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.

• \[ \dot{\varepsilon}^p = A \exp\left( -\frac{Q}{RT} \right) \left[ \sinh\left( \frac{j \sigma}{s} \right) \right]^{1/m} \]
• Plastic strain rate increases with stress and temperature.
• No explicit yield surface; flow occurs at all nonzero stresses.

• \[ s^* = \hat{s} \left( \frac{\dot{\varepsilon}^p}{A} \exp\left( \frac{Q}{RT} \right) \right)^n \]
• Defines the steady-state value that \( s \) evolves toward.
• Depends on strain rate and temperature.

• \[ \dot{s} = h_0 \left| 1 - \frac{s}{s^*} \right|^a \, \text{sign}\left(1 - \frac{s}{s^*}\right) \dot{\varepsilon}^p \]
• Describes dynamic hardening and softening of the material.
• \( s \) evolves depending on proximity to \( s^* \) and flow activity.

• Constant \( \dot{\varepsilon}^p \) and \( T \):
• \( s^* \) can be treated as approximately fixed
• Varying \( \dot{\varepsilon}^p \) or \( T \):
• \( s^* \) evolves and must be updated dynamically

In Wang's paper and Anand's original model, \( s^* \) is given by:

\[ s^* = \hat{s} \left( \frac{\dot{\varepsilon}^p}{A} e^{Q/RT} \right)^n \]

\( s^* \) explicitly depends on strain rate ( \( \dot{\varepsilon}^p \) ) and temperature ( \( T \) ).

Values are from correspond to 60Sn40Pb solder parameters used in Anand's model:

• \( S_0 \): Initial deformation resistance
• \( Q/R \): Activation energy over gas constant
• \( A \): Pre-exponential factor for flow rate
• \( \xi \): Multiplier of stress inside sinh
• \( m \): Strain rate sensitivity of stress
• \( h_0 \): Hardening/softening constant
• \( \hat{s} \): Coefficient for saturation stress
• \( n \): Strain rate sensitivity of saturation
• \( a \): Strain rate sensitivity of hardening or softening

• \( S_0 = 5.633 \times 10^7 \) Pa
• \( Q/R = 10830 \) K
• \( A = 1.49 \times 10^7 \) s\(^{-1}\)
• \( \xi = 11 \)
• \( m = 0.303 \)
• \( h_0 = 2.6408 \times 10^9 \) Pa
• \( \hat{s} = 8.042 \times 10^7 \) Pa
• \( n = 0.0231 \)
• \( a = 1.34 \)

These constants match Wang's paper for modeling 60Sn40Pb viscoplasticity.

• Internal variable \( s \) evolves dynamically:
• \( \dot{s} = h(\sigma, s, T) \dot{\varepsilon}^p - \phi(s, T) \)
• Describes both hardening and recovery processes.
• No fixed saturation \( s^* \) assumed.

• Defines a practical saturation stress \( s^* \):
• \( s^* = \hat{s} \left( \frac{\dot{\varepsilon}^p}{A} e^{Q/RT} \right)^n \)
• Relates \( s^* \) to strain rate and temperature.
• Simplifies parameter extraction for finite element simulations.

• At steady-state plastic flow, Wang assumes:
• \( s \approx \frac{\sigma}{\xi} \)
• \( \xi \) is the stress multiplier in the sinh function (called \( j \)).
• Provides a direct link between observed stress and internal variable \( s \).

• 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.

• 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.

• Defines a free energy \( \psi \) based on elastic strain and internal variables.
• Tracks entropy production \( \mathcal{D} \) to guarantee positive dissipation.
• Uses dissipation potential to derive flow rule and internal variable evolution.
• Explicit hardening and softening terms driven by thermodynamic forces.

• Introduces a saturation stress \( s^* \) that depends on \( \dot{\varepsilon}^p \) and \( T \).
• Directly evolves \( s \to s^* \) without explicitly tracking entropy or free energy.
• Positive dissipation guaranteed by construction (no free energy increase).
• Simplifies parameter extraction and implementation for FEA models.

• Anand: Thermodynamic forces drive \( \dot{s} \); complex but fundamental.
• Wang: \( s \) moves toward \( s^* \) with flow rate; simpler, but still thermodynamically consistent.
• Result: Wang’s model is easier to code and fit to data without losing physical realism.

• 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

• \(\dot{\psi} = \frac{\partial \psi}{\partial \mathbf{E}^e} : \dot{\mathbf{E}}^e + \frac{\partial \psi}{\partial s} \dot{s}\)
• \(\eta_r = -\frac{\partial \psi}{\partial \theta}\)
• \(\Rightarrow \dot{\psi} - \mathbf{T}:\dot{\mathbf{E}}^e - \eta_r\dot{\theta} \leq 0\)
• 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

• \( A \) – 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

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.