Michael Raba

Created: 2025-04-20 Sun 12:37

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<!– If the query includes ’print-pdf’, include the PDF print sheet –> <script> if( window.location.search.match( /print-pdf/gi ) ) { var link = document.createElement( ’link’ ); link.rel = ’stylesheet’; link.type = ’text/css’; link.href = ’https://cdn.jsdelivr.net/npm/reveal.js/css/print/pdf.css’; document.getElementsByTagName( ’head’ )[0].appendChild( link ); } </script> <script type=“text/javascript” src=“https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-AMS-MML_HTMLorMML”></script> </head> <body> <div class=“reveal”> <div class=“slides”> <section id=“sec-title-slide”><h1 class=“title”>Anand Model: Theoretical Forumation and Application to Solder Joints</h1><p class=“subtitle”></p> <h2 class=“author”>Michael Raba, MSc Candidate at University of Kentucky</h2><p class=“date”>Created: 2025-04-20 Sun 12:37</p> </section>

<section> <section id=“slide-org7628146”> <h2 id=“org7628146”>Source Paper</h2> <div style=“display: flex; gap: 2em; align-items: flex-start; font-family: ‘Segoe UI’, sans-serif;”>

<div style=“flex: 1; border-left: 6px solid #2ca02c; background: rgba(44, 160, 44, 0.05); padding: 1em 1.5em; border-radius: 8px;”> <div style=“font-size: 1.2em; font-weight: bold; color: #2ca02c; margin-bottom: 0.5em;”> <i>Constitutive Equations for Hot-Working of Metals</i> </div> <div><b>Author:</b> Lallit Anand (1985)</div> <div><b>DOI:</b> <a href=“https://doi.org/10.1016/0749-6419(85)90004-X”>10.1016/0749-6419(85)90004-X</a></div> <div style=“margin-top: 1em; font-size: 0.95em;”> <i>One of the foundational papers in thermodynamically consistent viscoplasticity modeling—especially significant in the context of metals subjected to large strains and high temperatures.</i> </div> </div>

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</section> </section> <section> <section id=“slide-org1c9de76”> <h2 id=“org1c9de76”>Introduction to Anand’s Unified Viscoplasticity Model (1985)</h2> <div style=“display: flex; gap: 2em; align-items: flex-start; font-family: ‘Segoe UI’, sans-serif; font-size: 1.05em;”>

<div style=“flex: 1; border-left: 6px solid #1f77b4; background: rgba(31, 119, 180, 0.05); padding: 1em 1.5em; border-radius: 8px;”> <div style=“font-weight: bold; color: #1f77b4; margin-bottom: 0.5em;”>Context & Motivation</div> <ul> <li>Many metals at high temperatures experience <b>creep</b> and <b>plasticity</b> simultaneously.</li> <li>Traditional plasticity models use yield surfaces and separation rules.</li> <li>Anand proposes a <i>unified framework</i> to capture both phenomena without a yield condition.</li> </ul> </div>

<div style=“flex: 1; border-left: 6px solid #ff7f0e; background: rgba(255, 127, 14, 0.05); padding: 1em 1.5em; border-radius: 8px;”> <div style=“font-weight: bold; color: #ff7f0e; margin-bottom: 0.5em;”>Core Contributions</div> <ul> <li>Introduces a smooth <b>viscoplastic flow model</b> with a single scalar resistance variable \( s \).</li> <li>Fully derived from thermodynamic principles (dissipation inequality).</li> <li>Applicable to <b>hot working</b>, <b>solder behavior</b>, and finite deformation problems.</li> </ul> </div>

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</section> </section> <section> <section id=“slide-orgeb3ff3a”> <h2 id=“orgeb3ff3a”>Primary Equations of Anand Model (1D)</h2> <div style=“display: flex; gap: 2em; align-items: flex-start; font-family: ‘Segoe UI’, sans-serif; font-size: 1.05em;”>

<div style=“flex: 1; border-left: 6px solid #2ca02c; background: rgba(44, 160, 44, 0.07); padding: 1em 1.5em; border-radius: 8px;”> <div style=“font-weight: bold; color: #2ca02c; margin-bottom: 0.5em;”> Stress & Flow Equations</div>

<p><b>Stress Equation</b><br/> Internal resistance to plastic flow:</p> \[ \sigma = s \cdot \sinh^{-1} \left( \frac{\dot{\varepsilon}^p}{A} \exp\left(\frac{Q}{RT}\right) \right)^{1/m} \cdot \frac{1}{\xi} \]

<p><b>Flow Equation</b></p> \[ \dot{\varepsilon}^p = A \cdot \exp\left(-\frac{Q}{RT} \right) \cdot \left[ \sinh \left( \frac{\xi \sigma}{s} \right) \right]^m \]

<p>This form enables smooth viscoplastic response based on thermal activation.</p> </div>

<div style=“flex: 1; border-left: 6px solid #d62728; background: rgba(214, 39, 40, 0.07); padding: 1em 1.5em; border-radius: 8px;”> <div style=“font-weight: bold; color: #d62728; margin-bottom: 0.5em;”>🔁 Evolution of Internal Variable</div>

<p><b>Evolution of \( s \)</b> (isotropic resistance):</p> \[ \dot{s} = h_0 \left| 1 - \frac{s}{s^*} \right|^a \cdot \text{sign}\left(1 - \frac{s}{s^*} \right) \cdot \dot{\varepsilon}^p \]

<p><b>Saturation Value:</b></p> \[ s^* = \hat{s} \cdot \left( \frac{\dot{\varepsilon}^p}{A} \cdot \exp(Q/RT) \right)^n \]

<p>This equation governs how hardening or softening evolves with time and temperature.</p> </div>

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</section> </section> <section> <section id=“slide-org7ec8fb4”> <h2 id=“org7ec8fb4”>Viscoelasticity: Stress Relaxation and Creep</h2> <div style=“display: flex; gap: 2em; align-items: flex-start; font-family: sans-serif;”>

<div style=“flex: 1; background: rgba(255, 210, 150, 0.15); padding: 1em; border-left: 4px solid #ff9c33;”> <h3 style=“margin-top: 0;”> Stress Relaxation</h3> <ul> <li>Occurs when strain is held constant and stress gradually decreases over time.</li> <li>Characteristic of viscoelastic materials that slowly release internal stress.</li> <li>Relevant in damping, cushioning, and biological tissues.</li> </ul> <p><i>Graph (left): Stress drops exponentially over time.</i></p> <img src=“creep00.png” alt=“Stress Relaxation” style=“width: 100%; border-radius: 4px; box-shadow: 0 0 6px rgba(0,0,0,0.1);”> </div>

<div style=“flex: 1; background: rgba(150, 200, 255, 0.15); padding: 1em; border-left: 4px solid #3399ff;”> <h3 style=“margin-top: 0;”> Creep</h3> <ul> <li>Strain increases slowly under constant stress, even if stress does not change.</li> <li>A slow, time-dependent deformation typical in metals at high temperature or polymers.</li> <li>Appears asymptotic — strain increases more slowly over time.</li> </ul> <p><i>Graph (right): Constant stress causes increasing strain.</i></p> <img src=“creep01.png” alt=“Creep Behavior” style=“width: 100%; border-radius: 4px; box-shadow: 0 0 6px rgba(0,0,0,0.1);”> </div>

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</section> </section> <section> <section id=“slide-org1829b12”> <h2 id=“org1829b12”>Time-Dependent Strain in Elastic, Viscous, and Viscoelastic Materials</h2> <div style=“background: linear-gradient(to right, rgba(255,255,255,0.05), rgba(255,255,255,0.15)); padding: 1.5em; border-left: 6px solid #3b82f6; border-radius: 10px; font-size: 1.05em;”>

<p><b>What This Shows:</b> The diagram below compares how materials respond to a constant applied shear stress, helping distinguish between:</p>

<ul style=“line-height: 1.6;”> <li><b>Pure Elastic:</b> Instantaneous strain recovery once stress is removed.</li> <li><b>Pure Viscous:</b> Strain grows linearly with time; stress removal halts strain, but does not reverse it.</li> <li><b>Viscoelastic:</b> Initial elastic jump followed by viscous creep. After stress is removed, material shows partial recovery (stress relaxation and memory effects).</li> </ul>

<p>Foundational to understanding <b>creep behavior</b> in time-dependent models like Anand’s, where inelastic strain is smooth, history-dependent, and thermal-rate controlled.</p>

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</section> </section> <section> <section id=“slide-orgeeb45a1”> <h2 id=“orgeeb45a1”>Breakthrough Features of Anand’s Viscoplastic Model</h2> <div style=“display: flex; gap: 2em; align-items: flex-start; font-size: 1.03em; font-family: ‘Segoe UI’, sans-serif;”>

<div style=“font-weight: bold; color: #333; margin-bottom: 0.5em; border-bottom: 2px solid #a5a5a5;”>1. No Yield Surface Needed</div> <ul> <li>Plastic flow occurs at <i>any stress level</i>.</li> <li>No von Mises yield or loading/unloading logic.</li> <li>Enables unified creep–plasticity modeling.</li> </ul>

<div style=“font-weight: bold; color: #333; margin: 1em 0 0.5em; border-bottom: 2px solid #a5a5a5;”>2. Scalar Internal Variable \( s \)</div> <ul> <li>Represents resistance to inelastic flow.</li> <li>Captures hardening, softening, and recovery.</li> <li>Governs evolution in Eq. (86).</li> </ul>

<div style=“font-weight: bold; color: #333; margin: 1em 0 0.5em; border-bottom: 2px solid #a5a5a5;”>3. Thermodynamic Consistency</div> <ul> <li>Grounded in reduced dissipation inequality (Eq. 28).</li> <li>Ensures entropy production and realism.</li> <li>Built from stress–strain conjugacy, energy balance.</li> </ul>

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<div style=“font-weight: bold; color: #333; margin-bottom: 0.5em; border-bottom: 2px solid #a5a5a5;”>4. Jaumann Rates Ensure Objectivity</div> <ul> <li>Uses Jaumann derivatives for stress and backstress.</li> <li>Maintains frame invariance (Eqs. 63, 65–66).</li> <li>Essential for rotating frames in FEA.</li> </ul>

<div style=“font-weight: bold; color: #333; margin: 1em 0 0.5em; border-bottom: 2px solid #a5a5a5;”>5. Practical for Experiments and FEA</div> <ul> <li>1D model extractable from uniaxial data.</li> <li>Wang (2001) shows direct parameter fitting.</li> <li>Equations (77–86) ready for FE implementation.</li> </ul>

<div style=“font-weight: bold; color: #3b3b3b; margin: 1em 0 0.5em;”>Key Idea</div> <p style=“margin: 0; color: #444;”> Anand’s model unifies physical laws, experiment, and computation in one robust viscoplastic framework. </p>

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</section> </section> <section> <section id=“slide-orgd423237”> <h2 id=“orgd423237”>How Anand’s Model Unifies Creep and Plasticity</h2> <div style=“display: flex; gap: 2em; align-items: flex-start; font-family: sans-serif;”>

<div style=“flex: 1; background: rgba(255, 235, 180, 0.15); padding: 1em; border-left: 4px solid #ffbb33;”> <h3 style=“margin-top: 0;”>🔥 Creep-Driven Terms</h3>

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

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

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

<p><b>Smooth rate-dependence:</b><br/> Enables creep-like flow even at low stress without a sharp yield point. </p> </div>

<div style=“flex: 1; background: rgba(200, 235, 255, 0.15); padding: 1em; border-left: 4px solid #3399ff;”> <h3 style=“margin-top: 0;”>🧱 Plasticity-Driven Terms</h3>

<p><b>Internal variable \( s \):</b><br/> Represents isotropic resistance; evolves with plastic strain. </p>

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

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

<p><b>No explicit yield surface:</b><br/> Still captures hardening and saturation as in classical models. </p> </div>

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</section> </section> <section> <section id=“slide-org5c1e720”> <h2 id=“org5c1e720”>Broad Strokes of Anand’s Unified Viscoplastic Model (1985)</h2> <div style=“display: flex; gap: 2em; align-items: flex-start; font-family: sans-serif;”>

<div style=“flex: 1; background: rgba(255, 235, 180, 0.15); padding: 1em; border-left: 4px solid #ffbb33;”> <h3 style=“margin-top: 0;”>🎯 1. Modeling Goal</h3> <ul> <li>Unify inelastic deformation: creep + plasticity</li> <li>Avoid yield surfaces and loading/unloading rules</li> <li>Support large deformation and high temperatures</li> </ul>

<h3>📦 2. State Variables</h3> \[ \{ \mathbf{T}, \theta, \mathbf{g}, \bar{\mathbf{B}}, s \} \]<br/>

Stress, temperature, and temperature gradient<br/>
Backstress-like tensor \( \bar{\mathbf{B}} \)<br/>
Scalar internal resistance \( s \)

<h3>📐 3. Reference Configuration Formulation</h3> <ul> <li>Switch to relaxed frame (material configuration)</li> <li>Formulate stress power and entropy production</li> <li>Arrive at dissipation inequality (Eq. 28)</li> </ul> </div>

<div style=“flex: 1; background: rgba(200, 235, 255, 0.15); padding: 1em; border-left: 4px solid #3399ff;”> <h3 style=“margin-top: 0;”>⚖️ 4. Thermodynamic Constraints</h3> <ul> <li>Apply (i)-(iv): entropy, energy, heat flow laws</li> <li>Use assumptions (a1)–(a5): small elastic stretch, isotropy, incompressibility</li> <li>Restrict response functions \( \bar{\mathbf{B}}, s, \dot{s} \)</li> </ul>

<h3>🧮 5. Simplified Constitutive Equations</h3> <ul> <li>Polynomial-based evolution for \( \bar{\mathbf{B}} \) and \( s \)</li> <li>Simplified plastic flow and hardening response</li> </ul>

<h3>🌐 6. Back to Current Configuration</h3> <ul> <li>Use small elastic stretch:</li> </ul> \[ \bar{\mathbf{T}} \approx \mathbf{R}^{eT} \mathbf{T} \mathbf{R}^e \] <ul> <li>Reformulate in spatial frame for FEA compatibility</li> </ul>

<h3>✅ 7. Final Model (Eqs. 77–86)</h3> <ul> <li>Includes stress rate, flow rule, and hardening law</li> <li>Unified viscoplastic response — smooth & thermally sensitive</li> <li>Ready for implementation in FEA solvers</li> </ul> </div>

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</section> </section> <section> <section id=“slide-org55408fb”> <h2 id=“org55408fb”>Summary of Anand’s Model</h2> <style> .ribbon-box { padding: 1em; border-left: 6px solid; margin-bottom: 1em; border-radius: 8px; backdrop-filter: blur(2px); }

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<div class=“ribbon-box ribbon-blue”> <b>Unification of Creep and Plasticity</b><br/> The model treats <i>rate-dependent creep</i> and <i>rate-independent plasticity</i> as a single, smooth phenomenon.<br/> Avoids arbitrary separation of strain types.<br/> Ideal for solder and hot-working cases. </div>

<div class=“ribbon-box ribbon-red”> <b>Single Internal Variable \( s \)</b><br/> Represents average isotropic resistance to plastic flow.<br/> Evolves with stress and temperature.<br/> Eliminates need for complex multi-surface rules. </div>

<div class=“ribbon-box ribbon-green”> <b>Hyperbolic Sine Flow Form</b><br/> Captures power-law breakdown and nonlinear rate sensitivity.<br/> Handles thermal-cycling hysteresis where traditional plasticity fails. </div>

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<div class=“ribbon-box ribbon-orange”> <b>Direct Parameter Fitting</b><br/> No need to distinguish creep from plastic experimentally.<br/> Parameters fit to total viscoplastic strain data.<br/> Simplifies experimental workflow. </div>

<div class=“ribbon-box ribbon-purple”> <b>Numerical Efficiency</b><br/> Uses stable backward Euler integration.<br/> No strict stability limit.<br/> Highly effective for long-term simulations in FEA. </div>

<div class=“ribbon-box ribbon-blue”> <b>Key Insight from Wang</b><br/> <q>The Anand model unifies both creep and plasticity into one smooth viscoplastic framework, enabling predictive modeling of time-dependent deformation with thermodynamic consistency and computational efficiency.</q> </div>

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</section> </section> <section> <section id=“slide-org6c162fc”> <h2 id=“org6c162fc”>Constitutive Equations for Isotropic Thermo-Elasto-Viscoplasticity</h2> <div style=“display: flex; gap: 2em; align-items: flex-start;”>

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

<b>(b) Flow Rule</b><br/> \[ \mathbf{D}^p = \dot{\varepsilon}^p \left\{ \frac{\mathbf{T}^r}{2 \bar{\tau}} \right\}, \quad \mathbf{W}^p = 0 \] 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)

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<b>(c) Evolution Equation</b><br/> \[ \dot{s} = \tilde{f}(\bar{\tau}, s, \theta) \] 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)

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</section> </section> <section> <section id=“slide-org55b4468”> <h2 id=“org55b4468”>Theoretical Stress-Strain Formulation</h2> <p> <b>Stress as a Function of Plastic Strain</b> \( \varepsilon^p \) Using the integrated model, the stress-strain relation becomes: \[ \sigma(\varepsilon^p) = \sigma^* - (\sigma^* - \sigma_0)(1 - \exp(-ch_0 (\varepsilon^p)^{1-a})) \] </p> <ul> <li>\( \sigma_0 \): initial yield stress</li> <li>\( \sigma^* \): saturation stress (UTS)</li> <li>\( c, h_0, a \): shape and evolution constants</li>

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</section> </section> <section> <section id=“slide-org5e15494”> <h2 id=“org5e15494”>Determination of Model Parameters</h2> <p> <b>Step 1: Saturation Stress Curve Fit</b> Fit \( \sigma^* \) using: \[ \sigma^* = \hat{s} \cdot \sinh^{-1}\left( \frac{\dot{\varepsilon}^p}{A} \exp\left(\frac{Q}{RT} \right) \right)^{1/m} \cdot \frac{1}{\xi} \] to get: \( \hat{s}, A, Q/R, m, \xi, n \) </p>

<p> <b>Step 2: Plastic Stress-Strain Fit</b> Fit: \[ \sigma(\varepsilon^p) = \text{(nonlinear form)} \] to get: \( s_0, h_0, a \) </p>

<p> <b>Notes</b> </p> <ul> <li>Use regression fitting (nonlinear least squares)</li> <li>Requires tests across a range of \( T \) and \( \dot{\varepsilon}^p \)</li>

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