Teaching Continuum Mechanics in a Mechanical Engineering Program Yucheng Liu University of Louisiana at Lafayette 1. Introduction Section 3 presents the detailed methodology and approach of teaching this course, which Abstract Continuum mechanics is a branch of me- include topics (concepts, principles, laws, and chanics that deals with the analysis of the ki- equations), class organization, and evaluation This paper introduces a gradu- nematics and the mechanical behavior of ma- instruments. Section 4 discusses the outcomes ate course, continuum mechan- terials modeled as a continuum. Modeling an of teaching this course at the University of Lou- ics, which is designed for and object as a continuum assumes that the sub- isville and indicates the modification of course taught to graduate students in stance of the object completely fills the space structure since then. Finally, this paper is con- a Mechanical Engineering (ME) it occupies, which is a simplifying assumption cluded and ended with section 5. program. The significance of con- for analysis purposes. As emphasized by tinuum mechanics in engineering Mase [1], continuum mechanics is the funda- education is demonstrated and mental basis upon which several graduate en- 2. Course Objectives the course structure is described. gineering courses are founded. These courses Continuum Mechanics is a three-credit Methods used in teaching this include elasticity, plasticity, viscoelasticity, and course which emphasizes the mathematics course such as topics, class or- fluid mechanics. Gollub [2] also demonstrated and analysis methods used in the study of the ganization, and evaluation instru- the important role that continuum mechanics behavior of a continuous medium. The course ments are explained. Based on plays in contemporary physics. Therefore, it is is designed to be taken by first-year graduate the student learning outcomes beneficial to teach the principles of continuum students of the ME department at UL Lafayette. and feedbacks, the course objec- mechanics to first-year graduate students or The primary objectives of this course are: tives were achieved. This paper upper-level undergraduates to provide them shows that ME program objec- 1. To study the conservation principles in the me- with the necessary background in continuum tives are well supported by this chanics of continua and formulate the equa- theory so that they can readily pursue a formal course. course in any of the aforementioned subjects. tions that describe the motion and mechanical behaviors of continuum materials, and Due to its importance in engineering edu- Keywords: continuum mechanics, 2. To present the applications of these equa- cation, continuum mechanics has been built mechanical engineering, first- tions to simple problems associated with into the engineering curriculum and taught at year graduate students many prominent universities, such as MIT, solid and fluid mechanics. Texas A&M University, Carnegie Mellon, etc. As mentioned before, this course is a prob- Lagoudas et al. [3] demonstrated the sig- lem solving course which focuses on a math- nificance of teaching continuum mechanics ematical study of mechanics of the idealized in their engineering program and designed a continuum mediums. As stated by Petroski [5] core continuum mechanics course for sopho- and Reddy [6], undergraduate mathematics more students at Texas A&M. In that course, plays an important role in continuum mechanics conservation laws and fundamental concepts research and the continuous material behavior of continuum mechanics were taught using can be precisely described using modern alge- computer tools [4]. This paper describes the bra. As indicated from the course description, a design of the course, Continuum Mechanics, fundamental basis in differential equations, me- for the Mechanical Engineering (ME) program chanics of materials, and fluid mechanics are at the University of Louisiana at Lafayette (UL required before taking this course. Lafayette). Compared to similar courses taught As stipulated by the Accreditation Board for in other universities, this course emphasizes Engineering and Technology (ABET) [7], the the application of mathematics in formulating Mechanical Engineering Department has a set and solving fundamental equations of fluid and of eleven educational objectives which are to solid mechanics, as detailed in the course syl- be satisfied by the curriculum. This course sup- labus. ports each of those objectives while emphasiz- The paper is organized as follows. Sec- ing the following: tion 2 describes the objectives of this course • Fundamentals. An ability to apply knowledge and briefly explains how the course objectives of mathematics, science, and engineering in correlate with the ME program objectives. the field of mechanical engineering. Journal of STEM Education Volume 12 • Issue 1 & 2 January-March 2011 17 • Problem Solving. An ability to identify, for- 3. Methodology mulate and solve problems in the field of This course has a well-defined schedule mechanical engineering. and syllabus so that the instructor will cover • Continuing Education. A recognition of the various topics in the class, which is required by need for, and an ability to engage in, life- the breadth of the material and subject matter. long learning in the field of mechanical engi- Visual and hands-on learning techniques will neering. be used wherever possible by the instructor of • Engineering Practice. An ability to use the this course. In many cases, the ME students techniques, skills, and modern tools neces- who take this course may have different back- sary for the practice of mechanical engi- grounds in mathematics. Students will be able neering. to use class time and available office hours to discuss special topics with instructor. Descrip- tions of a course syllabus are presented. As reflected from the syllabus, mathematical meth- ods and fundamental principles and laws play an important role in this class. Course Syllabus Continuum Mechanics Credit Hours: 3 Course Goals The goal of this course is to emphasize the formulation of problems in mechanics along with the fundamental principles that underlie the governing differential equations and boundary condi- tions. Prerequisites by Topic Differential equations; mechanics of materials; fluid mechanics Text Book Reading and homework assignments refer to the required textbook: T. J. Chung, General Continuum Mechanics, 2007, Cambridge University Press, New York, NY, USA. Organization This class is organized into three 50 minutes sessions per week devoted to lecture/discussion and problem solving. The difficulty of this course is such that a minimum weekly commitment of 8-10 outside study hours will be required. Quizzes Daily quizzes will be given, typically over the previous reading assignment or the material cov- ered in the last class. Missed quizzes cannot be made up. Exams Three closed-notes, closed-book exams will be given throughout the semester, which include two midterm exams and one final exam. Grading System Homework Assignments 20% Quizzes 15% Midterm Exams 2 @ 20% Final Exam 25% Total 100% Final grades will be calculated according to an absolute scale: ≤ 59 F Figure 1. Journal of STEM Education Volume 12 • Issue 1 & 2 January-March 2011 18 60-62 D- 63-67 = D 68-69 D+ 70-72 C- 73-77 = C 78-79 C+ 80-82 B- 83-87 = B 88-89 B+ 90-92 A- 93-97 = A 98-100 A+ Topic/Activity 1. Introduction to continuum mechanics (2 classes) - What is continuum mechanics? - Main themes and assumptions of continuum mechanics. 2. Vectors, tensors and essential mathematics (6 classes) - Vectors and tensors - Tensor algebra; summation convention; Kronecker delta; permutation symbol; ε-ε iden- tity. - Indicial notation; tensor product. - Matrix operation and linear algebra. - Transformation of Cartesian tensors. - Principal values (eigenvalue) and principal directions (eigenvector) of second order tensors. - Tensor fields and tensor calculus; partial differential operator; subscript comma. - Integral theorems of Gauss and Stokes 3. Stress principles (3 classes) - Body and surface forces; mass density. - Cauchy stress principles; Newton’s 2nd and 3rd laws. - Stress vector; stress tensor; state of stresses. - Force and moment equilibrium; stress tensor symmetry. - Stress transformation laws. 4. Principal stresses and principal axes (3 classes) - Principal stresses and principal stress directions. - Maximum and minimum stress values. - Mohr’s circle for stress. - Plane stress. - Spherical stress; deviator stress states. - Octahedral shear stress. 5. Analysis of deformation (3 classes) - Particles; configurations; deformation and motion. - Material and spatial coordinates; find velocity and acceleration for a given motion; Jacobian matrix. - Lagrangian and Eulerian descriptions. - Displacement field. - Material derivative. - Deformation gradients, Lagrangian and Eulerian finite strain tensors; Cauchy deformation tensor. 6. Velocity fields and compatibility conditions (3 classes) - Infinitesimal deformation theory; strain transformation; strain compatibility equations; normal strain and shear strain; cubical dilatation. - Calculate stretch ratios. - Rotation tensors and stretch tensors; polar decomposition. - Velocity gradient; rate of deformation and vorticity. 7. Fundamental law and equations (5 classes) - Material derivative of line elements, areas, and volumes; isochoric motion. - Balance laws; field equation; constitutive equation. - Material derivative of line, area, and volume integrals.