3. Relating Stress to Strain Using Matrices

Overview of the Stiffness Matrix in Cubic Systems

  • In a general system, there can be up to 36 potential independent terms in the stiffness matrix.

  • For a cubic system, symmetry reduces this to 3 independent terms.

    • This is seen in the stiffness matrix notation:

      • cij = c11 for the first three diagonal terms (c11, c22, c33)

      • Last three diagonal terms (c44, c55, c66) are equal to each other.

      • Off-diagonal terms (c12, c13, c23) also have specific relationships.

  • All terms not mentioned in this specific case are equal to zero.

Understanding Stress and Strain

  • Stress (σ) components:

    • Axial stress: σ1, σ2, σ3 (corresponding to x, y, z directions)

    • Shear stress: σ4, σ5, σ6

  • Strain (ε) components:

    • Axial strain: ε1, ε2, ε3

    • Shear strain: ε4, ε5, ε6

  • Relationship of stress to strain in cubic systems:

    • σ1 = c11ε1 + c12ε2 + c12ε3; showing interdependence between axial strains due to Poisson’s ratio effects.

    • Off-axis strains influence axial stresses, while shear terms do not affect axial stresses; shear terms operate independently.

Symmetry in a Cubic System

  • Each direction (x, y, z) is equivalent in a cubic crystal structure, leading to the following equivalences:

    • c11 = c22 = c33

    • c12 = c13 = c23

  • In low symmetry systems (e.g., triclinic), the matrix terms can be independent, but the relationships (like c12 = c21) hold based on thermodynamic principles.

  • Generally, cubic systems simplify the analysis due to their symmetry, making calculations manageable with linear algebra.

Compliance Matrix for Cubic Systems

  • The relation for compliance is given by:

    • εi = sijσj

    • Where sij represents the compliance matrix.

  • For cubic systems, simplifications yield:

    • Diagonal terms in sij: s11 for axial terms, s44 for shear terms.

    • Off-diagonal terms also defined, but many terms are 0 in this symmetry context.

  • Compliance terms are similarly reduced with independence reflecting symmetry: 21 independent terms for triclinic vs. 3 for cubic.

Stress and Strain in Different Orientations

  • Stress matrix is defined as applied load intensifies:

    • Helps describe a material’s elastic response under stress, typically assuming reversibility (the material returns to original shape upon load removal).

  • Defined principal directions for properties (cij, sij) are typically 100, 010, 001 for cubic but can analyze other orientations as well.

  • Process for analysis involves:

    • Begin with load in the defined single crystal direction (e.g., 111).

    • Convert stress tensor from original coordinates to the indexed coordinates where material properties are defined using rotation matrices.

    • Analyze stress in the primed coordinate system with cijs and sijs, followed by deriving strain.

    • Possibly convert strain back via tensor transformation to relate to the original coordinates for final analysis.

  • This stepwise tensor transformation helps manage complexity as it avoids unnecessary complications related to an 81-term matrix.