The Illusion of Strength—A Unified Theory of Mechanical Properties in Design

• Fundamental Mechanics: Stress and Strain
  • Defining Stress (\(\sigma\))
  • Defining Strain (\(\epsilon\))
  • The Constitutive Relationship: Hooke’s Law
• The Stress-Strain Curve: A Material’s Fingerprint
  • The Elastic Region: Stiffness vs. Strength
  • The Yield Point and Yield Strength
  • Ultimate Tensile Strength (UTS)
  • Fracture Strength and Ductility
  • Toughness: The Area Under the Curve
• Let’s Talk Strength in Engineering Applications
  • Tensile Strength
  • Compressive Strength
  • Flexural Strength
  • Shear Strength
  • Impact Strength
  • Hardness vs Strength
  • Advanced Mechanics: Time, Temperature, and Geometry
  • Testing Standards and Comparisons
• Conclusion

In the colloquial lexicon, the word “strength” is a catch-all term, a linguistic Swiss Army knife used to describe a material’s resistance to failure. We describe a bridge as strong if it supports heavy traffic; we call a diamond strong because it cannot be scratched; we consider a rubber band strong if it stretches significantly without snapping. However, to an engineer, a physicist, or a material scientist, the unqualified term “strength” is scientifically meaningless. It is a simplification that masks a complex multidimensional landscape of mechanical properties. When a client asks, “Is titanium stronger than steel?” or “Is carbon fiber stronger than aluminum?”, the silent internal rebuttal is, “Stronger in what mode? Under what conditions? Over what time scale?”

This ambiguity is not merely a semantic annoyance; it is a design liability. Relying on a vague notion of strength is the root cause of many catastrophic failures in product design, found across all engineered items. Herein, we solely seek to examine examples like electronics enclosures, printed circuit boards (PCBs), and additive manufacturing. A plastic enclosure that is “strong” against impact might fail miserably due to creep under the constant load of a screw. A PCB substrate that is “strong” in tension might delaminate under the thermal stress of reflow soldering. To design effectively, we must deconstruct the monolithic concept of strength into its constituent engineering properties: rigidity (stiffness), hardness, toughness, durability, and the various modes of yielding and fracture.

This post seeks to provide an examination of how these properties are defined, measured, and calculated, with a specific focus on the materials pertinent to modern electronics manufacturing—specifically PCB substrates like FR-4—and the diverse range of thermoplastics used in additive manufacturing (3D printing). By understanding the nuances of stress-strain relationships, viscoelasticity, and anisotropic behavior, we can try to navigate the trade-offs inherent in material selection, sailing carefully in the narrow strait between Charybdis and Scylla.

Fundamental Mechanics: Stress and Strain

To discuss material properties with precision, one must first establish the fundamental language of mechanics: stress and strain. These concepts allow engineers to normalize forces and deformations, making them independent of a component’s geometry. Without this normalization, we cannot compare a thick steel beam to a thin copper wire; with it, we can compare the intrinsic material properties of steel and copper directly.

Defining Stress (\(\sigma\))

Stress is the internal resistance of a material to an applied external load, normalized by the area over which that load acts. It is a measure of the intensity of internal forces. In the International System of Units (SI), stress is measured in Pascals (Pa), where \(1 \text{ Pa} = 1 \text{ N/m}^2\). Because one Pascal is a minuscule unit—roughly the weight of a single bank note spread over a table—engineering stress is typically expressed in MegaPascals (MPa) or GigaPascals (GPa).

The mathematical definition of average normal stress (\(\sigma\)) is:

\[\sigma = \frac{F}{A_0}\]

Where:

• \(F\) is the applied force (Newtons); and
• \(A_0\) is the original cross-sectional area (square meters).

This formula represents engineering stress, which assumes the cross-sectional area remains constant during loading. However, materials often deform and thin out (necking) under load. True stress (\(\sigma_t\)) accounts for the instantaneous cross-sectional area (\(A_i\)):

\[\sigma_t = \frac{F}{A_i}\]

For most engineering applications within the elastic limit, such as calculating the deflection of a PCB under the weight of a heatsink, the difference is negligible. However, in the plastic region where significant deformation occurs—such as in the thermoforming of a plastic enclosure or the crash structure of a drone—the distinction becomes critical.1

Defining Strain (\(\epsilon\))

Strain is the measure of deformation representing the displacement between particles in a material body relative to a reference length. It is a dimensionless quantity, often expressed as a decimal, a percentage, or in microstrain (\(\mu\epsilon\)).

Engineering strain (\(\epsilon\)) is defined as the change in length divided by the original length:

\[\epsilon = \frac{\Delta L}{L_0} = \frac{L_f - L_0}{L_0}\]

Where:

• \(\Delta L\) is the change in length;
• \(L_0\) is the initial gauge length; and
• \(L_f\) is the final length.

Similar to stress, there exists true strain (\(\epsilon_t\)), which integrates the incremental strain over the changing length:

\[\epsilon_t = \int_{L_0}^{L_f} \frac{dL}{L} = \ln\left(\frac{L_f}{L_0}\right)\]

Understanding strain is vital because it quantifies how much a material has stretched or compressed. In PCB design, the strain capabilities of copper traces determine whether they will crack during thermal cycling or board flexing. A difference in the Coefficient of Thermal Expansion (CTE) between the FR-4 substrate and the copper traces induces strain every time the board heats up; if this strain exceeds the ductility of the copper, the circuit fails.¹

The Constitutive Relationship: Hooke’s Law

For many materials, specifically metals and rigid plastics, stress and strain are linearly proportional at low loads. This relationship is governed by Hooke’s Law:

\[\sigma = E \cdot \epsilon\]

Here, the constant of proportionality \(E\) is the Young’s Modulus (or Modulus of Elasticity). This single variable encapsulates the concept of “stiffness.” It is an intrinsic property of the material, determined by the strength of the atomic bonds. A material with a high Young’s Modulus (like steel, approx. 200 GPa) is stiff; a material with a low Young’s Modulus (like rubber, approx. 0.01-0.1 GPa) is compliant.

This linear relationship allows us to predict behavior mathematically. If we know the stiffness of a polycarbonate sheet and the load applied to it, we can calculate exactly how much it will deflect without needing to break a sample. Furthermore, this predictive power is the foundation of all structural simulation and Finite Element Analysis (FEA) used in modern engineering.²

The Stress-Strain Curve: A Material’s Fingerprint

The stress-strain curve is the fundamental tool for analyzing material strength. Generated largely via the Uniaxial Tensile Test (ASTM D638 for plastics, ASTM E8 for metals), this curve reveals how a material behaves as load increases from zero to the point of fracture. Analyzing this curve allows us to distinguish between the different “strengths” that are often conflated.

The Elastic Region: Stiffness vs. Strength

The initial linear portion of the curve represents elastic deformation. In this region, the material acts like a spring: if the load is removed, the material returns to its exact original shape. The slope of this line is the Young’s Modulus (\(E\)).

Here is where the first major confusion arises: Stiffness is often confused with strength. A diamond is extremely stiff (high \(E\)), while a high-strength steel might be less stiff. Conversely, a titanium alloy might have extremely high yield strength but a lower modulus than steel. In PCB design, the stiffness of the FR-4 substrate (\(E \approx 24 \text{ GPa}\)) ensures the board remains flat during component placement. If the board were “strong” (hard to break) but “compliant” (low stiffness), it would sag under the placement head, causing misalignment of fine-pitch components.

The Yield Point and Yield Strength

As stress increases, the material eventually reaches a point where it can no longer recover its original shape. This is the Yield Point. Beyond this point, the material undergoes “plastic deformation”—a permanent change in shape. Yield Strength (\(\sigma_y\)) is the stress at which this transition occurs.

• For Metals: The yield point is often sharp. If not, an “offset yield strength” is calculated, typically at 0.2% strain.
• For Plastics: The yield point is often defined as the first point on the stress-strain curve where the slope (tangent) becomes zero (zero stiffness) or where distinct deviation from linearity occurs.

In almost all structural engineering and product design, Yield Strength is the true design limit. If a part yields, it has permanently bent or stretched, meaning the geometry has failed. A 3D printed bracket that has warped permanently under load has failed, even if it hasn’t snapped into two pieces. Therefore, when an engineer asks “is it strong enough?”, they usually mean “is the Yield Strength high enough to prevent permanent deformation?”.

Ultimate Tensile Strength (UTS)

Continuing past the yield point, the material undergoes strain hardening (in metals) or chain alignment (in polymers), requiring increasing stress to continue deforming. The curve eventually reaches a peak maximum stress. This peak is the Ultimate Tensile Strength (UTS) or simply Tensile Strength.

UTS represents the maximum load the material can carry. Beyond this point, necking begins—the cross-sectional area reduces locally, and the engineering stress drops until fracture. While UTS is the number most often marketed on 3D printing filament datasheets (e.g., “Carbon Fiber PLA has a Tensile Strength of 60 MPa”), it is relevant primarily for applications where one-time failure is the operating mode. Designing a reusable part based on UTS rather than Yield Strength will result in permanent deformation during use.

Fracture Strength and Ductility

The curve ends at the Fracture Point, where the material physically separates.

• Ductility: A measure of how much a material can deform plastically before fracturing. It is quantified by Percent Elongation at Break or Percent Reduction in Area. High ductility (e.g., Polypropylene, mild steel) implies the material gives ample warning before failure.
• Brittleness: The absence of ductility. Brittle materials (e.g., Acrylic, Ceramic, standard PLA) fracture with little to no plastic deformation. For these materials, the Yield Strength and UTS are often the same point.

Toughness: The Area Under the Curve

Here lies a critical distinction often missed in colloquialisms. Toughness is the ability of a material to absorb energy and plastically deform without fracturing. Mathematically, it is the integral of the stress-strain curve (the area underneath it).

\[\text{Modulus of Toughness} = \int_{0}^{\epsilon_f} \sigma \, d\epsilon\]

A material can be:

• Strong but Brittle: High UTS, low strain (e.g., Glass). The area under the curve is a tall, thin triangle. Low toughness.
• Weak but Ductile: Low UTS, very high strain (e.g., Chewing gum). The area is a long, flat rectangle. Low toughness.
• Tough: High yield, high ductility (e.g., Polycarbonate, Kevlar). The area is substantial. High toughness.

In 3D printing, standard PLA is “stronger” (higher UTS, ~60 MPa) than ABS (~35 MPa), but ABS is “tougher” (absorbs more impact energy). If you drop a PLA phone case, it might crack (brittle failure). If you drop an ABS case, it might dent or scratch but stay intact (ductile energy absorption). Thus, “strength” is synonymous with static load bearing, while “toughness” is synonymous with impact resistance.12

Let’s Talk Strength in Engineering Applications

While the generic term “strength” typically refers to tensile strength in casual conversation, engineering applications require a more nuanced understanding because materials behave differently depending on the loading configuration. A comprehensive analysis must categorize these distinct modes to ensure structural integrity.

Tensile Strength

Tensile strength measures a material’s resistance to being pulled apart and is generally split into yield and ultimate strength. This metric is particularly relevant for components like cables, hanging brackets, and pressurized vessels. To illustrate the variance in materials, standard FR-4 demonstrates a tensile strength of roughly 275-300 MPa along its fibers, whereas unreinforced polycarbonate sits much lower at around 60-70 MPa. This stark difference highlights the critical role of glass fiber reinforcement in PCB substrates, where the fibers carry the tensile load while the resin matrix transfers it.

Compressive Strength

Compressive strength is the resistance to being squeezed or crushed. The behavior under compression varies significantly by material type; ductile materials like metals and some plastics essentially squeeze out and expand laterally due to the Poisson effect without a clear fracture point, simply flattening under the load. In contrast, brittle materials such as concrete or cast iron fail catastrophically in compression, often via shear bands. Engineers must also account for asymmetry in material properties. For instance, concrete possesses immense compressive strength but negligible tensile strength, while unidirectional Carbon Fiber composites usually exhibit higher tensile capabilities than compressive ones due to fiber buckling mechanics. In the context of plastic enclosures, such as snap-fits or screw bosses, compressive yield is critical for maintaining torque retention. If the screw head exerts force exceeding the plastic’s compressive strength, the material crushes, resulting in torque loss and a loosened assembly.

Flexural Strength

Flexural strength measures the resistance to deformation under a bending load, typically tested via 3-point or 4-point bend tests (ASTM D790). Bending is mechanically unique because it creates a combination stress: the convex side of the beam is in tension while the concave side is in compression. Failure occurs on the surface where the stress, known as skin stress, is highest. The value is calculated using the formula:

\[\sigma_{flex} = \frac{3FL}{2bd^2}\]

Where \(L\) is the span, \(b\) is the width, and \(d\)is the depth.

While flexural strength is close to tensile strength for isotropic materials, plastics often exhibit higher flexural strength than tensile strength. This is because plastics are sensitive to surface defects; tensile tests expose the entire volume to stress, finding the weakest internal flaw, whereas flexural tests only highly stress the outer surface. In PCB design, flexural strength is the primary metric for stiffness to prevent component desoldering, as a board that flexes too much can break the solder joints of BGA components.

Shear Strength

Shear strength refers to the resistance to sliding forces or layers sliding past one another. For metals, the shear yield strength is often estimated as

\[0.577 \times \sigma_y\]

based on the Von Mises criterion. However, for plastics, this relationship is less linear and relies heavily on molecular structure. Shear strength is the specific “Achilles heel” of Fused Deposition Modeling (FDM) 3D printing. The tensile strength in the Z-axis of a printed part is actually a test of the shear or bond strength between layers. Consequently, a part printed flat might withstand a 50 MPa tensile load, but if printed upright, it might fail at only 15 MPa because the layer-to-layer fusion is weaker than the polymer chain strength. This anisotropy is a defining characteristic of printed strength.

Impact Strength

Impact strength measures resistance to sudden, high-velocity loading and is usually measured in Joules per meter (\(\frac{J}{m}\)) or \(\frac{kJ}/{m^2}\). Testing methods like Izod or Charpy often use notched samples to force crack initiation, revealing that some materials, like Polycarbonate, are incredibly tough when un-notched but become significantly weaker when notched due to stress concentrations. Materials vary widely in this regard: PLA has very low impact strength (~15-30 J/m) and shatters like glass; ABS has high impact strength (~200-400 J/m) and deforms while holding; and Polycarbonate boasts extreme impact strength (~600-800 J/m), making it suitable for bulletproof glass. In plastic enclosure design, “strength” against dropping is purely a function of impact strength and geometry, such as the use of fillets to reduce stress concentrations.

Hardness vs Strength

It is important to distinguish hardness from strength. Hardness is the resistance to localized plastic deformation, such as indentation or scratching, and is not a direct measure of overall structural strength. Different scales are used for different materials: Mohs for minerals, Shore for polymers/elastomers, and Rockwell or Vickers for metals. This distinction often causes confusion in consumer products. For example, a glass screen protector is harder (Mohs 6-7) than a plastic one (Mohs 2-3), meaning it resists scratches better. However, the plastic protector possesses higher toughness and tensile strength; the glass will shatter on impact where the plastic will absorb it. High hardness often correlates with brittleness.

Advanced Mechanics: Time, Temperature, and Geometry

Standard datasheets provide “static” values that act as snapshots in time at room temperature, but real-world engineering requires understanding how materials change with time and environment. This is particularly crucial for polymers due to viscoelasticity. While metals at room temperature are essentially elastic and act like springs, polymers act like a spring combined with a dashpot (shock absorber), meaning their response depends on the time and rate of loading.

This viscoelastic nature leads to creep, or cold flow. If a weight is hung on a rubber band, it stretches initially due to elasticity, but if left for a week, it stretches further due to viscous flow. Creep is defined as strain increasing over time under constant stress, caused by polymer chains slowly untangling and sliding past each other. In design, this impacts plastic enclosures; using a self-tapping screw applies constant stress to a plastic boss, and over months, the plastic creeps away from the stress, loosening the screw. This phenomenon explains why metal inserts are preferred for long-term holding strength.

The inverse of creep is stress relaxation. If a plastic snap-fit is deformed to a fixed position and held there, the stress required to maintain that deformation decreases over time. Consequently, a press-fit connector or plastic spring clip will lose its clamping force over time. While ABS and PC are susceptible to this, spring steel is not. This phenomenon invalidates the use of simple Hooke’s Law calculations for long-term plastic assemblies.

Furthermore, polymers are strain-rate dependent, meaning “how fast you hit it” matters. At low speeds, polymer chains have time to reorient and uncoil, making the material appear ductile and tough. At high impact speeds, chains do not have time to move, causing the material to act stiffer and significantly more brittle. This duality explains why “Impact Strength” is tested separately from “Tensile Strength,” as a material that looks ductile in a slow tensile test might shatter like glass in a drop test.

Testing Standards and Comparisons

Data is only useful if compared effectively, and standardized tests ensure consistency. ASTM D638 defines the “dogbone” specimen shape and pull speeds for plastics tensile testing, while ASTM D790 defines 3-point bend setups critical for determining flexural modulus. ASTM D256 covers the Izod impact test, where a pendulum swings to break a notched sample, and ASTM D2240 measures Shore hardness by gauging the depth of indentation of a spring-loaded pin.

When comparing impact testing specifically, the differences between Charpy and Izod are critical. In the Izod test (ASTM D256), the sample is held vertically as a cantilever with the notch facing the pendulum, which strikes the top tip. In the Charpy test (ASTM E23), the sample is held horizontally and supported at both ends, with the pendulum striking the center directly behind the notch. Mechanically, Izod concentrates stress at the root of the cantilever, making it very sensitive to notch sensitivity and surface imperfections. Charpy distributes stress slightly more due to the 3-point configuration. A material might have a high Charpy score but a mediocre Izod score because if the material is notch-sensitive (like Polycarbonate), the severe stress concentration in the Izod test will cause brittle failure earlier than in the Charpy test.

Analyzing this data reveals deeper second-order insights. There is generally a stiffness-toughness trade-off: as you increase stiffness (e.g., by adding glass fibers), you decrease toughness and the material becomes brittle. A carbon-fiber-filled nylon print is stiff but snaps under shock loading, whereas pure nylon bends and recovers. There is also a temperature-stiffness correlation; modulus is a function of temperature, and as temperature approaches \(T_g\) (the “Glass Transition Temperature”, stiffness plummets. A “strong” PLA bracket might fail in a hot car not because the load increased, but because the material strength evaporated. Finally, the processing-property link dictates that in 3D printing and PCB manufacturing, the process defines the properties. A PCB’s strength depends on the glass fiber weave and resin ratio, while a 3D print’s strength depends on thermal history variables like nozzle temperature and cooling fan speed. Ultimately, strength is a result of thermal history, not just chemistry.

Conclusion

The term “strength” is a linguistic placeholder for a spectrum of mechanical phenomena. For the electronics designer, the PCB manufacturer, or the 3D printing enthusiast, reliance on this vague term is a liability.

• Rigidity (Stiffness/Young’s Modulus) determines if a part bends under load.
• Yield Strength determines if that bend is permanent.
• Ultimate Strength determines when it breaks.
• Hardness determines wear resistance.
• Toughness determines impact survival.
• Creep/Relaxation determines long-term dimensional stability.

True engineering prowess lies in identifying the specific failure mode relevant to the application—whether it is the creep of a plastic enclosure screw boss, the thermal warping of a PCB during reflow, or the Z-axis delamination of a 3D printed bracket—and selecting the material property that specifically counteracts it. We must move beyond asking “Is it strong?” and instead ask: “Is it stiff enough to maintain tolerances? Is it tough enough to survive a drop? Is its yield strength high enough to handle the assembly torque?” Only by embracing this nuanced vocabulary can we design systems that endure the complex reality of the physical world.