Materials and Structures. Indian Institute of Technology Kanpur

Transcription

1 Introduction to Composite Materials and Structures Nachiketa Tiwari Indian Institute of Technology Kanpur

2 Lecture 36 Hygrothermal Stresses in Plates

3 Lecture Overview Introduction Hygrothermal Strains Stiffness Matrix for a Lamina Hygrothermal Forces and Moments

4 Introduction Hygrothermal strains, as defined in Eq can be written as: ε HT L= α L ΔT + β L Δc, ε HT T= α L ΔT + β T Δc (Eq ) γ LT = 0. Strains as defined in Eq are measured with respect to material axes L and T. When the same strains are measured with respect to x y coordinate system the relations for these strains are: (Eq. 36.1) Here, [T] 1 is the inverse of [T] as defined in Eq

5 Mechanical and Thermal Strains Hygrothermal strains by themselves cannot generate a force or a moment, unless the bodyis not completely free to deform due to temperature and moisture. Thus, at the level of a whole laminate, there are no resultant forces and moments solely due to hygrothermal effects. However, at the level of a lamina, the same may not be true. This is because a lamina by itself is not entirely free to bend ortwist or expand due to hygrothermal changes. These stresses in a lamina are attributable to strains which are in excess of hygrothermal strains as defined in Eq These excessive strains are known as mechanical strains, and are denoted by a superscript M. Thus: (Eq. 36.2)

6 Mechanical and Thermal Strains In Eq. 36.2, we see that mechanical strain is the difference of total strain, andhygrothermal strain. Further, hygro thermal stresses at individual ply level may be calculated by multiplying mechanical strain vector with lamina stiffness matrix. Thus: Eq However, in Eq. 36.3, 3 mid plane strains andmid plane curvatures arenot known. Equation may be integrated t over thickness (assuming temperature t and moisture contend do not change in thickness direction) to yield relations for force resultant vector. Similarly, Eq may be multiplied with z, and then integrated over thickness to yield relations for momentresultant vector.

7 Hygrothermal Force & Moment Resultants Thus, we get following relations for force and moment resultants. (Eq. 36.4) and (Eq. 36.5) Here, vectors {N HT } and {M HT } are internal hygrothermal stress and moment resultant vectors, respectively. If these are non zero, then even without application of external force and moment resultants, i.e. {N M } and {M M }, respectively, deflections and curvatures may develop in the composite.

8 Hygrothermal Forces and Moments Equations 36.6 provide the definition of hygrothermal force and moment resultants. (Eq. 36.6)

9 Hygrothermal Forces and Moments Equations 36.4 and 36.5 may be used in equilibrium equations as mentioned in Lecture 27to analyze composite plates subjected to hygrothermal stresses. Hygrothermal force and moment resultants can be physically interpreted as described below. Consider a laminate which is not subjected to external loads, but deforms due to hygrothermal effects. In such a case, hygrothermal force and moment resultants may be interpreted as those, yg y p force and moment resultants, which are required to produce deformations in the plate equaling those attributable to hygrothermal effects alone.

10 Rf References 1. Analysis and Performance of Fiber Composites, Agarwal, B.D.and Broutman, L. J., John Wiley & Sons. 2. Mechanics of Composite Materials, Jones, R. M., Mc Graw Hill. 3. Engineering Mechanics of Composite Materials, Daniel, I. M. and Ishai, O., Oxford University Press.