Mastering Titanium CNC Machining: Unleashing the Power of CNC with Titanium

It is accurate to state that titanium represents the apex in modern machining and its physically demanding form and unparalleled properties competitively synthesize well with modern technologies such as CNC or, computer numerical control. This technique is single-handedly transformative for many sectors like aerospace, healthcare, and automotive, to name a few. However, having only the best equipment does not guarantee proficiency in titanium CNC, it requires a nuanced mastery of CNC itself and an in-depth knowledge of the CNC universe. Titanium CNC machining is a part or straddles the boundary of science and art, which allows a plethora of unrestrained innovation, ranging from enhancing its strength and lowering its weight to fracturing the limitations the industry imposes. You will be equipped with practical suggestions on improving machining processes and managing the problems stemming from such a sophisticated raw element.

What Makes Titanium Unique in CNC Machining?

Due to its great strength payload ratio, extreme temperature resistance, and excellent corrosion resistance, titanium is unique in CNC machining. These qualities have made its use very favorable in the aerospace, medicine, and automobile industries. Issues, however, arise with its low cutting thermal conductivity and the tendency to work harden during cutting. To resolve this problem, precise tool selection, lower cutting value, and optimized cooling strategies need to be employed to achieve the most accurate and efficient results.

Understanding the Titanium Alloy Grades

Titanium alloys are divided into three main categories based on their microstructure, which affects their machinability in CNC Titanium Machining processes.

Alpha Alloys

These alloys are highly corrosion resistant and perform well under heat which is suitable for the marine and aerospace industry. They are also non-heat treatable and have better weldability.

Beta alloys possess unique properties which make them extremely difficult to machine when cutting processes are involved.

These are known for their excellent mechanical characteristics due to increased strength and formability as a result of being heat treated. They are widely used for medical implants and components in high-performance vehicles.

Alpha-Beta Alloys

These alloys are considered a combination of alpha and beta phases alloys, therefore versatile and widely used in aerospace and power generation. They provide better corrosion resistance while improving strength and toughness.

Titanium alloys are meant for specific performance characteristics and environmental condition requirements of the application at hand.

Exploring the Strength-to-Weight Ratio of Titanium

Used in the aerospace, automotive, and marine engineering industries, titanium is distinguished from other metals by its impressive strength-to-weight ratio. Density-wise, titanium is approximated to have a density of 4.5 g/cm³, specifically 60% of the density of steel. It provides similar strength to steel, even greater in some cases. Grade 5 titanium (Ti- 6Al- 4V) is one of the most common titanium alloys which has a density less than steel. It can maintain tensile strength over 900 MPa and is exceptionally lightweight. Thus, it can be used in applications requiring strong, but lightweight materials.

For example, ease of modification in titanium is presented with an elastic modulus of around 120 GPa which provides flexibility under stress. Compared to aluminum alloys, titanium is about 50% heavier, however, its strength is nearly doubled. This propels the use of titanium in regions in which weight savings should not sacrifice durability. Through the remarkable strength-to-weight ratio of titanium, engineers can reduce the volume of material to be used for a given structure while maintaining the expected performance. This is incredibly helpful when dealing with turbines and spacecrafts since they operate in high-stress environments.

Besides, the corrosion resistance and high-temperature structural stability of titanium strengthen the material’s resilience, enabling it to endure severe environmental conditions. This trait makes titanium even more useful in those design cases where weight savings and long-term dependability are primary concerns. Such qualities explain why titanium continues to be a material of choice for sophisticated engineering problems.

Delving into the Corrosion Resistance of Machined Titanium Parts

The exceptional corrosion resistance of titanium alloys can be ascribed primarily to the passivation induced by the dense, stable titanium dioxide (TiO₂) layer on the surface of the metals. This passive film protects the metal underneath from aggressive environments such as seawater, chlorides, and most acidic mediums. Even if the oxide layer is ruptured, it is guaranteed that the oxide layer restores itself immediately to ensure protection for its lifecycle.

Specific Regions of Interest

Marine Regions

The titanium components in marine machinery are invaluable due to their high resistance against seawater corrosion. Research indicates that Grade 2 and Ti-6Al-4V titanium alloys show nearly total immunity to pitting and crevice corrosion in chloride-laden seawater, which is considerably better than conventional alloys such as stainless steel or aluminum and even other metals. This is highly useful when it comes to the effectiveness and durability of marine machinery exposed to hostile marine conditions for extended periods.

Acidic & Chemical Regions

Titanium alloys are also able to efficiently resist most industrial chemicals as well such as nitric acid, sulfuric acid solutions, and organic acids. For example, titanium can withstand environments with a pH as low as one with practically no loss of material. This property enables titanium to be used in chemical processing plants, reactors, and other equipment that operate with highly aggressive materials.

High-Temperature Processes

Titanium has excellent thermal and chemical resistance, making it ideal for environments where thermal stability is critical. In oxidizing environments, titanium parts retain their protective oxide layer up to 500°C. This capability is especially advantageous for components used in the aerospace and power generation industries, which suffer frequent combined thermal and corrosive stresses.

Measuring Corrosion Resistance

Immersion tests involving 3.5% sodium chloride solutions (simulating seawater) demonstrate titanium’s superior performance against corrosion. Under the same conditions, titanium typically exhibits submicron annual corrosion rates, while stainless steel suffers several microns worth of corrosion. Crevice corrosion testing has further shown titanium’s ability to withstand concentrations of chlorides over one hundred thousand ppm.

Utilizing Titanium to Extend Service Life

An industry can reduce maintenance requirements by incorporating machined titanium parts into high-stress, corrosive environments, achieving extended service life for critical systems. Furthermore, advancements in CNC machining with titanium processes allow engineers to design components that are optimized for enhanced precision, performance, and environmental durability.

Titanium’s unparalleled resistance to corrosion reinforces its significance in advanced engineering applications, while its widespread use in industries where stability and longevity are desired makes it a versatile material.

How to Choose the Right CNC Machine for Titanium?

Evaluating Tool Wear and Cutting Tools for Titanium

When machining titanium parts, one of the problems is tool wear or tool failure. This problem arises because, titanium as a material is strong, does not conduct heat well, and reacts with cutting tools at higher temperatures. These properties of titanium create a lot of cutting forces which leads to excessive and premature wear of the cutting tools if adequate practices and tools are not employed.

Key Factors Contributing to Tool Wear

Another complication that arises from machining titanium with a CNC is elevated cutting temperatures that require effective cooling methods: The low thermal conductivity of titanium heats up the edge of the cutting tool which, in turn, increases the rate of tool wear. Tools are known to fail within a very short time under poor temperature management.

Adhesive and Diffusion Wear: Reactivity with heat makes titanium bond with the cutting tool which leads to material movement and tool breakage as well as erosion.

Resistance to Abrasion: Buried in titanium’s structure are its hardness and inclusions that aid in the abrasive wearing down of general-purpose cutting tools, thus, reducing the life of the tools.

Best Cutting Tools for Titanium

To increase productivity and maximize tool life, it is important to utilize cutting tools specific to titanium machining. Some recommendations are given below:

Cemented carbide tools are effective due to their high hardness and resistance to heat, making them better suited for dealing with the extreme conditions of titanium machining. Carbide tools are also required when performing CNC titanium machining as they are easily available and have reasonable tool life.

Coated Tools Usage of more advanced coated tools such as TiAlN or AlTiN grade tools for milling titanium results in less friction and heat generation at the cutting surfaces. Studies show that coated tools improve the life of the tool by 50% while operating in titanium environments.

Polycrystalline Diamond (PCD) Tools: PCD tools are ideal for operations where tool wear is a critical issue however their usage is mostly limited to non-ferrous titanium alloys.

Feeding Speed and Cutting Speed Guidelines

Studies show the necessity of proper cutting parameters being adhered to:

For titanium, it is suggested that cutting rates vary from 30 to 60 meters per minute (m/min).

It is important to manage feed rates in accordance with the material grade and tooling, as excessive feed rates can cause unneeded stress on the cutting edge. Moderate feed rates are generally recommended.

The selection of appropriate tools with sophisticated coatings must be done while controlling machining parameters in order to increase productivity and minimize tool wear. These strategies guarantee accuracy, efficiency, and cost-effective machining of components made of titanium. Managing this difficult material effectively is supported by regularly changing the cutting tools based on performance data.

Assessing Heat Buildup and Coolant Requirements

Heating issues emerge as a primary challenge in machining processes, especially when using materials such as titanium which have low thermal conductivity. Too much heat produced from the cutting processes increases the rate of tool deterioration while compromising surface quality which in turn leads to thermal distortion and loss of dimensional accuracy. Recent research indicates that titanium, retains 90% of heat while steel retains 45%. This proves the need for effective cooling to save the material and extend tool life.

To mitigate these issues, the adoption of high-pressure coolant systems has now become an industry standard. 70-100 bar systems are especially effective at removing heat from the cutting region and flushing out chips, thus decreasing friction and wear. Also, there have been changes in coolant designs where some add non-water-based temperature regulation materials to improve performance. With appropriate management of these advanced coolant systems, temperatures have been decreased by 40%, allowing faster cutting speeds while maintaining accuracy.

Also, coolant selection and distribution are important factors. A good example is direct jet cooling which provides liquid to the tool-chip interface, therefore delivering thermal management. If this method is used with sensors that monitor the temperature in real-time, then overheating can be prevented, and consistent heat reduction can be achieved. These methods will increase the efficiencies and reliabilities of machining high-performance titanium parts for the manufacturers.

Optimizing Feed Rate and Cutting Speed

Achieving an ideal feed rate and cutting speed is crucial for optimizing efficiency, product quality, and the longevity of the tool. These two parameters, feed rate, and cutting speed, are respectively defined as the distance advanced by the tool along the workpiece in unit time and the speed of the motion of the workpiece relative to the cutting edge of the tool.

New data in the field of metal cutting suggest that balancing set points should also include material, tool shape, and cooling method. For example, when aluminum alloys are being machined, cutting speeds of 200-400 meters per minute and feed rates of about 0.2-0.5 mm/revolution are typical. In contrast, stronger materials such as titanium or nickel-based superalloys tend to require controlling the overheating with lower cutting speeds (20-60 meters per minute) and reduced feed rates (0.1-0.2 mm/rev).

These parameters can also be adjusted accurately with the use of computer-aided manufacturing software which models the machining environment and predicts results. Moreover, using a changeable feed rate under dynamic loading conditions reduces tool wear and prevents error in the machining process. These refined methods enhance precision, shorten production time, and lower costs in manufacturing processes.

What are the Benefits of CNC Machining Services for Titanium Parts?

Enhancing Surface Finish of Custom Titanium Components

To accomplish an optimal finish on customized titanium workpieces, a combination of rigorous machining techniques alongside specialized tools is required. The use of high-quality cutting tools, tailored cutting fluids, and well-defined machining speeds greatly increases the quality of the surface. Furthermore, additional steps after machining such as surface treatments and polishing add to the quality of the finish, which guarantees that industry standards for functional and aesthetic quality are met.

Ensuring Tight Tolerances in Titanium CNC Machining

Maintaining close tolerances in CNC machining of titanium is achieved only with proper control of machining parameters and the use of sophisticated equipment. Important practices are tool calibration, maintaining constant cutting velocity, and minimizing thermal expansion through optimized cooling processes. Final inspection using appropriate quality verification tools, particularly high-accuracy measuring devices like CMM, confirms that the resulting parts are within the defined tolerances. Following these practices helps guarantee that titanium machining results are accurate and repeatable.

Leveraging Biocompatibility for Medical Implants

Due to unmatched biocompatibility, titanium is the preferred candidate for medically engineered implants, from dental to orthopedic ones. Its integration with bone tissue, known as osseointegration, facilitates enhanced stability and durability of implants. Recently developed titanium alloys such as Ti-6Al-4V possess better corrosion resistance, and fatigue strength, and integrate increased mechanical performance within the human body.

Research suggests a high success rate for titanium implants, especially dental ones, as their average survival rate tends to be higher than 95% over a decade. This is supported by other studies indicating low toxicity, Ti’s ability to chemically bond with oxygen as well as titanium’s low mass, which provides less fatigue to the surrounding tissues. Implants also benefit from surfaces being sandblasted or chemically etched since roughening the surface provides the titanium greater stability through bone bonding.

The expansion of options for titanium implementation in bespoke medical implants has greatly benefitted from 3D printing technology. Such technology allows for the creation of sophisticated designs to be put in place of the anatomy of the individual patient, leading to an increase in the quality of fit and function of the implants. Certain types of research show that the application of hydroxyapatite coatings, which are routinely put on titanium implants, generally helps to speed up the healing process owing to better bone-cell adhesion. All of these advances highlight the importance of titanium in improving patients’ outcomes in the medical field.

Why is Machining Titanium Considered Difficult to Machine?

Analyzing Low Thermal Conductivity and Its Impact

The low thermal conductivity of titanium, with a value of around 21.9 W/m·K, presents challenges when machining the material. Titanium, for instance, conducts heat roughly 10 times slower than aluminum (approximately 235 W/m·K) and significantly more than steel (which falls in the 50-60 W/m·K range). This characteristic leads to heat being retained in a cutting zone rather than being transferred to the workpiece or moved away through the cutting tool. As a result, this concentrated heat can lead to greater tool wear and deformation, which undermines machining productivity.

Moreover, the concentration of heat near the cutting edge considerably raises the likelihood of workpiece distortion, affecting both precision and stability. During high-speed machining, titanium tends to undergo a reaction with the cutting tools due to the extreme temperature ranges which leads to built-up edge formation. In modern machining, such difficulties are often solved by the use of advanced cooling methods like high-pressure coolant systems that effectively lower the cutting temperatures. Additionally, there has been the development of coated carbide and polycrystalline diamond (PCD) tools that better tolerate and endure the titanium’s heat behavior.

Optimizing processes in titanium machining is critical for combating the negative consequences of having low thermal conductivity. Research suggests that lower cutting speeds combined with an increased feed rate results in a reduction of the cutting temperature, thus improving the quality of the tool and workpiece. The integration of these methods, along with advancements in tool design, is greatly improving the stiff-efficiency titanium machining paradigm in aerospace and healthcare.

Investigating Radial Engagement Challenges

Machining metrics like radial engagement, defined as the segment of the cutting tool diameter actively working on a material, become even more important when dealing with materials, such as titanium, that are notoriously difficult to work with. Considerable radial engagement entails a higher cutting force and temperature that can, in turn, exacerbate wear of the tool, damage the surface, or deform the workpiece from the machining of the difficult-to-cut titanium materials. On the other hand, insufficient radial engagement may mitigate efficiency, and induce chatter or vibration, all of which reduce accuracy.

Recent studies suggest that, within certain limits of radial engagement, idle time is minimized while tool life is maximized, underscoring the need to optimize radial engagement. Specifically, there is evidence showing that 20-50% radial engagement for CNC titanium machining results in lower localized heat accumulation and better chip removal. By utilizing climb milling techniques along with specialized simulation software, control over radial engagement is improved to a degree that vibration frequencies and force fluctuations are drastically reduced, enabling better control.

Moreover, industrial data shows that adaptive toolpaths which change radial engagement values can increase the material removal rate by up to 25%, while increasing tool life by 15-20%. This is possible because these adaptive strategies allow for consistent engagement throughout the operation, leading to improved wear characteristics of the tool and better integrity of the machined surface.

Grasping the difficulties in radial engagement and optimizing machining methods are critical for industries dependent on accuracy and efficiency, like aerospace or medical device manufacturing, which have strict tolerance and surface quality requirements.

Understanding Tool Life and Cutting Forces

Tool life describes the amount of time a cutting tool can work effectively before it is deemed unusable or needs to be reconditioned. It is closely dependent on cutting velocity, feed per revolution, and the constituent materials. An optimum magnitude of cutting forces is necessary to prolong the tool life, as very high values could cause rapid tool degradation and eventual failure. Machining processes can improve the amount of material that can be removed while decreasing tool wear by controlling radial engagement, cooling, or lubrication. Knowing and dealing with these factors leads to better interference and lower costs of operations.

How Does CNC Milling Improve Machined Titanium Parts?

Exploring High-Pressure Coolant Systems

The implementation of high-pressure coolant systems along with special tools has proven to enhance the machining of titanium parts, yielding greater tool life and improving the efficiency of the entire process. These systems are reported, in my experience, to lower the heat produced during cutting, which is crucial for titanium owing to its low thermal conductivity. They also assist in clearing chips effectively, thus avoiding material re-cutting and guaranteeing a better surface finish. In addition, minimizing cooling enables lower cutting forces to be achieved, which, along with other aspects, increases the accuracy, reliability, and economy of CNC milling operations for titanium components.

Maximizing Mechanical Properties of Titanium Alloys

In order to maximize the mechanical properties of titanium alloys, it is essential to understand their distinct features. The impressive strength-to-weight ratio, excellent corrosion resistance, and remarkable fatigue performance of titanium alloys make them ideal for use in aerospace, medical, and industrial sectors. Nonetheless, achieving the desired mechanical properties requires skilled management of the alloy’s composition, heat treatment, and subsequently its manufacturing processes.

One important consideration is the transformation of phases in titanium alloys, which is mainly between the alpha (α) and beta (β) phases. For example, solution treatment and aging (STA) heat treatments are aimed at refining microstructure to achieve greater overall strength and ductility. Research demonstrates that the aging of titanium alloys at 480 to 600°C for certain periods leads to falling out of fine α-phase particles suspended in the β matrix thus raising the tensile strength as well as creep resistance.

Finer details involve the intentional addition of other alloying elements for tailoring material properties. For instance, aluminum and vanadium in the titanium alloyed with Ti-6Al-4V serve as strengthening and stabilizing agents for α and β phase, respectively. Research shows that one of the most widely used titanium alloys, Ti-6Al-4V has a tensile strength of over 900 MPa accompanied by an approximate elongation of 14%,  which indeed demonstrates the impressive properties of titanium.

Moreover, electron beam melting (EBM), an advanced form of additive manufacturing, has enhanced the control over the microstructures of titanium alloys. It has been established that this process achieves the best possible mechanical homogeneity at the lowest level of internal defects.

Through the use of advanced material processing techniques in combination with specific composition design, the mechanical properties of titanium alloys can be maximally optimized for use in high-performance applications across a multitude of industries.

Refining Custom Parts with Tight Tolerances

Advanced technologies and stringent quality checks are essential for achieving custom parts with refined and tight tolerances. The application of certain modern CNC machining and additive manufacturing techniques is critical to achieving tolerances of approximately ±0.001 inches or even greater. For example, well-designed CNC machines, equipped with accurate sensors and feedback loops, guarantee that very little deviation from the expected value occurs during the production process. Likewise, control over complex geometries by additive manufacturing techniques such as Selective Laser Melting (SLM) is superb with layer thicknesses of 20-50 microns.

The use of laser scanners and coordinate measuring machines (CMM) is an important improvement in accuracy during quality inspections for other types of parts. Dimensional checking using these devices can be done concerning the available CAD models which provides dependability and accuracy. For materials that are prone to distortion because of thermal expansion, stable thermal techniques are used to hold the dimensions during the whole manufacturing process and post-processes. Research indicates that the application of modern metrology can increase production yield rates by as much as thirty percent, especially for industries like aerospace and medical device manufacturing.

Merging super accurate equipment, extensive scrutiny along strong control of the material make the accomplishment of modern engineering applications possible. This principle is fundamental to the achievement of dependability and functionality in the aerospace, automotive, and medical industries.

Frequently Asked Questions (FAQs)

Q: What are the advantages of titanium CNC machining?

A: Titanium CNC machining comes with many benefits, such as increased strength-to-weight ratio and corrosion resistance along with biocompatibility. These characteristics make titanium-machined parts beneficial for the aerospace, automotive, and medical industries. CNC machining creates complex precise parts with tight tolerances. This is ideal for custom titanium parts for challenging geometries.

Q: What are the most common titanium grades used in CNC machining?

A: The most common titanium grades used in CNC machining include Grade 2 (commercially pure titanium), Grade 5 (Ti-6Al-4V), and Grade 23 (Ti-6Al-4V ELI). Grade 5 also known as Ti-6Al-4V is the titanium alloy with the biggest market share due to its high strength and light materials. Grade 2 titanium has excellent corrosion resistance along with great formability, which makes it a preferred choice in various applications. Grade 23 is a higher purity version of Grade 5 and is frequently used in medical implants.

Q: What issues come with titanium CNC machining?

A: Titanium CNC machining presents numerous challenges. First, titanium’s low thermal conductivity results in tool wear and built-up edge. Also, its high strength combined with low modulus of elasticity can cause chatter or vibration during CNC titanium machining. Furthermore, titanium’s reactivity with cutting tools at elevated temperatures can result in tool degradation. These issues complicate titanium machining more than aluminum or steel.

Q: In what ways does 5-axis CNC machining assist in the production of titanium parts?

A: In terms of titanium parts production, 5-axis CNC machining is particularly useful because complex geometries and intricate features can be machined in a single setup. This eliminates the need for multiple setups thereby increasing accuracy and saving time. This is particularly beneficial for aerospace components, medical implants, and other intricate parts that have stringent requirements in terms of precision and configuration.

Q: Which industries make use of CNC-machined titanium parts?

A: The aerospace and automotive industries, as well as the medical field, chemical processing, and marine industries, use CNC-machined titanium parts. In the aerospace industry, titanium parts such as aircraft engine components and structural sections are utilized. The automotive industry employs titanium for high-performance engine parts. The medical field uses them in implants and surgical instruments. Chemical processing industries such as use titanium parts to construct heat exchangers and reaction vessels because they are corrosion-resistant.

Q: What sets apart the process of machining titanium from other metals?

A: Machining titanium is distinctly different when compared to other metals because it demands specific considerations. With titanium’s characteristics, lower cutting speeds with greater feed rates are more typical. To minimize chatter and tool wear in challenging titanium materials, rigid setups coupled with sharp, coated cutting tools are required. Managing the build-up of heat necessitates abundant coolant, as with other metals. Trochoidal milling strategies are common as they assist in maintaining chip load consistency to avoid work hardening.

Q: Which surface treatments are best for CNC titanium machined components?

A: Surface treatment options like anodizing, which builds a protective oxide layer while adding color, in addition to titanium aluminum nitride (TiAlN) coating for better wear resistance, as well as shot peening to improve fatigue strength, are all common for CNC machined titanium part. Other surface treatment options like passivation improve corrosion resistance by creating a thin oxide layer, which also can be applied. The performance and appearance of titanium-machined parts can be significantly enhanced with these treatments.

Q: What criteria do you follow to select the right titanium grade for a particular CNC machining project?

A: It varies. Every titanium grade has its application characteristics, thus, the appropriate choice will depend on what is needed. Take into account the strength, weight, corrosion resistance, and biocompatibility required. For example, Grade 2 titanium is good for applications that need lower strength but excellent corrosion resistance. The same components with higher strength requirements in aerospace and automotive industries can use Grade 5 (Ti-6Al-4V) instead. In the case of medical implants, Grade 23 is preferred due to its high purity and biocompatibility. You may contact a titanium CNC machining service to help you choose the appropriate grade for your project.

Reference Sources

1. Review of Tool Wear Models concerning Cutting Conditions and Functional Parameters of Titanium Alloy on CNC Lathe Machine

• By: S. Ingle, Dadarao Raut
• Published Date: March 23, 2023
• Executive Summary: This study focuses on the tool wear mechanisms for CNC turning of titanium alloys, in particular, how various machining parameters impact tool wear and performance.
• Research Technique: The investigation carried out was based on experiments in which different cutting parameter combinations were used. Measurements and analysis were conducted to evaluate tool wear so that the wear rate could be modeled as a function of machining time(Ingle & Raut, 2023).

2. Integrated Energy Use Optimisation and Cutting Parameter Prediction Model – Assisting in the Process Planning of Ti6Al4V Machining on the CNC Lathe

• By: Tayisepi et al.
• Published: November 13, 2023
• Abstract: This study introduces a model that tactics both energy expenditure and cutting parameter selection while machining the titanium alloy, Ti6Al4V, on CNC lathes. This model seeks to improve efficiency in process planning.
• Research Approach: The author employed MATLAB and Visual Basic applications to develop a genetic algorithm-based tool for predicting optimal cutting parameters. A practical factorial experiment was performed to determine if the model was valid (Tayisepi et al., 2023).

3. Comparative Study of the Cutting Efficiencies of SiAlON Ceramic, Cubic Boron Nitride, and Carbide Tools in the Machining of Titanium

• By: S. Phokobye et al.
• Date of Issue: August 28, 2023
• Abstract: This study analyses the effectiveness of various cutting tools, including SiAlON ceramic, cubic boron nitride, and carbide in machining titanium alloys while evaluating the degree of tool wear and surface finish of the materials worked on.
• Procedure: The analysis was based on experimental machining tests wherein each tool variety was operated under similar machining conditions, and the resulting wear and surface quality were evaluated (Phokobye et al., 2023, pp. 3775–3786).