Warpage and thermal mismatch estimator
Estimate warpage and solder joint strain from thermal expansion mismatch between die, substrate, mould and board, with a fatigue life estimate.
ReliabilityFree to useHow it worksSourcesWhere it helps
Fills the inputs below with typical figures. Replace any of them with your own.
Fills the inputs below with typical figures. Replace any of them with your own.
- Bow over the span
- -132.3 µm
- Frown: the top face is convex and the corners drop. Span 30 mm.
- Radius of curvature
- 0.8506 m
- Curvature -1.176 per metre
- Free mismatch strain
- -1,596 ppm
- Over -140 K from the stress-free temperature
- Thickness and stiffness ratios
- 0.646 and 4.86
- Top over bottom layer
- Cycles to 50% failure
- 245
- 0.336 years at 2 cycles a day
- Shear strain range
- 3.928%
- Mismatch 500 ppm over a distance of 35.36 mm
- Ductility exponent
- -0.4529
- Mean joint temperature 50 °C
Sign in to save your inputs and come back to them later.
- Substrate 25% thinner (solid line, circles)
- As entered (dashed line, squares)
- Substrate 25% thicker (dotted line, triangles)
Show the data as a table
| Temperature (°C) | Substrate 25% thinner | As entered | Substrate 25% thicker |
|---|---|---|---|
| 25 | -144 | -132 | -118 |
| 46 | -122 | -112 | -99.9 |
| 68 | -100 | -91.9 | -81.9 |
| 89 | -78.1 | -71.7 | -63.9 |
| 110 | -56.1 | -51.5 | -45.9 |
| 132 | -34.1 | -31.3 | -28 |
| 153 | -12.2 | -11.2 | -9.96 |
| 175 | 9.81 | 9.02 | 8.04 |
| 196 | 31.8 | 29.2 | 26 |
| 217 | 53.7 | 49.4 | 44 |
| 239 | 75.7 | 69.6 | 62 |
| 260 | 97.7 | 89.8 | 80 |
| Temperature (°C) | Substrate 25% thinner (µm) | As entered (µm) | Substrate 25% thicker (µm) |
|---|---|---|---|
| 25 | -144 | -132.3 | -117.9 |
| 46.36 | -122 | -112.1 | -99.94 |
| 67.73 | -100 | -91.9 | -81.94 |
| 89.09 | -78.05 | -71.71 | -63.95 |
| 110.5 | -56.09 | -51.53 | -45.95 |
| 131.8 | -34.12 | -31.35 | -27.95 |
| 153.2 | -12.15 | -11.16 | -9.955 |
| 174.5 | 9.815 | 9.018 | 8.041 |
| 195.9 | 31.78 | 29.2 | 26.04 |
| 217.3 | 53.75 | 49.38 | 44.03 |
| 238.6 | 75.72 | 69.57 | 62.03 |
| 260 | 97.69 | 89.75 | 80.03 |
- Swing x 0.5 (solid line, circles)
- As entered (dashed line, squares)
- Swing x 1.5 (dotted line, triangles)
Show the data as a table
| Distance from the neutral point (mm) | Swing x 0.5 | As entered | Swing x 1.5 |
|---|---|---|---|
| 7.07 | 39,629 | 8,577 | 3,503 |
| 14.1 | 8,577 | 1,856 | 758 |
| 21.2 | 3,503 | 758 | 310 |
| 28.3 | 1,856 | 402 | 164 |
| 35.4 | 1,134 | 245 | 100 |
| 42.4 | 758 | 164 | 67 |
| 49.5 | 539 | 117 | 48 |
| 56.6 | 402 | 87 | 36 |
| 63.6 | 310 | 67 | 27 |
| 70.7 | 245 | 53 | 22 |
- Distance x 0.5 (solid line, circles)
- As entered (dashed line, squares)
- Distance x 1.5 (dotted line, triangles)
Show the data as a table
| Board temperature swing (K) | Distance x 0.5 | As entered | Distance x 1.5 |
|---|---|---|---|
| 25 | 24,212 | 5,240 | 2,141 |
| 44 | 6,797 | 1,471 | 601 |
| 64 | 3,050 | 660 | 270 |
| 83 | 1,696 | 367 | 150 |
| 103 | 1,068 | 231 | 94 |
| 122 | 728 | 158 | 64 |
| 142 | 526 | 114 | 46 |
| 161 | 396 | 86 | 35 |
| 181 | 308 | 67 | 27 |
| 200 | 245 | 53 | 22 |
Each package with its own geometry and cycle, using the fatigue constants entered above. Editorial examples, not data for a real part.
- Large flip-chip BGA on an organic substrate2450.336 years
- Wafer-level CSP on FR-41,5931.09 years
- QFN on FR-44700.322 years
- Ceramic package on FR-41,2531.71 years
- HBM interposer on an organic substrate, underfilled5230.239 years
Show the data as a table
| Item | Cycles to 50% failure |
|---|---|
| Large flip-chip BGA on an organic substrate | 245 |
| Wafer-level CSP on FR-4 | 1,593 |
| QFN on FR-4 | 470 |
| Ceramic package on FR-4 | 1,253 |
| HBM interposer on an organic substrate, underfilled | 523 |
| Package | Distance (mm) | Joint height (µm) | Mismatch (ppm) | Swing (K) | Strain range (%) | Cycles to failure | Life (years) |
|---|---|---|---|---|---|---|---|
| Large flip-chip BGA on an organic substrate | 35.36 | 450 | 500 | 100 | 3.928 | 245 | 0.336 |
| Wafer-level CSP on FR-4 | 1.697 | 200 | 2,211 | 165 | 1.876 | 1,593 | 1.09 |
| QFN on FR-4 | 3.111 | 80 | 825 | 165 | 3.208 | 470 | 0.322 |
| Ceramic package on FR-4 | 16.19 | 900 | 1,188 | 125 | 2.137 | 1,253 | 1.71 |
| HBM interposer on an organic substrate, underfilled | 19.8 | 80 | 798 | 70 | 2.962 | 523 | 0.239 |
| Material | Expansion (ppm/K) | Modulus (GPa) | Poisson's ratio | Typical glass transition (°C) | Note |
|---|---|---|---|---|---|
| Silicon | 2.6 | 130 | 0.28 | Expansion near room temperature; the modulus is for the <100> direction (about 180 GPa biaxial). | |
| Copper | 17 | 120 | 0.34 | Annealed copper; plated copper and alloys vary. | |
| BT laminate | 14 | 26 | 0.3 | Glass-reinforced bismaleimide triazine core, in plane. | |
| ABF build-up film | 23 | 7 | 0.3 | Filled epoxy build-up film. Grades differ widely: lower expansion goes with higher modulus. | |
| FR-4 board | 16 | 24 | 0.15 | 150 | In-plane figures. Through-thickness expansion is several times larger. |
| Mould compound | 10 | 20 | 0.3 | 150 | Expansion below the glass transition; it roughly triples above it. |
| Glass (borosilicate type) | 3.3 | 64 | 0.2 | Glass core and carrier grades range from about 3 to 9 ppm/K. | |
| Alumina (96%) | 6.5 | 300 | 0.21 | Mean expansion near room temperature to 200 °C; it rises with temperature. | |
| Solder (SnAgCu) | 22 | 50 | 0.36 | Lead-free tin-silver-copper. |
How it works
Two parts of one problem: parts of different materials expand by different amounts when the temperature changes. Part one finds how far a die or stiffener bonded to a substrate bends (Timoshenko's bi-material curvature). Part two finds how fast the solder joints under a package wear out when the package and the board expand differently through thermal cycles (Engelmaier's strain range and Coffin-Manson life).
- Mismatch strain. The layers want to change length by different amounts. The free mismatch strain is the difference in their expansion coefficients times the temperature change from the stress-free temperature (where the layers were bonded or cured) to the temperature you care about. Cooling gives a negative change.
mismatch strain = (CTE2 − CTE1) x (T use − T stress-free) - Curvature of two bonded layers. Layer 1 is the top layer (the die or stiffener) and layer 2 the bottom (the substrate). With thickness ratio m and modulus ratio n, Timoshenko's result for two layers gives the curvature. A positive curvature is a smile (the top face concave); negative is a frown. For a wide plate rather than a strip, each modulus E is replaced by E / (1 − nu), the equibiaxial modulus.
m = t1 / t2; n = E1 / E2; h = t1 + t2; curvature = 6 x mismatch strain x (1 + m)^2 / ( h x [ 3 x (1 + m)^2 + (1 + m x n) x (m^2 + 1 / (m x n)) ] ) - Radius and bow over a span. The radius of curvature is the reciprocal of the curvature. The bow, or sagitta, over a span L is the height of the arc over that chord; for small bends it is close to curvature x L^2 / 8.
radius = 1 / |curvature|; bow = radius x (1 − sqrt(1 − (curvature x L / 2)^2)) ≈ curvature x L^2 / 8 - Shear strain range of a solder joint. In a leadless solder joint the board and the component pull the two ends of the joint apart sideways. The joint furthest from the neutral point (the centre of the array) moves most. Dividing that movement by the joint height gives the shear strain range for one cycle. The distance from the neutral point (DNP) is half the diagonal of the array of outermost joints. If the component and the board swing through different temperatures, as in power cycling, each expansion uses its own swing.
strain range = F x DNP x |CTE board x swing board − CTE component x swing component| / joint height - Ductility exponent. Engelmaier's 1983 form makes the fatigue ductility exponent depend on the mean cyclic solder temperature Ts (°C) and the cycle frequency f (cycles a day): hotter and slower cycles let the solder creep and shorten the life.
c = −0.442 − 0.0006 x Ts + 0.0174 x ln(1 + f) - Cycles to failure and life in years. A Coffin-Manson relation between the strain range and the cycles at which half of a population has failed, with a fatigue ductility coefficient of 0.325 (fitted to tin-lead). Dividing by the cycles a day gives a simple life in years, without a Weibull spread.
N50 = 0.5 x (strain range / (2 x 0.325))^(1 / c); life in years = N50 / (cycles a day x 365.25)
Assumptions
- Part one is two perfectly bonded layers of uniform thickness, linear elastic, with constant properties, at a uniform temperature, free to bend (no clamping, no gravity). It is the classic bi-material result, not a multilayer or finite element model.
- The stress-free temperature is where the layers were bonded or cured (die attach cure, mould cure, solder solidification). Below it the stack is in a stressed state; the tool takes the stress to be zero there.
- Material values are typical figures for a first estimate. They change with temperature, direction, filler and supplier; replace them with data sheet values for your range.
- Part two applies to leadless or low-standoff joints in a regular array, where the shear from the expansion difference dominates. The distance from the neutral point is the distance to the furthest joint.
- The fatigue constants are the published tin-lead values. For lead-free solder enter recalibrated constants, and fit the non-ideality factor to your own simulation or test results.
Limits and warnings
- Stacks of three or more layers (die, mould and substrate), partial bonds, large deflections, viscoelastic creep of organic layers and temperature-dependent properties above the glass transition are outside the two-layer model. Use finite element analysis for a final answer.
- A strip (beam) result is stiffer in one direction than a plate. Use the plate setting for a die or panel that is wide compared with its thickness, and the beam setting for a long thin strip.
- The solder model gives the cycles to 50% failure for tin-lead joints. It does not give a failure distribution, and it is not calibrated for lead-free alloys, underfilled flip-chip bumps, compliant leads or power cycling with a large temperature gradient.
- Strain ranges above about 10% or below 0.1%, a mean joint temperature outside about 20 to 125 °C, and cycle rates outside about 1 to 1,000 a day are beyond the data the model was built on. The tool warns when it meets them.
- Joint height has a large effect and varies with solder volume, pad size and weight: a collapsed joint or a very low standoff gives a much shorter life. Use the measured standoff.
Sources
The methods and default figures rest on these public sources. The wording here is our own; read the source for its full text.
- Analysis of Bi-Metal Thermostats, S. Timoshenko, Journal of the Optical Society of America, volume 11, pages 233 to 255, 1925. The bending of two bonded layers of different expansion, thickness and stiffness. Citation confirmed from papers that cite it; the full text was not read.
- Fatigue Life of Leadless Chip Carrier Solder Joints During Power Cycling, W. Engelmaier, IEEE Transactions on Components, Hybrids, and Manufacturing Technology, CHMT-6, no. 3, pages 232 to 237, September 1983. The strain range and the temperature and frequency dependent ductility exponent. Citation confirmed from the DfR Solutions paper below; the constants used here are as quoted there.
- Predicting Fatigue of Solder Joints Subjected to High Number of Power Cycles (opens in a new tab), Craig Hillman, Nathan Blattau and Matt Lacy, DfR Solutions and Advanced Energy Industries, IPC APEX, 2015. Describes the distance-to-neutral-point model and gives the fatigue ductility coefficient 0.325 and the three constants of the exponent.
- An Acceleration Model for Lead-Free (SAC) Solder Joint Reliability under Thermal Cycling (opens in a new tab), Vasu Vasudevan and Xuejun Fan, Electronic Components and Technology Conference (ECTC), 2008. Reviews the Engelmaier model: the empirical non-ideality factor, half the maximum distance between joints, and the equivalent temperature range for different component and board swings.
- An evaluation of dwell time and mean cyclic temperature parameters in the Engelmaier model (opens in a new tab), Elviz George, Michael Osterman and Michael Pecht, Microelectronics Reliability, volume 55, pages 582 to 587, February to March 2015. Supports using cycle frequency and mean cyclic temperature, and notes the 1988 change to dwell time.
- Precise determination of lattice parameter and thermal expansion coefficient of silicon between 300 and 1500 K, Y. Okada and Y. Tokumaru, Journal of Applied Physics, volume 56, page 314, 1984. Silicon expansion of about 2.6 ppm/K near room temperature.
- Polymers in Electronic Packaging: Build-Up Films for Flip Chip Semiconductor Substrates, Part Two (opens in a new tab), Jeffrey Gotro, Polymer Innovation Blog, 23 April 2018. Build-up film trend: lower expansion goes with higher modulus across the grades. The ABF figure in the library is an editorial typical value; grades differ.
- Advanced Alumina: material properties table (AD-94, AD-96, AD-995) (opens in a new tab), CoorsTek, undated vendor datasheet. 96% alumina modulus near 303 GPa and Poisson's ratio 0.21. The library's 6.5 ppm/K is a mean over a room-temperature to 200 °C range, as in other 96% alumina data sheets; the CoorsTek table quotes a higher mean over 25 to 1,000 °C.
- BGA Material Properties (opens in a new tab), Topline, undated vendor reference sheet. Silicon die expansion of about 2.6 ppm/K and in-plane laminate moduli of roughly 23 to 30 GPa, used to place the silicon and FR-4 library values.
- IPC-9701 and IPC-D-279, IPC (Association Connecting Electronics Industries), current editions. Named for the industry's test method and design guideline for surface mount solder attachment reliability. Not copied or reproduced; check the edition your programme cites.
Where it helps
The lifecycle stages this tool serves: 5 stages in 4 lifecycles.
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Method notev1Formula, ranges, sources and limits
Estimates the curvature and bow of two bonded layers from their thickness, stiffness and thermal expansion, and the solder joint shear strain range, cycles to failure and life in years under temperature cycling.
Formula
mismatch strain = (CTE2 − CTE1) x (T use − T stress-free); m = t1 / t2; n = E1 / E2; h = t1 + t2; curvature = 6 x mismatch strain x (1 + m)^2 / ( h x [ 3 x (1 + m)^2 + (1 + m x n) x (m^2 + 1 / (m x n)) ] ); radius = 1 / |curvature|; bow = radius x (1 − sqrt(1 − (curvature x L / 2)^2)) ≈ curvature x L^2 / 8; strain range = F x DNP x |CTE board x swing board − CTE component x swing component| / joint height; c = −0.442 − 0.0006 x Ts + 0.0174 x ln(1 + f); N50 = 0.5 x (strain range / (2 x 0.325))^(1 / c); life in years = N50 / (cycles a day x 365.25)
| Input | Unit | Range it holds for | Comes from |
|---|---|---|---|
| Length unit for thickness and joint height | choice | um, mm, mil | Your figure |
| Length unit for span and distances | choice | mm, mil, in | Your figure |
| Modulus unit | choice | GPa, MPa, Msi | Your figure |
| Expansion unit | choice | ppmK, perK, ppmF | Your figure |
| Temperature unit | choice | C, F | Your figure |
| Bending model | choice | beam, plate | Your figure |
| Distance from the neutral point entered as | choice | direct, array | Your figure |
| Component and board share the temperature swing | choice | 1, 0 | Your figure |
| Solder alloy | choice | snpb, sac | Your figure |
| Top layer thickness | length unit | from 0.001 | Your figure |
| Top layer modulus | modulus unit | from 0.0001 | Your figure |
| Top layer Poisson's ratio | ratio | 0 to 0.5 | Your figure |
| Top layer expansion | expansion unit | from -50 | Your figure |
| Bottom layer thickness | length unit | from 0.001 | Your figure |
| Bottom layer modulus | modulus unit | from 0.0001 | Your figure |
| Bottom layer Poisson's ratio | ratio | 0 to 0.5 | Your figure |
| Bottom layer expansion | expansion unit | from -50 | Your figure |
| Stress-free temperature | temperature unit | any | Your figure |
| Use or reflow temperature | temperature unit | any | Your figure |
| Span for the bow | span unit | from 0.001 | Your figure |
| Distance from the neutral point | span unit | from 0.001 | Your figure |
| Joint array width | span unit | from 0.001 | Your figure |
| Joint array depth | span unit | from 0.001 | Your figure |
| Solder joint height | length unit | from 0.001 | Your figure |
| Component expansion | expansion unit | from -50 | Your figure |
| Board expansion | expansion unit | from -50 | Your figure |
| Board minimum temperature | temperature unit | any | Your figure |
| Board maximum temperature | temperature unit | any | Your figure |
| Component minimum temperature | temperature unit | any | Your figure |
| Component maximum temperature | temperature unit | any | Your figure |
| Thermal cycles a day | cycles | from 0.001 | Your figure |
| Non-ideality factor | ratio | 0.01 to 3 | Your figure |
| Fatigue ductility coefficient | ratio | 0.01 to 2 | Your figure |
| Exponent constant | ratio | -2 to -0.01 | Your figure |
| Exponent temperature coefficient | per °C | -0.01 to 0.01 | Your figure |
| Exponent frequency coefficient | ratio | -0.1 to 0.1 | Your figure |
Outputs: Curvature (1/m); Bow over the span (µm); Free mismatch strain (ppm); Solder joint shear strain range (fraction); Cycles to 50% failure (cycles); Life (years).
Assumptions
- Part one is two perfectly bonded layers of uniform thickness, linear elastic, with constant properties, at a uniform temperature, free to bend (no clamping, no gravity). It is the classic bi-material result, not a multilayer or finite element model.
- The stress-free temperature is where the layers were bonded or cured (die attach cure, mould cure, solder solidification). Below it the stack is in a stressed state; the tool takes the stress to be zero there.
- Material values are typical figures for a first estimate. They change with temperature, direction, filler and supplier; replace them with data sheet values for your range.
- Part two applies to leadless or low-standoff joints in a regular array, where the shear from the expansion difference dominates. The distance from the neutral point is the distance to the furthest joint.
- The fatigue constants are the published tin-lead values. For lead-free solder enter recalibrated constants, and fit the non-ideality factor to your own simulation or test results.
Uncertainty: Part one is exact for two perfectly bonded elastic layers and as good as the material values and the stress-free temperature entered. Part two is an empirical fatigue model; expect the real life of a joint to differ from the estimate by a factor of two or more, and more for lead-free solder.
Known limits
- Stacks of three or more layers (die, mould and substrate), partial bonds, large deflections, viscoelastic creep of organic layers and temperature-dependent properties above the glass transition are outside the two-layer model. Use finite element analysis for a final answer.
- A strip (beam) result is stiffer in one direction than a plate. Use the plate setting for a die or panel that is wide compared with its thickness, and the beam setting for a long thin strip.
- The solder model gives the cycles to 50% failure for tin-lead joints. It does not give a failure distribution, and it is not calibrated for lead-free alloys, underfilled flip-chip bumps, compliant leads or power cycling with a large temperature gradient.
- Strain ranges above about 10% or below 0.1%, a mean joint temperature outside about 20 to 125 °C, and cycle rates outside about 1 to 1,000 a day are beyond the data the model was built on. The tool warns when it meets them.
- Joint height has a large effect and varies with solder volume, pad size and weight: a collapsed joint or a very low standoff gives a much shorter life. Use the measured standoff.
Where the method comes from
Follows Timoshenko, Analysis of Bi-Metal Thermostats (Journal of the Optical Society of America, 1925), and Engelmaier, Fatigue Life of Leadless Chip Carrier Solder Joints During Power Cycling (IEEE CHMT, 1983) (opens in a new tab). The platform's version departs from it as follows.
- A plate setting replaces each modulus E with E / (1 − nu) for a wide plate, and the bow is the exact arc height over the span Why: Dies and panels are wide, not strips, so the equibiaxial modulus is closer to what they do; the strip form stays available as the beam setting.
- Different temperature swings are allowed for the component and the board, and every constant of the exponent can be edited Why: Power cycling heats the component more than the board, and lead-free solder needs recalibrated constants, which the original tin-lead fit does not provide.
- An empirical non-ideality factor multiplies the strain range Why: Underfilled bumps and stiff carriers do not behave like free leadless joints; the factor lets a person fit the simple model to their own simulation or test data.
Reference example, checked by test: Worked example in this method note: the default stack (775 µm silicon on 1,200 µm BT, cooled from 165 °C to 25 °C, plate model, 30 mm span) and the default large flip-chip BGA joint (50 mm array, 450 µm joint, 11 and 16 ppm/K, 0 to 100 °C, 2 cycles a day, tin-lead constants), hand-worked in the tool's unit test. Curvature -1.1757; Bow over the span -132.272; Free mismatch strain -1,596; Solder joint shear strain range 0.039284; Cycles to 50% failure 245.441; Life 0.33599.
Versions
- v1 (2026-10-25): First version, built from the tool idea.
Terms on this page
- Warpage
- Out-of-plane distortion of a die, substrate or finished package caused by mismatched thermal expansion between its layers, usually specified in micrometres across the body at several temperatures t...
- Die
- A single rectangular block of processed silicon cut from a finished wafer, carrying one complete circuit.
- Power cycling
- A reliability test in which a module is repeatedly self-heated by load current and cooled, producing large junction temperature swings.
- Coffin-Manson model
- The fatigue life model for cyclic stress, giving cycles to failure as a constant divided by the strain range raised to an exponent.
- Datasheet
- The authoritative customer-facing document giving a part's electrical characteristics, absolute maximum ratings, timing, pinout, package drawing and ordering information.
- Flip chip
- Mounting a die face down so its I/O bumps connect directly to the substrate or another die, instead of routing signals out through wires from the die edge.
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