4. Honeycombs: In-plane Behavior

Introduction to Honeycombs and Mechanical Properties

Overview of the Session

• The session begins with a reminder about supporting MIT OpenCourseWare, emphasizing its role in providing free educational resources.

• Lorna Gibson reviews previous discussions on honeycombs and introduces today's focus on deriving equations for their mechanical properties.

Stress-Strain Behavior

• Discussion includes stress-strain curves for compression (left side) and tension (right side), highlighting Young's modulus calculations.

• Different failure mechanisms are outlined: elastic buckling in elastomeric honeycombs, plastic yielding in metal honeycombs, and brittle crushing in ceramic honeycombs.

Deformation Mechanisms of Honeycombs

Linear Elastic Behavior

• Linear elastic behavior is linked to bending of cell walls; plateau regions relate to buckling or yielding depending on material type.

• Focus shifts to calculating Young's modulus starting with in-plane behavior, which involves bending of inclined cell walls under load.

Setup for Calculating Young's Modulus

• A unit cell is defined with specific dimensions (length h, length l, angle theta), establishing a framework for calculations.

• The global stress applied is denoted as sigma 1; depth b cancels out due to uniformity across the prismatic structure.

Calculating Unit Cell Dimensions

Length Definitions

• In the x1 direction, the unit cell length is defined as 2l cos(theta).

• In the x2 direction, the unit length accounts for both height h and an additional component from sine(theta).

Bending Analysis

• The inclined member’s deflection under load p is analyzed; moments at either end contribute to understanding deformation.

• Deflection delta is emphasized as critical for calculating both Young's modulus and Poisson's ratio.

Final Steps Towards Young's Modulus Calculation

Stress and Strain Relationships

Understanding Load and Deformation in Structural Members

Stress and Strain Calculations

• The stress in one direction is defined as the load p on a member divided by the unit cell length and the width of the board, leading to the formula:

[

sigma_1 = p/(h + l sin(theta))b

]

• The strain in that direction ( epsilon_1 ) is calculated as deformation divided by unit cell length:

[

epsilon_1 = Delta sin(theta)/l cos(theta)

]

• For hexagonal structures, geometrical factors are meticulously tracked, while for foams, these factors may be simplified.

Relating Load to Deformation

• To derive stiffness or modulus from load p and deformation delta, bending moment diagrams from previous studies (3032 course) are utilized.

• A shear diagram is constructed where Psin(theta) represents the perpendicular component of load. The shear diagram shows an increase to Psin(theta), followed by a horizontal line before decreasing back down.

Bending Moment Analysis

• The bending moment diagram indicates negative moments when tension is on top. This relationship can be expressed through integration of the shear diagram between two points.

• A comparison is made with cantilever beams under force F. The deflection for a cantilever beam can be represented as:

[

delta = F L^3/3EI

]

Application to Honeycomb Structures

• For honeycomb members modeled as two cantilevers, deflection ( delta) can be expressed using:

• Force: Psin(theta)

• Length: l/2

• Thus, for honeycomb members:

• Deflection becomes:

[

delta = Psin(theta)left(l/2right)^3

]

Finalizing Young's Modulus Calculation

• The modulus of solid cell wall material ( E_S) affects overall stiffness. The equation simplifies to yield:

delta = (P sin(theta)(l^3))/(12 E_S I)

• Here, I, the moment of inertia for inclined members, is given by:

I = BT^3/12

Conclusion on Young's Modulus

Understanding Cellular Solid Properties

Young's Modulus Derivation

• The discussion begins with defining cellular solid properties, specifically focusing on the relationship between stress (σ₁) and strain (ε₁), expressed as σ₁ over ε₁.

• The formula for δ (deflection) is introduced: δ = P sine(θ) l³ / 12E_s, where I (moment of inertia) is defined as bt³ / 12. This leads to simplifications in the equations.

• After canceling out terms like P, b, and 12, the equation rearranges to express Young's modulus in relation to thickness (t), length (l), and angles involved.

• Key parameters affecting stiffness are identified: Young's modulus of the material, a factor related to relative density or volume fraction of solids, and cell geometry defined by h/l ratio and θ.

• For regular hexagonal honeycombs, it’s noted that h/l = 1 and θ = 3 results in a specific modulus expression: 4/sqrt3 E_s cdot left(t/lright)^3 .

Poisson's Ratio Analysis

• Transitioning to Poisson's ratio analysis under uniaxial loading conditions; it examines how strain in one direction relates to strain in another direction.

• The strains ε₁ and ε₂ are derived from deflections involving angles θ; this sets up the framework for calculating Poisson's ratio.

• The derivation shows that both strains depend on δ but ultimately simplifies down as they cancel each other out when forming the ratio for Poisson’s ratio.

• It’s highlighted that Poisson’s ratio is solely dependent on cell geometry rather than material properties or relative density—an important distinction for structural analysis.

Understanding Honeycomb Structures and Their Mechanical Properties

Introduction to Honeycomb Mechanics

• The speaker introduces a honeycomb structure with a negative Poisson's ratio, demonstrating its properties by allowing the audience to physically interact with it.

• Participants are instructed on how to load the honeycomb correctly to observe its linear elastic behavior, emphasizing the importance of gentle loading.

Elastic Moduli and Poisson's Ratios

• Discussion on obtaining Young's modulus and Poisson's ratios in different directions, highlighting that formulas vary slightly but follow the same principles.

• The speaker notes that four moduli describe in-plane properties for anisotropic honeycombs, mentioning reciprocal relationships between Young’s moduli and Poisson’s ratios.

Compressive Strength Analysis

• Transitioning to compressive strength, the speaker outlines three failure modes: elastic buckling, yielding, and brittle crushing.

• Explanation of elastic buckling under specific loading conditions; when loaded in one direction, no buckling occurs—only bending deflections.

Stress-Strain Behavior

• Description of stress-strain curves during cell collapse due to elastic buckling; emphasizes that vertical struts buckle under compression.

• Introduction of plastic yielding where deformation localizes initially in one band of cells before propagating through neighboring cells as compression continues.

Failure Modes Overview

• Characterization of plastic failure as a process where localized yielding leads to wider bands collapsing over time.

• Discussion on brittle crushing resulting in serrated plateau stresses on the stress-strain curve due to individual cell wall fractures.

Elastic Buckling Failure Mechanism

• Focus on elastic buckling failure mode (denoted as sigma star el), which only occurs when loading is applied in the two-direction.

Understanding Buckling in Honeycomb Structures

The Concept of Critical Load and Constraints

• Lorna Gibson discusses the critical load that can cause vertical members to buckle, emphasizing the uncertainty of initial constraints on these members.

• The stiffness of adjacent inclined members affects the constraint on vertical members; thicker adjacent members provide more constraint compared to thinner ones.

• When buckling occurs, there is a rotation at the joints which contributes to the overall rotational stiffness of the structure.

Elastic Line Analysis

• Elastic line analysis is introduced as a method for calculating constraints (n), matching rotational stiffness between column h and inclined members.

• The relationship between n and the stiffness of adjacent inclined members is highlighted, with further details available in an appendix of a referenced book.

Ratios and Critical Buckling Load

• The value of n depends on the ratio h/l ; specific values are provided for different ratios:

• For h/l = 1 , n = 0.686

• For h/l = 1.5 , n = 0.76

• For h/l = 2 , n = 0.806

Calculating Buckling Stress

• To find buckling stress, divide critical load by unit cell area; this involves parameters like length in x1 direction and depth into the page.

• The formula for buckling stress incorporates various factors including modulus, moment of inertia (I), and dimensions related to cell geometry.

Dimensional Analysis for Regular Hexagonal Cells

• For regular hexagonal cells, buckling stress can be simplified to a function involving solid properties and relative density.

• It’s noted that Young's modulus remains isotropic across directions in regular hexagonal cells, leading to consistent strain calculations.

Transitioning from Elastic to Plastic Behavior

• Discussion shifts towards plastic collapse stress in metal honeycombs under loading conditions; initially elastic behavior transitions into yielding as loads increase.

• As deformation continues beyond yield stress, entire sections yield forming plastic hinges which allow further rotation without additional force required.

Understanding Plastic Hinge Formation in Structural Members

Mechanism of Plastic Hinge Formation

• The process of bending a structural member back and forth can lead to the formation of a plastic hinge, where it bends easily after sufficient deformation.

• Maximum moments occur at the ends of inclined members, indicating that plastic hinges initially form at these locations during loading in both x1 and x2 directions.

Stress Distribution and Yielding

• To determine the stress required for plastic hinge formation, we consider the yield strength (σ_ys) of cell walls, which signifies failure by yielding.

• In linear elastic conditions, stress distribution varies linearly across the cross-section with a neutral axis where no normal stress exists; as load increases, this stress approaches yield strength.

Progression of Yielding

• Once outer fiber stress reaches yield strength, yielding propagates through the entire cross-section. This transition alters the stress distribution significantly.

• After yielding begins, the stress distribution changes from linear to a more complex profile as loading continues.

Characteristics of Plastic Moment

• When full cross-section yielding occurs, a plastic hinge forms allowing rotation like a pin. The associated plateau stress can be calculated based on internal moments created during this process.

• The internal moment (M_p), related to plastic hinge formation, is derived from force distributions acting over specific distances within the member's geometry.

Equating Moments for Strength Calculation

• The calculation for M_p involves determining forces based on σ_ys and geometrical factors such as thickness (t).

Understanding Plastic Collapse and Elastic Buckling in Honeycombs

Internal Plastic Moment and Applied Moment

• The internal plastic moment is equal to the applied moment, leading to a relationship between stress and geometry.

• The equation for stress ( sigma_1 ) is expressed as P/(h + l sin(theta) cdot b) , allowing for substitution of P in terms of sigma_1 .

Calculating Plastic Collapse Stress

• Rearranging the equations leads to the calculation of plastic collapse stress, which depends on yield strength, thickness-to-length ratio (t/l), and geometric factors.

• For regular hexagonal cells, similar calculations can be performed for loading in different directions. Shear strength can also be derived.

Equating Stresses: Elastic Buckling vs. Plastic Yielding

• Two stresses are equated to establish criteria for buckling versus yielding: elastic buckling plateau stress and plastic collapse plateau stress.

• The buckling stress formula includes parameters like Young's modulus (E), while the plastic collapse stress incorporates yield strength ( sigma_ys ). This leads to a critical t/l value that determines failure mode.

Critical Thickness-to-Length Ratio

• If t/l < 12/(n^2pi^2)(h/l^2/cos(theta))(sigma_ys/E_s) , elastic buckling occurs first; otherwise, plastic yielding happens first. Specific values can be calculated for regular hexagonal honeycombs.

• For metals, the yield strength over modulus ratio is approximately 0.002, indicating that most metal honeycombs will buckle before yielding due to their density characteristics. Polymers may exhibit different behaviors based on their properties relative to this ratio.

Brittle Honeycomb Behavior

• Ceramic honeycombs can fail through brittle crushing modes when compressive forces exceed material limits defined by bending strengths or modulus of rupture. Initial cell wall bending precedes failure at maximum stress levels.

• Future discussions will focus on deriving equations similar to those used for plastic yielding but adapted for brittle materials with linear elastic distributions leading up to failure points instead of fully yielded states.

Upcoming Topics