Created: 2025-04-20 Sun 22:08
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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> </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 22:08</p> </section> <section> <section id=“slide-org40b571b”> <h2 id=“org40b571b”>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>
<div style=“flex: 1;”> <img src=“./anandPaper.png” alt=“Anand 1985 Paper” style=“max-width: 100%; border: 1px solid #ccc; border-radius: 6px;” /> </div>
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</section> </section> <section> <section id=“slide-orgfc69972”> <h2 id=“orgfc69972”>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-org281afaf”> <h2 id=“org281afaf”>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;”>
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<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 id=“slide-org237904c”> <h3 id=“org237904c”>Formulation pipeline for Anand’s viscoplastic model</h3> <div style=“border-left: 6px solid #2e86de; background: rgba(46, 134, 222, 0.05); padding: 1.2em 1.5em; border-radius: 8px; font-family: ‘Segoe UI’, sans-serif; font-size: 1.05em;”> <b style=“color: #2e86de;”>Visual Roadmap of Anand’s Model</b><br/><br/>
<p style=“margin-top: 1em;”>This flow ensures Anand’s model is thermodynamically consistent and computationally implementable.</p>
<div style=“margin-top: 1.5em;”> <img src=“anandFlow.png” style=“width: 100%; border: 1px solid #ccc; border-radius: 6px;”> </div> </div>
</section> <section id=“slide-org7e7fe56”> <h3 id=“org7e7fe56”>Broad Strokes of Anand’s Unified Viscoplastic Model (1985)</h3> <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/>
<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 id=“slide-orgf0f301a”> <h3 id=“orgf0f301a”>Thermodynamic Foundations of Anand’s Model</h3> <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 #ff7f0e; background: rgba(255, 127, 14, 0.07); padding: 1em 1.5em; border-radius: 8px;”> <div style=“font-weight: bold; color: #ff7f0e; margin-bottom: 0.5em;”>Key Constraints from Dissipation</div> <ul> <li>\(\dot{\psi} = \frac{\partial \psi}{\partial \mathbf{E}^e} : \dot{\mathbf{E}}^e + \frac{\partial \psi}{\partial s} \dot{s}\)</li> <li>\(\eta_r = -\frac{\partial \psi}{\partial \theta}\)</li> <li>\(\Rightarrow \dot{\psi} - \mathbf{T}:\dot{\mathbf{E}}^e - \eta_r\dot{\theta} \leq 0\)</li> <li>Result: All response functions must respect the second law of thermodynamics.</li> </ul> </div>
<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;”>Simplifying Assumptions (a1)–(a6)</div> <ul> <li>(a1) Objective stress measures (e.g., Jaumann rate)</li> <li>(a2) Isotropy in material response</li> <li>(a3) Incompressibility of plastic flow</li> <li>(a4) Free energy function is additively decomposed</li> <li>(a5) Temperature dependence enters through specific variables</li> <li>(a6) Separation of mechanical and thermal effects is approximated</li> </ul> </div>
</div> </section> </section> <section> <section id=“slide-org73a4b6f”> <h2 id=“org73a4b6f”>Primary Equations of Anand Model (1D)</h2>
</section> </section> <section> <section id=“slide-orge0c638d”> <h2 id=“orge0c638d”>Material Parameters in Anand’s Viscoplastic Model</h2> <div style=“display: flex; gap: 2em; align-items: flex-start; font-family: ‘Segoe UI’, sans-serif; font-size: 1.05em;”>
<!– Left column –> <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-weight: bold; color: #2ca02c; margin-bottom: 0.5em;”>Flow Parameters</div> <ul> <li><b>\( A \)</b> – Pre-exponential factor for flow rate.</li> <li><b>\( Q \)</b> – Activation energy (units of energy/mol).</li> <li><b>\( \xi \)</b> – Stress multiplier inside the sinh() law.</li> <li><b>\( m \)</b> – Strain rate sensitivity exponent.</li> <li><b>\( \dot{\varepsilon}^p \)</b> – Effective plastic strain rate.</li> <li><b>\( \bar{\sigma} \)</b> – Effective (von Mises) stress.</li> </ul>
<div style=“font-weight: bold; color: #2ca02c; margin: 1em 0 0.5em;”>Stress & Elasticity</div> <ul> <li><b>\( \mathbb{L} \)</b> – Elastic stiffness tensor.</li> <li><b>\( \Pi \)</b> – Stress-temperature coupling tensor.</li> <li><b>\( \bar{\mathbf{T}} \)</b> – Kirchhoff stress (reference frame).</li> <li><b>\( \mathbf{D}, \mathbf{D}^p \)</b> – Total and plastic strain rate tensors.</li> </ul> </div>
<!– Right column –> <div style=“flex: 1; border-left: 6px solid #d62728; background: rgba(214, 39, 40, 0.05); padding: 1em 1.5em; border-radius: 8px;”> <div style=“font-weight: bold; color: #d62728; margin-bottom: 0.5em;”>Internal Variable Evolution</div> <ul> <li><b>\( s \)</b> – Isotropic strength (scalar resistance variable).</li> <li><b>\( \hat{s} \)</b> – Saturation value for \( s \).</li> <li><b>\( n \)</b> – Sensitivity of \( \hat{s} \) to strain rate.</li> <li><b>\( h_0 \)</b> – Hardening modulus coefficient.</li> <li><b>\( a \)</b> – Exponent controlling recovery rate of \( s \).</li> </ul>
<div style=“font-weight: bold; color: #d62728; margin: 1em 0 0.5em;”>Backstress Evolution (Tensor \( \bar{\mathbf{B}} \))</div> <ul> <li><b>\( \xi_1, \xi_2 \)</b> – Coefficients for driving terms in \( \dot{\bar{\mathbf{B}}} \).</li> <li><b>\( \mathbf{W}^p \)</b> – Plastic spin tensor.</li> <li><b>\( b(\bar{\tau}_b) \)</b> – Oscillation control function (for shear stability).</li> </ul>
<div style=“font-size: 0.9em; color: #666; margin-top: 1em;”> Note: All parameters are temperature-dependent, and some (like \( A, Q, m \)) are fit to experimental data using the 1D simplification. </div> </div>
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</section> <section id=“slide-org6c86096”> <h3 id=“org6c86096”>How Anand’s Model Unifies Creep and Plasticity</h3> <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 id=“slide-orgf6987d6”> <h3 id=“orgf6987d6”>Interpretation of Intermediate Terms (S3 & S4)</h3> <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 #17becf; background: rgba(23, 190, 207, 0.07); padding: 1em 1.5em; border-radius: 8px;”> <div style=“font-weight: bold; color: #17becf; margin-bottom: 0.5em;”>Terms from Simplified Model</div> <ul> <li><b>\(\mathbf{L}^p = x_1 \tilde{\mathbf{T}}' + \eta_1(\tilde{\mathbf{T}}' \mathbf{B} - \mathbf{B} \tilde{\mathbf{T}}')\)</b></li> <li>Represents <i>viscoplastic flow direction</i> and includes <i>kinematic backstress effect</i>.</li>
<li><b>\(\dot{\mathbf{B}} = \xi_1 \tilde{\mathbf{T}}' + \xi_2 \mathbf{B}\)</b></li> <li>Linear evolution of internal backstress — similar to Armstrong–Frederick type models.</li>
<li><b>\(\dot{s} = h_0 \left|1 - \frac{s}{s^*} \right|^a \cdot \text{sign}\left(1 - \frac{s}{s^*} \right) \dot{\varepsilon}^p\)</b></li> <li>Captures isotropic hardening/softening and saturates toward \( s^* \).</li> </ul> </div>
<div style=“flex: 1; border-left: 6px solid #bcbd22; background: rgba(188, 189, 34, 0.07); padding: 1em 1.5em; border-radius: 8px;”> <div style=“font-weight: bold; color: #bcbd22; margin-bottom: 0.5em;”>Why It Matters</div> <ul> <li>Gives physical intuition: backstress = directional memory, \(s\) = isotropic “strength”.</li> <li>Helps map terms to graduate plasticity topics (e.g., hardening laws, associative flow).</li> <li>Facilitates debugging in FEA — parameters must align with observed behavior.</li> <li>Clarifies why Anand’s model is more than just a curve-fit: it encodes mechanics.</li> </ul> </div>
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</section> <section id=“slide-org611f8fd”> <h3 id=“org611f8fd”>Thermodynamic Foundations of Anand’s Model</h3> <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 #ff7f0e; background: rgba(255, 127, 14, 0.07); padding: 1em 1.5em; border-radius: 8px;”> <div style=“font-weight: bold; color: #ff7f0e; margin-bottom: 0.5em;”>Key Constraints from Dissipation</div> <ul> <li>\(\dot{\psi} = \frac{\partial \psi}{\partial \mathbf{E}^e} : \dot{\mathbf{E}}^e + \frac{\partial \psi}{\partial s} \dot{s}\)</li> <li>\(\eta_r = -\frac{\partial \psi}{\partial \theta}\)</li> <li>\(\Rightarrow \dot{\psi} - \mathbf{T}:\dot{\mathbf{E}}^e - \eta_r\dot{\theta} \leq 0\)</li> <li>Result: All response functions must respect the second law of thermodynamics.</li> </ul> </div>
<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;”>Simplifying Assumptions (a1)–(a6)</div> <ul> <li>(a1) Objective stress measures (e.g., Jaumann rate)</li> <li>(a2) Isotropy in material response</li> <li>(a3) Incompressibility of plastic flow</li> <li>(a4) Free energy function is additively decomposed</li> <li>(a5) Temperature dependence enters through specific variables</li> <li>(a6) Separation of mechanical and thermal effects is approximated</li> </ul> </div>
</div> </section> </section> <section> <section id=“slide-org47d6d76”> <h2 id=“org47d6d76”>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-orgb9621dd”> <h2 id=“orgb9621dd”>Case Study: Wang (2001)</h2> <div style=“display: flex; align-items: flex-start; gap: 2em; font-family: ‘Segoe UI’, sans-serif;”>
<div style=“flex: 1;”> <img src=“wangPaper.png” alt=“Wang Paper” style=“width:100%; border-radius: 6px; box-shadow: 0 0 8px rgba(0,0,0,0.2); margin-bottom: 1em;” /> <div style=“font-size: 0.9em; color: #666;”> Source: Wang, C. H. (2001). “A Unified Creep–Plasticity Model for Solder Alloys.” <br/> <b>DOI:</b> <a href=“https://doi.org/10.1115/1.1371781” target=“blank”>10.1115/1.1371781</a> </div> </div>
<div style=“flex: 2; border-left: 6px solid #1f77b4; background: rgba(31, 119, 180, 0.05); padding: 1.2em 1.5em; border-radius: 8px;”> <div style=“font-weight: bold; color: #1f77b4; font-size: 1.2em; margin-bottom: 0.5em;”>Why Wang’s Paper Matters</div> <ul style=“line-height: 1.6;”> <li>Applies Anand’s unified viscoplastic framework to model solder behavior.</li> <li>Focuses on thermal cycling fatigue and rate-dependent deformation.</li> <li>Demonstrates how Anand’s model can be reduced and fitted from experiments.</li> <li>Helps transition the theory into engineering-scale implementation.</li> </ul> </div> </div>
</section> <section id=“slide-org399e31e”> <h3 id=“org399e31e”>Comparing Anand Model Predictions at Two Strain Rates</h3> <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, 0.06); padding: 1em 1.5em; border-radius: 8px;”> <div style=“font-weight: bold; color: #2ca02c; margin-bottom: 0.5em;”>Observed Behavior</div> <ul> <li><b>Top Graph (a):</b> \( \dot{\varepsilon} = 10^{-2} \, \text{s}^{-1} \)</li> <li>High strain rate → higher stress</li> <li>Recovery negligible → pronounced hardening</li>
<li><b>Bottom Graph (b):</b> \( \dot{\varepsilon} = 10^{-4} \, \text{s}^{-1} \)</li> <li>Lower strain rate → lower stress at same strain</li> <li>Recovery and creep effects more significant</li> </ul> <p style=“margin-top: 1em;”><b>Model Accuracy:</b> Lines = model prediction, X = experimental data</p> </div>
<div style=“flex: 1; border-left: 6px solid #d62728; background: rgba(214, 39, 40, 0.06); padding: 1em 1.5em; border-radius: 8px;”> <div style=“font-weight: bold; color: #d62728; margin-bottom: 0.5em;”>Key Insights from Wang (2001)</div> <ul> <li>“At lower strain rates, recovery dominates… the stress levels off early.”</li> <li>“At high strain rates, hardening dominates, and the stress grows continuously.”</li> </ul> <p style=“margin-top: 1em;”>Anand’s model smoothly captures strain-rate and temperature dependence of solder materials.</p> </div>
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<div style=“text-align: center; margin-top: 1.5em;”> <img src=“wMPa.png” style=“width: 40%; margin-right: 2em;”> <img src=“wMPb.png” style=“width: 40%;”> </div>
</section> <section id=“slide-orgb41c332”> <h3 id=“orgb41c332”>Anand Approximation</h3> <div style=“display: flex; flex-direction: column; gap: 1.5em; font-family: ‘Segoe UI’, sans-serif; font-size: 1.05em;”>
<!– Section: Title and Image –> <div style=“display: flex; flex-direction: row; gap: 2em;”> <div style=“flex: 1;”> <img src=“wangHa.png” alt=“Wang Figure Comparison” style=“width: 100%; border: 1px solid #ccc; border-radius: 8px;”> </div> <div style=“flex: 1; border-left: 6px solid #2e86c1; background: rgba(46, 134, 193, 0.07); padding: 1em 1.5em; border-radius: 8px;”> <div style=“font-weight: bold; color: #2e86c1; margin-bottom: 0.5em;”>Anand Approximation</div> <ul> <li><b>FEA Ready:</b> Smooth equations, Jaumann derivatives, and rate-dependence make it suitable for cyclic thermal loads.</li> <li><b>Path Dependence & Hysteresis:</b> Anand’s model shows how evolving internal variables (like \( s \), \( \bar{\mathbf{B}} \)) naturally reproduce load history and hysteresis effects — a cornerstone of modern inelasticity.</li> </ul> </div> </div>
<!– Section: Graduate Plasticity Link –> <div style=“border-left: 6px solid #28b463; background: rgba(40, 180, 99, 0.07); padding: 1em 1.5em; border-radius: 8px;”> <div style=“font-weight: bold; color: #28b463; margin-bottom: 0.5em;”>Relation to Graduate Plasticity Course</div> <ul> <li><b>Path Dependence:</b> Internal variables like \( s \), \( \bar{\mathbf{B}} \) evolve, showing hysteresis and memory effects — core ideas in inelasticity.</li> <li><b>Rate Sensitivity:</b> The Anand model embodies a regularized flow rule, helping avoid ill-posedness</li> <li><b>Thermomechanical Coupling:</b> Graduate models often simplify heat effects; Anand incorporates temperature-dependent recovery and strain rates realistically.</li> </ul> </div>
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</section> <section id=“slide-orgafabf92”> <h3 id=“orgafabf92”>What If the Material Were Not Viscoplastic?</h3> <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;”>Expected Graphical Differences</div>
<ul> <li><b>No strain rate sensitivity</b>: All curves would collapse onto a single stress–strain curve, regardless of temperature.</li> <li><b>Sharp yield point</b>: Stress would remain low until a threshold is reached, then suddenly rise — no smooth buildup.</li> <li><b>Post-yield response</b>: Would likely show perfectly plastic or linear hardening behavior, independent of rate.</li> </ul> </div>
<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-weight: bold; color: #2ca02c; margin-bottom: 0.5em;”>Relation to Plasticity Course</div>
<ul> <li>This behavior mirrors <b>rate-independent J2 plasticity</b> with isotropic hardening.</li> <li>In graduate courses, it corresponds to models with <b>yield surfaces</b> and <b>flow rules</b> only activated above yield stress.</li> <li>Contrasts Anand’s approach, where flow begins <i>smoothly at any stress</i>, blending creep and plasticity into one.</li> </ul> </div>
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</section> </section> <section> <section id=“slide-org1b68630”> <h2 id=“org1b68630”>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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