Nopanom Kaewhanam

Secondary compression of soils shows a strain roughly linear in log time over several decades, with a slope—the coefficient of secondary compression C_α—that the Mesri law ties to the compression index C_c. Fractional creep elements are widely used for soil and rock creep; we ask how well they, and classical spectral laws, hold that slope within a stated tolerance. No single Scott-Blair springpot or fractional Kelvin–Voigt element yields an exactly constant C_α: the springpot slope grows by 10^α per decade and the Kelvin–Voigt slope decays as t^−α. A uniform retardation spectrum on a finite band [τ₁, τ₂]—the Becker law with an upper cutoff—gives the closed form C_α(t) = σ₀ A ln10 (e^−t/τ₂ − e^−t/τ₁), so the plateau height and tolerance-defined width follow exactly; holding C_α within 1% over N decades needs a band about N + 2.66 decades wide. For the same target slope and four-decade window under one amplitude rule (peak slope = C∗), a near-degenerate springpot and the Becker law need unbounded creep, while a Kelvin–Voigt element of order α ≈ 0.044 needs a bounded long-time compliance about six times that of the finite band. The finite-band element is thermodynamically admissible as a positive superposition of Voigt modes. The results support choosing a creep kernel by the window it sustains and the long-term settlement it implies.

We propose a joint response-based procedure to determine when one constant compression coefficient is sufficient for both pile drag increment and head settlement. Ordered compression-strain bounds are propagated through a finite-stiffness pile with fixed monotone shaft and toe laws; acceptable coefficient intervals are then intersected. Of 500 synthetic configurations, 339 pass a physical screen (upper strain at most 5% and soil–pile movement at most 0.3 m); in this subset, 335 admit a constant matching their prescribed nonlinear reference within 10%, whereas only 35 satisfy the conservative response-box test. Nine published compression curves provide approximation and withheld-stress checks. Explicit smooth whole-curve witnesses and curvature-constrained enclosures show which conclusions require additional shape information. A Bangkok stage-transfer reconstruction fails; a settlement-independent force-capacity bound identifies incompatibility of the frozen fitted transfer model with later observations. A worked single-pile example with explicit effective stress shows that groundwater lowering can change the decision and separates approximation accuracy from conservative design demands. The procedure supports conditional simplification decisions while making missing compression or transfer information explicit.

Pressure buildup in geological CO₂ storage depends on how pore volume changes with effective stress, yet the laboratory evidence behind simulator compressibility inputs is often sparse. We test when a constant logarithmic compression slope is adequate and what the choice changes in a caprock screen. An exact minimax bound identifies loading intervals where a constant slope must fail. Arc Joint via Optimum Parameters (AJOP) and five competitors were fitted to 19 published trajectories with trajectory-level statistics, information criteria and bootstrap tangent uncertainty, and an independent-specimen cohort was ranked before and after a pre-declared exclusion. Caprock laws based on two unloading moduli measured on one Draupne specimen were run in OPM Flow on a layered model and the published Smeaheia simulation model. AJOP5 gave a median range-normalized RMSE of 1.27% (95% CI 0.81–1.74%); the quadratic gave 0.26% but reversed in four cases. The curvature term was unsupported in 12 fits, and tangent-slope uncertainty reached 123%. High-stress rankings reversed after exclusion. The caprock law changed reservoir buildup by at most 19% but caprock pressure by up to 59% (layered model) and changed no screening class. The bound applies to primary loading only, and small curve errors do not ensure reliable tangent properties.

Finite element models of ultra-high-performance concrete (UHPC) deep beams are usually judged by peak load, yet a correct capacity can come with a wrong mechanism. Our earlier shell-element study of these beams reported 104–107% of the measured capacity for normal- and high-strength concrete (NSC, HSC) and 63% for UHPC, ratios that would be 0.32–0.54 if its predictions are applied loads, with flexural failure where the tests failed in shear and a code-derived UHPC tensile strength 38% below the measured value. A 2 × 2 design—layered shell versus solid elements, each with a code-derived and a measured tensile strength—separates the two causes in four beams. The shell model failed in flexure and over-predicted capacity 1.25–2.86 times; its element effect (with elastic transverse shear), −0.61 to −2.06, far exceeded the tension effect, −0.05 to +0.09. The measured tensile strength raised the solid UHPC predictions by 11–25%, yet left them at 0.72–0.73, consistent with missing fibre bridging but not separable from the mesh dependence of the solid model. The solid model came within 5–21% of the capacity of the beams without fibres, yet failed by diagonal cracking where the tests failed in diagonal compression. Peak-load agreement does not validate a deep-beam model.

A numerical restart of a Caputo operator must preserve its inherited history, whereas a reset deletes that history. We study the resulting sign-change locations under prescribed loading reversals. For a monotone prehistory with loading rate bounded between two positive constants, followed by a constant reverse rate, we prove uniqueness of the first reversal lag and sharp comparison bounds, whose endpoints are attained by constant-rate histories for each admissible rate-bound pair. For equal-spacing triangular loading, we prove that odd lags decrease and even lags increase toward a common limit, with an explicit separation bound. A positive exponential quadrature gives a computable uniform response-error budget that includes frequency truncation, rate approximation and restart-state errors. A residual estimate converts this budget into lag enclosures. Numerical checks include 72 nonlinear-prehistory cases and 40 lag enclosures verified with ball arithmetic. At fractional order 0.60, 768 states on the band [10^-10,10¹⁰ ] give a uniform response bound below 4.705×10^-4 over 0≤s≤9 and lag radii below 0.001044 for the first eight reversals. These enclosures separate continuous memory from a fixed window and reset for the stated protocol. The results concern scalar operators; material strain and dissipation require a separate constitutive embedding.

Fractional memory kernels provide compact descriptions of hereditary deformation, but a good fit does not establish that a continuous spectrum is needed. We construct a synthetic prediction benchmark by comparing local, Caputo-fractional and Prony memory in one associated elasto-viscoplastic host with directional hardening. The memory acts on the cumulative plastic multiplier. A positive spectral representation gives a common dissipation identity and finite-energy bounds. Preparations with matched mechanical variables, rate and memory force still separate during a common hold, for fractional and Prony memory alike. This establishes a role for hidden memory states, not a distinction between kernels. In separate fractional-versus-Prony-2 fit-and-predict tests, the generating family has the smaller error on both withheld histories in 20 of 20 low-noise cases and 16 of 18 high-noise cases. Four- and eight-mode Prony laws fit fractional-generated training data slightly better than the fractional law but predict the longer histories less accurately. The spread between fitted predictions is sensitive to the optimization budget and is not treated as proof of non-identifiability. Finer-grid generation and independently refined scoring reduce the prediction margin and expose discretization bias. These synthetic benchmarks distinguish predictive preference among tested candidates; they do not identify a physical spectrum or validate clay behaviour.

A fractional flow direction need not be determined by its yield boundary. We study a componentwise left Riemann–Liouville operator with fixed coordinates, scales, terminals and internal variables. Proposition 2 constructs two globally strictly convex polynomial functions. They define the same closed ellipse and have identical boundary derivatives through any prescribed finite order. Their normalized fractional directions nevertheless differ for every order between zero and one. A half-order circle example gives an exact separation of 30.008768 degrees. Thus, neither strict convexity nor finite boundary data restore representation invariance. Polynomial identities, adaptive quadrature and independent Grünwald–Letnikov refinement verify the construction. A published modified Cam-clay surface provides a further operator benchmark. A coupled AJOP-T diagnostic and Lode scans then illustrate possible mechanical consequences under stated admissibility conditions. The main result identifies constitutive information that cannot be recovered from finite boundary data alone. These synthetic tests verify the equations and implementation. They do not calibrate or validate a material model.

Fractional flow rules flexibly describe contraction and dilation, but three gaps arise when a scalar fractional rule is placed inside a three-dimensional bounding-surface model: its terminal may not coincide with the material zero-dilatancy state; a transformed-stress direction may not preserve physical plastic power; and a modified flow rule may be inconsistent with the loading-surface evolution. This study closes all three in one formulation. A normalized terminal-directed fractional average is constrained to terminate at the physical zero of any monotone local dilatancy seed, preserving its sign and bounds and recovering the local rule as the fractional order approaches one. A dual mapping is derived for any differentiable, positively homogeneous, mean-preserving stress transformation; a trace correction is shown to be necessary, without which a zero-dilatancy direction acquires spurious volume change. The AJOP-T compression map and the radial bounding-surface variable are coupled through one complete consistency equation yielding the plastic multiplier, loading modulus and continuum tangent directly. Non-negative plastic power follows when a single transformation coefficient is non-positive. Numerical audits confirm homogeneity, work conjugacy and consistency to machine precision, and twelve multistep paths exhibit first-order convergence with consistent algorithmic tangents. A restricted Kaolin stress-path comparison in the constant-slope limit quantifies unrefitted extension discrepancies; it does not validate curved AJOP-T hardening.

Checking maximum pile force can conceal relative uncertainty in drag because the head load dilutes its contribution. In 500 frozen synthetic cases, 29 of 105 joint total-force/settlement outer-envelope certificates at 10% are lost when drag increment replaces total force. We develop a procedure for deciding whether one constant compression coefficient meets both response tolerances. Ordered strain bounds are propagated through a finite-stiffness pile model with fixed monotone shaft and toe laws. Inverse-response intervals and a dimensionless tolerance-normalized index distinguish adequate simplification, insufficient information and incompatible response targets. A refined synthetic reference requires 16.124% minimum joint error; another case admits a 5.172% outer-envelope bound despite a 7.27 slope ratio. Nine published laboratory curves provide fit, shape and withheld-stress checks. AJOP reduces full-window and blocked RMSE relative to a constant logarithmic slope in all six Haihe specimens, while comparator and shape tests delimit that advantage. A Bangkok reconstruction tests transfer between monitored stages. The synthetic AJOP references are prescribed numerical benchmarks. The practical outcome is a shared admissible coefficient or a quantified reason to improve information or use a richer compression representation.

Yield-surface intersection is a scalar root problem in constitutive integration. Its reliability depends on the solver and the residual supplied by the material model. We distinguish residual-only stopping failure from endpoint misclassification caused by a discontinuous stress continuation. Safeguarded Pegasus combines unweighted computed-sign bracketing, bracket-width acceptance, and periodic bisection. Exact-arithmetic enclosure is distinguished from a conditional finite-precision bound involving evaluation error and a lower slope bound. An exact continuous extension of the spatially mobilised plane stress transform covers regular cone boundaries, rank-one edges, and the origin. Numerical guard bands are treated separately. A scalar arithmetic test demonstrates residual-only exhaustion. Twelve archived strip-footing runs show tolerance- and continuation-dependent outcomes. Under the legacy continuation, the reference policy stops during reload at normalized tolerance 10−12; bracket-width acceptance permits completion. Boundary continuation completes at both tested tolerances without forced global steps. Brent uses fewer evaluations on the continuous scalar benchmark, but all tested methods can converge to a discontinuity rather than a root. The contribution is a reproducible diagnosis of classification, stopping, and domain-evaluation failures, not a new general-purpose solver or unconditional finite-precision certification.