Notes on Lattice Structures and Their Mechanical Properties

Overview of Lattice Structures

• Lattice Structures:

  • Defined as open-celled arrangements of strut elements with specific connectivity at nodes.

  • Can also refer to cellular structures that fill space, distinguishing between closed cell (e.g., cork) and open cell structures (e.g., honeycomb).

Types of Foam

• Open Cell Foam:

  • Characterized by interconnected pores, with notable bending deformation under compression.

• Closed Cell Foam:

  • Contains cells that are not interconnected, providing different mechanical properties.

Common Truss Structures

• Roof Trusses:

  • Warren, King Post, Howe, Pratt, Fink.

• Bridge Trusses:

  • Pratt, Warren, Parker, K, Baltimore.

Truss Element and Stability

• Basic Truss Element:

  • Composed of three member triangular truss.

  • Formula for members (s) and joints (j):

  • $s = 3 + 2(j - 3) = 2j – 3$

• Internal Stability Conditions:

  • s < 2j – r: Internally unstable

  • $s = 2j – r$: Internally stable; geometric stability required

  • s > 2j – r: Indeterminate

Truss Stability Analysis

• Equilibrium and External Forces:

  • Maxwell stability criterion:

  • For each node: 2 free equilibrium equations for 2D, 3 for 3D.

  • Struts represent unknown forces, external forces must be balanced.

Mechanical Response of Lattice Structures

• Bending vs. Stretch Dominance:

  • Under-stiff (M < 0): Bending dominates, structures exhibit high compliance.

  • Just-stiff (M = 0): Determinate loading conditions.

  • Over-stiff (M > 0): Forces governed by tension/compression, no bending.

Design of Lattice Structures

• Hybrid Lattice Design:

  • Response dictated by a combination of the Maxwell number and loading types.

  • Design considerations include the use of under-stiff elements to mitigate excessive forces and over-stiff elements to control deflection.

Conformal and Periodic Structures

• Design Flexibility:

  • Hybrid and conformal lattice structures can be devised using a variety of unit cells, predominantly hexagonal for stability and mechanical response characterization.

Numerical and Experimental Studies

• Challenges in simulation due to computational costs in modeling lattice structures.

• Experimental Findings:

  • High ductility enables consideration of cell size effects on failure mechanisms.

  • Transition observed between bending-dominated and stretch-dominated responses based on geometrical configurations.

Conclusion on Lattice Structures

• Importance of understanding bending and stretch-dominated behaviors in the design of lattice structures for additive manufacturing (AM).

• Material choices and geometrical configurations critically affect manufacturability and mechanical performance.

Lattice Structures:
Defined as open-celled arrangements of strut elements, lattice structures exhibit specific connectivity at nodes, providing unique mechanical properties. Their design can optimize weight, strength, and material efficiency in various applications.

Lattice structures also encompass cellular structures that fill space, distinguishing between closed cell (e.g., cork) and open cell structures (e.g., honeycomb), which have diverse applications in engineering and architecture due to their mechanical and thermal insulating properties.

Types of Foam

Open Cell Foam:
Characterized by interconnected pores, open cell foam allows for air and moisture movement through its structure, resulting in significant bending deformation under compression. This type of foam typically offers cushioning and sound absorption applications.

Closed Cell Foam:
Contains cells that are not interconnected, thus providing different mechanical properties such as higher resistance to moisture and greater buoyancy. This makes closed cell foam ideal for insulation and structural applications where moisture ingress is a concern.

Common Truss Structures

Roof Trusses:
Warren, King Post, Howe, Pratt, and Fink trusses are common designs that distribute loads effectively over a roof structure, enhancing stability and allowing for open spaces below. These designs are critical in architectural applications for ensuring integrity and supporting complex shapes.

Bridge Trusses:
Pratt, Warren, Parker, K, and Baltimore trusses are used to distribute loads in bridge designs, allowing for efficient spanning of distances while maintaining structural integrity. The choice of truss type can significantly affect the material cost and durability of the structure.

Truss Element and Stability

Basic Truss Element:
Composed of three member triangular trusses, this fundamental component allows for efficient force distribution and stability under various loading conditions.

Formula for members (s) and joints (j):
$s = 3 + 2(j - 3) = 2j – 3$

Internal Stability Conditions:
s < 2j – r: Internally unstable, leading to potential structural failure. $s = 2j – r$: Internally stable, yet requires geometric stability to ensure performance under loads. s > 2j – r: Indeterminate, indicating the need for further analysis to understand stability.

Truss Stability Analysis

Equilibrium and External Forces:
Adhering to the Maxwell stability criterion, for each node, 2 free equilibrium equations must be satisfied for 2D structures and 3 for 3D structures.
Struts represent unknown forces, emphasizing the necessity of balancing external forces with internal responses for overall stability.

Mechanical Response of Lattice Structures

Bending vs. Stretch Dominance:
Under-stiff conditions (M < 0) result in bending dominance, where structures exhibit high compliance and are prone to deformation. Just-stiff conditions (M = 0) reflect determinate loading conditions, allowing for predictable responses. Over-stiff conditions (M > 0) indicate forces that are governed primarily by tension and compression, where bending plays a minimal role. This understanding is crucial for predicting performance under load.

Design of Lattice Structures

Hybrid Lattice Design:
The response of lattice structures is dictated by a combination of the Maxwell number and loading types, helping to optimize performance according to specific operational demands.
Design considerations necessitate the incorporation of under-stiff elements to mitigate excessive forces while using over-stiff elements to limit excessive deflection, achieving a balanced structural response.

Conformal and Periodic Structures

Design Flexibility:
Hybrid and conformal lattice structures can be devised utilizing a variety of unit cells, with hexagonal units being favored for their stability and effective mechanical response characterization. These design strategies allow for innovative applications in fields such as aerospace and automotive engineering where weight reduction is critical.

Numerical and Experimental Studies

Challenges in simulation:
Modeling lattice structures leads to computational costs due to the complexity of their geometries and behaviors during loading. Experimental findings reveal that high ductility enables a better understanding of cell size effects on failure mechanisms and overall strength attributes.
A noticeable transition between bending-dominated and stretch-dominated responses is observed based on the geometrical configurations, guiding the selection of lattice designs for specific applications.

Conclusion on Lattice Structures

Understanding the intricacies of bending and stretch-dominated behaviors is vital in the design of lattice structures tailored for additive manufacturing (AM).
Material choices, geometrical configurations, and their interplay critically affect manufacturability, performance, and longevity. Recognizing these relationships aids in creating optimized structures for varying engineering needs.