The Couple Stress/Strain Gradient Theory And Its Hybrid Stress Element Analysis

Classical continuum theory doesn’t care about the real micro structure and assumes that the material remains same from macroscale to micro-scale. However, with the material size scaling down to the order of microscales, the stiffness and the strength of metal materials can increase with the size decreasing, which is called size effects. Microstructure-dependent size effects have been exhibited by many micro-and nano-scale components and devices due to impurities, crystal lattice mismatch and micro cracks. The classical elasticity theory is not capable of predicting such size effects due to the lack of a material length scale parameter. To overcome this deficiency, theories for microstructures need to be developed.Theories for microstructures include couple stress/strain gradient theory, which can capture and explain the size effect by introducing material characteristic length constants to describe the impact of micro-defects in microscale. In this paper, couple stress/strain gradient theories, their convergence theories and finite element method are studied. The main works include as follows:(1) The author takes part in developing a new anisotropic modified couple stress theory, which let modified couple stress theory go on to the new stage of anisotropic. By introducing the hypothesis of the cross-section of plate, a new Reddy plate theory and its governing equations for anisotropic modified couple stress theory are developed. And then the model of composite laminated Reddy plate for couple stress theory is established. A cross-ply simply supported plate subjected to cylindrical bending loads is solved by directly applying the newly developed plate model.(2) The recent development of finite element methods for the couple stress/strain gradient theory is reviewed. The frame of the CO continuity and Cl continuity finite elements and convergence criterion of enhanced patch test are discussed. The enhanced patch test is used to test convergence of weak C1continuity element. And the capability of CO continuity penalty element for approaching C1continuity element is studied. Three couple stress penalty elements including (CST+BT3, ALT+BT3, MQ8) are adopted to compare with (RCT9+RT9). It is insufficient to explain the approachability of the CO model to C1theory by using only the example of pure bending beam.(3) Based on the couple stress theory and the hybrid stress element method, a refined18-DOF triangular hybrid stress element and its refined elements are established. The stress functions which satisfy the equilibrium equation in the domain of the element are chosen as the assumed stress interpolation, and boundary displacement interpolation functions are chosen as beam function. The characteristics of this element are possessing second-order accuracy and C0and C1continuity in the weak form, satisfying convergence condition for the enhanced patch test and does not exhibits extra zero energy modes. Moreover, in order to improve the element accuracy, a refined method is developed and the stress smooth technique is used, and they really work. The couple stress theory analytical solution for pure bending beam is also given in this chapter. Numerical examples show that the proposed model indeed possesses higher accuracy and can pass the enhanced patch test, and can capture the scale effects of microstructure.(4) Based on the hybrid stress element method and plane C1couple stress theory, a24-DOF quadrilateral hybrid stress element and reduced integration quadrilateral hybrid stress element for couple stress theory are proposed in this work. We study how to eshtablish the optimum assumed stress interpolation functions and proper displacement interpolation functions on the boundary. The two elements without spurious zero energy mode can satisfy weak C0continuity with second-order accuracy and weak C1continuity simultaneously. So the element can pass the enhanced patch test of a convergence condition. Moreover, the reduced integration and a stress smooth technique are introduced to improve the element accuracy, and really work.