4.1 Engineering vs. True Stress-Strain, Elastic Constants & Z-Direction Ductility

4.1 Stress, Strain, Elastic Modulus & Generalized Hooke's Law

Quick Answer: Uniaxial tension transforms external forces into internal stresses. Engineering stress ($s = P/A_0$) and strain ($e = \Delta L/L_0$) reference undeformed dimensions, whereas true stress ($\sigma = P/A = s(1+e)$) and true strain ($\varepsilon = \ln(1+e)$) account for instantaneous geometric reduction until necking occurs. In isotropic materials, elastic behavior is governed by Generalized Hooke's Law, coupling Young's modulus ($E$), shear modulus ($G$), bulk modulus ($K$), and Poisson's ratio ($\nu$) via $G = E / [2(1+\nu)]$ and $K = E / [3(1-2\nu)]$. In thick, restrained weldments, plane strain constraint ($\varepsilon_z = 0$) generates an out-of-plane stress $\sigma_z = \nu(\sigma_x + \sigma_y)$, creating a triaxial tensile stress state that suppresses plastic yielding and promotes brittle cleavage.

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1. Engineering Stress-Strain vs. True Stress-Strain

When a welded component or tensile specimen is subjected to an axial tensile force $P$, the material deforms elastically and subsequently plastically until fracture. Quantifying this mechanical response requires defining stress and strain.

   Stress (σ)
       ^
       |                          True Stress-Strain Curve: σ = s(1 + e)
  UTS -+ - - - - - - - - - - - - -...-------------------x (True Fracture: σ_f = P_f / A_f)
       |                     _--'  \
       |                 _--'       \ Engineering Stress-Strain Curve:
σ_ys -+             _--'            \ s = P / A_0
       |           /                  \ 
       |          /                    x (Engineering Fracture: s_f = P_f / A_0)
       |         / |                    
       |        /  |                    <-- Necking initiates at UTS (dP = 0)
       |       /   |                    
       +------+----+-------------------------------------> Strain (e, ε)
       0    0.002  e_uniform           e_total
             (0.2% Offset)

Definitions and Formulations

1. Engineering Stress ($s$ or $\sigma_{\text{eng}}$): The applied tensile load $P$ divided by the original, undeformed cross-sectional area $A_0$:

   $$s = \frac{P}{A_0}$$

   Units are expressed in Pascals ($\text{N/m}^2$), Megapascals ($1\text{ MPa} = 1\text{ N/mm}^2$), or pounds per square inch ($\text{psi}$, $\text{ksi} = 10^3\text{ psi}$).

2. Engineering Strain ($e$ or $\varepsilon_{\text{eng}}$): The change in gauge length $\Delta L = L - L_0$ divided by the initial gauge length $L_0$:

   $$\ne = \frac{\Delta L}{L_0} = \frac{L - L_0}{L_0} = \frac{L}{L_0} - 1$$

   Engineering strain is a dimensionless ratio, frequently reported as a percentage ($e \times 100\%$).

3. True Stress ($\sigma$ or $\sigma_{\text{true}}$): The applied load $P$ divided by the instantaneous cross-sectional area $A$ supporting the load at that exact moment:

   $$\sigma = \frac{P}{A}$$

4. True Strain ($\varepsilon$ or $\varepsilon_{\text{true}}$): The summation of incremental strain increments $dL / L$ as the gauge length stretches from $L_0$ to instantaneous length $L$:

   $$\varepsilon = \int_{L_0}^L \frac{dL}{L} = \ln\left( \frac{L}{L_0} \right)$$

Mathematical Conversion in the Uniform Plastic Regime

During plastic deformation prior to the onset of localized necking, plastic flow occurs at essentially constant volume (dilatational elastic volume changes are negligible, $\Delta V_{\text{plastic}} = 0$):

$$V = A_0 L_0 = A L \implies \frac{A_0}{A} = \frac{L}{L_0}$$

From the definition of engineering strain, $L / L_0 = 1 + e$. Therefore, the area ratio is:

$$\frac{A_0}{A} = 1 + e \implies A = \frac{A_0}{1 + e}$$

Substituting this into the definition of true stress yields the standard conversion identity:

$$\sigma = \frac{P}{A} = \frac{P}{A_0 / (1 + e)} = \left(\frac{P}{A_0}\right) (1 + e) = s (1 + e)$$

Similarly, converting engineering strain to true strain:

$$\varepsilon = \ln\left( \frac{L}{L_0} \right) = \ln(1 + e)$$

Critical Engineering Limitation: The analytical conversions $\sigma = s(1+e)$ and $\varepsilon = \ln(1+e)$ are strictly valid only up to the Ultimate Tensile Strength (UTS). At the UTS, plastic deformation concentrates into a localized neck. Once necking initiates, strain and stress cease to be uniform along the gauge length. Calculating post-necking true stress requires optical or contact measurement of the instantaneous minimum neck diameter ($d$) and application of the Bridgman correction factor to account for the triaxial tensile stress state induced by neck curvature:

$$\sigma_{\text{corrected}} = \frac{\sigma_{\text{avg}}}{\left( 1 + \frac{2R_c}{a_r} \right) \ln\left( 1 + \frac{a_r}{2R_c} \right)}$$

where $a_r$ is the radius of the minimum cross-section at the neck, and $R_c$ is the profile radius of curvature of the neck contour.

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2. Elastic Moduli & Generalized Hooke's Law

In the linear elastic regime, deformation is fully reversible; removing the applied load returns all atomic bonds to their equilibrium interatomic spacing $r_0$. Hooke's law states that stress is directly proportional to strain.

The Four Fundamental Elastic Constants

For a homogeneous, isotropic engineering material, elastic deformation is completely defined by two independent elastic constants, though four are standard in structural analysis:

1. Young's Modulus ($E$): The constant of proportionality between uniaxial normal stress $\sigma$ and normal strain $\varepsilon$:

   $$E = \frac{\sigma}{\varepsilon}$$

   Young's modulus represents the stiffness of the atomic bonds. It is a structure-insensitive property; heat treatment, cold work, and microstructural phase changes alter yield and tensile strength dramatically but leave $E$ virtually unchanged ($E_{\text{steel}} \approx 200\text{--}210\text{ GPa}$ across all plain carbon, HSLA, and quenched-and-tempered grades).

2. Shear Modulus ($G$ or Modulus of Rigidity): The constant of proportionality between shear stress $\tau$ and engineering shear strain $\gamma$:

   $$G = \frac{\tau}{\gamma}$$

3. Poisson's Ratio ($\nu$): The negative ratio of lateral (transverse) strain to axial (longitudinal) strain under uniaxial tensile loading:

   $$\nu = -\frac{\varepsilon_{\text{lateral}}}{\varepsilon_{\text{axial}}} = -\frac{\varepsilon_y}{\varepsilon_x} = -\frac{\varepsilon_z}{\varepsilon_x}$$

   Because axial elongation produces lateral contraction, the negative sign ensures $\nu > 0$ for typical engineering materials.

4. Bulk Modulus ($K$): The ratio of hydrostatic pressure (mean normal stress $\sigma_m = -p$) to volumetric strain $e_v$:

   $$K = -\frac{p}{\Delta V / V_0} = \frac{\sigma_m}{e_v}$$

Interrelationships Among Elastic Constants

Using tensor transformations and strain energy conservation in isotropic media, the four constants are coupled through exact analytical identities:

$$G = \frac{E}{2(1 + \nu)}$$ $$K = \frac{E}{3(1 - 2\nu)}$$ $$E = \frac{9 K G}{3 K + G}$$ $$\nu = \frac{3 K - 2 G}{2(3 K + G)}$$

Thermodynamic Bounds on Poisson's Ratio

For a material to remain mechanically stable, its strain energy density under arbitrary load must be positive definite ($U > 0$). This physical requirement dictates that both shear modulus $G$ and bulk modulus $K$ must be strictly positive ($G > 0, K > 0$):

• From $G = \frac{E}{2(1+\nu)} > 0$, we require $(1 + \nu) > 0 \implies \nu > -1.0$.
• From $K = \frac{E}{3(1-2\nu)} > 0$, we require $(1 - 2\nu) > 0 \implies \nu < 0.50$.

Thus, for isotropic media: $-1.0 < \nu < 0.50$. For structural metallic alloys, $\nu$ typically falls within the narrow band of $0.27\text{ to }0.33$. A material with $\nu = 0.50$ is completely incompressible ($K \to \infty$), undergoing zero volume change during elastic deformation (e.g., rubber, or metals during fully plastic flow).

Volumetric Strain ($e_v$)

Volumetric strain represents the fractional change in unit volume:

$$\ne_v = \frac{\Delta V}{V_0} = (1 + \varepsilon_x)(1 + \varepsilon_y)(1 + \varepsilon_z) - 1 \approx \varepsilon_x + \varepsilon_y + \varepsilon_z$$

Under a generalized triaxial state of normal stress $(\sigma_x, \sigma_y, \sigma_z)$:

$$\ne_v = \frac{1 - 2\nu}{E} (\sigma_x + \sigma_y + \sigma_z) = \frac{\sigma_x + \sigma_y + \sigma_z}{3 K} = \frac{\sigma_m}{K}$$

where $\sigma_m = \frac{1}{3}(\sigma_x + \sigma_y + \sigma_z)$ is the hydrostatic (mean) stress.

Three-Dimensional Generalized Hooke's Law

Under arbitrary multiaxial loading, each normal stress component induces an axial strain along its own direction and transverse Poisson contractions along the two perpendicular orthogonal axes:

$$\varepsilon_x = \frac{1}{E} \left[ \sigma_x - \nu(\sigma_y + \sigma_z) \right]$$ $$\varepsilon_y = \frac{1}{E} \left[ \sigma_y - \nu(\sigma_x + \sigma_z) \right]$$ $$\varepsilon_z = \frac{1}{E} \left[ \sigma_z - \nu(\sigma_x + \sigma_y) \right]$$

For isotropic media, shear stresses couple independently to shear strains without cross-coupling to normal components:

$$\gamma_{xy} = \frac{\tau_{xy}}{G}, \qquad \gamma_{yz} = \frac{\tau_{yz}}{G}, \qquad \gamma_{zx} = \frac{\tau_{zx}}{G}$$

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3. Tensile Properties, Yield Criteria & Ductility Metrics

Tensile testing conforming to ASTM E8/E8M or AWS B4.0 establishes the baseline mechanical properties used to qualify Welding Procedure Specifications (WPS) under AWS D1.1, ASME Section IX, or API 1104.

   Stress (σ)
       ^
       |               Upper Yield Point (σ_uy)
       |                  /\  Lüders Band Propagation
       |                 /  \------\----- Lower Yield Point (σ_ly)
       |  Proportional  /           \_____
       |  Limit (σ_pl) /                  \
       |              /                    \___ Plastic Flow
       |             / 
       |            /  Linear Elastic: σ = E · ε
       |           /
       +----------+-------------------------------------> Strain (ε)

Elastic and Yield Transition Points

• Proportional Limit ($\sigma_{pl}$): The greatest stress at which stress remains strictly linearly proportional to strain ($d\sigma/d\varepsilon = E$). Beyond this point, the curve deviates from linearity, though deformation may remain largely elastic.
• Elastic Limit: The greatest stress that the material can sustain without sustaining any measurable permanent (plastic) set upon complete unloading.
• Yield Point Phenomenon in Ferritic Steels: Hot-rolled, low-carbon structural steels exhibit distinct Upper ($\sigma_{uy}$) and Lower ($\sigma_{ly}$) yield points. This discontinuity is caused by Cottrell atmospheres: interstitial carbon and nitrogen atoms segregate to the strain fields of dislocations, pinning them securely in place. Once stress reaches $\sigma_{uy}$, dislocations tear free from their solute pin arrays simultaneously, causing stress to drop abruptly to $\sigma_{ly}$. Deformation then propagates along the gauge length at constant nominal stress via visible shear bands called Lüders bands (or stretcher-strain marks) until work hardening initiates.
• 0.2% Offset Yield Strength ($R_{p0.2}$ or $\sigma_{ys}$): High-strength steels (quenched and tempered, precipitation hardened), austenitic stainless steels, and aluminum alloys do not exhibit a sharp yield drop. Instead, yield strength is defined by the 0.2% offset method: a line is drawn parallel to the linear elastic modulus starting from an initial plastic strain of $\varepsilon = 0.002$ ($0.2\%$). The intersection of this line with the stress-strain curve defines $\sigma_{ys}$.

Standard Ductility Metrics

Ductility reflects a weldment's ability to deform plastically without fracture, redistributing localized stress concentrations at weld toes and roots.

1. Percentage Elongation ($EL\%$):

   $$EL\% = \frac{L_f - L_0}{L_0} \times 100\%$$

   where $L_f$ is the final gauge length measured after fitting the fractured halves together. Because total elongation includes both uniform elongation along the entire specimen and localized plastic strain inside the neck, $EL\%$ is strongly dependent on initial gauge length $L_0$. Standard ASTM E8 specimens specify $L_0 = 50\text{ mm}$ ($2.0\text{ in}$) or $4D$ ($L_0 = 4 \times \text{diameter}$). Comparing elongation percentages between specimens of different gauge-length-to-diameter ratios without correction formulas (such as the Oliver formula $EL_1 / EL_2 = (L_2/L_1)^n$) introduces severe error.

2. Percentage Reduction of Area ($RA\%$):

   $$RA\% = \frac{A_0 - A_f}{A_0} \times 100\%$$

   where $A_f$ is the minimum cross-sectional area measured at the fractured neck. Unlike elongation, $RA\%$ is a localized measurement independent of gauge length, providing the truest indication of severe plastic strain capacity before ductile void coalescence.

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4. Lamellar Tearing Susceptibility & Through-Thickness (Z-Direction) Ductility

In heavy structural steel weldments, rolling processes elongate non-metallic inclusions—predominantly manganese sulfides ($\text{MnS}$), silicates, and alumina—into flat, planar "stringers" parallel to the plate rolling plane ($X\text{-}Y$ surface). Consequently, rolled steel plates display severe mechanical anisotropy:

                    Y (Transverse)
                    ^
                    |       Z (Through-Thickness / Short-Transverse)
                    |      /
                    |     /   
                    +----+------------------> X (Rolling / Longitudinal)
                   /    /| 
                  /    / |   MnS Stringers flattened in X-Y plane
                 +----+  |   Low Z-direction tensile ductility (RA% < 15%)
                 |    |  +
                 |    | /
                 +----+

• Longitudinal ($L$, rolling direction): Highest tensile strength, impact toughness, and ductility ($RA\% \approx 60\text{--}70\%$).
• Transverse ($T$, width direction): Intermediate properties ($RA\% \approx 45\text{--}60\%$).
• Short-Transverse ($Z$, through-thickness direction): Severely degraded ductility ($RA\%$ can drop below $10\%$ in non-treated steels).

Lamellar Tearing Mechanism in Welded Joints

When heavy, highly restrained T-joints, corner joints, or cruciform connections are welded, transverse weld shrinkage exerts severe tensile stresses perpendicular to the plate surface ($Z$-direction). When these shrinkage stresses exceed the low through-thickness cohesion of the plate, micro-cracks initiate along the planar $\text{MnS}$ inclusions and link up via shear steps, producing catastrophic sub-surface separation known as lamellar tearing.

Code Mitigation Strategies

1. ASTM A770 / EN 10164 Z-Grade Steels: Steels specified for critical offshore or seismic nodal connections must satisfy through-thickness tensile testing. Grades designated $Z25$ or $Z35$ require an average minimum $Z$-direction reduction of area of $25\%$ or $35\%$, respectively.
2. Calcium Treatment & Low Sulfur: Utilizing modern ladle metallurgy, vacuum degassing, and calcium wire injection (CaSi) lowers sulfur content below $0.005\text{ wt}\%$ and spheroidizes remaining sulfides into benign, non-deformable calcium sulfides ($\text{CaS}$), preventing stringer formation.
3. Joint Detail Engineering: Replacing heavy corner fillet welds with forged transition corners, inserting soft "buttering" weld layers on the plate face before making the structural groove weld, or designing the joint so that shrinkage forces act parallel rather than perpendicular to the plate surface.

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