Search course

Use the search function to find more information about the study programmes and courses available at Chalmers. When there is a course homepage, a house symbol is shown that leads to this page.

Graduate courses

Departments' graduate courses for PhD-students.


Syllabus for

Academic year
MHA042 - Material mechanics  
Syllabus adopted 2014-02-17 by Head of Programme (or corresponding)
Owner: MPAME
7,5 Credits
Grading: TH - Five, Four, Three, Not passed
Education cycle: Second-cycle
Major subject: Mechanical Engineering, Civil and Environmental Engineering, Shipping and Marine Technology
Department: 42 - APPLIED MECHANICS

Teaching language: English
Open for exchange students
Block schedule: A

Course module   Credit distribution   Examination dates
Sp1 Sp2 Sp3 Sp4 Summer course No Sp
0107 Laboratory 6,0 c Grading: UG   6,0 c    
0207 Intermediate test 1,5 c Grading: UG   1,5 c   01 Jun 2015 pm H   Contact examiner,  Contact examiner

In programs

MPAME APPLIED MECHANICS, MSC PROGR, Year 1 (compulsory elective)


Professor  Kenneth Runesson
Professor  Ragnar Larsson
Professor  Magnus Ekh


MHA041   Constitutive models


In order to be eligible for a second cycle course the applicant needs to fulfil the general and specific entry requirements of the programme that owns the course. (If the second cycle course is owned by a first cycle programme, second cycle entry requirements apply.)
Exemption from the eligibility requirement: Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling these requirements.

Course specific prerequisites

Basic courses in mechanics and strength of materials equivalent to the course offered by the Mechanical Engineering program, TKMAS, are required. Mathematics, in particular, linear algebra, calculus , and numerical methods. A basic course in FEM is strongly recommended but not a hard prerequisite for attendance to the course.


Material mechanics is a key ingredient in the finite element modeling of advanced components, like engines, automotive and civil engineering structures etc. In fact, many engineering materials deviate from behaving perfectly elastic under service conditions. This occurs whether the loading is sustained, time-dependent of impact dynamic character, quasi-static with cyclic variation or due to elevated temperature. Typically, the material response becomes non-linear due to the development of inelastic deformations, which may be represented as elastic-plastic and/or viscous elastic-plastic, depending on the ambient temperature and the type of mechanical load.
One purpose of the course is to develop fundamental mathematical model concepts, also known as constitutive modeling, that describe the non-linear stress-strain response of the material under consideration. Examples of constitutive theories are those of elasticity, elasto-plasticity and elasto-viscoplasticity, which may be combined to model quite complex physically observed material behavior in conjunction with monotonic and cyclic loading. In fact, the observed behavior may be more or less well represented by the modeling concepts, thereby involving additional material parameters as compared to the perfectly elastic case. Special procedures of how to calibrate these parameters for a given loading are also outlined in the course.
Parallel with the conceptual model developments, another purpose of the course is to advance the computational handling of the constitutive models to facilitate the proper model calibration and to outline the link to finite element analysis of real world components. In this context, the finite element based analysis of a structure is emphasized with material mechanics being the heart of the modeling.

Learning outcomes (after completion of the course the student should be able to)

Describe, outline and implement the elasto-plastic stress-strain relation in the multiaxial stress-strain context based on the von Mises yield condition.
Formulate and implement the non-linear finite element analysis of elasto-plastic response with respect to multiaxial stress-strain states.
Formulate and implement calibration of material parameters based on experimental data based on concepts of optimization.
Explain the strong links between material modeling of observed physical phenomena, the fundamental principles of continuum mechanics, and the computational aspects (related to model calibration and finite element formulation) of the modeling.


Thermomechanical basis: Uniaxial bar problem and 2D elasticity
-Non-dissipative material: Linear and nonlinear elasticity
-Concepts of computational homogenization
CA1: Computational of homogenization of a material microstructure
Dissipative rate independent material response:
-Perfect and linear hardening plasticity
-Special evolution rules of inelastic deformation: non-linear hardening, mixed isotropic and kinematic hardening
-Computational aspects: temporal integration, linearization
Dissipative rate dependent material response
-Viscoplastic material behavior
-Computational aspects: integration, linearization
Stress and strain control
CA2a: Modeling residual stresses in column subjected to cyclic loading - Beam theory approach
Multiaxial stress problem: Basic concepts of stress and strain
-Second order tensor, eigenbases, stress spaces
-Formulation of yield criteria: mean stress (in)dependency
Multiaxial stress problem: Thermomechanical basis
-Isotropic elasticity
-Isotropic "J2-hardening plasticity"
-Computational aspects: temporal integration, linearization
-Non-linear FE problem
CA2b: Modeling residual stresses in column subjected to cyclic loading - Continuum mechanical approach
Calibration of constitutive equations
CA3: Calibration of plasticity model with linear mixed hardening


Teories and concepts are presented at the lectures. In the problem classes and the computer assignments the theories outlined are applied at specified problems. The computer assignments also trains your ability in programming and systematic analysis of a given problem.


Cf. the course homepage.


Mandatory graded Computer Assignments (CAs); Mandatory oral exam (dugga) and graded written exam.

Page manager Published: Thu 04 Feb 2021.