SAN JOSE, Calif. — When Damaris Triana, then 8, lost several “Little Critter” books that she had borrowed for her sister, the library here fined her $101 — including $40 in processing fees — a bill that was eventually turned over to an agency to collect from her parents.

The $101 is a small part of a whopping $6.8 million in unpaid fines at the San Jose Public Library, an amount that exceeds unpaid fees at some larger cities around the country. It also exceeds other Bay Area cities like Oakland, which has $3 million in outstanding fines, and San Francisco, which has $4.6 million. In San Jose, when the late fee hits $50, the library refers the debt to a collection agency.

As the total of overdue fines has increased, so has the number of cardholders who owe $10 or more and are prohibited from borrowing materials or using the library’s computers. Damaris, now 10, relies on her cousin’s card or uses her school’s library, where there are no fines for late or lost books.

The concept of free public libraries gained support in the 1830s and was popularized by the industrialist Andrew Carnegie , who helped build 1,689 libraries around the country in the late 1800s and early 1900s on the notion that all people should have an opportunity to improve themselves. But public libraries like San Jose’s are struggling to find money to pay for books and services.