Anand Model: Theoretical Forumation and Application to Solder Joints

• Many metals at high temperatures experience creep and plasticity simultaneously.
• Traditional plasticity models use yield surfaces and separation rules.
• Anand proposes a unified framework to capture both phenomena without a yield condition.

• Introduces a smooth viscoplastic flow model with a single scalar resistance variable \( s \).
• Fully derived from thermodynamic principles (dissipation inequality).
• Applicable to hot working, solder behavior, and finite deformation problems.

• Plastic flow occurs at any stress level.
• No von Mises yield or loading/unloading logic.
• Enables unified creep–plasticity modeling.

• Represents resistance to inelastic flow.
• Captures hardening, softening, and recovery.
• Governs evolution in Eq. (86).

• Grounded in reduced dissipation inequality (Eq. 28).
• Ensures entropy production and realism.
• Built from stress–strain conjugacy, energy balance.

• Uses Jaumann derivatives for stress and backstress.
• Maintains frame invariance (Eqs. 63, 65–66).
• Essential for rotating frames in FEA.

• 1D model extractable from uniaxial data.
• Wang (2001) shows direct parameter fitting.
• Equations (77–86) ready for FE implementation.

Anand's model unifies physical laws, experiment, and computation in one robust viscoplastic framework.

1. Modeling Goal

• Unify inelastic deformation: creep + plasticity
• Avoid yield surfaces and loading/unloading rules
• Support large deformation and high temperatures

2. State Variables

\[ \{ \mathbf{T}, \theta, \mathbf{g}, \bar{\mathbf{B}}, s \} \]
- Stress, temperature, and temperature gradient
- Backstress-like tensor \( \bar{\mathbf{B}} \)
- Scalar internal resistance \( s \)

3. Reference Configuration Formulation

• Switch to relaxed frame (material configuration)
• Formulate stress power and entropy production
• Arrive at dissipation inequality (Eq. 28)

️ 4. Thermodynamic Constraints

• Apply (i)-(iv): entropy, energy, heat flow laws
• Use assumptions (a1)–(a5): small elastic stretch, isotropy, incompressibility
• Restrict response functions \( \bar{\mathbf{B}}, s, \dot{s} \)

5. Simplified Constitutive Equations

• Polynomial-based evolution for \( \bar{\mathbf{B}} \) and \( s \)
• Simplified plastic flow and hardening response

6. Back to Current Configuration

• Use small elastic stretch:

\[ \bar{\mathbf{T}} \approx \mathbf{R}^{eT} \mathbf{T} \mathbf{R}^e \]

• Reformulate in spatial frame for FEA compatibility

7. Final Model (Eqs. 77–86)

• Includes stress rate, flow rule, and hardening law
• Unified viscoplastic response — smooth & thermally sensitive
• Ready for implementation in FEA solvers

Stress Relaxation

• Occurs when strain is held constant and stress gradually decreases over time.
• Characteristic of viscoelastic materials that slowly release internal stress.
• Relevant in damping, cushioning, and biological tissues.

Graph (left): Stress drops exponentially over time.

Creep

• Strain increases slowly under constant stress, even if stress does not change.
• A slow, time-dependent deformation typical in metals at high temperature or polymers.
• Appears asymptotic — strain increases more slowly over time.

Graph (right): Constant stress causes increasing strain.

In classical von Mises plasticity with isotropic hardening, the evolving yield stress is modeled as:

\[ \sigma_y = \sigma_0 + H \bar{\varepsilon}^p \]

where \( \sigma_y \) represents the resistance to plastic flow.

In Anand’s model, the internal variable \( s \) plays an analogous role to \( \sigma_y \), but it evolves continuously with strain rate and temperature, eliminating the need for yield surfaces and discrete flow rules.

• \( A \) – Pre-exponential factor for flow rate.
• \( Q \) – Activation energy (units of energy/mol).
• \( \xi \) – Stress multiplier inside the sinh() law.
• \( m \) – Strain rate sensitivity exponent.
• \( \dot{\varepsilon}^p \) – Effective plastic strain rate.
• \( \bar{\sigma} \) – Effective (von Mises) stress.

• \( \mathbb{L} \) – Elastic stiffness tensor.
• \( \Pi \) – Stress-temperature coupling tensor.
• \( \bar{\mathbf{T}} \) – Kirchhoff stress (reference frame).
• \( \mathbf{D}, \mathbf{D}^p \) – Total and plastic strain rate tensors.

• \( s \) – Isotropic strength (scalar resistance variable).
• \( \hat{s} \) – Saturation value for \( s \).
• \( n \) – Sensitivity of \( \hat{s} \) to strain rate.
• \( h_0 \) – Hardening modulus coefficient.
• \( a \) – Exponent controlling recovery rate of \( s \).

• \( \xi_1, \xi_2 \) – Coefficients for driving terms in \( \dot{\bar{\mathbf{B}}} \).
• \( \mathbf{W}^p \) – Plastic spin tensor.
• \( b(\bar{\tau}_b) \) – Oscillation control function (for shear stability).

Creep-Driven Terms

Eq. (84):
\[ \dot{\bar{\varepsilon}}^p = g(\bar{\sigma}, s, \theta) \]
Steady-state creep rate governed by stress and temperature.

Eq. (86):
\[ \dot{s} = h(\bar{\sigma}, s, \theta)\dot{\bar{\varepsilon}}^p - r(s, \theta) \]
Captures transient creep via thermal recovery.

Hyperbolic Sine Flow Law:
\[ \dot{\bar{\varepsilon}}^p \propto \sinh\left(\frac{\xi \sigma}{s}\right)^{1/m} \]
Models thermally activated dislocation motion.

Smooth rate-dependence:
Enables creep-like flow even at low stress without a sharp yield point.

Plasticity-Driven Terms

Internal variable \( s \):
Represents isotropic resistance; evolves with plastic strain.

Eq. (83):
\[ \mathbf{D}^p = \dot{\bar{\varepsilon}}^p \left\{ \bar{\sigma}^{-1} \mathbf{T}^r \right\} \]
Plastic flow direction set by stress deviator.

Eq. (85):
\[ \dot{s} = \tilde{g}(\bar{\sigma}, s, \theta) \]
Tracks hardening-like resistance from internal variable.

No explicit yield surface:
Still captures hardening and saturation as in classical models.

• \(\mathbf{L}^p = x_1 \tilde{\mathbf{T}}' + \eta_1(\tilde{\mathbf{T}}' \mathbf{B} - \mathbf{B} \tilde{\mathbf{T}}')\)
• Represents viscoplastic flow direction and includes kinematic backstress effect.
• \(\dot{\mathbf{B}} = \xi_1 \tilde{\mathbf{T}}' + \xi_2 \mathbf{B}\)
• Linear evolution of internal backstress — similar to Armstrong–Frederick type models.
• \(\dot{s} = h_0 \left|1 - \frac{s}{s^*} \right|^a \cdot \text{sign}\left(1 - \frac{s}{s^*} \right) \dot{\varepsilon}^p\)
• Captures isotropic hardening/softening and saturates toward \( s^* \).

• Gives physical intuition: backstress = directional memory, \(s\) = isotropic “strength”.
• Helps map terms to graduate plasticity topics (e.g., hardening laws, associative flow).
• Facilitates debugging in FEA — parameters must align with observed behavior.
• Clarifies why Anand’s model is more than just a curve-fit: it encodes mechanics.

• \(\dot{\psi} = \frac{\partial \psi}{\partial \mathbf{E}^e} : \dot{\mathbf{E}}^e + \frac{\partial \psi}{\partial s} \dot{s}\)
• \(\eta_r = -\frac{\partial \psi}{\partial \theta}\)
• \(\Rightarrow \dot{\psi} - \mathbf{T}:\dot{\mathbf{E}}^e - \eta_r\dot{\theta} \leq 0\)
• Result: All response functions must respect the second law of thermodynamics.

• (a1) Objective stress measures (e.g., Jaumann rate)
• (a2) Isotropy in material response
• (a3) Incompressibility of plastic flow
• (a4) Free energy function is additively decomposed
• (a5) Temperature dependence enters through specific variables
• (a6) Separation of mechanical and thermal effects is approximated

• Applies Anand’s unified viscoplastic framework to model solder behavior.
• Focuses on thermal cycling fatigue and rate-dependent deformation.
• Demonstrates how Anand's model can be reduced and fitted from experiments.
• Helps transition the theory into engineering-scale implementation.

Parameter	60Sn40Pb	62Sn36Pb2Ag	96.5Sn3.5Ag	97.5Pb2.5Sn
A (s⁻¹)	1.49e7	2.30e7	2.23e4	3.25e12
Q/R (K)	10830	11262	8900	15583
ξ	11	11	6	7
m	0.303	0.303	0.182	0.143
ŝ (MPa)	80.42	80.79	73.81	72.73
n	0.0231	0.0212	0.018	0.00437
h₀ (MPa)	2640.75	4121.31	3321.15	1787.02
a	1.34	1.38	3.73	3.73
s₀ (MPa)	56.33	42.32	39.09	15.09

• A: Pre-exponential factor in flow rate equation.
• Q/R: Activation energy over gas constant (K).
• ξ: Stress multiplier in hyperbolic sine term.
• m: Strain-rate sensitivity exponent.
• ŝ: Saturation value of internal strength variable \( s \).
• n: Exponent in evolution equation for \( s \).
• h₀: Hardening modulus-like coefficient (drives rate of evolution).
• a: Controls the sharpness of saturation behavior in \( s \).
• s₀: Initial value of internal strength \( s \).

• Top Graph (a): \( \dot{\varepsilon} = 10^{-2} \, \text{s}^{-1} \)
• High strain rate → higher stress
• Recovery negligible → pronounced hardening
• Bottom Graph (b): \( \dot{\varepsilon} = 10^{-4} \, \text{s}^{-1} \)
• Lower strain rate → lower stress at same strain
• Recovery and creep effects more significant

Model Accuracy: Lines = model prediction, X = experimental data

• “At lower strain rates, recovery dominates… the stress levels off early.”
• “At high strain rates, hardening dominates, and the stress grows continuously.”

Anand’s model smoothly captures strain-rate and temperature dependence of solder materials.

• Path Dependence: Internal variables like \( s \), \( \bar{\mathbf{B}} \) evolve, showing hysteresis and memory effects — core ideas in inelasticity.
• Rate Sensitivity: The Anand model embodies a regularized flow rule, helping avoid ill-posedness
• Thermomechanical Coupling: Graduate models often simplify heat effects; Anand incorporates temperature-dependent recovery and strain rates realistically.

• No strain rate sensitivity: All curves would collapse onto a single stress–strain curve, regardless of temperature.
• Sharp yield point: Stress would remain low until a threshold is reached, then suddenly rise — no smooth buildup.
• Post-yield response: Would likely show perfectly plastic or linear hardening behavior, independent of rate.

• This behavior mirrors rate-independent J2 plasticity with isotropic hardening.
• In graduate courses, it corresponds to models with yield surfaces and flow rules only activated above yield stress.
• Contrasts Anand’s approach, where flow begins smoothly at any stress, blending creep and plasticity into one.

• “\(s\) is a fixed yield stress” — Incorrect: \(s\) evolves dynamically with strain rate and temperature.
• “This is just a plasticity model” — Not true: Anand unifies creep and plasticity in a single viscoplastic flow rule.
• “Thermal effects are secondary” — False: temperature directly drives flow via \(\exp(-Q/RT)\) and affects recovery.
• “Classical loading/unloading rules still apply” — No: There’s no yield surface, so no return mapping or activation.

• \(s\) is an internal state variable → behaves like isotropic resistance, not a hard threshold.
• Plastic flow starts at any stress — model handles low-stress creep and high-stress yielding alike.
• Thermodynamic consistency governs structure: response functions must respect dissipation inequality.
• This model is most accurate under high-T and large-deformation conditions — not just small-strain metals.

• Hot working processes involve large inelastic strains, often combining creep and plasticity.
• Existing plasticity theories relied on separate yield surfaces and loading–unloading rules.
• Goal: develop a unified, thermodynamically consistent model to describe time-dependent inelastic deformation.

• Defined internal variable \( s \) to represent resistance to plastic flow.
• Formulated equations using continuum thermodynamics and the reduced dissipation inequality.
• Used objective rates (e.g., Jaumann) and decomposed velocity gradient to capture viscoplastic evolution.
• Developed a smooth flow law based on hyperbolic sine function of stress over \( s \).

• The model successfully captures steady-state and transient creep, plastic hardening, and softening.
• Yield surface and loading–unloading conditions become unnecessary.
• Formulation is compatible with finite deformation and FEA implementation.

• The Anand model offers a unified framework for viscoplasticity, blending creep and plasticity seamlessly.
• Its structure enables consistent thermodynamic modeling of large deformation in high-temperature materials.
• Laid groundwork for efficient numerical and experimental calibration in materials like solder and metals under thermal load.

• Lead-free and leaded solder joints exhibit time-dependent inelastic behavior under thermal cycling.
• Full viscoplastic models are computationally expensive and hard to calibrate.
• Goal: use Anand’s unified model to efficiently capture solder behavior during cyclic loading without compromising accuracy.

• Specialized Anand model for 1D finite strain formulation with internal variable \( s \).
• Calibrated parameters using constant strain-rate tests on solder alloys.
• Simulated thermal cycling to compare Anand vs. full creep models.

• Anand model accurately captured stress-strain hysteresis behavior across temperatures and strain rates.
• Reduced computation time by ~80% compared to full creep models.
• Good agreement with experimental cyclic behavior in solder joints.

• Anand's model provides a practical and thermodynamically sound alternative to detailed creep modeling.
• It is highly suitable for engineering simulations involving thermal cycling, such as electronics reliability analysis.

(a) Stress–Strain–Temperature Relation
\[ \dot{\mathbf{T}}^r = \mathbb{L}[\mathbf{D} - \mathbf{D}^p] - \Pi \dot{\theta} \] Equation (77)

(b) Flow Rule
\[ \mathbf{D}^p = \dot{\varepsilon}^p \left\{ \frac{\mathbf{T}^r}{2 \bar{\tau}} \right\}, \quad \mathbf{W}^p = 0 \] Equation (78) \[ \dot{\varepsilon}^p = f(\bar{\tau}, s, \theta) > 0, \quad f(0, s, \theta) = 0 \] Equation (79) \[ \bar{\tau} = \left[ \tfrac{1}{2} \text{tr}(\mathbf{T}^{r2}) \right]^{1/2} \] Equation (80)

(c) Evolution Equation
\[ \dot{s} = \tilde{f}(\bar{\tau}, s, \theta) \] Equation (81) \[ \bar{\sigma} := (\sqrt{3}) \bar{\tau}, \quad \dot{\bar{\varepsilon}}^p := \dot{\varepsilon}^p / \sqrt{3} \] Equation (82) \[ \mathbf{D}^p = \dot{\bar{\varepsilon}}^p \left\{ \bar{\sigma}^{-1} \mathbf{T}^r \right\} \] Equation (83) \[ \dot{\bar{\varepsilon}}^p = g(\bar{\sigma}, s, \theta) > 0, \quad g(0, s, \theta) = 0 \] Equation (84) \[ \dot{s} = \tilde{g}(\bar{\sigma}, s, \theta) \] Equation (85) \[ \dot{s} = h(\bar{\sigma}, s, \theta) \dot{\bar{\varepsilon}}^p - r(s, \theta) \] Equation (86)