Shear Modulus for Polymers and 3D-Printed Parts: Formula, Torsion Testing, Dynamic G, and Anisotropy

Shear modulus, G, measures resistance to elastic shear deformation, but a number labeled G is useful only when its test basis matches the design problem. ASTM E143-20 defines shear modulus from corresponding shear stress and shear strain below the proportional limit. For polymers and 3D-printed parts, temperature, rate or frequency, conditioning, and material direction can change the measured response. Static torsion G, dynamic storage modulus G′, and direction-specific FDM or FFF shear modulus values are therefore not interchangeable. This guide connects the shear modulus formula to torsion testing, polymer viscoelasticity, 3D printing anisotropy, and the material inputs needed for design and FEA.

The manufacturing context for build orientation, layer structure, and process selection is covered in EPOC CRAFTER’s additive manufacturing and 3D printing capabilities.

1. Shear Modulus: What G Actually Measures

The shear modulus definition is the ratio of shear stress to the corresponding shear strain within the proportional elastic range. ASTM E143-20 expresses this material property as G = τ/γ and also recognizes torsional modulus and modulus of rigidity as alternative names. A high shear modulus means less elastic shear strain at the same shear stress, within the applicable linear range. It does not tell you when the material will fail.

1.1 Shear Stress vs Shear Strain vs Shear Modulus

Quantity	Meaning	Decision use
Shear stress, τ	Tangential load divided by resisting area	Check the stress carried by the loaded plane.
Shear strain, γ	Angular distortion caused by shear loading	Quantify elastic or nonlinear deformation.
Shear modulus, G	Slope linking τ and γ in the applicable proportional range	Compare shear stiffness under a compatible test condition.
Shear strength	Stress associated with a defined failure or strength criterion	Check failure separately from stiffness.

Shear modulus vs shear strength is therefore a stiffness-versus-failure comparison, not two names for the same property. The article on shear stress and shear strength design checks covers pins, bolts, direct shear, and failure planes without duplicating that failure analysis.

1.2 Modulus of Rigidity, Rigidity Modulus, and Torsion Modulus

Modulus of rigidity, rigidity modulus, torsional modulus, and shear modulus are commonly used for G. The terminology becomes risky when torsional rigidity is used as if it were also a material constant. Torsional rigidity includes section geometry, while G describes material response. That distinction drives the rest of the article.

2. Shear Modulus Formula: Calculation Routes and Assumptions

The shear modulus formula is simple; choosing a valid input state is harder. Use G = τ/γ only in the applicable proportional response, and use G = E/[2(1 + ν)] only when an isotropic linear elastic model is justified. Both equations can return precise numbers while representing the wrong material state.

2.1 How to Calculate Shear Modulus from Stress and Strain

For a linear elastic shear response, the shear modulus equation is G = τ/γ. If τ = 20 MPa and γ = 0.010, G = 2.0 GPa. The same calculation should not be applied to a point after yielding, interlayer separation, local slip, or another nonlinear event. ASTM E143-20 places the shear modulus definition below the proportional limit and treats tangent or chord shear modulus for nonlinear behavior as outside its formal scope.

2.2 Young’s Modulus to Shear Modulus: The Isotropy Boundary

For an isotropic linear elastic material, G = E/[2(1 + ν)]. This shear modulus vs Young’s modulus relationship also depends on Poisson’s ratio. A Young’s modulus to shear modulus conversion is valid only when the isotropic linear elastic assumption fits the material. With E = 3.0 GPa and ν = 0.35, the calculated G is 1.11 GPa. ASTM E143-20 warns that anisotropic materials, including many plastics, can have direction-dependent G and should not be forced into this relationship.

For the tensile elastic side of that relationship, Young’s modulus and part stiffness explains why a material modulus and finished-part stiffness are separate quantities. The corresponding volumetric property is covered in bulk modulus and shear modulus in elastic models.

For orthotropic material properties, a model can require E1, E2, E3, G12, G13, G23, and directional Poisson ratios. One scalar G cannot represent every shear plane. This is the key limitation when using shear modulus in 3D printing or composite-like printed structures.

3. Torsion Testing: From Torque and Twist to G

ASTM E143-20 determines room-temperature shear modulus from controlled torsion when creep is negligible relative to the immediate strain. The method uses cylindrical or tubular specimens, measures torque and angle of twist at the same loading state, and requires uniform twist through the gauge region.

3.1 ASTM E143 Shear Modulus Equation

For a circular specimen, G = TL/(Jθ), where T is torque, L is gauge length, J is polar moment of inertia, and θ is gauge-length twist in radians. For a solid cylinder, J = πD⁴/32. For a tube, J = π(Do⁴ − Di⁴)/32. Because diameter enters the solid-cylinder equation to the fourth power, specimen dimensional error can materially shift the calculated modulus.

3.2 Gauge Length, Alignment, and Measurement

ASTM E143 specifies a gauge length of at least 4D and requires specimen geometry that remains straight and uniform around the measurement region. Fixtures should avoid unintended bending and axial force. Twist is measured across the gauge region, not assumed from total actuator rotation. That matters for polymers because grip compliance, end deformation, and fixture flexibility can contaminate the apparent response.

Torque and twist should be recorded simultaneously, and G is determined from the linear stress-strain response rather than from a peak value. The test speed must also be high enough to make creep negligible. A torsion test shear modulus from this regime is not a creep modulus, storage modulus G′, loss modulus G″, or complex shear modulus G*.

4. Material G Is Not Part Torsional Stiffness

For a circular elastic member, θ = TL/(GJ). The material term G and geometric term J therefore control different parts of the response. A shear stiffness calculation that changes diameter, wall thickness, bore size, or section shape can change J without changing G. This is why torsional stiffness or torsional rigidity should not be copied from a material property table.

A finished printed component adds another layer of complexity. Shells, raster paths, layer interfaces, infill, internal voids, hubs, and local section changes all influence the torque-to-rotation response. On sparse-infill or lattice parts, a whole-part torsion test may represent effective structural stiffness rather than a homogeneous bulk material G.

A useful shear modulus table therefore needs a clear purpose. Material values can screen candidate materials only when the test method and state are compatible. They cannot replace the part geometry, boundary conditions, and allowable twist used in the actual design.

5. Dynamic Shear Modulus in Polymers

The shear modulus of polymers can depend on loading time scale, temperature, and material state. Polymer viscoelasticity makes dynamic shear modulus testing different from quasi-static torsion. Dynamic mechanical analysis separates the recoverable and dissipative parts of a cyclic response instead of reporting one condition-independent G.

5.1 Storage Shear Modulus, Loss Shear Modulus, and G*

The complex shear modulus is G* = G′ + iG″. The shear storage modulus G′ represents the recoverable part of the cyclic response, while the shear loss modulus G″ represents the dissipative part. The loss factor is tan δG = G″/G′. A DMA shear modulus value is incomplete unless the reported quantity and test condition are clear.

5.2 ISO 6721-6 vs ISO 6721-7

Method	Loading and output	Conditions that stay with the result	Design boundary
ASTM E143-20	Static or quasi-static torsion; material G from torque and twist	Room temperature; loading speed that keeps creep negligible; specimen geometry and linear range	Use for compatible elastic torsion. Do not relabel it as G′, G″, or G*.
ISO 6721-6:2019	Forced shear vibration, non-resonance; G′, G″, tan δG	Frequency, temperature, conditioning, shear strain amplitude, apparatus and resonance limits	Use for dynamic shear response. Typical frequency range is about 0.01–100 Hz.
ISO 6721-7:2019	Torsional vibration, non-resonance; G′, G″, tan δG	Frequency, temperature, conditioning, dynamic amplitude, specimen and transducer resonance limits	Use for torsional dynamic response. Do not substitute a single G′ for a full viscoelastic law.

Source note: ASTM E143-20 and ISO 6721-6:2019 / ISO 6721-7:2019 as summarized in the supplied technical standard extraction. The table distinguishes test outputs; it does not equate them.

Frequency and temperature are part of the result definition for viscoelastic shear modulus. ISO 6721-6 gives a typical frequency range of 0.01–100 Hz and requires dynamic shear strain amplitude to be reported with the test state. Once the material leaves the linear strain range, dynamic properties can become amplitude dependent. No universal temperature or frequency correction factor can be taken from these standards for PLA, ABS, PA12, or another specific polymer.

6. 3D-Printed Parts: Orientation and Orthotropic Shear Modulus

3D printed shear modulus can be direction dependent because deposited roads, layer interfaces, perimeters, and voids create anisotropic material properties. Build orientation mechanical properties should therefore stay attached to the test result. An XY value should not be assumed to represent XZ or YZ without evidence.

The same boundary appears in 3D-printed tensile stress and anisotropy, where orientation and specimen state also control what a coupon result means. The shear problem adds direction-specific G and torsional loading.

6.1 FDM Shear Modulus, FFF Shear Modulus, and Build Direction

FDM anisotropy and FFF shear modulus data depend on more than feedstock identity. Raster direction, layer interfaces, perimeter count, infill, conditioning, temperature, and specimen orientation all affect the tested state. ASTM E143-20 supports the general anisotropy warning, but it does not prescribe FDM build directions, raster angles, or an additive-manufacturing test matrix.

6.2 G12, G13, and G23 for Orthotropic FEA

An orthotropic shear modulus model can require G12, G13, and G23 instead of one scalar G. This is anisotropic shear modulus modeling: the constants describe different material planes and are valid only when the coordinate system is mapped correctly to the printed part. Using one direction-dependent modulus in all three planes erases the anisotropy the model was meant to capture.

7. PA12-CF Case: From Coupon Data to a Torsion-Critical Coupler

EPOC CRAFTER used a PA12-CF15 printed coupler for a compact robotic indexing axis to test whether orientation-specific shear data changed the release decision. The part had to transmit 8.0 N·m while keeping peak-load rotation below 0.55° without changing the external envelope.

7.1 Printed and Test State

The material was PA12-CF15 with 15 wt.% chopped carbon fiber. Filament was dried 8 h at 80°C, then printed by FFF with a 0.6 mm nozzle, 0.20 mm layers, ±45° alternating raster, 100% rectilinear infill, and six perimeters. XY, XZ, and YZ groups contained five specimens each. The specimens were conditioned for 24 h at 23°C and 50% RH before testing.

7.2 Quasi-Static and Dynamic Results

Orientation	Mean quasi-static G	CV from 5 specimens	G′ at 1 Hz, 23°C	tan δG
XY	2.104 GPa	1.60%	2.26 GPa	0.052
XZ	1.426 GPa	1.89%	1.55 GPa	0.065
YZ	1.308 GPa	1.83%	1.43 GPa	0.069

Source note: EPOC CRAFTER shop-floor case data. Quasi-static G came from an internal method based on ASTM E143-20 geometry and torsion-calculation principles, not a claim of full ASTM E143 compliance. Dynamic data used ISO 6721-7:2019 at 1.0 Hz, 23°C, and 0.05% dynamic shear strain amplitude. CV values above are recalculated from the five listed raw G values for each orientation.

Relative to the XY mean, XZ was about 32% lower and YZ about 38% lower in this case. Those percentages are case-specific, not generic FDM knockdown factors. XZ and YZ specimens also showed first whitening and local interlayer separation outside the fitted linear range near the gauge-to-grip transition.

7.3 Orthotropic FEA and Release Decision

The orthotropic linear elastic model used G12 = 2.10 GPa, G13 = 1.43 GPa, and G23 = 1.31 GPa. With the motor-side hub fixed and 8.0 N·m applied to the output hub across the same 48 mm free torsion span used on the bench, FEA predicted 0.43° rotation. The measured XY part response was 0.46°. The XY build stayed below the 0.55° design limit, while XZ and YZ were rejected for this torsion-critical coupler.

That workflow fits DFM design guidelines for load-bearing geometry because build orientation and structural load paths are design inputs, not post-processing details. For early hardware verification, functional rapid prototyping validation provides the manufacturing route for checking the same fit and load assumptions before release.

8. Using Shear Modulus in FEA and Data Review

Material properties for FEA must match the constitutive model. A simple isotropic linear elastic analysis can use one G when isotropy is defensible. An orthotropic printed-part model may need G12, G13, and G23. Harmonic, damping, or frequency-dependent analysis can require dynamic shear data over the relevant frequency and temperature range rather than one static value.

8.1 Minimum Data Before You Use a Published G

Check	Minimum record	Decision
Material state	Exact grade, reinforcement, processing and conditioning	Reject mixed-grade or mixed-state comparisons.
Direction	Build orientation or material coordinate system	Do not reuse one direction as all orthotropic shear constants.
Method	Static torsion, ISO 6721-6, ISO 6721-7, or another defined method	Match the measured property to the analysis type.
Test state	Temperature, rate or frequency, and dynamic amplitude where applicable	Keep viscoelastic conditions attached to polymer data.
Specimen and architecture	Geometry, gauge region, raster, perimeters, infill when relevant	Separate material response from structural architecture.
Statistics	Replicates plus scatter	Do not treat one specimen as a complete material characterization.
Model purpose	Static stiffness, torsion, vibration, damping, or another load case	Use only the property needed by that constitutive model.

Source note: Editorial checklist built from ASTM E143-20, ISO 6721-6:2019, ISO 6721-7:2019, and the supplied EPOC CRAFTER case. It contains no generic material-property values.

When you are screening grades before a test program, EPOC CRAFTER’s materials and mechanical properties guide can narrow the material family first. A published G still needs the method and state checks above before it becomes a design input.

9. Questions Engineers Ask About Shear Modulus

9.1 Is Shear Modulus the Same as Modulus of Rigidity?

Yes. Shear modulus and modulus of rigidity refer to the same elastic property G, and ASTM E143-20 also recognizes torsional modulus as an alternative term. Do not confuse modulus of rigidity with torsional rigidity, because torsional rigidity includes section geometry through GJ.

9.2 Can Shear Modulus Be Calculated from Young’s Modulus?

Yes, when isotropic linear elasticity is a valid assumption: G = E/[2(1 + ν)]. For anisotropic material properties, including many printed or fiber-reinforced systems, one tensile modulus does not define all shear directions. Direct shear, torsion, off-axis, or another suitable characterization route may be needed for the material model.

9.3 How Is Shear Modulus Measured?

Static or quasi-static G can be determined from controlled torsion by measuring torque and gauge-length twist on a defined specimen. Dynamic shear modulus is measured under oscillatory loading, where amplitude and phase are needed to separate G′ and G″. The correct method depends on whether your design needs static stiffness, dynamic stiffness, damping, or a frequency-dependent material law.

9.4 Is Shear Modulus the Same as Shear Strength?

No. Shear modulus measures elastic stiffness in shear; shear strength refers to a defined strength or failure limit. A material or printed orientation can be relatively stiff and still fail at a different stress because modulus and failure mechanism answer different questions.

9.5 Does Polymer Shear Modulus Change with Temperature or Frequency?

Yes. Two practical questions are whether shear modulus changes with temperature and whether shear modulus depends on frequency. For polymers under dynamic loading, both can matter, which is why ISO 6721 methods keep temperature and frequency with G′ and G″. The standards do not provide a universal correction factor. Use data for the actual polymer grade, conditioning state, frequency, temperature, and strain amplitude relevant to the load case.

10. Match the Property to the Load Case

A shear modulus value is useful when the property definition matches the load case. Use static G for compatible elastic torsion, directional G12, G13, and G23 when orthotropic response matters, and dynamic G′ and G″ when frequency-dependent polymer behavior is part of the problem. For a finished 3D-printed part, carry the analysis through geometry, build orientation, boundary conditions, and allowable twist instead of stopping at a material-table value.