing the bolt from loosening; at the same time, the pre-set stress concentration area prevents stress from transferring to the middle of the bolt shank (which can easily lead to overall fracture).
Application Scenarios: Automotive engine bolts, connecting components in aerospace equipment. 3. Building Structure: Energy Dissipation Design of Seismic Joints
Principle: In buildings in earthquake-prone areas (such as frame structures), beam-column joints are intentionally designed as locally weak areas (e.g., reducing joint cross-sections, setting expansion joints). Stress concentration causes the joints to preferentially undergo plastic deformation under seismic loads, absorbing seismic energy ("energy dissipation"), thereby protecting the main structural components such as beams and columns from brittle fracture and improving the building's seismic resistance.
Application Scenarios: Seismic design of high-rise buildings and bridges.
III. Special Functional Devices: Performance Regulation Using Stress Concentration
In precision devices or functional materials, stress concentration can be used to regulate the material's physical properties (such as electrical and optical properties) to achieve specific functions.
1. Sensors: Sensitive Element Design of Stress Sensors
Principle: The core of a stress sensor (such as a strain gauge or pressure sensor) is the "sensitive element" (such as a metal foil or semiconductor material), whose surface is designed with a mesh-like structure or a structure with tiny notches. When subjected to external pressure or strain, stress concentration at the notch amplifies the material's deformation (or resistance change), making the sensor more sensitive to minute stresses and improving detection accuracy.
Application Scenarios: Automotive tire pressure sensors, pressure monitoring in industrial equipment, pulse sensors in the medical field.
2. Microelectronic Devices: Flexible Electronics' "Stretchable Design"
Principle: Flexible electronics (such as circuits in wearable devices) need to maintain functionality when bent and stretched. Metal wires in the circuit are designed with wavy or micro-inflection points. Stress concentration at these points disperses the overall stress during stretching, preventing the wires from breaking due to excessive stretching. Simultaneously, localized deformation in the stress concentration area allows the wires to adapt to the deformation of the flexible substrate, ensuring circuit continuity.
Application Scenarios: Circuit design for smart bracelets and flexible displays.
3. Fracture Mechanics Research: "Controllable Guidance" of Crack Propagation
Principle: In fracture mechanics experiments, by pre-fabricating cracks of specific shapes (such as penetrating cracks or surface cracks) on the material surface, the stress concentration at the crack tip (the stress at the crack tip theoretically tends to infinity) is used to study the crack propagation law. This research provides a theoretical basis for "structural life prediction" in aerospace, nuclear power, and other fields (such as predicting the propagation rate of cracks in aircraft wings to avoid sudden fractures).
IV. Core Principles of Application: "Controllability" and "Avoiding Negative Effects"
Although stress concentration has many applications, all applications are premised on **"proactive design and precise control"**, and it is necessary to avoid "unintended stress concentration" caused by improper design (such as sharp corners in the structure or unpolished welds, which may lead to premature structural failure). The core principles include:
**Defining Stress Concentration Areas:** Using tools such as Finite Element Analysis (FEA), accurately calculate the stress concentration factor to ensure that stress concentration only occurs in predetermined locations;
**Matching Material Properties:** Brittle materials (such as glass and ceramics) are suitable for using stress concentration to achieve fracture (e.g., cutting), while ductile materials (such as metals) are suitable for using stress concentration to achieve plastic deformation (e.g., seismic joints);
Avoiding Excessive Concentration: Even in predetermined stress concentration areas, the stress gradient needs to be "mitigated" using methods such as rounded corners and transitional structures to prevent premature material failure under normal operating conditions.
In summary, the essence of applying stress concentration is "turning adversity into advantage"-through precise structural design, stress is guided to a controllable area, achieving both processing, safety, and functional goals while ensuring the overall structural reliability. This is one of the indispensable core ideas in modern engineering design.
In daily life, stress concentration is a very common phenomenon, both as a "natural phenomenon" caused by structural design and in scenarios where people actively utilize its principles to solve problems. These examples essentially involve local structural elements (such as notches, sharp corners, and holes) altering the stress distribution, causing stress to concentrate in specific areas, leading to deformation, fracture, or specific functionalities. The following analysis, categorized into three types-"Use of Everyday Items," "Phenomena in Daily Life Scenarios," and "Active Utilization Scenarios"-uses specific case studies:
I. Everyday Items: Stress Concentration Due to Structural Design (Easily Overlooked)
In these examples, the local structure of the item (such as notches, holes, and sharp corners) is the "source" of stress concentration, often causing wear and breakage in specific areas. This may also be intentionally designed by the designer to achieve a specific function.
1. Plastic Bottles/Cans: "Easy-to-Open Design" at the Bottle Neck and Pull-Out Tab
Stress Concentration Points
: The "tear strip" connecting the cap and body of a plastic bottle (with a small notch); the area below the pull tab of a can (a small, pre-compressed groove).
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Principle: The notch in the tear strip concentrates stress at the notch-when we pull the tear strip, we don't need to use too much force; the plastic at the notch will break due to stress exceeding its strength limit, easily opening the bottle cap. The same principle applies to the groove under the pull tab of a can; when the tab is pressed, stress concentrates at the groove, causing the aluminum sheet to "break," making it easy to open.
Life Experience: If the tear strip has no notch (or the notch is worn), opening a plastic bottle becomes very difficult because it lacks the "assistance" of stress concentration.
2. Paper/Plastic Bags: The "Easy-Tearing Property" of Edge Notches
Stress Concentration Points: The "serrated notch" on the handle of a supermarket plastic bag, the "tear lines" (a row of small holes) on the edge of a notebook paper.
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Principle: Paper or plastic bags are flexible materials, but the notches/holes at their edges change the stress distribution-when we pull along the notch, the stress concentrates at the tip of the notch (or the weak area between the holes), causing the material to break along a predetermined path, avoiding a "crooked" tear.
Counterexample
If the plastic bag has no notches, pulling directly on the handle will distribute the stress across the entire handle area, making it more prone to tearing the handle as a whole (rather than breaking cleanly along the edge).
3. Clothing/Fabric: "Easy Wear and Tear Issues" at Buttonholes and Seams
Stress Concentration Points
Buttonholes in clothing (with perforated edges) and the junction of seams and fabric ("localized concentration points" formed by the seams).
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Principle
Buttonholes are "holes" in the fabric. When putting on or taking off buttons, the pressure of the button on the edge of the hole concentrates stress around the hole; at seams, due to friction and pulling between the thread and fabric, stress concentrates near the needle hole through which the thread passes. Over time, these areas are prone to wear and tear (e.g., enlarged buttonholes, pilling or holes in the fabric at the seams).
Remedies
Many clothes have "lining" sewn around the buttonholes, essentially increasing the local thickness, reducing the stress concentration coefficient, and minimizing wear.
4. Phone Cases/Eyeglass Frames: "Easily Cracked at Corners and Openings"
Stress Concentration Points
The four right angles (sharp corners) of phone cases, and the small screw holes connecting the temples and lenses of eyeglass frames.
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Principle
When a phone case is dropped, the corners (sharp corners) hit the ground first. The impact concentrates stress at these points-plastic or silicone phone cases are prone to cracking at sharp corners due to stress exceeding their strength. The screw holes in eyeglass frames are "hole structures," and the opening and closing of the temples concentrates stress around the holes. Over time, the metal/plastic near these holes is prone to deformation and breakage.
Designer's Solution
Many phone cases now replace right angles with rounded corners, increasing the radius of curvature to reduce the stress concentration coefficient at sharp corners and decrease the probability of cracking.
II. Everyday Scenarios: Naturally Occurring Stress Concentration Phenomena
In these cases, stress concentration is "naturally formed," usually related to the shape of the object and the manner in which external forces are applied. This is common in everyday "fracture and deformation" scenarios.
Image 1. Trees: Tree trunks are prone to breakage at forks and scars.
Stress Concentration Points:
The junctions between the trunk and branches (the smaller the fork angle, the more pronounced the stress concentration) and scars on the trunk (such as cuts or insect holes).
Principle: When a tree trunk is subjected to wind loads, the "sharp angle structure" at forks causes stress concentration-the smaller the fork angle (e.g., acute fork), the higher the stress concentration coefficient, making it easier to break at the fork in strong winds; scars are "local weak points" (equivalent to gaps) on the trunk, where stress concentrates at the edge, making the trunk more prone to cracking and breaking.
2. Glass/Tile: "Easily broken" after scratches.
Stress Concentration
Midpoint
: Tiny scratches on glass surfaces (such as scratches on a phone screen from a key), and chipped edges on tiles.
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Principle
: Glass and tiles are brittle materials. Scratches on their surfaces are equivalent to "tiny chips," where stress concentrates sharply at the tip (theoretically, stress at the tip tends to infinity). Even slight external force (such as a phone screen accidentally hitting a table) can cause stress to exceed the glass's fracture limit, leading to cracking at the scratch or even the entire glass shattering.
Life Tip
: Applying a tempered glass screen protector to your phone not only prevents scratches but also reduces stress concentration at scratches through the film's cushioning, lowering the probability of breakage.
3. Chopsticks/Spoons: "Easily Broken Joint" Between Handle and Head
Stress Concentration Points
: The "narrow section" of wooden chopsticks (the transition section between the handle and the head, where the diameter decreases), and the "sharp corner" where the handle and head of a plastic spoon connect.
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Principle: When chopsticks are used to pick up food, the external force mainly acts on the tip. The "waist" section, due to its smaller diameter (equivalent to "local cross-sectional contraction"), concentrates stress. Over time, this narrow section is prone to breakage due to fatigue stress (repeated stress). The same principle applies to the pointed corners of plastic spoons; stress concentrates at these corners during stirring, making them prone to breakage at the joint.
III. Proactive Utilization: "Turning Harm into Benefit" Applications of Stress Concentration in Daily Life
These examples demonstrate how people proactively utilize the principle of stress concentration to solve everyday problems. The essence is consistent with engineering application logic (controllable breakage, ease of operation).
1. Sticky Notes/Tape: "Easy-Tear Lines" on the Edge
Application Principle: The top of sticky notes and the sides of tape are designed with "serrated easy-tear lines" (a row of tiny notches). Utilizing the stress concentration at these notches-when we pull along the easy-tear lines, the stress concentrates at the tip of the notch, allowing the sticky note/tape to break neatly along a predetermined path, without the need for scissors.
Comparison
1. If the tape lacks an easy-tear line, pulling it directly will cause stress dispersion, resulting in uneven tears or even making it impossible to tear.
2. Food Packaging: "Tear-Off Openings" (e.g., snack bags, milk cartons)
Application Principle: The "tear-off opening" of snack bags (with a small protruding plastic strip and a notch at the bottom) and the "triangular opening" of milk cartons (pre-pressed creases + tiny notches) both create stress concentration through the notches-when pulling the plastic strip, the stress is concentrated at the notch, and the plastic film is easily torn; the crease of the milk carton acts as a "local weak point," where pressure is concentrated, causing the cardboard at the crease to break, facilitating the pouring of milk.
Image 3. Nail Clippers/Scissors: The "Sharp Angle" of the Blade
Application Principle: The blade of nail clippers is a "sharp angle structure," and the blade of scissors is also a "wedge-shaped angle"-when cutting nails or paper, the angle concentrates stress at the contact point between the blade and the object. With less force, the local stress on the nail/paper can exceed the breaking limit, achieving the "cutting" function.
Essence: The sharp blade is essentially a "tiny notch," reducing the external force required for cutting through stress concentration, making the tool more effortless.
Image Summary: The Core Characteristics of Stress Concentration in Daily Life
These examples reveal that stress concentration in daily life is essentially "uneven stress distribution caused by local structural changes," with both positive and negative effects:
The "negative" side:
It can cause wear and breakage in specific areas of items (e.g., a cracked phone case, worn buttonholes in clothing). Design optimization (e.g., rounded corners, adding lining) is needed to reduce these negative impacts.
The "positive" side:
It can be actively utilized to achieve "ease of operation and opening" (e.g., tear-out edges, easy-tear seams), making daily use more convenient.
Understanding these examples can also help us use items better-for example, avoiding direct impact of sharp corners on the ground with phone cases (reducing cracking caused by stress concentration), and tearing plastic bags along the perforations (easier and neater).
